re latest theory
James Whitehead
news:artfldodgr-D45C2...@news.fu-berlin.de...
> In article <bad37d$2ih$1...@news6.svr.pol.co.uk>,
> "James Whitehead" <Abx4...@jjh76g7856gh.com> wrote:
>
> > A proposition of mathematics does not express a thought. The logic of
the
> > world, which is shown in tautologies by the propositions of logic, is
shown
> > in equations by mathematics. Proof in logic is merely a mechanical
expedient
> > to facilitate the recognition of tautologies in complicated cases. The
> > propositions of logic are tautologies. Therefore the propositions of
logic
> > say nothing. We feel that even when all possible scientific questions
have
> > been answered , the problems of life remain completely untouched. The
urge
> > towards the mystical comes of the non-satisfaction of our wishes by
science.
> > Words are like the film on deep water. To pray is to think about the
> > meaning of life. I am my world. The World and life are one. How things
stand
> > , is God. Mine is the first and only world! Art is a kind of expression.
> > Good art is complete expression. A proposition of mathematics does not
> > express a thought.
>
> This reads like a compendium of fortune cookie wisdom. Have you been
> eating a lot of Chinese food lately?
Dear John-
no actually i have been reading a couple of books by a guy called
Wittgenstein- (all these above are from Notebooks and Tractatus) not a
particularly Chinese sounding name - and before you make another mistake it
was not where i got my ideas re tautology from - see my reply to your pal.
'From someone who seems to have a profound misunderstanding of both
Mathematics &
Science, and Art.' - i assume you mean Wittgenstein, Russell, Moore, Ayer,
Kosuth.......?.
Stephen Hawkings
"James Whitehead" <Abx4...@jjh76g7856gh.com> wrote in message
news:bafhn4$eqh$1...@newsg1.svr.pol.co.uk...
> "John Dawkins" <artfl...@aol.com> wrote in message
> news:artfldodgr-D45C2...@news.fu-berlin.de...
> > In article <bad37d$2ih$1...@news6.svr.pol.co.uk>,
> > "James Whitehead" <Abx4...@jjh76g7856gh.com> wrote:
> >
> > > A proposition of mathematics does not express a thought.
On the contrary, each distinct mathematical proposition expresses a distinct
thought. The burden of proof belongs to the naysayers.
> > > The logic of the world, which is shown in tautologies by the
propositions of logic, is
> > > shown in equations by mathematics.
> > > Proof in logic is merely a mechanical
> > > expedient to facilitate the recognition of tautologies in complicated
cases. The
> > > propositions of logic are tautologies.
> > > Therefore the propositions of
> > > logic say nothing.
There are at least three types of mathematical statements: tautologies,
contraditions and propositions which are neither tautologies nor
contraditions. These propositions, can suffer from various defects, which
make their results meaningless; a single example being the excluded middle.
Tautologies typically are always true because they are so contructed - and
thus tautologies by simple inspection.
It is commonplace to hear various philosophers in history make the statement
that mathematics is simply tautologies. This is a false notion and horribly
naive. A proposition being true doesn't necessarily make it a tautology.
A typically mathematical proposition is a non-tautology, and makes clear a
symmetry or geometrical relation in the system of discourse not at all
obvious from the original problem. This is equally true for physics as it is
for mathematics.
Lets take a simple, yet elegant mathematical proposition, that all people
should be able to follow:
1 + 2 + 3 + ...+ n-1 + n = n(n+1)/2 ; n >= 1, n = integer
A proof of this mathematical proposition is as follows:
Observe the sum of 1 + 2 + ... + n-1 + n
Clearly there are n terms, from the least on the left to the greatest on the
right.
Now repeat this sum, but now in decreasing order:
n + n-1 + ... + 2 + 1
Again there are n terms, now from the greatest on the left to the least on
the right.
Now place these two sums one above the other so as to place numbers of the
same position in the same colums:
1 + 2 + ..... + n-1 + n
n + n-1 + ..... + 2 + 1
Notice that there are still n columns, but that the numbers in each column
sum to n+1 (!). Therefore, the sum of these two expressions is clearly
n(n+1) - there are n columns, where each column sums to n+1. Since the
original expression is simply half of this new, repeated expression, the
result is
1 + 2 + .... + n-1 + n = n(n+1)/2 quod erat demonstrandum
now upon close examination, it is clear that the above expressionis not a
tautology. Its proof mas not mechanical, but required a creativity of a
sort to yield its inherent symetry and form. Different propositions will
require different proofs, depending on the underlying structure that is to
be exposed.
In brief, the notion that mathematics is simply tautologies is false, and is
an extremely lazy sort of arguement. The fame or "authority" of the author
of such nonsense is irrelevant - the most briliant minds say idiotic things
at times, just as do dull minds. However no person who has had the great
fortume to have created a modest amount of original work has based it on
"authority" or "fame", but on the logic of their ideas.
> > > ...We feel that even when all possible scientific questions
> > > have been answered , the problems of life remain completely untouched.
The
> > > urge towards the mystical comes of the non-satisfaction of our wishes
by
> > > science.
I would urge you to consider that if truth is found to be insufficient, then
one is deceiving oneself.
> > > Words are like the film on deep water. To pray is to think about the
> > > meaning of life. I am my world. The World and life are one. How things
> > > stand
> > > , is God. Mine is the first and only world! Art is a kind of
expression.
> > > Good art is complete expression.
I've had Buddhist friends express similar thoughts. Though they would say
things such as:
- "your're only profound on the surface - below that you're shallow"
or
- "your depth is only on the surface"
Actually it has been shown empirically that the "act of praying" reduces
ones serotonine levels and makes one intentionally of lower social rank as a
consequence. Prayer can be chemically countered by taking chemicals that
elevate ones seratonine levels, such as prozac, zoloft, and the like. It has
been shown that monkeys of socially low rank, when given prozac,
automatically increase their social rank.
> > > A proposition of mathematics does not
> > > express a thought.
False - actually more on order of a lie.
> >
> > This reads like a compendium of fortune cookie wisdom. Have you been
> > eating a lot of Chinese food lately?
>
> Dear John-
> no actually i have been reading a couple of books by a guy called
> Wittgenstein- (all these above are from Notebooks and Tractatus) not a
> particularly Chinese sounding name - and before you make another mistake
it
> was not where i got my ideas re tautology from - see my reply to your pal.
>
> 'From someone who seems to have a profound misunderstanding of both
> Mathematics &
> Science, and Art.' - i assume you mean Wittgenstein, Russell, Moore, Ayer,
> Kosuth.......?.
>
One does and understands mathethatics and science by doing mathematics and
science, not by doing philosophy; certainly not by worshiping philosophical
authority.
James Whitehead
"Stephen Hawkings" <Stephen_...@damtp.cam.ac.uk> wrote in message
news:C3Mya.620$bO6.5...@news1.news.adelphia.net...
>
> "James Whitehead" <Abx4...@jjh76g7856gh.com> wrote in message
> news:bafhn4$eqh$1...@newsg1.svr.pol.co.uk...
> > "John Dawkins" <artfl...@aol.com> wrote in message
> > news:artfldodgr-D45C2...@news.fu-berlin.de...
> > > In article <bad37d$2ih$1...@news6.svr.pol.co.uk>,
> > > "James Whitehead" <Abx4...@jjh76g7856gh.com> wrote:
> > >
> > > > A proposition of mathematics does not express a thought.
>
> On the contrary, each distinct mathematical proposition expresses a
distinct
> thought. The burden of proof belongs to the naysayers.
I've tried to answer this elsewhere - but its is for my part saying
something of the difference between mathematical propositions and other
propositions, or not do I particularly agree with Wittgenstein & Russell
(who i thought was a mathematician) on the matter. I was merely pointing out
to someone who accused me of having a profound misunderstanding of both
Mathematics & Science, and *Art*, that the misunderstanding is not mine but
if it is its of Wittgenstein Russell & co. I've yet to be shown how
mathematics - its propositions can tell me something new about the world. So
back to you for the proof.
>
> > > > The logic of the world, which is shown in tautologies by the
> propositions of logic, is
> > > > shown in equations by mathematics.
> > > > Proof in logic is merely a mechanical
> > > > expedient to facilitate the recognition of tautologies in
complicated
> cases. The
> > > > propositions of logic are tautologies.
> > > > Therefore the propositions of
> > > > logic say nothing.
>
> There are at least three types of mathematical statements:
But we are talking about propositions! Not about statements.
tautologies,
> contraditions and propositions which are neither tautologies nor
> contraditions. These propositions, can suffer from various defects, which
> make their results meaningless; a single example being the excluded
middle.
>
> Tautologies typically are always true because they are so contructed - and
> thus tautologies by simple inspection.
Typical ? You mean there are rare kinds which are not! I might one day
stumble on a tautology which isnt true?
>
> It is commonplace to hear various philosophers in history make the
statement
> that mathematics is simply tautologies. This is a false notion and
horribly
> naive. A proposition being true doesn't necessarily make it a tautology.
>
> A typically mathematical proposition is a non-tautology, and makes clear a
> symmetry or geometrical relation in the system of discourse not at all
> obvious from the original problem. This is equally true for physics as it
is
> for mathematics.
Youve slipped in the word obvious - i dont think i or the above philosophers
and mathematicians have done so - we go back to the greek root which is
"same". Again your being very clumsy - of course a proposition being true
doesnt make it a tautology - who said that!. Its logical propositions -
which propose a truth which are tautologies - according to W&R et al.
>
> Lets take a simple, yet elegant mathematical proposition, that all people
> should be able to follow:
>
> 1 + 2 + 3 + ...+ n-1 + n = n(n+1)/2 ; n >= 1, n = integer
>
> A proof of this mathematical proposition is as follows:
>
> Observe the sum of 1 + 2 + ... + n-1 + n
> Clearly there are n terms, from the least on the left to the greatest on
the
> right.
>
> Now repeat this sum, but now in decreasing order:
> n + n-1 + ... + 2 + 1
> Again there are n terms, now from the greatest on the left to the least on
> the right.
>
> Now place these two sums one above the other so as to place numbers of the
> same position in the same colums:
>
> 1 + 2 + ..... + n-1 + n
> n + n-1 + ..... + 2 + 1
>
> Notice that there are still n columns, but that the numbers in each column
> sum to n+1 (!). Therefore, the sum of these two expressions is clearly
> n(n+1) - there are n columns, where each column sums to n+1. Since the
> original expression is simply half of this new, repeated expression, the
> result is
>
> 1 + 2 + .... + n-1 + n = n(n+1)/2 quod erat demonstrandum
By this' = 'do you mean the two sides are the same? Or only similar?
>
> now upon close examination, it is clear that the above expressionis not a
> tautology. Its proof mas not mechanical, but required a creativity of a
> sort to yield its inherent symetry and form. Different propositions will
> require different proofs, depending on the underlying structure that is to
> be exposed.
Not mechanical - then what! Where in this proof is there something not
mechanical? You dont think mechanisms can be creative? Have you ever used a
hammer?
>
> In brief, the notion that mathematics is simply tautologies is false, and
is
> an extremely lazy sort of arguement.
talk of lazy! Again we/they were talking of propositions of logic. And even
more lazy and sloppy thinking - science or physics just jogs along - and art
is simply left out altogether. If you really think the truth of a logical
proposition is not a tautology then so be it. If you think an act of
creativity is the reason i'd also have to let you have your way, i'd simply
restate my own question which is if i'm using the word tautology to mean "is
the same" then are you saying that an equals sign does not mean "is the same
as"
The fame or "authority" of the author
> of such nonsense is irrelevant -
I quite agree, though it should cause one to think. But i do think the
nonsense might be of your own making.
the most briliant minds say idiotic things
> at times, just as do dull minds.
Then whats the difference? If both say idiotic things at times.
However no person who has had the great
> fortume to have created a modest amount of original work has based it on
> "authority" or "fame", but on the logic of their ideas.
How naive you are! "If there's something to be stolen, i steal it,"
Picasso
(then there was all that bitching about the calculus!)
>
>
> > > > ...We feel that even when all possible scientific questions
> > > > have been answered , the problems of life remain completely
untouched.
> The
> > > > urge towards the mystical comes of the non-satisfaction of our
wishes
> by
> > > > science.
>
> I would urge you to consider that if truth is found to be insufficient,
then
> one is deceiving oneself.
Your urging Wittgenstein who is dead. But its probably not worth making the
point that deception also plays a part in life.
There is quite a deal more on this subject- but we must i fear pass over
this..
>
> > > > Words are like the film on deep water. To pray is to think about
the
> > > > meaning of life. I am my world. The World and life are one. How
things
> > > > stand
> > > > , is God. Mine is the first and only world! Art is a kind of
> expression.
> > > > Good art is complete expression.
>
> I've had Buddhist friends express similar thoughts. Though they would say
> things such as:
> - "your're only profound on the surface - below that you're shallow"
> or
> - "your depth is only on the surface"
>
> Actually it has been shown empirically that the "act of praying" reduces
> ones serotonine levels and makes one intentionally of lower social rank as
a
> consequence.
Sorry? Prayer makes one of 'lower social rank' - do you know about the
Queen! She lives nearby and can been seen at prayer most Sundays in
Sandringham's Church.
Prayer can be chemically countered by taking chemicals that
> elevate ones seratonine levels, such as prozac, zoloft, and the like. It
has
> been shown that monkeys of socially low rank, when given prozac,
> automatically increase their social rank.
Your beautiful! presumably they were first taught to pray. Conversely there
is evidence that religion elevates ones social rank - George W and Blair
both pray regularly, Methodism which began amongst the working classes soon
became middle-class due to the sobriety of its adherents... etc etc.
>
>
> > > > A proposition of mathematics does not
> > > > express a thought.
>
> False - actually more on order of a lie.
>
> > >
> > > This reads like a compendium of fortune cookie wisdom. Have you been
> > > eating a lot of Chinese food lately?
> >
> > Dear John-
> > no actually i have been reading a couple of books by a guy called
> > Wittgenstein- (all these above are from Notebooks and Tractatus) not a
> > particularly Chinese sounding name - and before you make another mistake
> it
> > was not where i got my ideas re tautology from - see my reply to your
pal.
> >
> > 'From someone who seems to have a profound misunderstanding of both
> > Mathematics &
> > Science, and Art.' - i assume you mean Wittgenstein, Russell, Moore,
Ayer,
> > Kosuth.......?.
> >
>
> One does and understands mathethatics and science by doing mathematics and
> science, not by doing philosophy; certainly not by worshiping
philosophical
> authority.
One may do mathematics by doing it! but it doesnt follow that one
understands it. If so it follows one understands life by living it, and the
only way of understanding Stephen Hawkings is by being (i wont say doing)
Stephen Hawkings.
Tom Adams
> "Stephen Hawkings" <Stephen_...@damtp.cam.ac.uk> wrote in message
> news:C3Mya.620$bO6.5...@news1.news.adelphia.net...
> >
> > "James Whitehead" <Abx4...@jjh76g7856gh.com> wrote in message
> > news:bafhn4$eqh$1...@newsg1.svr.pol.co.uk...
> > > "John Dawkins" <artfl...@aol.com> wrote in message
> > > news:artfldodgr-D45C2...@news.fu-berlin.de...
> > > > In article <bad37d$2ih$1...@news6.svr.pol.co.uk>,
> > > > "James Whitehead" <Abx4...@jjh76g7856gh.com> wrote:
> > > >
> > > > > A proposition of mathematics does not express a thought.
> >
> > On the contrary, each distinct mathematical proposition expresses a
> distinct
> > thought. The burden of proof belongs to the naysayers.
>
>
> I've yet to be shown how
> mathematics - its propositions can tell me something new about the world.
About the world?
It one thing to be told something new.
> So back to you for the proof.
I am not sure new things come with separate proofs. They just happen.
Does tic-tac-toe express something new?
I think so. The rules do at least.
James Whitehead
"Tom Adams" <tada...@yahoo.com> wrote in message
news:ea44f5a1.0306...@posting.google.com...
I'm a little lost in all the headers - if the world includes mathematics
then statements are created which are new - the discussion - for what it was
worth was -regarding the tautological nature of logic/mathematics. I asked
if mathematics was no longer tautology - i did not say pejoratively it was -
there was a great deal of abuse thrown in my general direction - and in
Wittgenstein's - though being dead he wasnt bothered. Even a Stephen
Hawkings from Cambridge poped up - but not i assume - "THE" . The only idea
of the non-tautology which i can think of is the Godel equation. Now that's
interesting as it de-stabilises mathematics in that it brings with it an
uncertainty. The looping/Turing problem is undecideable - and of course
un-decideabilty is a feature of Derrida's deconstruction. Engineers do not
bother with such things - which are mathematics- as they simply want to use
it, so they ignore the consequences of Godel, Turing et al. To quote from
'about a boy' they assume there is a man down the essex road who can fix
things. What is surprising is that recently THE Stephen Hawking has put
forward similar ideas - re undeciability in *science*. Mike & the mechanics
are on an ice flow drifting towards the equator noticing how a) the flow is
getting smaller and b) the penguins are leaving.
I noticed a book "What Philosophers think" includes in its contributors Alan
Sokal, Don Cupitt, Richard Dawkins, John Searle,.... One assumes because
one can put a new washer on a tap that building a space shuttle is just a
matter of scale.
Post Script (a good derridianesq feature!)
On running the spell checker
Godel - Godless
Derrida - Dread's
Sokal - Soak
Cupitt - Cupid
Dawkins - Dawns
Searle - Seattle
.
Mr. Vibrating
As you can see, I am crossposting this from alt.postmodern. The issue we
have been discussing requires clarification from a knowledgeable
mathematician.
The statement we are discussing is the following:
> >A proposition of mathematics does not express a thought. The logic of
the world,
> > which is shown in tautologies by the propositions of logic, is shown
> > in equations by mathematics. Proof in logic is merely a mechanical
expedient
> > to facilitate the recognition of tautologies in complicated cases. The
> > propositions of logic are tautologies. Therefore the propositions of
> > logic say nothing.
So evidently some authoritative philosophers and mathematicians claim that
(1) all of mathematics is "tautology" and (2) makes no claims about the real
world.
So what is the definition of a tautology, when is a mathematical statement a
tautology and when is it no, and does mathematics make any claims about the
world (what is the subject/object matter or mathematics?)?
We need some help here from real mathematicians.
Thanks!
Mr. Vibrating
"James Whitehead" <Abx4...@jjh76g7856gh.com> wrote in message
news:bcepod$esh$1...@news8.svr.pol.co.uk...
Mr. Vibrating
James Whitehead
"Mr. Vibrating" <eastwood...@yahoo.com> wrote in message
news:QNOGa.6769$Jw6.3...@news1.news.adelphia.net...
Though it would be interesting to hear from "real" mathematicians i would
not depend totally on their help. We are discussing the ontology of
mathematics - in much the same way as we could discuss the ontology of the
US military - though the Pentagon will no doubt provide "real" military and
some definitive statements - they will not end the matter - or close the
question as to what they are doing in Iraq for instance.
My guess is mathematicians will be doing maths - and not thinking about its
ontology- and so we will not get any reply- but we shall see.
George Cox
>
> Dear recreational mathematicians!
>
> As you can see, I am crossposting this from alt.postmodern. The issue we
> have been discussing requires clarification from a knowledgeable
> mathematician.
>
> The statement we are discussing is the following:
> > >A proposition of mathematics does not express a thought. The logic of
> the world,
> > > which is shown in tautologies by the propositions of logic, is shown
> > > in equations by mathematics. Proof in logic is merely a mechanical
> expedient
> > > to facilitate the recognition of tautologies in complicated cases. The
> > > propositions of logic are tautologies. Therefore the propositions of
> > > logic say nothing.
>
> So evidently some authoritative philosophers and mathematicians claim that
> (1) all of mathematics is "tautology" and (2) makes no claims about the real
> world.
> So what is the definition of a tautology, when is a mathematical statement a
A tautology is something like "p or not-p" where "p" is a statement that
is true or false. "p or not-p" where "p" is true no matter whether "p"
is true or false. That is not meant to be a definition--just a sloppy
example. Tautologies (I'm still not offering a definition) are true
_without regard to the facts_. Tautologies are the subject of a branch
of mathematics called Propsitional Calculus (which you could Google
for). Some will claim that mathematics makes no claim about the real
world (but I don't like the use of the word "real" here) and hence is
true (if it is true) independently of the way the world is--it is
tautological in a broader sense of tautology than Propsitional Calculus
deals with. But if this is true, how come mathematics is so successful
in its applications ranging from accounting to physics?
GC
> tautology and when is it no, and does mathematics make any claims about the
> world (what is the subject/object matter or mathematics?)?
>
> We need some help here from real mathematicians.
>
> Thanks!
>
> Mr. Vibrating
>
...
Jesse F. Hughes
> Dear recreational mathematicians!
>
> As you can see, I am crossposting this from alt.postmodern. The issue we
> have been discussing requires clarification from a knowledgeable
> mathematician.
>
> The statement we are discussing is the following:
>> >A proposition of mathematics does not express a thought. The logic of
> the world,
>> > which is shown in tautologies by the propositions of logic, is shown
>> > in equations by mathematics. Proof in logic is merely a mechanical
> expedient
>> > to facilitate the recognition of tautologies in complicated cases. The
>> > propositions of logic are tautologies. Therefore the propositions of
>> > logic say nothing.
>
> So evidently some authoritative philosophers and mathematicians claim that
> (1) all of mathematics is "tautology" and (2) makes no claims about the real
> world.
> So what is the definition of a tautology, when is a mathematical statement a
> tautology and when is it no, and does mathematics make any claims about the
> world (what is the subject/object matter or mathematics?)?
The idea that mathematical theorems are tautological (or, the related
claim that they are "true by convention") was particularly common in
logical positivism. You can see this in a Hans Hahn article, although
I don't have the title of the article handy. Sorry. It can be found
in Ayer's collection titled (I think) "Logical Positivism".
Quine has a critical examination of this view titled, "Truth by
Convention". It can be found in the Benacerraf and Putnam text,
"Philosophy of Mathematics: Selected Readings".
I won't say too much about the reasonableness of the claim that
mathematics has no content, aside from a personal reaction to the Hahn
article. It may not be all that relevant to your question, but I
prefer not to speculate on big philosophical claims. Instead, I'll
give a reaction not to the broad claim that math has no content, but
to one particular account of this claim.
Hahn claims that mathematical statements are merely consequences of
our conventions regarding mathematical terms. Thus, they convey no
real information. However, whether space is Euclidean or not appears
to be an empirical fact. Thus, it is not apparent that geometric
theorems are tautologous. They may very well follow from the
conventions that we have adopted when we speak of line, etc., but when
applied to physical observations, geometric deductions are not
necessarily truth-deserving (since whether or not space is Euclidean
seems to be a consideration that convention alone cannot settle).
In this respect, mathematics seems very different than logic. If you
read the Hahn article, however, you will see that he believes the two
are completely analogous. Just as logical rules reflect conventions
surrounding our use of certain connectives (and, or, etc.), so
(according to Hahn) do the mathematical axioms reflect our conventions
for using mathematical terms.
Note: Hahn makes broad claims about mathematics, but always
illustrates those claims with arithmetic. It is instructive, I think,
to consider whether his illustrations would work with geometry,
instead of arithmetic, for the reasons I sketched above.
I hope this is useful. If you'd like more references, let me know and
perhaps I can dig them up.
--
"[N]ow for once I might actually have an audience that realizes that
[my proof of Fermat's Last Theorem is correct], because you see,
they'll finally know what's in it for them--cold, hard cash."
--James Harris embarks on a new mathematical strategy.
James Whitehead
"George Cox" <georg...@btinternet.com> wrote in message
news:3EEC9920...@btinternet.com...
it is
> tautological in a broader sense of tautology than Propsitional Calculus
> deals with. But if this is true, how come mathematics is so successful
> in its applications ranging from accounting to physics?
Success is not a guarantee of truth - often the reverse.
i.e. painting cherries to fool the birds - or invading another country.
Maradona's hand of god or for that matter the split boxing glove which
cheated Henry Cooper out of the world title.
I remember in maths once having to use a quadratic equation to calculate the
height of a bridge - which resulted in two answers - one a negative this we
were told could be ignored as it was obvious we were after a positive value.
G*rd*n
> > tautological in a broader sense of tautology than Propsitional Calculus
> > deals with. But if this is true, how come mathematics is so successful
> > in its applications ranging from accounting to physics?
"James Whitehead" <Abx4...@jjh76g7856gh.com>:
> Success is not a guarantee of truth - often the reverse.
>
> i.e. painting cherries to fool the birds - or invading another country.
> Maradona's hand of god or for that matter the split boxing glove which
> cheated Henry Cooper out of the world title.
Well, yes, but if the success isn't "truth" you can just call
it something else -- the fact is there's an alignment between
mathematics and the physical universe which doesn't seem to
arise from any noticeable direct connection or common path,
and some people find that curious. As I noted previously,
_i_ seems to have been arrived at by thinking, "There's no
such number as __ but what if there were? Let's just pretend
there is." Hence arise zero, negative numbers, fractional
and negative exponents, and eventually the square root(s) of
-1, which turns out to be useful in representing the behavior
of electricity because formulae using it correspond to electical
phenomena. Electricity, as far as I know, did not investigate
the possibilities of extending number systems.
> I remember in maths once having to use a quadratic equation to calculate the
> height of a bridge - which resulted in two answers - one a negative this we
> were told could be ignored as it was obvious we were after a positive value.
They didn't want you to know about the _other_ bridge.
--
(<><>) /*/
}"{ G*rd*n }"{ g...@panix.com }"{
{ http://www.etaoin.com | latest new material 1/19/03 <-adv't
Proginoskes
>
> A tautology is something like "p or not-p" where "p" is a statement that
> is true or false. "p or not-p" where "p" is true no matter whether "p"
> is true or false. That is not meant to be a definition--just a sloppy
> example. Tautologies (I'm still not offering a definition) are true
> _without regard to the facts_. Tautologies are the subject of a branch
> of mathematics called Propsitional Calculus (which you could Google
> for).
A tautology is a propositional calculus formula (involving OR, AND, and
implications, and variables whose values are Boolean values -- true or
false) which is always true due to its intrinsic _form_. For instance,
"p or not-p" is a tautology. It doesn't matter what p represents; we could
even let p be the statment "2 + 2 = 5". Then the statement
"2 + 2 is 5 or 2 + 2 is not 5"
is true. It doesn't say anything useful, though.
There are other mathematical results which are true because of their
_content_. For instance, the statement "p implies q" is not a tautology--
its value is false if p is true and q is false--but there are true statements
that take this form. For instance, the 5-Color* Theorem:
If G is a planar graph, then the vertices G can be colored with 5
colors (so that no two adjacent vertices have the same color).
This is a statement of the form "p implies q" which has some substance to it.
> Some will claim that mathematics makes no claim about the real
> world (but I don't like the use of the word "real" here) and hence is
> true (if it is true) independently of the way the world is--it is
> tautological in a broader sense of tautology than Propsitional Calculus
> deals with. But if this is true, how come mathematics is so successful
> in its applications ranging from accounting to physics?
A lot of people ask this question, and the answer is that it is the reverse
which is true. Sure, you can define some weird addition and numbers, saying
that 2 + 2 = 5, and 2 + 0 = 1 (as written in "standard" mathematics) and
live your life based on that, but you won't be able to live in the real world
with "truths" like these.
Mathematics has evolved the other way: People have developed ideas and
formulas that work, and these ideas have been generalized to create
mathematics. One thing I tell my math students (I teach at ASU) is that
the formulas, concepts, etc., difficult as they are, are designed to make
life easier, not more difficult, and also to solve problems that people
have stumbled on. For instance: multiplication of decimals is more difficult
than addition of decimals, so logarithms were developed, which turn
a multiplicative expression into an additive one.
-- Christopher Heckman
* I'm avoiding using the 4-Color Theorem as an example because there are some
some people claim that neither of the two proofs -- Appel and Haken, and
Robertson--Sanders--Seymour--Thomas -- is an actual proof.
Jesse F. Hughes
> There are other mathematical results which are true because of their
> _content_. For instance, the statement "p implies q" is not a
> tautology-- its value is false if p is true and q is false--but
> there are true statements that take this form. For instance, the
> 5-Color* Theorem:
>
> If G is a planar graph, then the vertices G can be colored with
> 5 colors (so that no two adjacent vertices have the same
> color).
>
> This is a statement of the form "p implies q" which has some
> substance to it.
Those who advocate the view that mathematics has no content would
claim that your statement above is a mere shorthand for the proper
statement of the 5-Color theorem. The "proper" statement would be:
If (conjunction of the axioms of graph theory) then if G is a
planar graph then the vertices G can be colored with 5 colors (so
that no two adjacent vertices have the same color).
So, this is not (in their view) an example of a statement with content
at all, since it is tautologous when stated "properly".
--
Jesse Hughes
"Surround sound is going to be increasingly important in future
offices."
-- Microsoft marketing manager displays his keen insight
Will Twentyman
> Dear recreational mathematicians!
>
> As you can see, I am crossposting this from alt.postmodern. The issue we
> have been discussing requires clarification from a knowledgeable
> mathematician.
>
> The statement we are discussing is the following:
>
>>>A proposition of mathematics does not express a thought. The logic of
>
> the world,
>
>>>which is shown in tautologies by the propositions of logic, is shown
>>>in equations by mathematics. Proof in logic is merely a mechanical
>
> expedient
>
>>>to facilitate the recognition of tautologies in complicated cases. The
>>>propositions of logic are tautologies. Therefore the propositions of
>>>logic say nothing.
>
>
> So evidently some authoritative philosophers and mathematicians claim that
> (1) all of mathematics is "tautology" and (2) makes no claims about the real
> world.
> So what is the definition of a tautology, when is a mathematical statement a
> tautology and when is it no, and does mathematics make any claims about the
> world (what is the subject/object matter or mathematics?)?
>
> We need some help here from real mathematicians.
>
> Thanks!
>
> Mr. Vibrating
>
Ok, this is my take on things for what it's worth.
Mathematics takes a collection of axioms (statements that are assumed to
be true) and attempts to determine what (if any) conclusions can be
drawn from them.
Things that are important include: are the axioms consistent? If you
can prove the negation of one of the axioms using the axioms, then it is
not consistent. Using that set of axioms anything can be "proven" true,
which makes it uninteresting.
With a consistent set of axioms: any statement of the form
If (axioms) then (provable theorem)
is a tautology (a statement that is always true).
Similarly
If (axioms) then (false theorem)
is a contradiction (a statement that is never true).
Notice that there is no claim made about "reality". There are three
major forms of Geometry: Euclidean, perspective, and geometry on a
sphere. Which one describes "reality"? Certainly not all three, yet
they are all studied in mathematics.
Math can be used as a model for reality, but that does not mean its
purpose is to model reality. It is a tool, nothing more, nothing less.
--
Will Twentyman
email: wtwentyman at copper dot net
G. A. Edgar
> So evidently some authoritative philosophers and mathematicians claim that
> (1) all of mathematics is "tautology" and (2) makes no claims about the real world.
> So what is the definition of a tautology, when is a mathematical statement a
> tautology and when is it no, and does mathematics make any claims about the
> world (what is the subject/object matter or mathematics?)?
Some claim that numbers, patterns, sets, relationships, functions,
etc. are "real". So instead of talking about "the real world", let's
talk about "the physical world". Connections between mathematics and
the physical world are what mathematicians call "applications" of
mathematics. (Some part of) mathematics is a "model" of (some aspect
of) the physical world. This is an important study, but it is not
itself mathematics. Those who confuse the model (in mathematics) with
the application (in the physical world) are making a mistake.
So... to what extent is mathematics relevant to the physical world? It
is sometimes said that mathematics is the science of patterns. To the
extent that the physical world exhibits patterns, it is natural that
mathematics can be applied to the physical world. [But, see above, not
that mathematics IS the physical world.] On the other hand, to the
extent that the physical world does NOT exhibit patterns, can it be
studied at all?
When someone says, for example, "mathematics shows that black holes
behave in a certain way", they should subdivide this into two claims:
(a) mathematics behaves a certain way; and (b) this mathematics
accuratately models the physical world.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Jesse F. Hughes
> When someone says, for example, "mathematics shows that black holes
> behave in a certain way", they should subdivide this into two claims:
> (a) mathematics behaves a certain way; and (b) this mathematics
> accuratately models the physical world.
You may want a look at Quine, especially his "Two Dogmas of
Empiricism" article for a criticism of this view. Quine claims that
the traditional distinction between analytic and synthetic
propositions is an illusion. I think that this is applicable to your
suggestion here, though I may be mistaken.
--
"Sorry, wakeup to the real world. You're on your own dependent on me
as your guide. Luckily for you, I'm self-correcting to a large extent,
so if the proof were wrong, I'd tell you. It's not wrong."
--- James Harris confirms that his proof is correct.
George Cox
>
> ...
>
> A tautology is something like "p or not-p" where "p" is a statement that
> is true or false. "p or not-p" where "p" is true no matter whether "p"
The line above should read
`is true or false. "p or not-p" no matter whether "p"'
> is true or false. That is not meant to be a definition--just a sloppy
> example. ...
Proginoskes
> progi...@email.msn.com (Proginoskes) writes:
>
> > If G is a planar graph, then the vertices G can be colored with
> > 5 colors (so that no two adjacent vertices have the same
> > color).
> >
> > This is a statement of the form "p implies q" which has some
> > substance to it.
>
> Those who advocate the view that mathematics has no content would
> claim that your statement above is a mere shorthand for the proper
> statement of the 5-Color theorem. The "proper" statement would be:
>
> If (conjunction of the axioms of graph theory) then if G is a
> planar graph then the vertices G can be colored with 5 colors (so
> that no two adjacent vertices have the same color).
>
> So, this is not (in their view) an example of a statement with content
> at all, since it is tautologous when stated "properly".
The form of this statment is
(A(1) AND A(2) AND A(3) AND ... A(n)) --> B
A statement with this FORM need not be true. Consider replacing the 5 with
a 3:
If (conjunction of the axioms of graph theory) then if G is a
planar graph then the vertices G can be colored with 3 colors (so
that no two adjacent vertices have the same color).
This statement is FALSE, even though it has the same form:
(A(1) AND A(2) AND A(3) AND ... A(n)) --> B
A tautology is a boolean formula which is ALWAYS true, no matter what you
replace the variables with. This was my point.
-- Christopher Heckman
P.S. If anyone claims that mathematics has no content, I dare them to provide
a (correct) proof of the Four Color Theorem.
Proginoskes
>
> With a consistent set of axioms: any statement of the form
> If (axioms) then (provable theorem)
> is a tautology (a statement that is always true).
No; it just says that (provable theorem) can be proved by (axioms).
A tautology is a boolean formula which is always true, no matter what
values the variables have. For instance, P OR NOT P.
This means that (2 + 2 = 5) OR NOT (2 + 2 = 5) is true. (FALSE OR TRUE =
TRUE.) This statement is true because of its FORM, not its content.
(It doesn't matter what P actually IS.)
P -> Q (implication) is NOT a tautology; the value of P -> Q depends
on the values of P and Q. It depends on the CONTENT of P and Q.
Elsewhere in this thread, I use the 5-Color Theorem* as an example:
If G is a planar graph, then G can be 5-colored. (The conclusion
means: For each vertex v of G, a number in the set {1,2,3,4,5} can
be assigned to v, and these assignments can be made so that no two
adjacent vertices (a pair with an edge between them) have the same
color).
Compare this with the following statement, which is FALSE:
If G is a planar graph, then G can be 3-colored.
My challenge is: How does changing a single number change the value of
the statement? Nothing in the _form_ of the statement has been changed.
> Notice that there is no claim made about "reality". There are three
> major forms of Geometry: Euclidean, perspective, and geometry on a
> sphere. Which one describes "reality"? Certainly not all three, yet
> they are all studied in mathematics.
Mathematics comes out of reality, not vice versa.
> Math can be used as a model for reality, but that does not mean its
> purpose is to model reality. It is a tool, nothing more, nothing less.
Right. Mathematics can be used to work out "what if ..." statements.
-- Christopher Heckman
* I'm avoiding the 4-Color Theorem (which is also true), because there is
not complete agreement that either of the proofs (Appel-Haken, or Robertson-
Sanders-Seymour-Thomas) is actually valid. However, the 5-Color Theorem
is undoubtedly true, proven by Kempe in the 1880s.
Will Twentyman
> Will Twentyman <wtwen...@read.my.sig> wrote in message news:<3eede248_4@newsfeed>...
>
>>With a consistent set of axioms: any statement of the form
>>If (axioms) then (provable theorem)
>>is a tautology (a statement that is always true).
>
>
> No; it just says that (provable theorem) can be proved by (axioms).
>
> A tautology is a boolean formula which is always true, no matter what
> values the variables have. For instance, P OR NOT P.
And those axioms, if taken as true, will always result in the provable
theorem being true. Therefor: you have either T->T, F->T, or F->F, all
of which are true. Therefor, it is a tautology.
> This means that (2 + 2 = 5) OR NOT (2 + 2 = 5) is true. (FALSE OR TRUE =
> TRUE.) This statement is true because of its FORM, not its content.
> (It doesn't matter what P actually IS.)
>
> P -> Q (implication) is NOT a tautology; the value of P -> Q depends
> on the values of P and Q. It depends on the CONTENT of P and Q.
>
You are assuming that P and Q are independent of each other. The fact
that the example mentions provability implies this is not the case.
> Elsewhere in this thread, I use the 5-Color Theorem* as an example:
>
> If G is a planar graph, then G can be 5-colored. (The conclusion
> means: For each vertex v of G, a number in the set {1,2,3,4,5} can
> be assigned to v, and these assignments can be made so that no two
> adjacent vertices (a pair with an edge between them) have the same
> color).
>
> Compare this with the following statement, which is FALSE:
>
> If G is a planar graph, then G can be 3-colored.
Definition: a graph is planar if it can be 3-colored. This change in
the definition of coloring/planarity would make your false statement
true. It must exist in a context of axioms/definitions.
> My challenge is: How does changing a single number change the value of
> the statement? Nothing in the _form_ of the statement has been changed.
You have missed the key in the form: namely the fact that a theorem is
_provable_. "If (axioms) then (provable theorem)" represents a
collection of tautologies since the original post I saw mentioned them.
when you change from 5 to 3 you changed the consequent from a provable
theorem to a false conclusion. Thus, you changed the form.
Note: without including the axioms/definitions of graph theory that we
accept, we are NOT discussing tautologies, but merely provable theorems.
>>Notice that there is no claim made about "reality". There are three
>>major forms of Geometry: Euclidean, perspective, and geometry on a
>>sphere. Which one describes "reality"? Certainly not all three, yet
>>they are all studied in mathematics.
>
>
> Mathematics comes out of reality, not vice versa.
>
Many pieces of math were been invented before there was a practical
application. i as the square root of -1 was invented long before it had
any apparent use. Unless you want to argue that math IS reality, I have
to disagree.
Jesse F. Hughes
> jes...@cs.kun.nl (Jesse F. Hughes) wrote in message
> news:<87n0gis...@phiwumbda.localnet>...
>
>> progi...@email.msn.com (Proginoskes) writes:
>>
>> > If G is a planar graph, then the vertices G can be colored with
>> > 5 colors (so that no two adjacent vertices have the same
>> > color).
>> >
>> > This is a statement of the form "p implies q" which has some
>> > substance to it.
>>
>> Those who advocate the view that mathematics has no content would
>> claim that your statement above is a mere shorthand for the proper
>> statement of the 5-Color theorem. The "proper" statement would be:
>>
>> If (conjunction of the axioms of graph theory) then if G is a
>> planar graph then the vertices G can be colored with 5 colors (so
>> that no two adjacent vertices have the same color).
>>
>> So, this is not (in their view) an example of a statement with content
>> at all, since it is tautologous when stated "properly".
>
> The form of this statment is
>
> (A(1) AND A(2) AND A(3) AND ... A(n)) --> B
>
> A statement with this FORM need not be true.
Actually, it should be universally quantified. It's essential that we
look at the first- or perhaps higher-order formalization. The
propositional formalization is far too crude to give any hint of what
one means when he says math is tautologous. See below.
> Consider replacing the 5 with
> a 3:
>
> If (conjunction of the axioms of graph theory) then if G is a
> planar graph then the vertices G can be colored with 3 colors (so
> that no two adjacent vertices have the same color).
>
> This statement is FALSE, even though it has the same form:
>
> (A(1) AND A(2) AND A(3) AND ... A(n)) --> B
>
> A tautology is a boolean formula which is ALWAYS true, no matter what you
> replace the variables with. This was my point.
I agree that's the commonest meaning of the word tautology, but that's
not at all the relevant meaning for the claim "mathematical theorems
are mere tautologies". The relevant meaning there is that
mathematical theorems are necessary truths. In this case, it means
that they express a truth of first-order logic.
The statement is trivially false (and stupid) if we try to understand
it in terms of propositional logic.
--
Jesse Hughes
"And a journal can beg me for the right to publish it [...] because
I'd rather see it in "People" magazine [...]"
--James Harris on his simple proof of Fermat's last theorem
James Whitehead
"Proginoskes" <progi...@email.msn.com> wrote in message >
[...]
P.S. If anyone claims that mathematics has no content, I dare them to
provide
> a (correct) proof of the Four Color Theorem.
The crosspost arose out of confusion as to the *subject* of mathematics.
Lets compare this to physics - where some empirical data can refute a
proposition of physics, can this occur in mathematics- i thought not. The
subject may not be empty? bit in which case what - other than itself is the
subject of mathematics? Isn't the only content/subject the idea of equality?
And isnt this dodgey?
Mr. Vibrating
Knowledge of nature" by Hans Hahn (vienna, 1933). This looks the source of
the notion that all math is "tautologies".
I reproduce it here for your convenience. Its an easy read.
Logic, Mathematics and Knowledge of Nature
BY HANS HAHN
(TRANSLATED BY ARTHUR PAP)
EVEN A CURSORY glance at the statements of physics shows that they are
obviously of a very diverse character. There are statements like "if a
stretched string is plucked, a tone is heard" or "if a ray of sunlight is
passed through a glass prism, then a colored band, interspersed with dark
lines, is visible on a screen placed behind the prism," which can be tested
at any time by observation. We also find statements like "the sun contains
hydrogen," "the satellite of Sirius has a density of about 60,000," "a
hydrogen atom consists of a positively charged nucleus around which a
negatively charged electron revolves," which cannot by any means be tested
by immediate observation, but which is made only on the basis of theoretical
considerations and likewise are testable only with the help of theoretical
considerations. And thus we are confronted by the urgent question: what is
_the relationship between observation and theory in physics - and not just
in physics but in science generally. For there is but one science, and
wherever there is scientific investigation it proceeds ultimately according
to the same methods; only we see everything with the greatest clarity in the
case of physics, because it is the most advanced, neatest, most scientific
of all the sciences. And in physics, indeed, the interaction of observation
and theory is
[footnote: This contribution comprises the first four sections of the
pamphlet "Logik, Mathematik und Naturerkennen," published in Vienna in 1933
as the second volume of the series entitled "Einheitswissenschaft." It is
reproduced here with the kind permission of Mrs. Lilly Hahn, Gerold & Co.,
Vienna, and Professor Rudolf Carnap, the coeditor of Einheitswissenschafl.
The last two sections of Hahn's pamphlet which are omitted do not deal with
the nature of logical or mathematical propositions.]
[147]
[ 148 ] HANS HAHN
especially pronounced, even officially recognized by the institution of
special professorships for experimental physics and for theoretical physics.
Now, presumably the usual conception is roughly speaking the following: we
have two sources of knowledge, by means of which we comprehend "'the world,"
"the reality" in which we are "placed": experience, or observation on the
one hand, and thinking on the other. For example, one is engaged in
experimental physics or in theoretical physics according to one's using the
one or the other of these sources of knowledge in physics.
Now, in philosophy we find a time-honored controversy about these two
sources of knowledge: which parts of our knowledge derive from observation,
are "a posteriori," and which derive from thinking, are "a priori"? Is one
of these sources of knowledge superior to the others, and if so, which?
From the very beginning philosophy has raised doubts about the reliability
of observation (indeed, these doubts "are perhaps the 'source of all
philosophy). It is quite understandable that such doubts arose: they spring
from the belief that sense-perception is frequently deceptive. At sunrise or
at sunset the snow on distant mountains appears red, but "in reality" it is
surely white! A stick which is immersed in water appears crooked, but "in
reality" it is surely straight! If a man recedes from me, he appears smaller
and smaller to me, but surely he does not change size "in reality"!
Now, although all the phenomena to which we have been referring have long
since been accounted for by physical theories, so that nobody any longer
regards them as deceptions caused by sense-perception, the consequences
which flow from this primitive, long discarded conception still exert a
powerful influence. One says: if observation is sometimes deceptive, perhaps
it is always so! Perhaps everything disclosed by the senses is mere
illusion! Everybody knows the phenomenon of dreams, and everybody knows how
difficult it is at
times to decide whether a given experience was "real life" or "a mere
dream." Perhaps, then, whatever we observe is merely a dream object!
Everybody knows that hallucinations occur, and that they can be so vivid
that the subject cannot be dissuaded from taking his hallucination for
reality. Perhaps then, whatever we observe is only a hallucination! If we
look through appropriately polished lenses, everything appears distorted;
who knows whether perhaps we do not always, unknowingly, look at the world
as it were through distorting glasses, and therefore see everything
distorted, different from what it really is! This is one of the basic themes
of the philosophy of Kant.
Logic, Mathematics and Knowledge of Nature [ 149 ]
But let us return to antiquity. As we said, the ancients believed that they
were frequently deceived by observation, But nothing of this kind ever
happened in the case of thought: there were plenty of delusions of sense,
but no delusions of thought! And thus, as confidence in observation got
shaken, the belief may have arisen that thinking is a method of knowledge
which is absolutely superior to observation, indeed the only reliable
method of knowledge: observation discloses mere appearance, thought alone
grasps true being.
This, "rationalistic," doctrine that thinking is a source of knowledge
which surpasses observation, that it is indeed the only reliable source" of
knowledge, has remained dominant from the climax of Greek philosophy until
modem times. I cannot even intimate what peculiar fruits matured on the tree
of such knowledge. At any rate, they proved to have extraordinarily little
nourishing value; and thus the "empiricist" reaction, originating in
England, slowly gained the upper hand, supported by the tremendous success
of modem natural
science-the philosophy which teaches that observation is superior to
thought^ indeed is the only source of knowledge; nihil est in intellectu,
quod non prius fuerit in sensu; in English: "nothing is in the intellect
which was not previously in the senses."
But at once this empiricism faces an apparently insuperable difficulty: how
is it to account for the real validity of logical and mathematical
statements? Observation discloses to me only the transient, it does not
reach beyond the observed; there is no bond that would lead from one
observed fact to another, that would compel future observations to have the
same result as those already made.
The laws of logic and mathematics, however, claim absolutely universal
validity: that the door of my room is now closed, I know by observation;
next time I observe it it may be open. That heated bodies expand, I know by
observation; yet the very next observation may show that some heated body
does not expand; but that two and two make four, holds not only for the case
in which I verify it bycounting I know with certainty that it holds always
and everywhere. Whatever I know by observation could be otherwise: the door
of my room might have been open now, I can easily imagine it; and I can
easily imagine that a body does not expand on being heated; but two and two
could not occasionally make five, I cannot imagine in any way what it would
be like for twice two to equal five. The conclusion seems inevitable: since
the propositions of logic and mathematics have absolutely universal
validity, are
apodeictically certain, since it must be as they say and cannot be
otherwise, these propositions cannot be derived from experience. In view of
the tre-
[ 150] HANS HAHN
mendous importance of logic and mathematics in the system of our knowledge,
empiricism, therefore, seems to be irrevocably refuted. To be sure, in spite
of all this the older empiricists haye attempted to found logic and
mathematics upon experience. According to them we now believe that something
must be this way and cannot be otherwise simply because the relevant
experience is so old and the relevant observations have been repeated
innumerable times. On this view, therefore, it is entirely conceivable that,
just as an observation might show that a heated body does not expand, two
and two might sometimes make five. This is alleged to have escaped our
notice so far because it happens with such extraordinary rarity, like
finding a piece of four-leaved clover which for superstitious people is a
sign of good luck, an occurrence which is not so very rare - how much more
promise of fortune would there be in the discovery of a case where two and
two make five! One can safely say that on closer sight these attempts to
derive logic and mathematics from experience are fundamentally
unsatisfactory, and it is doubtful whether anybody seriously holds this view
today.
Rationalism and empiricism having thus, as it were, suffered
shipwreck-rationalism, because its fruits lacked nourishing value,
empiricism, because it could not do justice to logic and
mathematics -dualistic conceptions gained the upper hand, with the view that
thinking and observation are equally legitimate sources of knowledge which
are both indispensable to our comprehension of the world and play a
distinctive role in the system of our knowledge. Thought grasps the most
general laws of all being, as formulated perhaps in logic and mathematics;
observation provides the detailed filling of this framework. As regards the
limits set to the two sources of knowledge, opinions diverge.
Thus it is, for instance, disputed whether geometry is a priori or a
posteriori, whether it is based on pure thinking or on experience. And the
same dispute is encountered in connection with the most fundamental physical
laws, e.g. the law of inertia, the laws of the conservation of mass and
energy, the law of attraction of masses: all of them have already been
acclaimed as a priori, as necessities of thought, by various
philosophers-but always after they had been established and well confirmed
as empirical laws in physics. This was bound to lead to a skeptical
attitude, and as a matter of fact there is probably a prevalent tendency
among physicists to regard the framework which can be grasped by pure
thinking as being as wide and general as possible, and to acknowledge
Logic, Mathematics and Knowledge of Nature [ 151 ]
experience as the source of our knowledge of everything that is somehow
concrete. The usual conception, then, may be described roughly as follows:
from experience we learn certain facts, which we formulate as "laws of
nature"; but since we grasp by means of thought the most general lawful
connections (of a logical and mathematical character) that pervade reality,
we can control nature on the basis of facts disclosed by observation to a
much larger extent than it has actually been observed. For we know in
addition that anything which can be deduced from observed facts by
application of logic and mathematics must be found to exist. According to
this view, the experimental physicist provides knowledge of laws of nature
by
direct observation. The theoretical physicist thereafter enlarges this
knowledge tremendously by thinking, in such a way that we are in a position
also to assert propositions about processes that occur far from us in space
and time and about processes which, on account of their magnitude or
minuteness, are not directly observable but which are connected with what is
directly observed by the most general laws of being, grasped by thought, the
laws of logic and mathematics. This view seems to be strongly supported by
numerous discoveries that have been made with the help of theory, like-to
mention just some of the best known-the calculation of the position of the
planet Neptune by Leverrier, the calculation of electric waves by Maxwell,
the calculation of the bending of light rays in the gravitational field of
the sun by Einstein and the calculation of the red-shift in the solar
spectrum, also by Einstein.
Nevertheless we are of the opinion that this view is entirely untenable.
For on closer analysis it appears that the function of thought is
immeasurably more modest than the one ascribed to it by this theory. The
idea that thinking is an instrument for learning more about the world than
has been observed, for acquiring knowledge of something that has absolute
validity always and everywhere in the world, an instrument for grasping
general laws of all being, seems to us wholly mystical. Just how should it
come to pass that we could predict the necessary outcome of an observation
before having made it? Whence should our thinking derive an executive power,
by which it could compel an observation to have this rather than that
result? Why should that which compels our thoughts also compel the course of
nature? One would have to believe in some miraculous pre-established harmony
.between the course of our thinking arid the course of nature, an idea which
is highly mystical and ultimately theological.
There is no way out of this situation except a return to a purely
[ 152 ] HANS HAHN
empiricist standpoint, to the view that observation is the only source of
knowledge of facts: there is no a priori knowledge about matters of fact,
there is no "material" a priori. However, we shall have to avoid the error
committed by earlier empiricists, that of interpreting the propositions of
logic and mathematics as mere facts of experience. We must look out for a
different interpretation of logic and mathematics.
II
Let us begin with logic. The old conception of logic is approximately as
follows: logic is the account of the most universal properties of things,
the account of those properties which are common to all things; just as
ornithology is the science of birds, zoology the science of all animals,
biology the science of all living beings, so logic is the science of all
things, the science of being as such. If this were the case, it would remain
wholly unintelligible whence logic derives its certainty. For we surely do
not know all things. We have not observed everything and hence we cannot
know how everything behaves. Our thesis, on the contrary, asserts: logic
does not by any means treat of the totality of things, it does not treat of
objects at all but only of our way of speaking about objects; logic is
first generated By language. The certainty and universal validity, or
better, "the Irrefutability of a proposition of logic derives just from the
fact that It says nothing about objects of any kind. Let us clarify the
point by an example. I talk about a well-known plant: I describe it, as is
done in botanical reference books, in
terms of the number, color and form of its blossom leaves, its calyx leaves,
its stamina, the shape of its leaves, its stem, its root, etc., and I make
the stipulation: let us call any plant of this kind "snow rose," but let us
also call it "helleborus niger." Thereupon I can pronounce with absolute
certainty the universally valid proposition: "every snow rose is a
helleborus niger." It is certainly valid, always and everywhere; it is not
refutable by any sort of observation; but it says nothing at all about
facts. I learn nothing from it about the plant in question, when it is in
bloom, where it may be found, whether it is common or rare. It tells me
nothing about the plant; it cannot be disconfirmed by any observation. This
is the basis of its certainty and universal validity. The statement merely
expresses a convention concerning the way we wish to talk about the plant in
question.
Logic, Mathematics and Knowledge of Nature [ 153 ]
Similar considerations apply to the principles of logic. Let us make the
point with reference to the two most famous laws of logic: the law of
contradiction and the law of the excluded middle. Take, for example, colored
objects. We learn, by training as I am tempted to say, to apply the
designation "red" to some of these objects, and we stipulate that the
designation "not red" be applied to all other objects. On the basis of this
stipulation we now can assert with absolute certainty the proposition that
there is no object to which both the designation "red" and the designation
"not red" is applied. It is customary to formulate this briefly by saying
that nothing is both red and not red. This is the law of contradiction. And
since we have stipulated that the designation "red" is to be applied to some
objects and the designation "not red" to all other objects, we can likewise
pronounce with absolute certainty the proposition:
everything is either designated as "red" or as "not red," which it is
customary to formulate briefly by saying that everything is either red or
not red. This is the law of the excluded middle. These two propositions, the
law of contradiction and the law of the excluded middle, say nothing at all
about objects of any kind. They do not tell me of any of them whether they
are red or not red, which color they have, or anything else. They merely
stipulate a method for applying the designations "red" and "not red" to
objects, i.e. they prescribe a method of speaking about things. And their
universal validity and certainty, their irrefutability,
just derives from the fact that they say nothing at all about objects. The
same is to be said of all the other principles of logic. We shall presently
return to this point. But first let us insert another consideration. We have
previously maintained that there can be no material a priori, i.e. no a
priori knowledge about matters of fact. For we cannot know the outcome of an
observation before the latter takes place. We have made clear to ourselves
that no material a priori is contained in the laws of contradiction and of
excluded middle, since they say nothing about facts. There are those,
however, who would perhaps admit that the nature of the laws of logic is as
described, yet would insist that there is a material a priori
elsewhere, e.g. in the statement "nothing is both red and blue" (of course
what is meant is: at the same time and place) which is alleged to express
real a priori knowledge about the nature of things. Even before having made
any observation, they say, one can predict with absolute certainty that it
will not disclose a thing which is both blue and red; and it is maintained
that such a priori knowledge is obtained by "eidetic insight" or an
intuitive grasp of
[ 154 ] HANS HAHN
the essence of colors. If one desires to adhere to our thesis that there is
no kind of material a priori, one must somehow face statements like "nothing
is both blue and red." I want to attempt this in a few suggestive words,
though they cannot by any means do full justice to this problem which is not
easy. It surely is correct that we can say with complete certainty before
having made any observations: the latter will not show that a thing is both
blue and red-just as we can say with complete certainty that no observation
will yield the result that a thing is both red and not red, or that a snow
rose is not a helleborus niger. The first statement, however, is not a case
of a material a priori any more than the second and third. Like the
statements "every snow rose is a helleborus niger": and "nothing is both red
and not red," the statement "nothing is both blue and red" says nothing at
all about the nature of things; it
likewise refers only to our proposed manner of speaking about objects, of
applying designations to them. Earlier we said: there are some objects that
we call "red," every other object we call "not red," and from this we derive
the laws of contradiction and excluded middle. Now we say: some objects we
call "red," some other objects we call "blue," and other objects again we
call "green," etc. But if it is in this way that we ascribe color
designations to objects, then we can say with certainty in advance: in this
procedure no object is designated both as "red" and as "blue," or more
briefly: no object is both red and blue. The reason why we can say this with
certainty is that we have regulated the ascription of color designations to
objects in just this way.
We see, then, that there are two totally different kinds of statements:
those which really say something about objects, and those which do not say
anything about objects but only stipulate rules for speaking about objects.
If I ask "what is the color of Miss Ema's new dress?" and get the answer
"Miss Ema's new dress is not both red and blue (all over)," then no
information about this dress has been given to me at all. I have been made
no wiser by it. But if I get the answer "Miss Erna's new dress is red," then
I have received
some genuine information about the dress.
Let us clarify this distinction in terms of one more example. A statement
which really says something about the objects which it mentions, is the
following: "If you heat this piece of iron up to 800°, it will turn red, if
you heat it up to 1300°, it will turn white." What makes the difference
between this statement and the statements cited above, which say nothing
about facts? The application of temperature designations to objects is
independent of the appli-
Logic, Mathematics and Knowledge of Nature [ 155 ]
cation of color designations, whereas the color designations "red" and "not
red," or "red" and "blue" are applied to objects in mutual
dependence. The statements "Miss Erna's new dress is either red or not red"
and "Miss Erna's new dress is not both red and blue"
merely express this dependence, hence make no assertion about that dress,
and are for that reason absolutely certain and irrefutable.
The above statement about the piece of iron, on the other hand, relates
independently given designations, and therefore really says
something about that piece of iron and is for just that reason not certain
nor irrefutable by observation.
The following example may make the difference between these two kinds of
statements particularly clear. If someone were to tell
me; "I raised the temperature of this piece of iron to 800° but it did not
turn red," then I would test his assertion; the result of
the test may be that he was lying, or that he was the victim of an illusion,
but perhaps it would turn out that-contrary to my previous
beliefs-there are cases where a piece of iron heated to 800° does not become
red-hot, and in that case I would just change my opinion
about the reaction of iron to heating. But if someone tells me "I raised the
temperature of this piece of iron to 800°, and this made
it turn both red and not red" or "it became both red and white,"then I will
certainly make no test whatever. Nor will I say "he has
told me a lie," or "he has become the victim of an illusion" and it is quite
certain that I would not change my beliefs about the
reaction of iron to heating. The point is-it is best to express it in
language which any card player is familiar with-that the man has
revoked: he has violated the rules in accordance with which we want to
speak, and I shall refuse to speak with him any longer. It
is as though one attempted in a game of chess to move the bishop
orthogonally. In this case too, I would not make any tests, I would
not change my beliefs about the behavior of things, but I would refuse to
play chess with him any longer.
To sum up: we must distinguish two kinds of statements: those which say
something about facts and those which merely express
the way in which the rules which govern the application of words to facts
depend upon each other. Let us call statements of the latter
kind tautologies: they say nothing about objects and are for this very
reason certain, universally valid, irrefutable by observation;
whereas the statements of the former kind are not certain suild are
refutable by observation. The logical laws of contradiction and
of the excluded middle are tautologies, likewise, e.g., the statement
"nothing is both red and blue."
[ 156 ] HANS HAHN
And now we maintain that in the same way all the other laws of logic are
tautologies. Let us, therefore, return to logic once more
in order to clarify the matter by an example. As we said, the designation
"red" is applied to certain objects and the convention is
adopted of applying the designation "not red" to any other object. It is
this convention about the use of negation which is expressed
by the laws of contradiction and of the excluded middle. Now we add the
convention-still taking our examples from the domain of
colors-that any object which is called "red" is also to be called "red or
blue," "blue or red," "red or yellow," "yellow or red," etc.,
that every object which is called "blue," is also called "blue or red", "red
or blue," "blue or yellow," "yellow or blue," etc., and so on.
On the basis of this convention, we can again assert with complete certainty
the proposition: "every red object is either red or blue."
This is again a tautology. We do not speak about the objects, but only about
our manner of talking about them.
If once more we remind ourselves of the way in which the designations
"red," "not red," "blue," "red or blue," etc. are applied
to objects, we can moreover assert with complete certainty and
irrefutability: everything to which both designations "red or blue"
and "not red" are applied, is also designated as "blue"-which is usually put
more briefly: if a thing is red or blue and not red, then
it is blue. Which is again a tautology. No information about the nature of
things is contained in it, it only expresses the sense in
which the logical words "not" and "or" are used.
Thus we have arrived at something fundamental: our conventions regarding
the use of the words "not" and "or" is such that in
asserting the two propositions "object A is either red or blue" and "object
A is not red," I have implicitly already asserted "object A
is blue." This is the essence of so-called logical deduction. It is not,
then, in any way based on real connections between states of
affairs, which we apprehend in thought. On the contrary, it has nothing at
all to do with the nature of things, but derives from our manner
of speaking about things. A person who refused to recognize logical
deduction would not thereby manifest a different belief from mine
about the behavior of things, but he would refuse to speak about things
according to the same rules as I do. I could not convince
him, but I would have to refuse to speak with him any longer, just as I
should refuse to play chess with a partner who insisted on
moving the bishop orthogonally.
What logical deduction accomplishes, then, is this: it makes us
Logic, Mathematics and Knowledge of Nature [ 157 ]
aware of all that we have implicitly asserted-on the basis of conventions
regarding the use of language-in asserting a system of
propositions, just as, in the above example, "object A is blue" is
implicitly asserted by the assertion of the two propositions "object
A is red or blue" and "object A is not red."
In saying this we have already suggested the answer to the question, which
naturally must have forced itself on the mind of every
reader who has followed our argument: if it is really the case that the
propositions of logic are tautologies, that they say nothing about
objects, what purpose does logic serve?
The logical propositions which were used as illustrations derived from
conventions about the use of the words "not" and "or" (and
it can be shown that the same holds for all the propositions of so called
prepositional logic). Let us, then, first ask for what purpose
the words "not" and "or" are introduced into language. Presumably the reason
is that we are not omniscient. If I am asked about the
color of the dress worn by Miss Erna yesterday, I may not be able to
remember its color. I cannot say whether it was red or blue or
green; but perhaps I will be able to say at least "it was not yellow." Were
I omniscient, I should know its color. There would be no need
to say "it was not yellow": I could say "it was red." Or again: my daughter
has written to me that she received a cocker-spaniel as a
present. As I have not seen it yet, I do not know its color; I cannot say
"it is black" nor "it is brown"; but I am able to say "it is black
or brown." Were I omniscient, I could do without this "or" and could say
immediately "it is brown."
Thus logical propositions, though being purely tautologous, and logical
deductions, though being nothing but tautological trans-
formations, have significance for us because we are not omniscient. Our
language is so constituted that in asserting such and such prop-
ositions we implicitly assert such and such other propositions-but we do not
see immediately all that we have implicitly asserted in this
manner. It is only logical deduction that makes us conscious of it. I
assert, e.g., the propositions "the flower which Mr. Smith wears
in his buttonhole, is either a rose or a carnation," "if Mr. Smith wears a
carnation in his buttonhole, then it is white," "the flower
which Mr. Smith wears in his buttonhole is not white." Perhaps I am not
consciously aware that I have implicitly asserted also "the
flower which Mr. Smith wears in his buttonhole is a rose"; but logical
deduction brings it to my consciousness. To be sure, this does
not mean that I know whether the flower which Mr. Smith wears
[158 ] HANS HAHN
in his buttonhole really is a rose; if I notice that it is not a rose, then
I must not maintain my previous assertions-otherwise I sin
against the rules of speaking, I revoke.
III
If I have succeeded in clarifying somewhat the role of logic, I may now be
quite brief about the role of mathematics. The proposi-
tions of mathematics are of exactly the same kind as the propositions of
logic: they are tautologous, they say nothing at all about the
objects we want to talk about, but concern only the manner in which we want
to speak of them. The reason why we can assert apodeictically
with universal validity the proposition: 2 + 3 == 5, why we can say even
before any observations have been made, and can say it with
complete certainty, that it will not turn out that 2 -)- 3 = 7, is that by
"2 + 3" we mean the same as by "5"-just as we mean the same by "helleborus
niger" as by "snow rose." For this reason no botanical investigation,
however subtle, could disclose that an instance of the species "snow
rose" is not a helleborus niger. We become aware of meaning the same by "2 +
3" and by "5," by going back to the meanings of "2," "3," "5,"
"+," and making tautological transformations until we just see that "2 +3"
means the same as "5." It is such successive tautological
transformation that is meant by "calculating"; the operations of addition
and multiplication which are learnt in school are directives for
such tautological transformation; every mathematical proof is a succession
of such tautological transformations. Their utility, again, is due
to the fact that, for example, we do not by any means see immediately that
we mean by "24 X 31" the same as by "744"; but if we calculate
the product "24 X 31," then we transform it step by step, in such a way that
in each individual transformation we recognize that on
the basis of the conventions regarding the use of the signs involved (in
this case numerals and the signs "+" and "X") what we mean
after the transformation is still the same as what we meant before it, until
finally we become consciously aware of meaning the same by
"744" as by "24 X 31."
To be sure, the proof of the tautological character of mathematics is not
yet complete in all details. This is a difficult and
arduous task; yet we have no doubt that the belief in the tautological
character of mathematics is essentially correct.
There has been prolonged opposition to the interpretation of mathematical
statements as tautologies; Kant contested the tauto-
Logic, Mathematics and Knowledge of Nature [ 159 ]
logical character of mathematics emphatically, and the great mathematician
Henri Poincare, to whom we are greatly indebted also for
philosophical criticism, went so far as to argue that since mathematics
cannot possibly be a huge tautology, it must somewhere contain an
a priori principle. Indeed, at first glance it is difficult to believe that
the whole of mathematics, with its theorems that it cost such
labor to establish, with its results that so often surprise us, should admit
of being resolved into tautologies. But there is just one little
point which this argument overlooks: it overlooks the fact that we are not
omniscient. An omniscient being, indeed, would at once
know everything that is implicitly contained in the assertion of a few
propositions. It would know immediately that on the basis of the
conventions concerning the use of the numerals and the multiplication sign,
"24X31" is synonymous with "744." An omniscient
being has no need for logic and mathematics. We ourselves, how ever, first
have to make ourselves conscious of this by successive
tautological transformations, and hence it may prove quite surprising to us
that in asserting a few propositions we have implicitly also as-
serted a proposition which seemingly is entirely different from them, or
that we do mean the same by two complexes of symbols which are
externally altogether different.
IV
And now let us be clear what a world-wide difference there is between our
conception and the traditional-perhaps one may say:
platonizing-conception, according to which the world is made in accordance
with the laws of logic and mathematics ("God is perennially doing
mathematics"), and our thinking, a feeble reflection of God's omniscience,
is an instrument given to us for comprehending the eternal laws of
the world. No! Our thinking cannot give insight into any sort of reality. It
cannot bring us information of any fact in the world. It only
refers to the manner in which we speak about the world. All it can do is to
transform tautologically what has been said. There is no
possibility of piercing through the sensible world disclosed by observation
to a "world of true being": any metaphysics is impossible!
Impossible, not because the task is too difficult for our human thinking,
but because it is meaningless, because every attempt to do
metaphysics is an attempt to speak in a way that contravenes the agreement
as to how we wish to speak, comparable to the attempt to capture
the queen (in a game of chess) by means of an orthogonal move of the bishop.
[ 160 ] HANS HAHN
Let us return now to the problem which was our point of departure: what is
the relationship between observation and theory
in physics? We said that the usual view was roughly this: experience teaches
us the validity of certain laws of nature, and since our
thinking gives us insight into the most general laws of all being, we know
that likewise anything which is deducible from these laws of
nature by means of logical and mathematical reasoning must be found to
exist. We see now that this view is untenable; for thinking does not
grasp any sort of laws of being. Never and nowhere, then, can thought supply
us with knowledge about facts that goes beyond the observed. But
what, then, should we say about the discoveries made by means of theory on
which, as we pointed out, the usual view so strongly relies for
its support? Let us ask ourselves, e.g., what was involved in the
computation of the position of the planet Neptune by Leverrier! Newton
noticed that the familiar motions, celestial as well as terrestrial, can be
well described in a unified way by the assumption that between
any two mass points a force of attraction is exerted which is proportional
to their masses and inversely proportional to the square of their
distance. And it is because this assumption enables us to give a
satisfactory description of the familiar motions, that he made it, i.e. he
asserted tentatively, as an hypothesis, the law of gravitation: between any
two mass points there is a force of attraction which is
proportional to their masses and inversely proportional to the square of
their distance. He could not pronounce this law as a certainty, but
only as an hypothesis. For nobody can know that such is really the behavior
of every pair of mass points nobody can observe all mass points.
But having asserted the law of gravitation, one has implicitly asserted many
other propositions, that is, all propositions which are
deducible from the law of gravitation (together with data immediately
derivable from observation) by calculation and logical inference. It is
the task of theoretical physicists and astronomers to make us conscious of
everything we implicitly assert along with the law of gravitation.
And Leverrier's calculations made people aware that the assertion of the law
of gravitation implies that at a definite time and definite
place in the heavens a hitherto unknown planet must be visible. People
looked and actually saw that new planet-the hypothesis of the law of
gravitation was confirmed. But it was not Leverrier's calculation that
proved that this planet existed, but the looking, the observation.
This observation could just as well have had a different result. It could
just as well have happened that nothing was visible at the computed
place in the heavens-in which case the law of gravitation would
Logic, Mathematics and Knowledge of Nature [ 161 ]
not have been confirmed and one would have begun to doubt whether it is
really a suitable hypothesis for the description of the
observable motions. Indeed, this is what actually happened later: in
asserting the law of gravitation, one implicitly asserts that at a
certain time the planet Mercury must be visible at a certain place in the
heavens. Whether it would actually be visible at that time at
that place, only observation could disclose; but observations showed that it
was not visible at exactly the required position in the heavens.
And what happened? They said: since in asserting the law of gravitation we
implicitly assert propositions which are not true, we cannot
maintain the hypothesis of the law of gravitation. Newton's theory of
gravitation was replaced by Einstein's.
It is not the case, then, that we know through experience that certain laws
of nature are valid, and-since by our thinking we
grasp the most general laws of all being-therefore also know that whatever
is deducible from these laws by reasoning must exist. On
the contrary, the situation is this: there is not a single law of nature
which we know to be valid; the laws of nature are hypotheses which
we assert tentatively. But in asserting such laws of nature we implicitly
assert also many other propositions, and it is the task of
thinking to make us conscious of the implicitly asserted propositions. So
long, now, as these implicitly asserted propositions, to the extent
that they are about the directly observable, are confirmed by observation,
these laws of nature are confirmed and we adhere to them; but if
these implicitly asserted propositions are not confirmed by observation,
then the laws of nature have not been confirmed and are replaced by
others.
*****
James Whitehead
> The following is a scan of the original article : Logic, Mathematics and
> Knowledge of nature" by Hans Hahn (vienna, 1933). This looks the source of
> the notion that all math is "tautologies".
Well the Tractatus was written from about 1914 onwards - publishd 1921-
"There are no such things as analytic propositions" L.W. Notebooks
30.10.1914 and even earlier in letters to Russell "All propositions of
logic are generalisations of tautologies..." 1913 -
But perhaps he got the notion from elsewhere....
but the quote was interesting...
Jesse F. Hughes
> The following is a scan of the original article : Logic, Mathematics and
> Knowledge of nature" by Hans Hahn (vienna, 1933). This looks the source of
> the notion that all math is "tautologies".
I wouldn't guess that it's the source of that notion. It's just the
presentation with which I am most familiar.
--
"[I]t's good for the economy to charge for intellectual property, so
open source software cannot be good, while Microsoft is the most
far-thinking company around and is doing it all for the good of the
public." -- Linus Torvalds paraphrases Microsoft VP Craig Mundie
|-|erc
> open source software cannot be good, while Microsoft is the most
> far-thinking company around and is doing it all for the good of the
> public." -- Linus Torvalds paraphrases Microsoft VP Craig Mundie
OT : those BASTARDS at microsoft took JAVA out of Windows,
XP no longer has JAVA.
Want to run a program from a browser, uh, not on this planet.
Put back the JVM, include VRML in it (reading a book on it now
its just intuitive), then I can program my 3D virtual chess room
with 50 lines of code to run instantly on all browsers. UH someone
write a free OS thats backwards compatible to MS.
If you've tried to program anything on MS, you soon realise all
the technology is rigged to stop users programming anything. If
the new fangled sandbox object high level program_what_we_want_
you_to archiology starts to do useful things, just wait for the upgrade
to introduce a new high level that disables this with a new flora of
useless fruit. Draw a FUCKING POLYGON, takes 100 levels of
function calls with 3 plug ins. they're fucking jokers.
They're to you like the truman company is to me. Every domain I start
to accomplish anything in gets yanked, oh you do so well lets start
from scratch AGAIN see what you can do
</rant>
Herc
G*rd*n
>
> "Proginoskes" <progi...@email.msn.com> wrote in message >
>
> [...]
>
> P.S. If anyone claims that mathematics has no content, I dare them to
> provide
> > a (correct) proof of the Four Color Theorem.
>
> The crosspost arose out of confusion as to the *subject* of mathematics.
> Lets compare this to physics - where some empirical data can refute a
> proposition of physics, can this occur in mathematics-
We observe conjectures which may seem true but not be proven
for a long time, for example the map-coloring problem or
Fermat's famous theorem. If mathematics were informally
tautologous in the sense that every consequence of every axiom
were immediately obvious, then these problems shouldn't take
centuries to solve. So what is this space that has to be
taken up before the solution comes into view?
> i thought not. The
> subject may not be empty? bit in which case what - other than itself is the
> subject of mathematics? Isn't the only content/subject the idea of equality?
> And isnt this dodgey?
Mr. Vibrating
Knowledge of nature" by Hans Hahn (vienna, 1933). This looks the source of
the notion that all math is "tautologies".
I reproduce it here for your convenience. Its an easy read.
[ 148 ] HANS HAHN
[ 150] HANS HAHN
[ 152 ] HANS HAHN
II
does not by any means treat of the totality of things, it does not treat of
Mr. Vibrating
>
> "Mr. Vibrating" <eastwood...@yahoo.com> wrote in message
> news:pEFHa.11829$Jw6.4...@news1.news.adelphia.net...
> > The following is a scan of the original article : Logic, Mathematics and
> > Knowledge of nature" by Hans Hahn (vienna, 1933). This looks the source
of
> > the notion that all math is "tautologies".
>
> Well the Tractatus was written from about 1914 onwards - publishd 1921-
>
>
Wittgenstein -from Tractatus Logico-Philosophicus
relevant occurrences of text fragment "Tautol"
4.46 Among the possible groups of truth-conditions there are two extreme
cases. In one of these cases the proposition is true for all the
truth-possibilities of the elementary propositions. We say that the
truth-conditions are tautological. In the second case the proposition is
false for all the truth-possibilities: the truth-conditions are
contradictory . In the first case we call the proposition a tautology; in
the second, a contradiction.
4.461 Propositions show what they say; tautologies and contradictions show
that they say nothing. A tautology has no truth-conditions, since it is
unconditionally true: and a contradiction is true on no condition.
Tautologies and contradictions lack sense. (Like a point from which two
arrows go out in opposite directions to one another.) (For example, I know
nothing about the weather when I know that it is either raining or not
raining.)
4.46211 Tautologies and contradictions are not, however, nonsensical. They
are part of the symbolism, much as '0' is part of the symbolism of
arithmetic.
4.462 Tautologies and contradictions are not pictures of reality. They do
not represent any possible situations. For the former admit all possible
situations, and latter none . In a tautology the conditions of agreement
with the world--the representational relations--cancel one another, so that
it does not stand in any representational relation to reality.
4.463 The truth-conditions of a proposition determine the range that it
leaves open to the facts. (A proposition, a picture, or a model is, in the
negative sense, like a solid body that restricts the freedom of movement of
others, and in the positive sense, like a space bounded by solid substance
in which there is room for a body.) A tautology leaves open to reality the
whole--the infinite whole--of logical space: a contradiction fills the whole
of logical space leaving no point of it for reality. Thus neither of them
can determine reality in any way.
4.464 A tautology's truth is certain, a proposition's possible, a
contradiction's impossible. (Certain, possible, impossible: here we have the
first indication of the scale that we need in the theory of probability.)
4.465 The logical product of a tautology and a proposition says the same
thing as the proposition. This product, therefore, is identical with the
proposition. For it is impossible to alter what is essential to a symbol
without altering its sense.
4.466 What corresponds to a determinate logical combination of signs is a
determinate logical combination of their meanings. It is only to the
uncombined signs that absolutely any combination corresponds. In other
words, propositions that are true for every situation cannot be combinations
of signs at all, since, if they were, only determinate combinations of
objects could correspond to them. (And what is not a logical combination has
no combination of objects corresponding to it.) Tautology and contradiction
are the limiting cases--indeed the disintegration--of the combination of
signs.
4.4661 Admittedly the signs are still combined with one another even in
tautologies and contradictions--i.e. they stand in certain relations to one
another: but these relations have no meaning, they are not essential to the
symbol .
5.101 The truth-functions of a given number of elementary propositions can
always be set out in a schema of the following kind: (TTTT) (p, q) Tautology
(If p then p, and if q then q.) (p z p . q z q) (FTTT) (p, q) In words : Not
both p and q. (P(p . q)) (TFTT) (p, q) " : If q then p. (q z p) (TTFT) (p,
q) " : If p then q. (p z q) (TTTF) (p, q) " : p or q. (p C q) (FFTT) (p, q)
" : Not g. (Pq) (FTFT) (p, q) " : Not p. (Pp) (FTTF) (p, q) " : p or q, but
not both. (p . Pq : C : q . Pp) (TFFT) (p, q) " : If p then p, and if q then
p. (p + q) (TFTF) (p, q) " : p (TTFF) (p, q) " : q (FFFT) (p, q) " : Neither
p nor q. (Pp . Pq or p | q) (FFTF) (p, q) " : p and not q. (p . Pq) (FTFF)
(p, q) " : q and not p. (q . Pp) (TFFF) (p,q) " : q and p. (q . p) (FFFF)
(p, q) Contradiction (p and not p, and q and not q.) (p . Pp . q . Pq) I
will give the name truth-grounds of a proposition to those
truth-possibilities of its truth-arguments that make it true.
5.1362 The freedom of the will consists in the impossibility of knowing
actions that still lie in the future. We could know them only if causality
were an inner necessity like that of logical inference.--The connexion
between knowledge and what is known is that of logical necessity. ('A knows
that p is the case', has no sense if p is a tautology.)
5.142 A tautology follows from all propositions: it says nothing.
5.143 Contradiction is that common factor of propositions which no
proposition has in common with another. Tautology is the common factor of
all propositions that have nothing in common with one another.
Contradiction, one might say, vanishes outside all propositions: tautology
vanishes inside them. Contradiction is the outer limit of propositions:
tautology is the unsubstantial point at their centre.
5.152 When propositions have no truth-arguments in common with one another,
we call them independent of one another. Two elementary propositions give
one another the probability 1/2. If p follows from q, then the proposition
'q' gives to the proposition 'p' the probability 1. The certainty of logical
inference is a limiting case of probability. (Application of this to
tautology and contradiction.)
5.525 It is incorrect to render the proposition '(dx) . fx' in the words,
'fx is possible ' as Russell does. The certainty, possibility, or
impossibility of a situation is not expressed by a proposition, but by an
expression's being a tautology, a proposition with a sense, or a
contradiction. The precedent to which we are constantly inclined to appeal
must reside in the symbol itself.
6.1 The propositions of logic are tautologies.
6.11 Therefore the propositions of logic say nothing. (They are the analytic
propositions.)
6.12 The fact that the propositions of logic are tautologies shows the
formal--logical--properties of language and the world. The fact that a
tautology is yielded by this particular way of connecting its constituents
characterizes the logic of its constituents. If propositions are to yield a
tautology when they are connected in a certain way, they must have certain
structural properties. So their yielding a tautology when combined in this
shows that they possess these structural properties.
6.1262 Proof in logic is merely a mechanical expedient to facilitate the
recognition of tautologies in complicated cases.
6.127 All the propositions of logic are of equal status: it is not the case
that some of them are essentially derived propositions. Every tautology
itself shows that it is a tautology.
6.2 Mathematics is a logical method. The propositions of mathematics are
equations, and therefore pseudo-propositions.
6.21 A proposition of mathematics does not express a thought.
6.211 Indeed in real life a mathematical proposition is never what we want.
Rather, we make use of mathematical propositions only in inferences from
propositions that do not belong to mathematics to others that likewise do
not belong to mathematics. (In philosophy the question, 'What do we actually
use this word or this proposition for?' repeatedly leads to valuable
insights.)
6.22 The logic of the world, which is shown in tautologies by the
propositions of logic, is shown in equations by mathematics.
6.23 If two expressions are combined by means of the sign of equality, that
means that they can be substituted for one another. But it must be manifest
in the two expressions themselves whether this is the case or not. When two
expressions can be substituted for one another, that characterizes their
logical form.
6.234 Mathematics is a method of logic.
6.2341 It is the essential characteristic of mathematical method that it
employs equations. For it is because of this method that every proposition
of mathematics must go without saying.
6.24 The method by which mathematics arrives at its equations is the method
of substitution. For equations express the substitutability of two
expressions and, starting from a number of equations, we advance to new
equations by substituting different expressions in accordance with the
equations.
6.241 Thus the proof of the proposition 2 t 2 = 4 runs as follows: (/v)n'x =
/v x u'x Def., /2 x 2'x = (/2)2'x = (/2)1 + 1'x = /2' /2'x = /1 + 1'/1 + 1'x
= (/'/)'(/'/)'x =/'/'/'/'x = /1 + 1 + 1 + 1'x = /4'x. 6.3 The exploration of
logic means the exploration of everything that is subject to law . And
outside logic everything is accidental.
Mr. Vibrating
"Will Twentyman" <wtwen...@read.my.sig> wrote in message
> Proginoskes wrote:
> > Will Twentyman <wtwen...@read.my.sig> wrote in message
news:<3eede248_4@newsfeed>...
> >
> >>With a consistent set of axioms: any statement of the form
> >>If (axioms) then (provable theorem)
> >>is a tautology (a statement that is always true).
> >
>> >
> >
>
> Many pieces of math were been invented before there was a practical
> application. i as the square root of -1 was invented long before it had
> any apparent use. Unless you want to argue that math IS reality, I have
> to disagree.
>
> --
> Will Twentyman
> email: wtwentyman at copper dot net
>
I've got a favor from everybody (or at least somebody). Could you please
show me how the following well know (and easily understood - unlike coloring
problems) mathematical expression:
1+ 2 + ... + (N-1) + N = N(N+1)/2 for N = integer, N>= 1
is a tautology. I personally don;t think this is a tautology at all, but I'm
always willing to learn something new. Lets use this as a test bed (if you
agree) to see if all mathematical epxressions (proofs) are tautologies or
not. P.S. Ground rules: appeals to authority (be it Hans Hahn or Wittgensten
or Oscar Wilde or God) don't "count".
SHOW ME!
C. Bond
It seems to me you have more than one choice about what you regard as an
acceptable answer. If (big IF) there is a commonly accepted definition about
what a tautology is, then the only problem is to determine whether your equation
fits the definition. If there is some distinction between a tautology, an
identity, an equality, etc. then the distinctive properties of each should be
expressed in order to determine the appropriate classification for your
equation.
On the other hand, you can engage in endless dialogues, similar to those of the
six blind men from Hindustan, who each focus on a different concept and all come
to different conclusions.
By the way, citing a reference is not necessarily appealing to authority --
unless the message is held to be subordinate to the messenger.
--
There are two things you must never attempt to prove: the unprovable -- and the
obvious.
--
Democracy: The triumph of popularity over principle.
--
http://www.crbond.com
Proginoskes
>
> The crosspost arose out of confusion as to the *subject* of mathematics.
> Lets compare this to physics - where some empirical data can refute a
> proposition of physics, can this occur in mathematics- i thought not.
Yes it can. Suppose I conjecture (this is the mathematical version of the
physicist's "proposition"; in mathematics, a proposition is something which
is true):
Every graph which can be drawn in the plane can be 3-colored.
This is false, and all I would need to show it would be one counterexample.
Consider the graph with 4 vertices, and an edge between every pair. This
graph can be drawn in the plane; crudely, it could look like:
A------- (Letters represent the vertices; - / \ | represent edges
/ \ | as drawn in the plane.)
/ \ |
B-----C | This graph cannot be 3-colored. Since A is adjacent to B,
\ / | A and B must be given different colors. Same for A and C,
\ / | same for the pairs A and D, B and C, B and D, C and D. In
D------- fact, no two vertices can receive the same color; hence
this graph requires 4 colors to be "properly colored".
However, if I change 3 to 6, I get the "conjecture"
Every graph which can be drawn in the plane can be 6-colored.
This statement IS true, and it can be shown that it is true.* In fact, if
you have a planar graph, it is an easy matter to construct a 6-coloring
(labelling each vertex with one of the numbers 1, 2, ..., 6, so that no
two adjacent vertices have the same color) if given a graph.
* To open up a can of worms: Kurt G\"odel proved that if you make any given
set of assumptions (axioms), then one of two things happens: Either you can
prove something which is actually not true, or there is some statement which
is true, but you can't prove it (only given your assumptions).
-- Christopher Heckman
Proginoskes
> Proginoskes wrote:
> > Will Twentyman <wtwen...@read.my.sig> wrote in message news:<3eede248_4@newsfeed>...
> >
> >>With a consistent set of axioms: any statement of the form
> >>If (axioms) then (provable theorem)
> >>is a tautology (a statement that is always true).
> >
> > No; it just says that (provable theorem) can be proved by (axioms).
> >
> > A tautology is a boolean formula which is always true, no matter what
> > values the variables have. For instance, P OR NOT P.
>
> And those axioms, if taken as true, will always result in the provable
> theorem being true.
This is true, but the converse is not. At the risk of repeating myself
within this thread:
To open up a can of worms: Kurt G\"odel proved that if you make any given
set of assumptions (axioms), then one of two things happens: Either you can
prove something which is actually not true, or there is some statement which
is true, but you can't prove it (only given your assumptions).
Hence, it is possible for a thing to be true, but not provable. (Check out
_Godel, Escher, Bach: An Eternal Golden Braid_ by Douglas Hofstadter if you
want a specific example.)
> Therefor [sic]: you have either T->T, F->T, or F->F, all
> of which are true. Therefor [sic], it is a tautology.
How do you know you have one of these three things? In order to deduce that
you do, you need some information first.
> > This means that (2 + 2 = 5) OR NOT (2 + 2 = 5) is true. (FALSE OR TRUE =
> > TRUE.) This statement is true because of its FORM, not its content.
> > (It doesn't matter what P actually IS.)
> >
> > P -> Q (implication) is NOT a tautology; the value of P -> Q depends
> > on the values of P and Q. It depends on the CONTENT of P and Q.
> >
>
> You are assuming that P and Q are independent of each other.
In a tautology, they CAN BE; that's my main point. That's why (some)
mathematical theorems have actual content to them; in order for P -> Q to
be true, P must be related to Q, in some (nonobvious) way. [See my response
below.]
> > Elsewhere in this thread, I use the 5-Color Theorem* as an example:
> >
> > If G is a planar graph, then G can be 5-colored. (The conclusion
> > means: For each vertex v of G, a number in the set {1,2,3,4,5} can
> > be assigned to v, and these assignments can be made so that no two
> > adjacent vertices (a pair with an edge between them) have the same
> > color).
> >
> > Compare this with the following statement, which is FALSE:
> >
> > If G is a planar graph, then G can be 3-colored.
>
> Definition: a graph is planar if it can be 3-colored. This change in
> the definition of coloring/planarity would make your false statement
> true. It must exist in a context of axioms/definitions.
Yes, if you define "planar" that way, then the statement is a tautology,
because it is a case where I'm substituting two equivalent things for P
in the boolean formula P -> P (which is a tautology; true no matter what
you put in place of P).
But I'm not (and the rest of the mathematical community is not) defining
"planar" and "3-color[able]" in that way. In fact, planarity is a
topological property of the graph -- it relates to how the loops
(technically, cycles) within a graph are related to each other.
I am defining 3-colorability (or 5-colorability) in terms of setting up a
function and trying to figure out how to assign various numbers to each
vertex so that some property is true. In particular, I'm never looking at
cycles.
In short (and this is my point, so forgive the capitals): THESE DEFINITIONS
HAVE NOTHING WHATSOEVER TO DO WITH EACH OTHER. It's like the implication
P -> Q, not P -> P. (actually, for all x, P(x) -> Q(x).) You CAN make the
implication P -> Q false, because there are choices for P and Q which make
P -> Q false (namely make P true and Q false).
HOWEVER: If you let P(x) be the statement "x is planar" and Q(x) the
statement "x is 5-colorable", then P(x) -> Q(x) is ALWAYS true.
Whenever P(x) is true, so is Q(x); it is impossible to make P(x) true and
Q(x) false. The incredible thing is that somehow a graph being planar also
makes it 5-colorable.
How do I know this? How do I know you won't find a graph G which is planar
and cannot be 5-colored? (In fact, I'll give you a million dollars if you
can ... and open it up to Usenet.*) Well, that is where the INFORMATION
content of the statement "every planar graph is 5-colorable" comes in. And
without this information content (i.e., the proof), there is no reason to
suspect that the statement "every planar graph is 5-colorable" is true,
because the FORM of the statement (P(x) -> Q(x)) is not ALWAYS true, as it
would be for a tautology (a statement whose FORM is ALWAYS true).
* I may regret the second half of this sentence, not because I'll have to
pay up, but because I might receive a huge number of e-mails about this.
> > My challenge is: How does changing a single number change the value of
> > the statement? Nothing in the _form_ of the statement has been changed.
>
> You have missed the key in the form: namely the fact that a theorem is
> _provable_. "If (axioms) then (provable theorem)" represents a
> collection of tautologies since the original post I saw mentioned them.
> when you change from 5 to 3 you changed the consequent from a provable
> theorem to a false conclusion. Thus, you changed the form.
The FORM of the statement "If (axioms) then (provable theorem)" is P -> Q,
which is the same for "every planar graph is 5-colorable" and "every planar
graph is 3-colorable". Hence the difference in one being true and the other
being false is due to CONTENT (whether a certain number is 3 or 5). Q.E.D.
> Note: without including the axioms/definitions of graph theory that we
> accept, we are NOT discussing tautologies, but merely provable theorems.
Tautologies have nothing to do with graph theory. (Well, very little, but
it only goes in one direction: left to right, and not right to left, so
they're not the same thing.) Take an Introduction to Logic course (or skim
through the textbook) and you'll see what I mean; you'll never even learn
what a graph IS, so you can't possibly work with one.
> >>Notice that there is no claim made about "reality". There are three
> >>major forms of Geometry: Euclidean, perspective, and geometry on a
> >>sphere. Which one describes "reality"? Certainly not all three, yet
> >>they are all studied in mathematics.
> >
> >
> > Mathematics comes out of reality, not vice versa.
>
> Many pieces of math were been invented before there was a practical
> application. i as the square root of -1 was invented long before it had
> any apparent use. Unless you want to argue that math IS reality, I have
> to disagree.
How convenient! You've left out my next paragraph, which reads:
> > Right. Mathematics can be used to work out "what if ..." statements.
... which is its other main virtue. (I should have inserted "also" between
"can" and "be" in my response.)
-- Christopher Heckman
And one more can of worms: What IS reality?
Jesse F. Hughes
I don't want to write a strictly formal proof of that claim (*), but I
assume that you agree that there is a proof P of (*) in Peano
Arithmetic.
P uses only finitely many axioms of PA, let's call them A_1, ..., A_n.
Then, it is easy to see that one can transform the proof P in PA into a
proof
(A_1 & ... & A_n) -> (*)
in pure first order logic.
This is what (at least some) mean when they say that mathematics is
tautologous. They mean that, when I assert (*), I'm really asserting
that a finite subset of axioms of PA imply (*) and that, furthermore,
a proof of this claim does not require any extra-logical axioms or
rules of inference.
--
"If you *still* believe that [my proof is wrong], then I have to think
that your mind is limited [...], and it may be the case that not
everyone *can* achieve that, as the mental wiring may not be there for
the task." -- James Harris, on faculties needed to accept his proof.
Jesse F. Hughes
> "James Whitehead" <Abx4...@jjh76g7856gh.com>:
>>
>> "Proginoskes" <progi...@email.msn.com> wrote in message >
>>
>> [...]
>>
>> P.S. If anyone claims that mathematics has no content, I dare them to
>> provide
>> > a (correct) proof of the Four Color Theorem.
>>
>> The crosspost arose out of confusion as to the *subject* of mathematics.
>> Lets compare this to physics - where some empirical data can refute a
>> proposition of physics, can this occur in mathematics-
>
> We observe conjectures which may seem true but not be proven
> for a long time, for example the map-coloring problem or
> Fermat's famous theorem. If mathematics were informally
> tautologous in the sense that every consequence of every axiom
> were immediately obvious, then these problems shouldn't take
> centuries to solve. So what is this space that has to be
> taken up before the solution comes into view?
Advocates of the view that mathematics is tautologous do not claim
that "tautologous" is synonymous with "immediately obvious".
I could write down a propositional tautology as large as you wish.
It need not be immediately obvious that the formula I write down is
true.
With predicate calculus, the situation may be even more difficult.
One cannot expect the fact that a particular first order formula is
provable to be "immediately obvious".
So, your use of "tautologous in the sense that every consequence of
every axiom were immediately obvious" isn't really relevant to
evaluating the claim as it's intended.
--
"I've ... contacted [some of the...] highest I.Q.'s in the country...
I've even helped the FBI out a few times... I've met at least one
governor..., a senator... and I've had some really good seats at
sports games. My experiences are not your experiences." --JSH != you
James Whitehead
(There is no appeal to authority on my part but it would be wrong to present
the idea as mine.) Your formula above is not easily understood - this is an
empirical proposition that is provable by observation. Do you think 2+2= 4
is not a tautology? The point is this - that the proof of a mathematical
truth is the realisation of the equality. This may be immediate or take
years or forever, but that is what the proof is. If you are mistaken in your
proof in mathematics its because you have failed to notice that 2+2 is not
the same as 4. The two things are different. (There is a philosophical
problem here - the identity of indiscernables - Leibniz)
Clearly -empirically "2+2" is not identical with "4" but also clearly
mathematically they can be. *they can be* And this mathematical equality is
"perfect". i.e. if i give one person 4 apples and another (2+2) apples they
have regarding the "number" of apples 'the same' - the absolute same number.
Now lets compare two apples - are they the same - no- they will occupy
different life histories - as well as the more trivial thing of being
slightly different in taste and colour. Now what authority am i appealing
to - not Wittgenstein's or God's but yours! - See how you have elevated
yourself above these?
2. By appeal to God.
Lets appeal to God - i pray that you will in the next two days come into
some money - and this will be a sign from God that mathematics is a
tautology.
James Whitehead
>
> HOWEVER: If you let P(x) be the statement "x is planar" and Q(x) the
> statement "x is 5-colorable", then P(x) -> Q(x) is ALWAYS true.
> Whenever P(x) is true, so is Q(x); it is impossible to make P(x) true and
> Q(x) false. The incredible thing is that somehow a graph being planar also
> makes it 5-colorable.
>
> How do I know this? How do I know you won't find a graph G which is planar
> and cannot be 5-colored? (In fact, I'll give you a million dollars if you
> can ... and open it up to Usenet.*) Well, that is where the INFORMATION
> content of the statement "every planar graph is 5-colorable"
I dont see how you can *restrict* yourself to five colours without
tautology.
> And one more can of worms: What IS reality?
You've already signed up to the reality of the metaphysical idea of "being"
in "IS" which generates your "reality" worms and all.
James Whitehead
But this strikes me as a logical property of such graphs - and i wouldn't
require drawing them to see this property - unlike saying that all arm
chairs are brown. Its a logical property of these graphs that they cannot
*all* be coloured in less than 4 or is it 5 colours. I'm set up with set of
propositions out of which comes such a result. You wont allow me to bend
the graph or shade the colours - so that what is green shades
(imperceptibly) to something which could be blue- you given me no choice -
or rather you have said nothing about shapes or colours. (you are treating
these as logically neutral) We can compare this to the noble gases so called
because the cant be (now they can) oxidised. Or heavier than air flying
machines...etc. And if we illustrate your graph on TV we will be colouring
it with only 3 colours - and be able to do so for all the graphs - even the
one above.
>
> * To open up a can of worms: Kurt G\"odel proved that if you make any
given
> set of assumptions (axioms), then one of two things happens: Either you
can
> prove something which is actually not true, or there is some statement
which
> is true, but you can't prove it (only given your assumptions).
>
> -- Christopher Heckman
I was aware of the unproveable true - but not the other?
James Whitehead
"G*rd*n" <g...@panix.com> wrote in message
news:bco3tc$o02$1...@panix1.panix.com...
> "James Whitehead" <Abx4...@jjh76g7856gh.com>:
> >
> > "Proginoskes" <progi...@email.msn.com> wrote in message >
> >
> > [...]
> >
> > P.S. If anyone claims that mathematics has no content, I dare them to
> > provide
> > > a (correct) proof of the Four Color Theorem.
** how does this proof work - not like proof of perpetual motion machines -
which given certain physics might be possible? The proof of the colour
theorem would lie in its being presented as a tautology from which we cannot
escape.
> >
> > The crosspost arose out of confusion as to the *subject* of mathematics.
> > Lets compare this to physics - where some empirical data can refute a
> > proposition of physics, can this occur in mathematics-
>
> We observe conjectures which may seem true but not be proven
> for a long time, for example the map-coloring problem or
> Fermat's famous theorem. If mathematics were informally
> tautologous in the sense that every consequence of every axiom
> were immediately obvious, then these problems shouldn't take
> centuries to solve. So what is this space that has to be
> taken up before the solution comes into view?
A tautology need not be immediate as Jesse points out - and i think also as
did Gautama!
Will Twentyman
In and of itself, it is NOT a tautology. The tautology would be:
IF (laws/definitions of algebra AND weak induction) THEN (if (n is an
integer and n>=1) then 1+2+...+(n-1)+n = n(n+1)/2).
Here's the trick: a theorem can only be shown to be true in a larger
context of axioms/definitions. The tautology lies in the fact that if
the axioms and definitions are true then that theorem must be true.
Note: We are currently swimming in the topic of mathematical logic
courses, which tend to include very convoluted/detailed reasoning. The
closer you look at these ideas the more attention you must pay to the
semantics of what's being said.
G*rd*n
> A tautology need not be immediate as Jesse points out - and i think also as
> did Gautama!
Yet people, including the august Wittgenstein, seem to be
saying -- or implying -- that mathematics is tautologous and
_therefore_ one will observe nothing new in its productions.
The first meanings of the word given in both the OED and the
American Heritage Dictionary explicitly mention "needless
repetition" or the like. Witt (and I believe you) _claim_
to be doing philosophy, not mathematics, where the term
_tautology_ has a somewhat different definition -- evidently
the "repetition" isn't thought to be superfluous and anyway,
isn't repetition in the experiential sense. It seems to me
that there's a sort of shuttling between one sense of the
word and another going on here.
What about Gautama?
Jesse F. Hughes
> "James Whitehead" <Abx4...@jjh76g7856gh.com>:
>> A tautology need not be immediate as Jesse points out - and i think also as
>> did Gautama!
>
> Yet people, including the august Wittgenstein, seem to be
> saying -- or implying -- that mathematics is tautologous and
> _therefore_ one will observe nothing new in its productions.
I won't comment on Wittgenstein, but Hahn's claim is not that we will
observe nothing new. He claims, indeed, that applications of
mathematics (like applications of logic) say nothing about objects --
in this sense, they convey no new *content*. But this does not mean
that they convey no observations which are new to *us*.
Explicitly (thanks to Mr. Vibrating's probable infraction of copyright
law):
Thus logical propositions, though being purely tautologous, and
logical deductions, though being nothing but tautological trans-
formations, have significance for us because we are not
omniscient. Our language is so constituted that in asserting such
and such prop- ositions we implicitly assert such and such other
propositions-but we do not see immediately all that we have
implicitly asserted in this manner. It is only logical deduction
that makes us conscious of it. I assert, e.g., the propositions "the
flower which Mr. Smith wears in his buttonhole, is either a rose or
a carnation," "if Mr. Smith wears a carnation in his buttonhole,
then it is white," "the flower which Mr. Smith wears in his
buttonhole is not white." Perhaps I am not consciously aware that I
have implicitly asserted also "the flower which Mr. Smith wears in
his buttonhole is a rose"; but logical deduction brings it to my
consciousness. To be sure, this does not mean that I know whether
the flower which Mr. Smith wears in his buttonhole really is a rose;
if I notice that it is not a rose, then I must not maintain my
previous assertions-otherwise I sin against the rules of speaking, I
revoke.
(While this paragraph is literally about logic, not mathematics per
se, it is clear that Hahn intends it to apply to mathematical
reasoning as well.)
--
"That's all the legacy I ever wanted, to have people remember me like
a shooting star streaking across their Life sky, illuminating, for
just one moment, unparalleled beauty unique to itself."
-- Weblogs are a particularly humble medium, unique to themselves.
James Whitehead
> "James Whitehead" <Abx4...@jjh76g7856gh.com>:
> > A tautology need not be immediate as Jesse points out - and i think also
as
> > did Gautama!
>
> Yet people, including the august Wittgenstein, seem to be
> saying -- or implying -- that mathematics is tautologous and
> _therefore_ one will observe nothing new in its productions.
> The first meanings of the word given in both the OED and the
> American Heritage Dictionary explicitly mention "needless
> repetition" or the like. Witt (and I believe you) _claim_
> to be doing philosophy, not mathematics, where the term
> _tautology_ has a somewhat different definition -- evidently
> the "repetition" isn't thought to be superfluous and anyway,
> isn't repetition in the experiential sense. It seems to me
> that there's a sort of shuttling between one sense of the
> word and another going on here.
You need to ask why the dictionary maintains its "needless" its because it
implies 'the same' - and though post-moderns might argue that this is not
true the idea of "the same" seems important in a mathematical proof. So I
dont think needless repetition will do - why say 2+2 and 4 - that is
needless if you say 2+2 you have 4 to say both is needless - in a sense -
though its useful to know perhaps - better the source from the Latin tauto =
"same" doesnt a proof depend on (something) = ( something else) being
reduced to 1=1 or a = a which is saying that a is the same as a - further
the proof is in that if the equals sign in an equation is correct then we
can apply the same operation to both sides giving 0=0 or nothing! The
content of the proposition is therefore empty.
Its amusing because if the propositions did have a content then they would -
like the inside of your fridge - change over time - be subject to
deconstruction.
> What about Gautama?
It was blindingly obvious? (but took a while to see this)
James has a wife and two cats.
James is not a bachelor and has one cat and another cat.
James is a husband and has (6-4) cats
James is married and has 1+1 cats.
James is not single and has (insert any proposition which yields 2) cats.
With each proposition is anything more given in terms of information about
James, which leads us to the conclusion that all the sentences following the
first are tautologies.
That the rosseta stone contained such tautologies and was thought extremely
useful - but!
'To be very schematic I would say that the difficulty of defining an
therefore also of translating the word "deconstruction" stems from the fact
that all the predicates, all the defining concepts, all the lexical
significations, and even the syntactic articulations, which seem at one
moment to lend themselves to this definition or to that translation, are
also deconstructed or deconstructible, directly or otherwise, etc. And that
goes for the word deconstruction, as for every word. *Of Grammatology*
questioned the unity "word" and all the privileges with which is was
credited, especially in its nominal form. It is therefore only a discourse
or rather a writing that can make up for the incapacity of the word to be
equal to a "thought". All sentences of the type "deconstruction is X" or
"deconstruction is not X" a priori miss the point, which is to say that they
are at least false. As you know, one of the principal things at stake in
what is called in my texts "deconstruction" is precisely the delimiting of
ontology and above all of the third person present indicative: S is P.'
James tem uma esposa e dois gatos.
Maybe then offers something else?
James Whitehead
>
> P.S. If anyone claims that mathematics has no content, I dare them to
provide
> a (correct) proof of the Four Color Theorem.
So http://www.math.gatech.edu/~thomas/FC/fourcolor.html is wrong?
G*rd*n
> >> A tautology need not be immediate as Jesse points out - and i think also as
> >> did Gautama!
g...@panix.com (G*rd*n) writes:
> > Yet people, including the august Wittgenstein, seem to be
> > saying -- or implying -- that mathematics is tautologous and
> > _therefore_ one will observe nothing new in its productions.
jes...@cs.kun.nl (Jesse F. Hughes):
> I won't comment on Wittgenstein, but Hahn's claim is not that we will
> observe nothing new. He claims, indeed, that applications of
> mathematics (like applications of logic) say nothing about objects --
> in this sense, they convey no new *content*. But this does not mean
> that they convey no observations which are new to *us*.
> ...
In the case of Hahn I have a somewhat different objection,
this time around the concept of, in fact, "content". I don't
see the validity of the distinction he is trying to make. To
persist with the metaphor, let us say we have two boxes, one
labeled "mathematics" and the other "the physical universe".
We can reach into either box and take out objects which have
never been seen before. In fact, in the case of the box
labeled "mathematics" we have good reason to believe we can
take out infinitely many objects which have never been seen
before, whereas the physical-universe box may well be finite.
That being the case, it seems to me that both boxes must be
said to have content, and that the mathematics box seems to
possibly have more content than the physical-universe box --
infinitely more content.
Now, it may be said that, yes, the mathematics box has a lot
of things in it, but in principle we can reduce these to
objects we put into the box in the first place, axioms,
definitions, procedures and so forth, so there's really nothing
in the box but what we put in it. But this doesn't correspond
with our actual experience of the box, in which the combination
of the things we put in the box produces things we didn't
put in the box. Just yesterday I was looking at a piece of
the Mandelbrot set I am pretty sure no one has ever seen
before, as who hasn't?
I realize I am arguing with a metaphor which is perhaps more
the realm of poetry than philosophy and is certainly non-
mathematical, but Hahn started it.
James Whitehead
i can draw a straight line and equally be sure that no one has ever seen it
before- but its not maths! Or was your nice picture.
>
> I realize I am arguing with a metaphor which is perhaps more
> the realm of poetry than philosophy and is certainly non-
> mathematical, but Hahn started it.
>
There is i think something very strange about your metaphor - firstly i
think that one box should be an object inside the other :-) And the
objects you take out of one will be significantly *different* to the other.
The objects you take out of the box labelled physical universe will be
strange fuzzy things which appear and disappear - which the audience will
shout oooooo! a rabbit - naaaaa its a hare etc. And now its my lunch.....
whereas the objects you take out of the maths box will be just that
"objects" - now a final question - in which box will you find 'words'?
(of course you won't be able to show us any of the 'objects' from the maths
box- e.g circles - straight lines prime numbers etc. - i guess it will
look to all intents empty! )
Ned Ludd
>
> (of course you won't be able to show us any of the 'objects'
> from the maths box- e.g circles - straight lines prime numbers
> etc. - i guess it will look to all intents empty! )
>
As opposed to the other box? Look closer. Every component of
every thing in that box is 99.99999% empty.
Like the math box, the qualities that things have in that box
are the qualities we bring to them.
Ned
Mr. Vibrating
that the logical term "tautology" has a definition different from that of
"any true statement". Just like a contradiction is different then a false
statement. So those, such as Hans Hahn, that reduce mathematics to
tautology, assume that every mathematical expression, being true, must be a
tautology. The purpose of this is to, like all reductionism, disguise
ignorance.
Science seems to START with clear and precise definitions, and move forwards
from there. Postmodernism never does. Its goal seems to be to seek
definitions, but starting from hoped-for results, rather then definitions.
Thus the "conclusion" that logc and math are tautologies.
The reason that
1 + 2 + (N-1) + N = N (N+1)/2
is not because 2 + 2 = 4, but because other reasons. "2+2=4" is true because
it is a definition: the definition of the names of particular numbers and
the definition of the operator "+". Is
" x*x +1 = 0 "
a Tautology? Well, the expression is true for some values of x, and false
for other values of x. It so happens that it is true for +i and -i, and
false for all others. According to Hahn, its a tautology if x= i or -i, and
a contradiction, other wise.
Lets assume that my real name is "Mr. Vibrating" (on my birth certificate).
Is the statement "My name is Mr Vibrating" either a tautology or
contradition?
Suppose that I'm a physician and want to take the temperature of my patient
at 4:00pm. I take the temperature at 4:00pm and the thermometer reads 98.6
degrees F. I record: "The patients temperature at 4:00pm is 98.6 degree F".
Is this a tautology?
In other words, Tautology has lost its restrictive definition and become
synonymous with "TRUE".
In electrical engineering, it is common to construct "logical expressions"
or digital circuits. These "logical expressions" have inputs and outputs,
and the outputs change as a function of the inputs.
It is also possible to have a "circuit" who output doesn't change with
input, but is always either T or F - these are refered to as generators: a
generator that always has an output of 'T' would be a tautology, and a
generator with a constant output of 'F' would be a contradition.
Now, you can plug in generators into the inputs of the logical expression
(circuit) - thus the outputs get certain values. Take for example, the
humble inverter ( the "NOT" function):
NOT(T) = F
NOT(F) = T
Is the NOT() function a tautology or a contradition? It is obviously
neither, because its output depends on input.
Now, is the following a tautology?
NOT(F) = T
Is the following a contradition?
NOT(T) = F
What about the following?
NOT(T) = T
or this?
NOT(F) = F
Obviously, the logical expression
NOT() depends on its inputs for its outputs.
Logical expressions have a "truth table" (even Wittgenstein knew this). The
individual entries of the truth table are NOT by definition TAUTOLOGIES,
but, by definitions, TRUE. Even the expression "NOT(T) = F", being a member
of the expressions truth table, is "TRUE" - i.e. "A TRUTH". The statement
"NOT(T) = T" is not a member of the truth table, and thus "FALSE" - i.e. "A
FALSEHOOD".
Now a TAUTOLOGY is a particular kind of truth table, one in which
reguardless of the state of the inputs, the output is TRUE.
A CONTRADICTION is another kind of truth table, one in which reguardless of
the state of inputs, the output is FALSE.
We can construct a truth table from sctach, or from combining previously
defined truth tables.
We can construct a TAUTOLOGY from, lets say, the truth table for NOT() and
the truth table for OR(). Let this new truth table be called TAUTO1(), where
"TAUTO1(x) = OR (NOT(X), X)" (prefix notation)
That is, "TAUTO1 = NOT(x) OR X" (infix notation)
The resulting truth table for Tauto1 has only "TRUE" as outputs and
therefore is a TAUTOLOGY by definition.
Similarly one can construct a truth table for a CONTRADITION, which only
have FALSE as allowable outputs.
Truth Tables are associated with the definition of Operations, not with
specific instances, although specific instances are elements of of the truth
table of the operation.
Taking a temperature with a specific thermometer is an operation whose
allowable outputs is the continuous range of allowable temperatures that the
thermometer can support, within its inherent tolerance. A specific
temperature taken is simply an instance, or element of thermometer's truth
table, and cann't ever, by definition, be a tautology. One can design a
specific digital thermometer which only have three output values in its
truth table: "Normal Temperature" and "High temperature" and "low
temperature".
Truth Tables of operators can consist of one or more entries of binary
values or real values, or complex values or discrete "fuzzy" (or linguistic)
values. In all cases, the truth table of the operator specifies "TRUE" or
"valid" behavior. Values outside of the truth table indicate "FALSE" or
"broke" or whatever, depending on context.
The english word "IS" has a number of different, context dependant,
meanings. Does not necessarily mean the same ting as the "=" equal sign.
a. "X is Y"
In this sense, the implication is that (x,y) belongs to the truth table of
the operator associated with X. Example: "my name is Mr. Vibrating."
b. "X is"
In this, unary, sense, "IS" is the exisitential operator, which "includes" X
in the universe, inductively. If 'U' is the universe, then saying "X is"
means
U = U OR X
"X is not" means
U = U AND NOT(X)
"Neutiquam erro."
Mr. Vibrating
aka TL
aka Mounard le Fougueux
"C. Bond" <cb...@ix.netcom.com> wrote in message
news:3EEFCA80...@ix.netcom.com...
Mr. Vibrating
The reason that
"Neutiquam erro."
news:3EEFCA80...@ix.netcom.com...
"James Whitehead" <Abx4...@jjh76g7856gh.com> wrote in message
news:bcpb9o$a44$1...@news7.svr.pol.co.uk...
Jesse F. Hughes
> jes...@cs.kun.nl (Jesse F. Hughes):
>> I won't comment on Wittgenstein, but Hahn's claim is not that we will
>> observe nothing new. He claims, indeed, that applications of
>> mathematics (like applications of logic) say nothing about objects --
>> in this sense, they convey no new *content*. But this does not mean
>> that they convey no observations which are new to *us*.
>> ...
Before I begin, let me state this clearly: I am not defending Hahn's
view, but I hope to clarify it a bit.
In particular, we have lately been speaking of mathematics as
tautologous on the grounds that theorems can be represented as
implications provable by pure logic. This is only a weak sense of
Hahn's claim, however. The stronger sense is this: Given any true set
of observations, and any application of mathematics deriving a new
claim from the given, that new claim is true-by-convention --- such
derivations only make explicit the conventions in which we have chosen
to use the mathematical terminology.
This is a different claim, and is considerably more controversial. As
I mentioned at the start, one evident problem with this claim is the
competing geometric models of the universe. I don't know how Hahn can
acknowledge that the claim "the universe is (non-)Euclidean" is an
empirical claim, while still maintaining that mathematical axioms
simply set a convention for how certain technical terms (like lines)
can be used. On his view, it seems like one ought to make a choice on
whether to adopt the fifth postulate or not, and that once that choice
is made, mathematical deductions/calculations are truth-preserving
transformations.
> In the case of Hahn I have a somewhat different objection,
> this time around the concept of, in fact, "content". I don't
> see the validity of the distinction he is trying to make. To
> persist with the metaphor, let us say we have two boxes, one
> labeled "mathematics" and the other "the physical universe".
> We can reach into either box and take out objects which have
> never been seen before. In fact, in the case of the box
> labeled "mathematics" we have good reason to believe we can
> take out infinitely many objects which have never been seen
> before, whereas the physical-universe box may well be finite.
> That being the case, it seems to me that both boxes must be
> said to have content, and that the mathematics box seems to
> possibly have more content than the physical-universe box --
> infinitely more content.
>
> Now, it may be said that, yes, the mathematics box has a lot
> of things in it, but in principle we can reduce these to
> objects we put into the box in the first place, axioms,
> definitions, procedures and so forth, so there's really nothing
> in the box but what we put in it. But this doesn't correspond
> with our actual experience of the box, in which the combination
> of the things we put in the box produces things we didn't
> put in the box. Just yesterday I was looking at a piece of
> the Mandelbrot set I am pretty sure no one has ever seen
> before, as who hasn't?
But, Hahn didn't claim that deductions yield obvious results. He only
claimed that they yielded results which were implicit from our
linguistic conventions and that these conventions are most explicitly
expressed in our axioms and rules of deduction.
--
"Evariste Galois was clearly a passionate man. He tried to kill the
king of France for instance... Remember that the French Revoloution
*did* happen, so he wasn't really out of his times." -- JSH on Galois
(1811 - 1832) prefiguring the French Revolution (1789)
Mr. Vibrating
The reason that
"Neutiquam erro."
(more below)
"C. Bond" <cb...@ix.netcom.com> wrote in message
> > SHOW ME!
> 1. By appeal to your authority
> (There is no appeal to authority on my part but it would be wrong to
present
> the idea as mine.) Y
> Do you think 2+2= 4
> is not a tautology?
As I point out above - I think the statement "2+2=4" is TRUE but not a
"TAUTOLOGY". It is an element of the truth table of the binary
operator '+' (as defined by abelian rings and galois fields und so weider).
The truth table of an operator may be a tautology, byt in the
case of "+" it is not.
>The point is this - that the proof of a mathematical
> truth is the realisation of the equality. This may be immediate or take
> years or forever, but that is what the proof is. If you are mistaken in
your
Why so much focus on the equal sign?? It seems more fruitful to focus on the
proof itself.
> proof in mathematics its because you have failed to notice that 2+2 is not
> the same as 4. The two things are different. (There is a philosophical
> problem here - the identity of indiscernables - Leibniz)
> Clearly -empirically "2+2" is not identical with "4" but also clearly
> mathematically they can be. *they can be* And this mathematical equality
is
It seems the notion of "truth tables" both stabilizes the notion of
logical/mathematical expressions as well as saves it from reductionism to
"tautology".
In this I am chiefly motivated by my suspicion of linguistic reductionism -
that is, any attempt to reduce a clearly complex and
multidimensional field of discourse to "Brand X" - for example
"Logic/Mathematics is just Tautologies". This intuitive suspicion hasn't
failed me yet, and seems to me an honorable enough way to heat my pentium's
heatsink.
> "perfect". i.e. if i give one person 4 apples and another (2+2) apples
they
> have regarding the "number" of apples 'the same' - the absolute same
number.
> Now lets compare two apples - are they the same - no- they will occupy
> different life histories - as well as the more trivial thing of being
> slightly different in taste and colour.
>
> 2. By appeal to God.
> Lets appeal to God - i pray that you will in the next two days come into
> some money - and this will be a sign from God that mathematics is a
> tautology.
?????????????
"Nihil curo de ista tua stulta superstitione. "
Russell Blackadar
[snip]
> Success is not a guarantee of truth - often the reverse.
[snip]
> I remember in maths once having to use a quadratic equation to calculate the
> height of a bridge - which resulted in two answers - one a negative this we
> were told could be ignored as it was obvious we were after a positive value.
(You did this only once?!)
I hope you're not trying to suggest you did anything "untrue"
in this procedure. A number that satisfies *some* of the
necessary conditions of a solution can still fail to be a
solution on other grounds. Nothing at all unusual about that.
G*rd*n
> > > A tautology need not be immediate as Jesse points out - and i think also
> as
> > > did Gautama!
"G*rd*n" <g...@panix.com>
> > Yet people, including the august Wittgenstein, seem to be
> > saying -- or implying -- that mathematics is tautologous and
> > _therefore_ one will observe nothing new in its productions.
> > The first meanings of the word given in both the OED and the
> > American Heritage Dictionary explicitly mention "needless
> > repetition" or the like. Witt (and I believe you) _claim_
> > to be doing philosophy, not mathematics, where the term
> > _tautology_ has a somewhat different definition -- evidently
> > the "repetition" isn't thought to be superfluous and anyway,
> > isn't repetition in the experiential sense. It seems to me
> > that there's a sort of shuttling between one sense of the
> > word and another going on here.
"James Whitehead" <Abx4...@jjh76g7856gh.com>:
> You need to ask why the dictionary maintains its "needless" its because it
> implies 'the same' - and though post-moderns might argue that this is not
> true the idea of "the same" seems important in a mathematical proof. So I
> dont think needless repetition will do - why say 2+2 and 4 - that is
> needless if you say 2+2 you have 4 to say both is needless - in a sense -
> though its useful to know perhaps - better the source from the Latin tauto =
> "same" doesnt a proof depend on (something) = ( something else) being
> reduced to 1=1 or a = a which is saying that a is the same as a - further
> the proof is in that if the equals sign in an equation is correct then we
> can apply the same operation to both sides giving 0=0 or nothing! The
> content of the proposition is therefore empty.
It is not. When you remove the a's from 'a = a', by
exerting a rule for '=', you still have '='. How are you
going to get rid of it? But this is beside the point.
> Its amusing because if the propositions did have a content then they would -
> like the inside of your fridge - change over time - be subject to
> deconstruction.
That's a new rule you've just made up, in fact, two new
rules: "content -> change" and "change -> deconstruction".
_Tautology_ is etymologically "saying the same thing" and was
originally a term of rhetoric. If one had said "X!" it seemed
unnecessary to say it again, "X! X!" since one "X!" would do
(logically; we pass over its uses for emphasis, derision,
decoration, rhythm, etc.) Clearly, the term has acquired a
different meaning in mathematics. Yet saying that tautologies,
including mathematical ones, are "empty" seems to look back
to this meaning: "X!", "X!" => "X!" as the logicians observed.
But in any case this is not how we actually experience
mathematics.
All this seems very different to me from the Buddhist notion
of emptiness, but maybe I'm just being stubbornly attached
to detachment or something.
Proginoskes
>
> But this strikes me as a logical property of such graphs - and i wouldn't
> require drawing them to see this property
Well, some people want proof. Something that will stand up in a court of law,
"beyond the shadow of a doubt."
> You wont allow me to bend
> the graph or shade the colours - so that what is green shades
> (imperceptibly) to something which could be blue- you given me no choice -
> or rather you have said nothing about shapes or colours. (you are treating
> these as logically neutral) We can compare this to the noble gases so called
> because the cant be (now they can) oxidised. Or heavier than air flying
> machines...etc. And if we illustrate your graph on TV we will be colouring
> it with only 3 colours - and be able to do so for all the graphs - even the
> one above.
The things I am calling "colors" are the numbers 1, 2, 3, ... . I never
mention "blue".
> > * To open up a can of worms: Kurt G\"odel proved that if you make any
> given
> > set of assumptions (axioms), then one of two things happens: Either you
> can
> > prove something which is actually not true, or there is some statement
> which
> > is true, but you can't prove it (only given your assumptions).
> >
> > -- Christopher Heckman
>
> I was aware of the unproveable true - but not the other?
Suppose you are given two assumptions:
(A1) 2 + 2 is 4.
(A2) 2 + 2 is not 4.
There is no reason to suppose that both of these are true, but I am not asking
you to do that. I'm asking what the CONSEQUENCES of accepting those two
statements are.
Once you've done that, then you have to accept that 1 + 1 = 3 as well. Here's
why:
(1) The statement (P and NOT P) -> Q is a tautology; no matter what P and Q
are, it is true.
(2) If we replace P with "2 + 2 is 4" and Q with "1 + 1 = 3", then we still
have a true statement. Thus ((2 + 2 is 4) AND (2 + 2 is not 4)) ->
(1 + 1 = 3) is a true statement.
(3) Since (A1) and (A2) are true, the statement A1 AND A2 is also true; thus
(2 + 2 is 4) AND (2 + 2 is not 4) is a true statement.
(4) If P -> Q is assumed to be true, and P is assumed to be true, Q is also
true. (This is called "modens ponens".) Thus 1 + 1 = 3 is also true.
(5) This is what we wanted to show. QED = "quod erat demonstratum".
I've thus managed to prove a false statement.
If you want to say I already have a false statement, namely (A2), then consider
the other statements:
(A1') Euler's constant is rational.
(A2') Euler's constant is irrational.
I can use the method from (1)-(5) to show that 1 + 1 = 3 is a consequence of
accepting (A1') and (A2'), and I have a specific false statement which I have
managed to show is true.
(It is not known which of (A1') and (A2') is true, but most people
suspect (A2') is true. Euler's constant (denoted by the symbol gamma) is
defined to be the limit of the following expression, as n goes to infinity:
1/1 + 1/2 + 1/3 + ... + 1/n - ln(n)
It is known as the most important obscure constant of mathematics.)
-- Christopher Heckman
Proginoskes
> "G*rd*n" <g...@panix.com> wrote in message
> news:bco3tc$o02$1...@panix1.panix.com...
> > "James Whitehead" <Abx4...@jjh76g7856gh.com>:
> > >
> > > "Proginoskes" <progi...@email.msn.com> wrote in message >
> > >
> > > provide
> > > > a (correct) proof of the Four Color Theorem.
>
> ** how does this proof work - not like proof of perpetual motion machines -
> which given certain physics might be possible?
Check out the Four-Color Theorem homepage!
http://www.math.gatech.edu/~thomas/FC/fourcolor.html
> The proof of the colour theorem would lie in its being presented as a
> tautology from which we cannot escape.
If it's possible to do this, it's extremely difficult. Lots of people have
attempted to prove the 4CT and failed.
Another thought: If this could be done, then why prove the 4CT the way it
has been? Why mention "Kempe chains", "reduction", "unavoidable sets", and
all sorts of terminology, if you don't need to?
> > > The crosspost arose out of confusion as to the *subject* of mathematics.
> > > Lets compare this to physics - where some empirical data can refute a
> > > proposition of physics, can this occur in mathematics-
> >
> > We observe conjectures which may seem true but not be proven
> > for a long time, for example the map-coloring problem or
> > Fermat's famous theorem. If mathematics were informally
> > tautologous in the sense that every consequence of every axiom
> > were immediately obvious, then these problems shouldn't take
> > centuries to solve. So what is this space that has to be
> > taken up before the solution comes into view?
>
> A tautology need not be immediate as Jesse points out - and i think also as
> did Gautama!
Nope; it's the other way around. That a statement is a tautology is immediate.
(Consider the formula ((P -> Q) AND (Q -> R)) -> (P -> R). You can show this
is a tautology by going over all possibilities for P, Q, and R (all possible
choices as TRUE or FALSE for these variables), substituting them into the
formula, and evaluating it. In every case the value of the whole expression
is TRUE. For instance, if P is TRUE, Q is FALSE, and R is TRUE, then the
value of ((P -> Q) AND (Q -> R)) -> (P -> R) is
((TRUE -> FALSE) AND (FALSE -> TRUE)) -> (TRUE -> TRUE)
(FALSE AND FALSE) -> TRUE
FALSE -> TRUE
TRUE.)
That a statement is true takes a lot more work.
-- Christopher Heckman
G*rd*n
> ...
> i can draw a straight line and equally be sure that no one has ever seen it
> before- but its not maths! Or was your nice picture.
A drawn straight line is not different from another drawn
straight line of the same length in the same experiential
sense that one section of the Mandelbrot set is different
from a another section of the Mandelbrot set, at least not
to me.
> > I realize I am arguing with a metaphor which is perhaps more
> > the realm of poetry than philosophy and is certainly non-
> > mathematical, but Hahn started it.
> There is i think something very strange about your metaphor - firstly i
> think that one box should be an object inside the other :-)
Well, that raises other problems. For the moment these two
boxes are categories of experience, and I'm not privileging
the ontology of one over the ontology of the other. If you
wanted to do that, it could go either way.
> And the
> objects you take out of one will be significantly *different* to the other.
Yes, this is why it's remarkable to me that there seems to be
a correspondence between the contents of one box and the other.
> The objects you take out of the box labelled physical universe will be
> strange fuzzy things which appear and disappear - which the audience will
> shout oooooo! a rabbit - naaaaa its a hare etc. And now its my lunch.....
> whereas the objects you take out of the maths box will be just that
> "objects" - now a final question - in which box will you find 'words'?
I think that may be yet a different box.
> (of course you won't be able to show us any of the 'objects' from the maths
> box- e.g circles - straight lines prime numbers etc. - i guess it will
> look to all intents empty! )
You can "see" some mathematical objects if you want to, through
well-known techniques of visualization. Others, like seven-
dimensional hypercubes, may be somewhat resistant. Of
course, if you _don't_ want to see them, you won't. But
this is also true of physical objects.
Note that when you close your eyes, however, neither set of
objects goes away -- a test of "reality".
James Whitehead
"Ned Ludd" <ned...@ix.netcom.com> wrote in message
news:bcq7c0$gsk$1...@slb9.atl.mindspring.net...
talking about one box with the objects from the other.
|-|erc
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Proof of numerology at www.adamskingdom.com
name one billionaire?
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Tiger :: golf :: ?
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Nic Cage stars in what kind of movies ?
Who is the smartest man, Haw.... ?
"G. A. Edgar" <gzda...@sneakemail.com> wrote >
> > So evidently some authoritative philosophers and mathematicians claim that
> > (1) all of mathematics is "tautology" and (2) makes no claims about the real world.
> > So what is the definition of a tautology, when is a mathematical statement a
> > tautology and when is it no, and does mathematics make any claims about the
> > world (what is the subject/object matter or mathematics?)?
>
> Some claim that numbers, patterns, sets, relationships, functions,
> etc. are "real". So instead of talking about "the real world", let's
> talk about "the physical world". Connections between mathematics and
> the physical world are what mathematicians call "applications" of
> mathematics. (Some part of) mathematics is a "model" of (some aspect
> of) the physical world. This is an important study, but it is not
> itself mathematics. Those who confuse the model (in mathematics) with
> the application (in the physical world) are making a mistake.
>
> So... to what extent is mathematics relevant to the physical world? It
> is sometimes said that mathematics is the science of patterns. To the
> extent that the physical world exhibits patterns, it is natural that
> mathematics can be applied to the physical world. [But, see above, not
> that mathematics IS the physical world.]
neural net post : is intelligence mathematical
>> >I agree. I don't think it's possible to come up with a magic mathematical
>> >algorithm that will all of a sudden create artificial intelligence. Math is
>>
>> thats funny, not use maths?
>> many prominent physicists consider plato to be the real universe
>> and space time a component. the physical world behaves to only
>> a fraction of known mathematics. anything can be abstracted and
>> hence mathematically modeled, nothing escapes maths, even
>> nothing is mathematical.
>>
>> intelligence is a such a major component of the universe it *has* to
>> have a fundamental mechanism. this forum toys around it all the time,
>> computers are temporal functions from a function (program) to a dataset,
>> a neural net is the reverse, from the dataset the function is derived.
>> by all accounts intelligence can be defined as inverse computation,
>> neural nets already accomplish this *process, what is the *algorithm*?
>>
>> Herc
>>
>>
>While computation refers to mathematics, the use of mathematics to describe
>an ordinary algorithm would be tedious and trivial. There might be a strong
>mathematical framework for computer algorithm some day, but that day has not
>yet arrived. Math is in describing a computer algorithm, but it is not
>adequate to express the ideas that programmers think of when they are
>programming.
>
>Mathematics could be described with pure words without a reference to a
>technical term, but who would want to try? Mathematics is a subset of human
>endeavor; it is not the other way around.
>
>
There is our discipline of maths, and there is the totality of maths. We do
have a myriad of theories of the mind it would be odd for them not to
come together in time.
The universe doesn't run one way one day and another the next, we are
components of a precise precise machine. Calculus wasn't invented
at the same time by two people on opposite sides of the world, it was
discovered. Our knowledge of events has a counterpart - a timeless
labyrinth of information, we can describe and manipulate it, never touch
or change it.
is there a relationship between
these 2 disparate types of knowledge? if we ceased to
exist could maths exist? yes. if maths ceased to exist
could we exist? no.
Herc
_____________________________
Jesse F. Hughes
>> * To open up a can of worms: Kurt G\"odel proved that if you make any
> given
>> set of assumptions (axioms), then one of two things happens: Either you
> can
>> prove something which is actually not true, or there is some statement
> which
>> is true, but you can't prove it (only given your assumptions).
>>
>> -- Christopher Heckman
>
> I was aware of the unproveable true - but not the other?
The other, that you can prove something which is actually not true, is
a bad way of expressing that the system is inconsistent. It's bad
because it presupposes that every mathematical theory has an intended
model.
This characterization of Goedel's theorem is also misleading because
it claims that the incompleteness theorems apply to any set of
axioms. They most certainly do not. The axioms must be sufficient to
represent a certain amount of arithmetic before Goedel's constructions
can apply.
--
Jesse Hughes
"Casting [Demi] Moore as a woman who has come to the New World so that
she can 'worship without fear or persecution' in _The_Scarlet_Letter_
is like casting Bruce Willis as Young Rene Descartes." -Joe Queenan
Jesse F. Hughes
> Nope; it's the other way around. That a statement is a tautology is
> immediate. (Consider the formula ((P -> Q) AND (Q -> R)) -> (P ->
> R). You can show this is a tautology by going over all possibilities
> for P, Q, and R (all possible choices as TRUE or FALSE for these
> variables), substituting them into the formula, and evaluating
> it. In every case the value of the whole expression is TRUE. For
> instance, if P is TRUE, Q is FALSE, and R is TRUE, then the value of
> ((P -> Q) AND (Q -> R)) -> (P -> R) is ((TRUE -> FALSE) AND (FALSE
> -> TRUE)) -> (TRUE -> TRUE) (FALSE AND FALSE) -> TRUE FALSE -> TRUE
> TRUE.) That a statement is true takes a lot more work.
This is *NOT* the relevant sense of tautology here. You're applying
the definition that a propositional sentence Phi is tautalogous if,
however we substitute truth values for the propositional variables of
Phi, the resulting expression evaluates to true.
This is not the meaning intended when one claims that mathematics is
tautologous.
In any case, you have a funny sense of immediate. If I give you a
long expression consisting of 64 propositional variables, then you
must consider 2^64 substitutions in order to determine whether this is
a tautology or not. I can't imagine calling such a judgment
"immediate".
--
"Just because you're ... in a Ph.d program it does not mean that
you're up to the challenge of being a real mathematician. Only those
who have a purity of mind and dedication to the truth as the highest
ideal have a chance." --James Harris, as Sir Galahad the Pure.
Ned Ludd
>>> (of course you won't be able to show us any of the 'objects'
>>> from the maths box- e.g circles - straight lines prime numbers
>>> etc. - i guess it will look to all intents empty! )
>> As opposed to the other box? Look closer. Every component of
>> every thing in that box is 99.99999% empty.
>> Like the math box, the qualities that things have in that box
>> are the qualities we bring to them.
> in this case 0.000001 is infinitely bigger than nothing. But
> we are talking about one box with the objects from the other.
>
They are only objects because you haven't looked at them
closely enough. In the vast emptiness of the atom there is
an outer shell (or cloud) of vibration that accepts and throws
off light in the range that your eye can detect. You settle
for this as the "object" and don't look any deeper.
In the math box you accept certain postulates about numbers
and how they interact because those attributes have been assigned
to them (by you, or whoever you accept as creator of the math box).
Ned
Mr. Vibrating
> "James Whitehead" <Abx4...@jjh76g7856gh.com>:
> > ...
> > i can draw a straight line and equally be sure that no one has ever seen
it
> > before- but its not maths! Or was your nice picture.
>
>
> A drawn straight line is not different from another drawn
> straight line of the same length in the same experiential
> sense that one section of the Mandelbrot set is different
> from a another section of the Mandelbrot set, at least not
> to me.
>
If you look at a point on a line, and magnify that scale , you still see a
line at that point. In other words, look inside a line and you see - the
line,
Now look at mandelbrot set - say a julia set (
http://aleph0.clarku.edu/~djoyce/julia/explorer.html ). If you magnify parts
of the Julia set, yu see more (infinitely more) julia sets of various
orientations , of - and more julia set within those - etc.
The difference between the two is the greater structure of the Julia set
compare to a line.
The commonality between the two is "contransmagnificandjewbangtantiality".
It was explained to me that the universe rests on a back of a turtle. The
turtle rests on the back of another turtle, and so forth. In other words, it
"turtles all the way down".
And now your telling me that there's been a paradign shift?! Is it now
"boxes within boxes - all the way in"?
G*rd*n
> ...
> "boxes within boxes - all the way in"?
I thought it had been shown that the mathematics box could
not contain itself.
"It's turtles all the way down" was once the punch line of a
joke, like "It's the only game in town." But in the flawed
reign of the Demiurge, these become universal truisms.
Proginoskes
> "James Whitehead" <Abx4...@jjh76g7856gh.com> writes:
>
> > I was aware of the unproveable true - but not the other?
>
> The other, that you can prove something which is actually not true, is
> a bad way of expressing that the system is inconsistent.
I wanted to avoid dragging terminology into the discussion. (It's gone on
long enough, in my opinion; someone should volunteer to go through the whole
thread and write up a summary. But what else should I expect from Usenet?
(shrug).)
> It's bad because it presupposes that every mathematical theory has an
> intended model.
No, it doesn't; I never claimed such a thing.
> This characterization of Goedel's theorem is also misleading because
> it claims that the incompleteness theorems apply to any set of
> axioms. They most certainly do not. The axioms must be sufficient to
> represent a certain amount of arithmetic before Goedel's constructions
> can apply.
Well, if you can't say "23 is prime" in your system, you certainly can't
prove it, can you? For instance, in group theory (where you only have one
operation), you can prove everything that's true, but everything falls back
on your one operation (usually denoted *).
-- Christopher Heckman
Proginoskes
>
> You're applying
> the definition that a propositional sentence Phi is tautalogous if,
> however we substitute truth values for the propositional variables of
> Phi, the resulting expression evaluates to true.
>
> This is not the meaning intended when one claims that mathematics is
> tautologous.
How is "tautologous" defined then? My assumption (which I think was a valid
one) is that "mathematics is tautologous" meant "everything in mathematics
can be proven via tautologies". Evidently lots of other people think the
same way.
According to Merriam-Webster (http://www.m-w.com/cgi-bin/dictionary ),
tautologous is defined to be:
1 : involving or containing rhetorical tautology : REDUNDANT
2 : true by virtue of its logical form alone
... which agrees with "my" definition. What's yours?
> In any case, you have a funny sense of immediate. If I give you a
> long expression consisting of 64 propositional variables, then you
> must consider 2^64 substitutions in order to determine whether this is
> a tautology or not. I can't imagine calling such a judgment
> "immediate".
It's "immediate" in the sense that you don't have to think long about how
to prove it. As I like to say, "the devil's in the details."
-- Christopher Heckman
Proginoskes
That was a bit hasty on my part. I should have said "If anyone claims that
mathematics has no content, I dare them to provide a (correct) proof of the
Four Color Theorem _using only the definition of a planar graph and of a
4-coloring_.
In other words, no Kempe chains, no discharging, no reduction.
-- Christopher Heckman
Proginoskes
> progi...@email.msn.com (Proginoskes) writes:
>
> > A tautology is a boolean formula which is ALWAYS true, no matter what you
> > replace the variables with. This was my point.
>
> I agree that's the commonest meaning of the word tautology, but that's
> not at all the relevant meaning for the claim "mathematical theorems
> are mere tautologies". The relevant meaning there is that
> mathematical theorems are necessary truths.
Well then, a theorem is by definition true. However, to show that a statement
is true, you may need more than the definitions of the terms in it. So it
might be a difference in perspective; if you're asking why a statement is
true, you wouldn't say theorems are tautologous, but if you accept it without
questioning why it's true or how to show it, then it is a necessary consequence
of certain axioms and definitions, and so could be seen as 'tautologous',
but that's not how 'tautologous' usually defined.
In short, the claim "mathematical theorems are [mere] tautologies" is false.
The claim "mathematical 'theorems' are [mere] tautologies" is true.
-- Christopher Heckman
P.S. If you don't understand the difference between the two statements in the
last paragraph, don't reply. Think about it for about an hour.
Jesse F. Hughes
> jes...@cs.kun.nl (Jesse F. Hughes) wrote in message
> news:<87el1qa...@phiwumbda.localnet>...
>> "James Whitehead" <Abx4...@jjh76g7856gh.com> writes:
>>
>> > I was aware of the unproveable true - but not the other?
>>
>> The other, that you can prove something which is actually not true, is
>> a bad way of expressing that the system is inconsistent.
>
> I wanted to avoid dragging terminology into the discussion. (It's gone on
> long enough, in my opinion; someone should volunteer to go through the whole
> thread and write up a summary. But what else should I expect from Usenet?
> (shrug).)
Whoever writes a summary will undoubtedly be giving his take on the
numerous unsettled points of the discussion. I don't see what good
that serves.
>> It's bad because it presupposes that every mathematical theory has an
>> intended model.
>
> No, it doesn't; I never claimed such a thing.
Unless there is an intended interpretation, it makes no sense to claim
that a formal sentence is true or false.
Here's a sentence in a theory.
For all x, f(x) = g(x,f(x)).
Is that sentence true? Is it false? Is it even reasonable to say
that it has a truth value without an interpretation for the function
symbols f and g?
>> This characterization of Goedel's theorem is also misleading because
>> it claims that the incompleteness theorems apply to any set of
>> axioms. They most certainly do not. The axioms must be sufficient to
>> represent a certain amount of arithmetic before Goedel's constructions
>> can apply.
>
> Well, if you can't say "23 is prime" in your system, you certainly can't
> prove it, can you? For instance, in group theory (where you only have one
> operation), you can prove everything that's true, but everything falls back
> on your one operation (usually denoted *).
No, that's not the point.
The point is that Goedel's theorem only applies in theories which are
capable of representing (in a technical sense) the metamathematical
notions. For this, it is sufficient that one can conservatively
introduce a fragment of primitive recursive arithmetic. This is all
fairly technical for the casual reader, but it is important to note
that Goedel's theorem does not apply to arbitrary sets of axioms,
contrary to your synopsis.
--
"So, at this time, I'd like to assure you that I am not interested in
making sure mathematicians worldwide get fired."--JSH Apr 28, 2003
"I'll have prosecutors knocking on your doors. I have no problem with
any number of mathematicians spending time in jail."--JSH Jun 10, 2003
Jesse F. Hughes
> jes...@cs.kun.nl (Jesse F. Hughes) wrote in message
> news:<87brwua...@phiwumbda.localnet>...
>>
>> You're applying
>> the definition that a propositional sentence Phi is tautalogous if,
>> however we substitute truth values for the propositional variables of
>> Phi, the resulting expression evaluates to true.
>>
>> This is not the meaning intended when one claims that mathematics is
>> tautologous.
>
> How is "tautologous" defined then? My assumption (which I think was a valid
> one) is that "mathematics is tautologous" meant "everything in mathematics
> can be proven via tautologies". Evidently lots of other people think the
> same way.
>
> According to Merriam-Webster (http://www.m-w.com/cgi-bin/dictionary ),
> tautologous is defined to be:
>
> 1 : involving or containing rhetorical tautology : REDUNDANT
> 2 : true by virtue of its logical form alone
>
> ... which agrees with "my" definition. What's yours?
The second definition is fine, as long as we interpret in terms of
first order logic, and *not* propositional logic. Propositional logic
is hopelessly inadequate for representing mathematical proofs.
Consider the following statement:
For all x, there is a y such that x < y.
In first order logic, it is easy to formalize this statement. Indeed,
it's almost already formalized. You write
(A x)(E y)(x < y).
In propositional logic, how would we formalize it. We would have no
recourse but to formalize it thus:
P
That is not adequate.
In propositional logic, one can determine whether a sentence is
tautologous by truth tables. This is not the case for first order
logic. There, one must either provide a proof (in "pure" first order
logic) of the statement or else show that in every interpretation, the
statement is true. Since there is no advantage in the latter, one may
reasonably say: To show that Phi is tautologous (in the sense that
matters to Hahn and others), one must show that there is a proof of
Phi using only logical axioms and rules of inference.
If you continue to argue that mathematics is not tautologous, because
it is not tautologous in the sense of propositional logic, then you
are simply wasting your time on a straw man. No one would ever assert
such a ludicrous claim.
Certainly, Hans Hahn was a mathematician. He was under no illusion
that mathematical theorems are obvious. He knew a thing or two about
mathematics more than you and me.
--
"If you *still* believe that [my proof is wrong], then I have to think
that your mind is limited [...], and it may be the case that not
everyone *can* achieve that, as the mental wiring may not be there for
the task." -- James Harris, on faculties needed to accept his proof.
Jesse F. Hughes
> jes...@cs.kun.nl (Jesse F. Hughes) wrote in message
> news:<87znkhp...@phiwumbda.localnet>...
>> progi...@email.msn.com (Proginoskes) writes:
>>
>> > A tautology is a boolean formula which is ALWAYS true, no matter what you
>> > replace the variables with. This was my point.
>>
>> I agree that's the commonest meaning of the word tautology, but that's
>> not at all the relevant meaning for the claim "mathematical theorems
>> are mere tautologies". The relevant meaning there is that
>> mathematical theorems are necessary truths.
>
> Well then, a theorem is by definition true. However, to show that a
> statement is true, you may need more than the definitions of the
> terms in it. So it might be a difference in perspective; if you're
> asking why a statement is true, you wouldn't say theorems are
> tautologous, but if you accept it without questioning why it's true
> or how to show it, then it is a necessary consequence of certain
> axioms and definitions, and so could be seen as 'tautologous', but
> that's not how 'tautologous' usually defined.
You want to dispute the use of the word tautologous in the statement
"theorems are tautologies"? You think it's an inappropriate meaning
for that word? Fine, what do I care? It's only the choice of a word.
If we want to interpret the claim "theorems are tautologies", then we
need to discover what the original utterer meant (or what *somebody*
means, anyway). If he's using a word differently than you think is
appropriate, then one may criticize his diction. But it hardly serves
to criticize his claim, which is the relevant aim.
Back to your claim above:
However, to show that a statement is true, you may need more than
the definitions of the terms in it.
No, this is not the correct understanding. You're misunderstand
*what* statement is necessarily true.
The statement "there are an infinite number of primes" is *not* a
necessary truth.
The statement "[Some finite fragment of PA] implies there are an
infinite number of primes" is a necessary truth, provable on purely
logical grounds. There is absolutely no additional material needed.
The statement of Fermat's last theorem is not a necessary truth. The
statement "[Whatever long list of axioms Wiles used] implies FLT" is a
necessary truth. That long list includes more than mere PA, I
understand, but that fact doesn't have any real relevance here.
> In short, the claim "mathematical theorems are [mere] tautologies" is false.
> The claim "mathematical 'theorems' are [mere] tautologies" is true.
>
> -- Christopher Heckman
>
> P.S. If you don't understand the difference between the two statements in the
> last paragraph, don't reply. Think about it for about an hour.
Thanks, but I haven't an hour.
Is this how you help others reach enlightenment? Write something
cryptic (or obtuse) and then tell them to think about it for an hour?
Why not simply be explicit in what you mean?
--
"Destiny is a funny thing. Once I thought I was destined to become
Emperor of Greenland, sole monarch over its 52,000 inhabitants. Then
I thought I was destined to build a Polynesian longship in my garage.
I was wrong then, but I've got it now." -- The Tick
James Whitehead
I dont need to get rid of it - but to say nothing equals nothing i would
suggest doesnt say much - and has therefore no content. One might just as
well say nothing - to say this equals itself is logically tautologous or
needless. Not that the realisation of this is needless- that is essential to
seeing how statements of logic are different from statements about medium
sized dry goods - ethics et al. Which was i think Wittgenstein's point.
>
> > Its amusing because if the propositions did have a content then they
would -
> > like the inside of your fridge - change over time - be subject to
> > deconstruction.
>
>
> That's a new rule you've just made up, in fact, two new
> rules: "content -> change" and "change -> deconstruction".
No new rules at all, the properties of four cats is more than number, this
is an observation. Are you saying that mathematical statements have a
content which is fixed? This i would say are imaginary - and not needed -
and as real as father christmas or the idea of an unchanging God. I'm
probably repeating myself (needlessly?) but i think that rather than use
the term tautologous pejoratively its the other case. Facts of nature do
change - and Popper would say any proposition of science should have a
possible experiment which would refute it. Is this the case of a
proposition, x=y, of mathematics? Maybe - but the whole drift of science
and analysis is to get things into a formal language which seems to gurantee
truth (by virtie of tautology) or may it now not? - one day 2+2 will be
found to be 6. At any rate Wittgenstein i think was pointing out our
confidence in logic - and noting that its unchangeable - because in fact
nothing new is stated within the proposition, the only new thing is our
*recognition*. Are you saying thats not the case and a staement of
mathematics is like any other?
>
> _Tautology_ is etymologically "saying the same thing" and was
> originally a term of rhetoric. If one had said "X!" it seemed
> unnecessary to say it again, "X! X!" since one "X!" would do
> (logically; we pass over its uses for emphasis, derision,
> decoration, rhythm, etc.) Clearly, the term has acquired a
> different meaning in mathematics. Yet saying that tautologies,
> including mathematical ones, are "empty" seems to look back
> to this meaning: "X!", "X!" => "X!" as the logicians observed.
> But in any case this is not how we actually experience
> mathematics.
It is a good definition of how we do experience mathematics - as a system
whose subject is itself. This is the light that Wittgenstein was shining on
it. We seem stuck here more because of a pejorative idea which was never
there, in my case and Ws i think not. That God is contentless is not to
demean God, in negative theology, and makes logical sense - otherwise god
would either be dividable and/or occupy everything. 2+2=4 is more
tautologous than i think therefore i am - or i doubt therefore i doubt...?
this is not pejorative? but the foundation of a philosophy? Maybe bloddy
obvious but not in philosophy till the 17c. Now which side are you on? the
maths guys seem to think that a=a is not to be demeaned with the word
tautology - OK - then they mistake how i and he are using it. Unless they
are saying that mathematical statements should be treated like any other?
like statements of literary criticism! - i dont think they do - maybe you
do - i'd like to move on and see what this would mean. If you put language
in the box with maths - and i dont see how you can avoid this after all "two
plus two is four" then we should be able to deconstruct it. Maths'
logocentricism is based on the idea of tautology. If the repetition in maths
isnt needless then it rhetorical - like a poem? "a poem!"
>
> All this seems very different to me from the Buddhist notion
> of emptiness, but maybe I'm just being stubbornly attached
> to detachment or something.
>
Its from India that zero came.... like enlightenment.
James Whitehead
> "James Whitehead" <Abx4...@jjh76g7856gh.com> wrote in message
news:<bcpier$8vf$3...@newsg4.svr.pol.co.uk>...
> > "G*rd*n" <g...@panix.com> wrote in message
> > news:bco3tc$o02$1...@panix1.panix.com...
> > > "James Whitehead" <Abx4...@jjh76g7856gh.com>:
> > > >
> > > > "Proginoskes" <progi...@email.msn.com> wrote in message >
> > > >
> > > > P.S. If anyone claims that mathematics has no content, I dare them
to
> > > > provide
> > > > > a (correct) proof of the Four Color Theorem.
> >
> > ** how does this proof work - not like proof of perpetual motion
machines -
> > which given certain physics might be possible?
>
> Check out the Four-Color Theorem homepage!
> http://www.math.gatech.edu/~thomas/FC/fourcolor.html
>
> > The proof of the colour theorem would lie in its being presented as a
> > tautology from which we cannot escape.
>
> If it's possible to do this, it's extremely difficult. Lots of people have
> attempted to prove the 4CT and failed.
>
> Another thought: If this could be done, then why prove the 4CT the way it
> has been? Why mention "Kempe chains", "reduction", "unavoidable sets", and
> all sorts of terminology, if you don't need to?
I'm confused - i did do a google and found two sites claiming it had been
proved? As for how or why this is done thats up to you- but one proof was
done on a computer? which presumably used binary to arrive at either 0 or 1?
>
> > > > The crosspost arose out of confusion as to the *subject* of
mathematics.
> > > > Lets compare this to physics - where some empirical data can refute
a
> > > > proposition of physics, can this occur in mathematics-
> > >
> > > We observe conjectures which may seem true but not be proven
> > > for a long time, for example the map-coloring problem or
> > > Fermat's famous theorem. If mathematics were informally
> > > tautologous in the sense that every consequence of every axiom
> > > were immediately obvious, then these problems shouldn't take
> > > centuries to solve. So what is this space that has to be
> > > taken up before the solution comes into view?
> >
> > A tautology need not be immediate as Jesse points out - and i think also
as
> > did Gautama!
>
> Nope; it's the other way around. That a statement is a tautology is
immediate.
Then we differ over words - firstly it cannot be immediate - as it would not
be a (needless) repetition- elsewhere Gordon makes it a claim of Rhetoric -
but repetition is a very useful tool of rhetoric - the needless bit arrive
from a logical not rhetorical anaylsis - Jesus says Verily Verily, Tennyson
half an leauge half a league hal a leauge onwards - why not say a league and
a half...
What you imply here is that a staement like 2+2=4 is a tautology to a maths
graduate but to a five year old it isnt! So your
"((P -> Q) AND (Q -> R)) -> (P -> R). " might be a tautolgy to someone - who
immeditely recognises it - but not to me!
So as i learn my tables what were once not tautologies they become
tautologies - we are discussing my mental state - my psychology and not the
nature of a mathematical statement - you've missed the point of what
Wittgenstein was on about - which was the nature of mathematics - not the
psychology of the person parsing the mathematical statement. (he maybe moves
on to this). It follows therefore that - your argument leads to a
psychological explanation of maths? Well in that case i think ((P -> Q) AND
(Q -> R)) -> (P -> R). sucks!
> (Consider the formula ((P -> Q) AND (Q -> R)) -> (P -> R). You can show
this
> is a tautology by going over all possibilities for P, Q, and R (all
possible
What Wittgenstein is saying is that consider ALL of mathematical
propositions - in principle they all are nothing more than tautologies and
this is GOOD as is makes them more reliable than poetry in building bridges
etc, so once we get our empirical observations into maths success is
guaranteed - the failure occurring in the world of observation and
translation, not that some formulae every so often fails. Or is there a
margin of error in a calculation by virtue of logic itself being not
definite.
In which case mathematics is a kind of Astrology ? Maybe the shuttle failed
due to one of the calculations failing just at the moment of re-entry?
That's weird and very interesting but not what you mean?
> choices as TRUE or FALSE for these variables), substituting them into the
> formula, and evaluating it. In every case the value of the whole
expression
> is TRUE. For instance, if P is TRUE, Q is FALSE, and R is TRUE, then the
> value of ((P -> Q) AND (Q -> R)) -> (P -> R) is
> ((TRUE -> FALSE) AND (FALSE -> TRUE)) -> (TRUE -> TRUE)
> (FALSE AND FALSE) -> TRUE
> FALSE -> TRUE
> TRUE.)
> That a statement is true takes a lot more work.
> -- Christopher Heckman
All you seem to be saying is that speed is at work - thats parsing two
strings for tautology is dependent on how fast the operation takes. You are
welcome to this - i think the point Wittgenstein was making is that there
was not a difference- and realising this was significant in logic. From a
post modern standpoint rhetoric becomes important and of course we arrive at
difference- always difference....
James Whitehead
i find. The world is full of things because we go round naming them - "there
is no signified" - Genesis is spot on "the Lord God formed every beast ....
and brought them unto Adam to see what he would call them..."
James Whitehead
> "James Whitehead" <Abx4...@jjh76g7856gh.com>:
> > ...
> > i can draw a straight line and equally be sure that no one has ever seen
it
> > before- but its not maths! Or was your nice picture.
>
>
> A drawn straight line is not different from another drawn
> straight line of the same length in the same experiential
> sense that one section of the Mandelbrot set is different
> from a another section of the Mandelbrot set, at least not
> to me.
>
Picasso out of a pencil?
>
> > > I realize I am arguing with a metaphor which is perhaps more
> > > the realm of poetry than philosophy and is certainly non-
> > > mathematical, but Hahn started it.
>
> > There is i think something very strange about your metaphor - firstly i
> > think that one box should be an object inside the other :-)
>
>
> Well, that raises other problems. For the moment these two
> boxes are categories of experience, and I'm not privileging
> the ontology of one over the ontology of the other. If you
> wanted to do that, it could go either way.
But youve put yourself outside and are logiaclly contructing arguements? - i
think you must be talking from out of one of the boxes :-)
>
>
> > And the
> > objects you take out of one will be significantly *different* to the
other.
>
>
> Yes, this is why it's remarkable to me that there seems to be
> a correspondence between the contents of one box and the other.
Not to me - that there isn't one - if they are all the same kinds of things
then why have two boxes. Otherwise i could worry a bit of calculus until it
goes off its food and dies, i could lock up quadratic equations - or murder
them in the night while they sleep. Fractions could pull back jobs and
triangles molest young children. Conic sections could become Anglican
Bishops.
(the later is probably true - you might have a point!)
>
>
> > The objects you take out of the box labelled physical universe will be
> > strange fuzzy things which appear and disappear - which the audience
will
> > shout oooooo! a rabbit - naaaaa its a hare etc. And now its my
lunch.....
> > whereas the objects you take out of the maths box will be just that
> > "objects" - now a final question - in which box will you find 'words'?
>
>
> I think that may be yet a different box.
Now you have three boxes! Lets try to see not how a "verb" is like a
butterfly or a fraction, but why should they be put in separate boxes. How
do you if one night i (or some demon) empties all the boxes on the living
room floor - how when you wake do you sort them out and put them back? Thats
the help i think Wittgenstein was offering you. (he also had three boxes,
Science, Logic and Nonsense)
>
>
> > (of course you won't be able to show us any of the 'objects' from the
maths
> > box- e.g circles - straight lines prime numbers etc. - i guess it will
> > look to all intents empty! )
>
>
> You can "see" some mathematical objects if you want to, through
> well-known techniques of visualization. Others, like seven-
> dimensional hypercubes, may be somewhat resistant. Of
> course, if you _don't_ want to see them, you won't. But
> this is also true of physical objects.
Try as i might the bill for the electricity i cannot make it invisible- i
can using 'well-known techniques of visualization' see pixies and Platonic
ideals - but i havent yet managed to see one for real.
>
> Note that when you close your eyes, however, neither set of
> objects goes away -- a test of "reality".
>
Then what does go away?
James Whitehead
news:<bcpieq$8vf$2...@newsg4.svr.pol.co.uk>...
> >
> > But this strikes me as a logical property of such graphs - and i
wouldn't
> > require drawing them to see this property
>
> Well, some people want proof. Something that will stand up in a court of
law,
> "beyond the shadow of a doubt."
>
> > You wont allow me to bend
> > the graph or shade the colours - so that what is green shades
> > (imperceptibly) to something which could be blue- you given me no
choice -
> > or rather you have said nothing about shapes or colours. (you are
treating
> > these as logically neutral) We can compare this to the noble gases so
called
> > because the cant be (now they can) oxidised. Or heavier than air flying
> > machines...etc. And if we illustrate your graph on TV we will be
colouring
> > it with only 3 colours - and be able to do so for all the graphs - even
the
> > one above.
>
> The things I am calling "colors" are the numbers 1, 2, 3, ... . I never
> mention "blue".
Sigh... colours are not numbers, numbers are logical objects represented by
the symbols 1, 2, 3 - and i think one such property of this logic is
1=1=1=1=1=1=1. You would have me take this as a property?
>
>
> > > * To open up a can of worms: Kurt G\"odel proved that if you make any
> > given
> > > set of assumptions (axioms), then one of two things happens: Either
you
> > can
> > > prove something which is actually not true, or there is some statement
> > which
> > > is true, but you can't prove it (only given your assumptions).
> > >
> > > -- Christopher Heckman
> >
> > I was aware of the unproveable true - but not the other?
>
I'm not sure if you want to spend the time talking me through this but i'd
say all along if you do you will want me to accept some basic premises? - or
rules? - which someone already having these rules can follw them clearly.
Ever willing to be the fool here goes...
> Suppose you are given two assumptions:
>
> (A1) 2 + 2 is 4.
> (A2) 2 + 2 is not 4.
>
> There is no reason to suppose that both of these are true, but I am not
asking
> you to do that. I'm asking what the CONSEQUENCES of accepting those two
> statements are.
The consequeces that they are both true? I cant do that without accepting
that logic is illogical.
>
> Once you've done that, then you have to accept that 1 + 1 = 3 as well.
Here's
> why:
But i cant do that? And why do i *have* to accept 1+1 = 3
> (1) The statement (P and NOT P) -> Q is a tautology; no matter what P and
Q
> are, it is true.
(its not immediate to me at all!)
You need to let me know what -> does?
> (2) If we replace P with "2 + 2 is 4" and Q with "1 + 1 = 3", then we
still
> have a true statement. Thus ((2 + 2 is 4) AND (2 + 2 is not 4)) ->
> (1 + 1 = 3) is a true statement.
im stuck at -> here.
> (3) Since (A1) and (A2) are true,
you said you were not asking me to do this - now you are saying they are?
this -> need explaining....
the statement A1 AND A2 is also true; thus
> (2 + 2 is 4) AND (2 + 2 is not 4) is a true statement.
> (4) If P -> Q is assumed to be true, and P is assumed to be true, Q is
also
> true. (This is called "modens ponens".) Thus 1 + 1 = 3 is also true.
> (5) This is what we wanted to show. QED = "quod erat demonstratum".
>
> I've thus managed to prove a false statement.
>
not to me yet...
> If you want to say I already have a false statement, namely (A2), then
consider
> the other statements:
>
> (A1') Euler's constant is rational.
> (A2') Euler's constant is irrational.
>
> I can use the method from (1)-(5) to show that 1 + 1 = 3 is a consequence
of
> accepting (A1') and (A2'), and I have a specific false statement which I
have
> managed to show is true.
> (It is not known which of (A1') and (A2') is true, but most people
> suspect (A2') is true. Euler's constant (denoted by the symbol gamma) is
> defined to be the limit of the following expression, as n goes to
infinity:
>
> 1/1 + 1/2 + 1/3 + ... + 1/n - ln(n)
>
> It is known as the most important obscure constant of mathematics.)
>
i cant begin to think about this last bit.
James Whitehead
"Proginoskes" <progi...@email.msn.com> wrote in message
news:953c225f.03061...@posting.google.com...
> "James Whitehead" <Abx4...@jjh76g7856gh.com> wrote in message
news:<bcptth$n5r$2...@news7.svr.pol.co.uk>...
> > "Proginoskes" <progi...@email.msn.com> wrote in message
> > news:953c225f.03061...@posting.google.com...
> >
> > >
> > > P.S. If anyone claims that mathematics has no content, I dare them to
> > provide
> > > a (correct) proof of the Four Color Theorem.
> >
> > So http://www.math.gatech.edu/~thomas/FC/fourcolor.html is wrong?
>
> That was a bit hasty on my part. I should have said "If anyone claims that
> mathematics has no content, I dare them to provide a (correct) proof of
the
> Four Color Theorem _using only the definition of a planar graph and of a
> 4-coloring_.
>
> In other words, no Kempe chains, no discharging, no reduction.
Was it you who was prepared to give a million dollars? Such a hasty bet
would be a serious danger to the health if placed with certain bookmakers.
But what you mean by the definition of 4-coloring / planar graphs strikes me
as a clue, but i dont see how that somehow gives maths a content - i'd say
subject other than itself. What then is the content of the propositions of
mathematics - what is signified?
James Whitehead
"Russell Blackadar" <rus...@mdli.com> wrote in message
news:3EF0C75B...@mdli.com...
> James Whitehead wrote:
>
> [snip]
>
> > Success is not a guarantee of truth - often the reverse.
>
> [snip]
>
> > I remember in maths once having to use a quadratic equation to calculate
the
> > height of a bridge - which resulted in two answers - one a negative this
we
> > were told could be ignored as it was obvious we were after a positive
value.
>
> (You did this only once?!)
Well it should not surprise people here that it was just the once! My
education in mathematics was poor - though i liked the subject we were
taught by someone who had been a prisoner of war i think - by the Japanese
and was quite mad, until he was replaced by a real mathematician, though not
before terrorising us into a fear of mathematics - well i liked the
subject - but it was rather like liking a vicious dog. I remember the said
POW stabbing a friend in the hand with a pair of compasses - and hurling a
board rubber at another for calling a decimal point a "dot".
And i lied about doing it - (the bridge equation!) we were rather dragged
through the equation and felt cheated at the end. My secondary school was
an art school and as such academic subjects not taken seriously - science
was an option- the geography teacher an alcoholic and the science teacher
polish. We had lessons by well qualified and excellent teachers in
'Lettering' 'art history' 'Design' 'Metal work' - 'silver smithing'
'woodwork' 'Analytical Drawing' 'painting', 'graphic design', 'pottery'
,'3D' and 'printmaking (relief - wood engraving etching silk screen and
lithography.) A popular past time amongst third year boys being copying
confederate bank notes and drawing girls from the tabloid newspapers without
their swimwear.
So maths had tough competition.
>
> I hope you're not trying to suggest you did anything "untrue"
> in this procedure. A number that satisfies *some* of the
> necessary conditions of a solution can still fail to be a
> solution on other grounds. Nothing at all unusual about that.
I said we felt cheated. I'm hoping to find out what -> is - but have the
similar feeling creeping up on one.
James Whitehead
"|-|erc" <che...@yahoo.com> wrote in message
news:5hcIa.100$Fz3....@nnrp1.ozemail.com.au...
>
>
> --
> www.winternet.com/~mikelr/flame76.html
> __
> / /\
> / / \
> / / /\ \
> / / /\ \ \
> / /__/__\ \ \
> /________\ \ \
> \___________\/
> Proof of numerology at www.adamskingdom.com
>
Why is it that maths and science guys like escher?
> name one billionaire? The Queen
> what did Lady Di do? Shag
> Tiger :: golf :: ? Woods
> star wars program was introduced by which president ? Ray Gun
> Nic Cage stars in what kind of movies ? Dont Know -
> Who is the smartest man, Haw.... ? The Duke of Edinburgh
>
>
[..]
> The universe doesn't run one way one day and another the next,
how do you know it doesnt? It might have been in existence only 10 minutes..
we are
> components of a precise precise machine.
Then why do i feel an outsider?
Calculus wasn't invented
> at the same time by two people on opposite sides of the world,
Germany is not on the other side of the world from England.
it was
> discovered.
So was Hamlet discovered by Shakespeare and Beethoven found his 5th Symphony
down the back of the sofa?
Fire was discovered, the zippo lighter invented - strange? What about the
wheel or 78 rpm records? Was adultary discovered or invented or both but
not in the order here?
Our knowledge of events has a counterpart - a timeless
> labyrinth of information, we can describe and manipulate it, never touch
> or change it.
I feel maniuplated at times - has it changed me - i'm not sure?
>
> is there a relationship between
> these 2 disparate types of knowledge? if we ceased to
> exist could maths exist? yes. if maths ceased to exist
> could we exist? no.
Do numbers age?
|-|erc
Hey good answers, I recently thought of a defn for numbers and time.
Counting is a function from space to time,
X X X > 1 2 3
-->
t
[monospaced]
Counting is a function from space to time,
X X X > 1 2 3
-->
t
[arial]
This is from my introduction at www.adamskingdom.com that
explains why the wheel is discovered, it also goes into your
view about changing physical laws if interested.
Phonetics is the primitive of language, and language is the tool
of meaning. In a purposeful universe words, names and symbols
are all linked in a complex network. Pythagarus didn't invent the
circle, most people can comprehend that it transcends time, but
the culmination of an entire language, the associated derivations
from people, places, animals and biology, chemistry, all jargon
and all symbolic objects, is difficult to believe that it is more than
a historically developed facet. Movie makers carefully make up
names to add a dimension to each character, the script is calculated,
very carefully at certain points to ensure flow in each scene and
everything ties together at the finish. Could not the universe itself
play out life, the story of the most advanced man, each scene
culminating into events to lead him to the most advanced woman?
The climax would be the progress of life, something we all know to
exist but not accommodated in any laws of physics.
The universe isn't a big space with moving particles, that
doesn't count for the existence of a concept, a desire.
it is intricate, a structure of meaning and purpose that
drives our physical world.
Such a pity, the wooden axle never able to bear its required load, the
tribe returned to the old technology, the proven rolling logs method,
the spark of ingenuity destined to remain one man's dream. We live
in a strange universe, we think of moving ourselves and other objects
and shortly after things happen, its a comfortable understanding, but
how does it work? Things didn't always happen, and things don't exist
when they are made. If the tribe failed in its invention, another tribe would
take its place, in a sense the wheel existed before it was made, and
will exist after they are all destroyed. We have one understanding of
ourselves with our subcomposition, but what we find aren't things that
move, physics describes matter like a magic show, vanishing, blurring,
jumping, doubling. Time is the mystery, on one hand a universe with
a beginning and ending of time, just a still object we flow through, and
another still world that we can't even touch, void of all events yet
visible, described as the platonic world, knowledge.
But things do have a temporal existence, our reality the physical world,
and there is only one universe, how it all blends together, this is my story.
Would be interesting to see alien civililisations if the 5th symphony or Hamlet
have 'coungerparts', I suggest they would, I even suspect the english
language would be uniform, and even extend that historic figures would
follow the same history, maybe Vitali would play DaVinci's role !
Herc
James Whitehead
"Mr. Vibrating" <eastwood...@yahoo.com> wrote in message
news:zb2Ia.12568$Jw6.4...@news1.news.adelphia.net...
> I agree. The fundamental issue in this "debate", as I understand it, is
> that the logical term "tautology" has a definition different from that of
> "any true statement". Just like a contradiction is different then a false
> statement. So those, such as Hans Hahn, that reduce mathematics to
> tautology, assume that every mathematical expression, being true, must be
a
> tautology. The purpose of this is to, like all reductionism, disguise
> ignorance.
>
I really need to get some bread and clean the kitchen but....
> Science seems to START with clear and precise definitions, and move
forwards
> from there.
Science seems to want to express the many complex things by simple
expressions - it *is* reductionist - "Science is the search for compressions
of strings of data into briefer encodings ('Laws of Nature ') which contain
the same information"
Postmodernism never does. Its goal seems to be to seek
> definitions,
That is way way off - quite the reverse - and not -
"All sentences of the type "deconstruction is X" or "deconstruction is not
X" a priori miss the point, which is to say that they are at least false. "
>but starting from hoped-for results, rather then definitions.
Again i dont think so -
> Thus the "conclusion" that logc and math are tautologies.
No - the conclusion is from in my case Wittgenstein in the Tractatus - not
investigations! and cited by Victor Bugin in Art After Philosophy as the
idea behind Modern (ART). Post Moderninsm i think would challenge the idea
of tautology and decidability.
>
> The reason that
>
> 1 + 2 + (N-1) + N = N (N+1)/2
>
> is not because 2 + 2 = 4, but because other reasons. "2+2=4" is true
because
> it is a definition: the definition of the names of particular numbers and
> the definition of the operator "+". Is
> " x*x +1 = 0 "
> a Tautology? Well, the expression is true for some values of x, and false
> for other values of x. It so happens that it is true for +i and -i, and
> false for all others. According to Hahn, its a tautology if x= i or -i,
and
> a contradiction, other wise.
>
> Lets assume that my real name is "Mr. Vibrating" (on my birth
certificate).
> Is the statement "My name is Mr Vibrating" either a tautology or
> contradition?
Youve turned a sentence about the world into a logical proposition - with
this very difficult and interesting text "Lets assume that my real name is".
Let me try to be post-modern here.
<PM>
First, and this is not to say to prioritorize in any way as we shall see,
you 'say' (or do you certify before the certification?) 'real name'. Real
and not false - but then false is not wanted to be here - present -in its
omission -(and how erased written out un-scribed?) - a spectre, which
implies a falsity, even the possibility of a false name? And how are we to
know which of these 'false' and real names are real, and not ghosts (or are
we to see your false names- what are they?) by recourse to your birth
certificate - but i have to find this, and find the real certificate - and
not the possibility of a false copy or mistake? also- also - (a double
falsehood?) and a not a copy. (A good copy!) And a real name distinguished
by being given and not taking a false name? - taken from who by what means.
One takes a name so to speak or is given a name - a nick name a stolen name.
But leaving this for a moment - and not to complete this important enquiry
or line of investigation we must perhaps address - address the correct name
of the addressee? or 'name' another problematic- in naming itself -
'name', 'does name, name?' Can name name - obviously so- and how can name
name, which is perhaps more interesting to us. And though i would put this
on oneside - by an act of convenience - an act of naming - if i could find a
name for it - or find its 'true' given name - the real name- that i doubt-
is beyond or before me. One would signify in writing - that is 'certify'.
And here the whole interability of certification must not be ignored, this
certificate which inscribes repeatably the naming of speech, the naming of
names, some true some however false? Truth like the new-born child which is
born. It repeats the 'real' birth and certifies it? And carries with it
title - ownership of the un-born? Or re-born? To quickly. Which is
presented to as a proof - - to the name or birth by a repetition of birth
which is at once impossible and possible - legitimate and illegitimate. And
now i must confess my poor ability here - this is a written certification of
it's truth - (not false) and fear we touch or stumble in the metaphysic of
being - is to fall back to a heideggerian epoch of "being" and Dasein. (?)
And a Naming of Dasein! And this i would not wish to do - to fall back
without certification for doing this - given by who or taken from who? As
certified by the science of writing, or writing of a science, which is
another question or (false) name. However we lack not only competence, the
certification perhaps - but perhaps not, but time and so i would repeat -
re-certify "And being here means that beings, are, and are not
non-existent" And so rush at this carelessly - and without certification!,
certified by a valid document of science, that they are not non-existent - a
double negative! And already i have - in order too fulfil your command
perhaps begun - and it is only a beginning - to assume what - to assume too
much - how so? At your command? By what certification?
>
> Suppose that I'm a physician and want to take the temperature of my
patient
> at 4:00pm. I take the temperature at 4:00pm and the thermometer reads 98.6
> degrees F. I record: "The patients temperature at 4:00pm is 98.6 degree
F".
> Is this a tautology?
Would you name this so - by a certification? Am i done with taking on my
self the arrogance of supposition which does not require a certificate. Am i
masquerading or are you under a false - assumed name as a doctor - without
certificate or licence to practice.
>
> In other words, Tautology has lost its restrictive definition and become
> synonymous with "TRUE".
We are yet to find - we are now an arrogant detective looking for the
'other' certificate- and then suspect it of a forgery and masquerade, of
pracricisng without licence - other than what? Of Alias?
>
> In electrical engineering, it is common to construct "logical expressions"
> or digital circuits. These "logical expressions" have inputs and outputs,
> and the outputs change as a function of the inputs.
>
> It is also possible to have a "circuit" who output doesn't change with
> input, but is always either T or F - these are refered to as generators:
a
> generator that always has an output of 'T' would be a tautology, and a
> generator with a constant output of 'F' would be a contradition.
>
> Now, you can plug in generators into the inputs of the logical expression
> (circuit) - thus the outputs get certain values. Take for example, the
> humble inverter ( the "NOT" function):
>
> NOT(T) = F
> NOT(F) = T
>
> Is the NOT() function a tautology or a contradition? It is obviously
> neither, because its output depends on input.
>
> Now, is the following a tautology?
> NOT(F) = T
>
> Is the following a contradition?
> NOT(T) = F
>
> What about the following?
> NOT(T) = T
>
> or this?
> NOT(F) = F
>
> Obviously, the logical expression
> NOT() depends on its inputs for its outputs.
>
> Logical expressions have a "truth table" (even Wittgenstein knew this).
The
> individual entries of the truth table are NOT by definition TAUTOLOGIES,
> but, by definitions, TRUE. Even the expression "NOT(T) = F", being a
member
> of the expressions truth table, is "TRUE" - i.e. "A TRUTH". The statement
> "NOT(T) = T" is not a member of the truth table, and thus "FALSE" - i.e.
"A
> FALSEHOOD".
>
> Now a TAUTOLOGY is a particular kind of truth table, one in which
> reguardless of the state of the inputs, the output is TRUE.
>
> A CONTRADICTION is another kind of truth table, one in which reguardless
of
> the state of inputs, the output is FALSE.
>
> We can construct a truth table from sctach, or from combining previously
> defined truth tables.
>
> We can construct a TAUTOLOGY from, lets say, the truth table for NOT() and
> the truth table for OR(). Let this new truth table be called TAUTO1(),
where
>
> "TAUTO1(x) = OR (NOT(X), X)" (prefix notation)
> That is, "TAUTO1 = NOT(x) OR X" (infix notation)
>
> The resulting truth table for Tauto1 has only "TRUE" as outputs and
> therefore is a TAUTOLOGY by definition.
>
> Similarly one can construct a truth table for a CONTRADITION, which only
> have FALSE as allowable outputs.
>
> Truth Tables are associated with the definition of Operations, not with
> specific instances, although specific instances are elements of of the
truth
> table of the operation.
>
> Taking a temperature with a specific thermometer is an operation whose
> allowable outputs is the continuous range of allowable temperatures that
the
> thermometer can support, within its inherent tolerance. A specific
> temperature taken is simply an instance, or element of thermometer's truth
> table, and cann't ever, by definition, be a tautology. One can design a
> specific digital thermometer which only have three output values in its
> truth table: "Normal Temperature" and "High temperature" and "low
> temperature".
>
> Truth Tables of operators can consist of one or more entries of binary
> values or real values, or complex values or discrete "fuzzy" (or
linguistic)
> values. In all cases, the truth table of the operator specifies "TRUE" or
> "valid" behavior. Values outside of the truth table indicate "FALSE" or
> "broke" or whatever, depending on context.
>
> The english word "IS" has a number of different, context dependant,
> meanings. Does not necessarily mean the same ting as the "=" equal sign.
>
> a. "X is Y"
> In this sense, the implication is that (x,y) belongs to the truth table of
> the operator associated with X. Example: "my name is Mr. Vibrating."
>
> b. "X is"
> In this, unary, sense, "IS" is the exisitential operator, which "includes"
X
> in the universe, inductively. If 'U' is the universe, then saying "X is"
>
> means
> U = U OR X
> "X is not" means
> U = U AND NOT(X)
>
> "Neutiquam erro."
>
> Mr. Vibrating
> aka TL
> aka Mounard le Fougueux
>
The guilty may be before the fact - that is before the equation - was
written - how so?
</PM>
G*rd*n
>
> "G*rd*n" <g...@panix.com> wrote in message
> news:bcq848$k8i$1...@panix2.panix.com...
> > "James Whitehead" <Abx4...@jjh76g7856gh.com>:
> > > ...
> > > i can draw a straight line and equally be sure that no one has ever seen
> it
> > > before- but its not maths! Or was your nice picture.
> >
> >
> > A drawn straight line is not different from another drawn
> > straight line of the same length in the same experiential
> > sense that one section of the Mandelbrot set is different
> > from a another section of the Mandelbrot set, at least not
> > to me.
> >
> How does "experiential" work here - without difference you couldnt get a
> Picasso out of a pencil?
I agree that there is a difference between every actually
drawn line and every other actually drawn line, but if both
lines are 2" straight lines on a piece of white paper I tend
to experience them in the same way, whereas two different
sections of the Mandelbrot set, graphically visualized, seem
to me to contain forms which look different in a way the two
lines don't. The difference seems to have something to do
with the complexity involved in the two different images.
It's possible that the local cats would see it differently,
finding subtleties in the straight lines which the
Mandelbrot set does not possess. But I'm not a cat.
Picasso is not at issue here, I think, because in art one is
allowed to draw in things from the "outside world" often
through rather elaborate and subtle chains of reference,
sometimes by omission -- for instance, Picasso could draw a
face with one eye and it might Mean Something. The present
questions are difficult enough, for me, anyway, without
dragging Picasso into them.
> > > > I realize I am arguing with a metaphor which is perhaps more
> > > > the realm of poetry than philosophy and is certainly non-
> > > > mathematical, but Hahn started it.
> >
> > > There is i think something very strange about your metaphor - firstly i
> > > think that one box should be an object inside the other :-)
> >
> >
> > Well, that raises other problems. For the moment these two
> > boxes are categories of experience, and I'm not privileging
> > the ontology of one over the ontology of the other. If you
> > wanted to do that, it could go either way.
>
> But youve put yourself outside and are logiaclly contructing arguements? - i
> think you must be talking from out of one of the boxes :-)
No. Again, you can conceptually put me in either of the
boxes, or both of them, one inside the other, but at this
point I'm not considering that possibility. They're just two
categories of experience.
> > > And the
> > > objects you take out of one will be significantly *different* to the
> other.
> >
> >
> > Yes, this is why it's remarkable to me that there seems to be
> > a correspondence between the contents of one box and the other.
>
> Not to me - that there isn't one - if they are all the same kinds of things
> then why have two boxes. Otherwise i could worry a bit of calculus until it
> goes off its food and dies, i could lock up quadratic equations - or murder
> them in the night while they sleep. Fractions could pull back jobs and
> triangles molest young children. Conic sections could become Anglican
> Bishops.
> (the later is probably true - you might have a point!)
I didn't say they had the same kinds of things in both
boxes, only that there are evident correspondences between
objects in one box and objects in the other.
> > > The objects you take out of the box labelled physical universe will be
> > > strange fuzzy things which appear and disappear - which the audience
> will
> > > shout oooooo! a rabbit - naaaaa its a hare etc. And now its my
> lunch.....
> > > whereas the objects you take out of the maths box will be just that
> > > "objects" - now a final question - in which box will you find 'words'?
> >
> >
> > I think that may be yet a different box.
>
> Now you have three boxes! Lets try to see not how a "verb" is like a
> butterfly or a fraction, but why should they be put in separate boxes. How
> do you if one night i (or some demon) empties all the boxes on the living
> room floor - how when you wake do you sort them out and put them back? Thats
> the help i think Wittgenstein was offering you. (he also had three boxes,
> Science, Logic and Nonsense)
Well, "nonsense" is nonsense. It's not a very useful
category. Warum man nicht sprechen kann, darueber musst er
singen.
In any case I am not ready to go on to a third
box.
The physical-world box and the mathematics box are
categories of experience. I am not claiming some deeper
ontology for them, so the idea of a demon emptying the two
boxes on the floor and mixing up the contents, while amusing,
doesn't mean much -- it's possible that I could suffer some
kind of brain damage which would make it impossible for me to
distinguish between a quadratic equation and a cat, but that
doesn't say anything about my previous experience where I
could easily distinguish between them.
> > > (of course you won't be able to show us any of the 'objects' from the
> maths
> > > box- e.g circles - straight lines prime numbers etc. - i guess it will
> > > look to all intents empty! )
> >
> >
> > You can "see" some mathematical objects if you want to, through
> > well-known techniques of visualization. Others, like seven-
> > dimensional hypercubes, may be somewhat resistant. Of
> > course, if you _don't_ want to see them, you won't. But
> > this is also true of physical objects.
>
> Try as i might the bill for the electricity i cannot make it invisible- i
> can using 'well-known techniques of visualization' see pixies and Platonic
> ideals - but i havent yet managed to see one for real.
Heh. What do you mean by "real"?
> > Note that when you close your eyes, however, neither set of
> > objects goes away -- a test of "reality".
> >
> Then what does go away?
Your attention.
G*rd*n
"G*rd*n" <g...@panix.com>
> > It is not. When you remove the a's from 'a = a', by
> > exerting a rule for '=', you still have '='. How are you
> > going to get rid of it? But this is beside the point.
"James Whitehead" <Abx4...@jjh76g7856gh.com>:
> I dont need to get rid of it - but to say nothing equals nothing i would
> suggest doesnt say much - and has therefore no content. One might just as
> well say nothing - to say this equals itself is logically tautologous or
> needless. Not that the realisation of this is needless- that is essential to
> seeing how statements of logic are different from statements about medium
> sized dry goods - ethics et al. Which was i think Wittgenstein's point.
Saying "x = x" doesn't tell us much about x, whether x "is"
something or nothing. But it does tell us something about
'='. '=', especially in concert with its little buddies,
turns out to be very productive, in fact. Neither you nor
Witt can get rid of it by waving it off. It's _there_.
Saying that all that sort of thing is nothing is simply
privileging one sort of ontology over another, and I don't
see where you get the charter to do that when so many people
have experiences which run directly counter to it.
"James Whitehead" <Abx4...@jjh76g7856gh.com>:
> > > Its amusing because if the propositions did have a content then they
> would -
> > > like the inside of your fridge - change over time - be subject to
> > > deconstruction.
"G*rd*n" <g...@panix.com>
> > That's a new rule you've just made up, in fact, two new
> > rules: "content -> change" and "change -> deconstruction".
"James Whitehead" <Abx4...@jjh76g7856gh.com>:
> No new rules at all, the properties of four cats is more than number, this
> is an observation. Are you saying that mathematical statements have a
> content which is fixed? This i would say are imaginary - and not needed -
> and as real as father christmas or the idea of an unchanging God. I'm
> probably repeating myself (needlessly?) but i think that rather than use
> the term tautologous pejoratively its the other case. Facts of nature do
> change - and Popper would say any proposition of science should have a
> possible experiment which would refute it. Is this the case of a
> proposition, x=y, of mathematics? Maybe - but the whole drift of science
> and analysis is to get things into a formal language which seems to gurantee
> truth (by virtie of tautology) or may it now not? - one day 2+2 will be
> found to be 6. At any rate Wittgenstein i think was pointing out our
> confidence in logic - and noting that its unchangeable - because in fact
> nothing new is stated within the proposition, the only new thing is our
> *recognition*. Are you saying thats not the case and a staement of
> mathematics is like any other?
It's conceptually easy to propose a modeling scheme of the
physical world in which nothing changes. Transform time into
a fourth spatial dimension. If you insist on indeterminacy,
I give you many-worlds. So now we can say that a sufficiently
intelligent being could "see" the physical world as an unchanging
mathematical object. Of course, we don't experience the world
in this way, and I'm not saying it "is" an u.m.o., only that
such a modeling appears hypothetically possible. So there
can be a correspondence between the apparently unchanging
facts of mathematics and the changing facts of the physical
world. Therefore, if the physical world has content, the
mathematical world can also be said to have content, whatever
"having content" means -- I guess I would say it means supporting
experience.
Nevertheless, I do not experience mathematics as merely
repeating itself, that is, as _rhetorically_ tautologous. In
mathematics and logic, as I said, _tautology_ appears to have
other meanings and connotations.
I'm not ready to go on to language.
> > All this seems very different to me from the Buddhist notion
> > of emptiness, but maybe I'm just being stubbornly attached
> > to detachment or something.
> >
> Its from India that zero came.... like enlightenment.
Well, that's been debated. I believe the zero came into use
because of positional notation, which I seem to recall was
an invention of the Sumerians or Babylonians. They may have
used an empty column between drawn lines indicating the
positions of the digits, or a point -- I don't remember now.
Of course in India they claim it was invented there. But once
you invent zero it's no longer "nothing" and you've blown your
enlightenment out the window.
Martin Eisenberg
> "James Whitehead" <Abx4...@jjh76g7856gh.com>:
[snip]
>> Its from India that zero came.... like enlightenment.
>
>
> Well, that's been debated. I believe the zero came into use
> because of positional notation, which I seem to recall was
> an invention of the Sumerians or Babylonians. They may have
> used an empty column between drawn lines indicating the
> positions of the digits, or a point -- I don't remember now.
Here's a nice article explaining that:
http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Zero.html
Mr. Vibrating
Mediation of something else, which is not material, operate upon, and
effect other Matter without mutual Contact, as it must be, if Gravitation
in the Sense of Epicurus, be essential and inherent in it. And this is one
Reason why I desired you would not ascribe innate Gravity to me. That
Gravity should be innate, inherent and essential to Matter, so that one Body
may act upon another at a distance thro' a Vacuum, without the Mediation of
any thing else, by and through which their Action and Force may be conveyed
from one to another, is to me so great an Absurdity, that I believe no Man
who has in philosophical Matters a competent Faculty of thinking can ever
fall into it."
Issac Newton Letter is dated February 25, 1692/3.
Here's another onteresting quote from a famous wise guy
"Jesse F. Hughes" <jes...@cs.kun.nl> wrote in message
news:87ptlco...@phiwumbda.localnet...
> "Mr. Vibrating" <eastwood...@yahoo.com> writes:
>
> > The following is a scan of the original article : Logic, Mathematics and
> > Knowledge of nature" by Hans Hahn (vienna, 1933). This looks the source
of
> > the notion that all math is "tautologies".
>
> I wouldn't guess that it's the source of that notion. It's just the
> presentation with which I am most familiar.
>
> --
> "[I]t's good for the economy to charge for intellectual property, so
> open source software cannot be good, while Microsoft is the most
> far-thinking company around and is doing it all for the good of the
> public." -- Linus Torvalds paraphrases Microsoft VP Craig Mundie
Proginoskes
>
> You want to dispute the use of the word tautologous in the statement
> "theorems are tautologies"? You think it's an inappropriate meaning
> for that word? Fine, what do I care? It's only the choice of a word.
But that word is what this whole thread (81 articles and counting) is about.
> Back to your claim above:
>
> However, to show that a statement is true, you may need more than
> the definitions of the terms in it.
>
> No, this is not the correct understanding. You're misunderstand
> *what* statement is necessarily true.
Yes, you do need to put the statement into context; I'll grant that. But
the context alone might not be enough; that is my point.
> > P.S. If you don't understand the difference between the two statements
> > in the last paragraph, don't reply. Think about it for about an hour.
>
> Is this how you help others reach enlightenment? Write something
> cryptic (or obtuse) and then tell them to think about it for an hour?
> Why not simply be explicit in what you mean?
I have been explicit. I have also replied about 10 times to this thread,
and I'm tired of people not understanding what I have to say, forcing me
to repeat it. It's an old Usenet rule: READ THE REST OF THE THREAD BEFORE
YOU RESPOND.
-- Christopher Heckman
Proginoskes
> "Proginoskes" <progi...@email.msn.com> wrote in message
> news:953c225f.03061...@posting.google.com...
> > "James Whitehead" <Abx4...@jjh76g7856gh.com> wrote in message
> news:<bcptth$n5r$2...@news7.svr.pol.co.uk>...
> > > "Proginoskes" <progi...@email.msn.com> wrote in message
> > > news:953c225f.03061...@posting.google.com...
> > >
> > > >
> > > > P.S. If anyone claims that mathematics has no content, I dare them to
> provide
> > > > a (correct) proof of the Four Color Theorem.
> > >
> > > So http://www.math.gatech.edu/~thomas/FC/fourcolor.html is wrong?
> >
> > That was a bit hasty on my part. I should have said "If anyone claims that
> > mathematics has no content, I dare them to provide a (correct) proof of
> the
> > Four Color Theorem _using only the definition of a planar graph and of a
> > 4-coloring_.
> >
> > In other words, no Kempe chains, no discharging, no reduction.
>
> Was it you who was prepared to give a million dollars?
Yes I was.
> Such a hasty bet
> would be a serious danger to the health if placed with certain bookmakers.
No, it's perfectly okay, because if such a proof existed, someone would have
found it decades ago (maybe even a century or more ago).
The exact parameters of my bet is the following. You must prove the 4-Color
Theorem, using only the following facts:
(1) The plane is a 2-dimensional Euclidean plane. (In fact, I'll let you use
anything from Euclid's _Geometry_.)
(2) The definition of a function, and continuity. All of Set Theory.
(3) A graph G consists of a finite set of vertices (usually denoted V) and
a set of edges (usually denoted by E), where every element of E is an
ordered pair, where each component is an element of V.
(4) An embedding of a graph G consists of two functions, f: V -> P, and
g: E x [0,1] -> P, where P is the Euclidean plane. Furthermore, an
embedding must be one-to-one (f(u) =/= f(v), if u and v are two distinct
vertices), g must be a continuous function, and if e is an edge (u,v),
then g((u,v),0) = f(u) and g((u,v),1) = v.
(5) A graph G is planar if there is an embedding so that each point p in P
is f(v) for some vertex v, or there is at most one pair (e, t) so that
g(e, t) = p. In other words, "the edges don't cross".
(6) A 4-coloring of G is a function c:V -> {1, 2, 3, 4}, such that for every
edge (u,v) in G, c(u) =/= c(v).
Now, the Four Color Theorem says: If G is a planar graph with no loops (no
edges are of the form (v,v)), then there is a 4-coloring of G.
You (anyone) are supposed to show that the Four Color Theorem is true, using
ONLY the items mentioned in the list above ((1)-(6)).
-- Christopher Heckman
Proginoskes
>
> So evidently some authoritative philosophers and mathematicians claim that
> (1) all of mathematics is "tautology" and (2) makes no claims about the real
> world.
> So what is the definition of a tautology, when is a mathematical statement a
> tautology and when is it no, and does mathematics make any claims about the
> world (what is the subject/object matter or mathematics?)?
After having responded 10 times to this thread (at least), I feel that no
progress has been made, at least in my part. What is needed is the answer to
the following question:
* What do you mean by "tautology"?
To answer your second question, what is needed is the answer to the following
question:
* What is the real world?
These are questions that YOU, Mr. Vibrating, need to answer.
Until the definitions for this discussion have been established, further
meaningful communication is not possible (and I will not respond to this
thread until then).
-- Christopher Heckman
Jesse F. Hughes
> jes...@cs.kun.nl (Jesse F. Hughes) wrote in message
> news:<87wufhq...@phiwumbda.localnet>...
>>
>> You want to dispute the use of the word tautologous in the statement
>> "theorems are tautologies"? You think it's an inappropriate meaning
>> for that word? Fine, what do I care? It's only the choice of a word.
>
> But that word is what this whole thread (81 articles and counting)
> is about.
Nonsense. The thread is about the content of the claim "math is
tautological" and its truth value. Whether the word "tautological"
has an unusual meaning in this context is beside the point. One
should read that claim with the meaning that its advocates have in
mind and judge accordingly.
>> Is this how you help others reach enlightenment? Write something
>> cryptic (or obtuse) and then tell them to think about it for an hour?
>> Why not simply be explicit in what you mean?
>
> I have been explicit. I have also replied about 10 times to this thread,
> and I'm tired of people not understanding what I have to say, forcing me
> to repeat it. It's an old Usenet rule: READ THE REST OF THE THREAD BEFORE
> YOU RESPOND.
Sorry, I won't read the whole thread. There's far too much postmodern
discussion (no surprises) and other kinds of philosophy in which I
have no interest. Thus, I have stuck to those articles which are
clearly and explicitly about philosophy of mathematics in a
recognizable form. It is on this topic that I am competent and
interested to contribute.
Therefore, my life will be the poorer for not understanding the deep
wisdom contained in your punctuation (tick marks! wow!). Darn the
luck.
As far as repetition goes: How many times have I explicitly said (in
response to you) that one shouldn't interpret the word "tautology" in
terms of propositional logic[1]? You have a PhD in math, right? You
know the difference between propositional and first order logic?
(Sorry, but I'm really not sure whether the distinction is well known
to your average mathematician.)
Of course, even if you weren't clear on the distinction, you could
have asked why I kept repeating the claim that *no* *one* thinks math
is tautological in the sense of propositional logic.
Footnotes:
[1] I'm petty. I counted. I repeated this claim in reply to your
posts four times. I don't ask that you read the whole thread to know
what I've said, but reading my replies to you seems reasonable --
given that you responded to them and all.
--
"I am one of the more important discoverers in mathematical history,
but future students will have the luxury of knowing that, and may be
puzzled by your behavior now." -- James Harris
(At least I have the foresight to quote his pearls of wisdom.)
Jesse F. Hughes
> "Mr. Vibrating" <eastwood...@yahoo.com> wrote in message
> news:<QNOGa.6769$Jw6.3...@news1.news.adelphia.net>...
>>
>> So evidently some authoritative philosophers and mathematicians
>> claim that (1) all of mathematics is "tautology" and (2) makes no
>> claims about the real world. So what is the definition of a
>> tautology, when is a mathematical statement a tautology and when is
>> it no, and does mathematics make any claims about the world (what
>> is the subject/object matter or mathematics?)?
>
> After having responded 10 times to this thread (at least), I feel
> that no progress has been made, at least in my part. What is needed
> is the answer to the following question:
Why ask what he means? Wiser men than he is have made a claim such as
this, so let's try to deduce what *they* meant in order to evaluate
the claim.
> * What do you mean by "tautology"?
The only reasonable meaning is the same one I've given repeatedly.
Namely, a theorem of first order logic, i.e., a statement provable via
pure logic alone, the strongest reasonable form of necessary truth in
this context.
> To answer your second question, what is needed is the answer to the
> following question:
>
> * What is the real world?
What a silly and awful question.
To understand part 2, I would follow Hahn's presentation. What he
meant is that, when mathematical calculations are performed in a
scientific setting, the results of those calculations add no
information about the objects of the theory at all. Rather, in
exactly the same way that if I tell you the dress is blue or not blue,
you have gained no new information about the dress, if I perform a
calculation, I have merely transformed the information in the existing
observations into a new form, without adding information.
Of course, as beings with tiny minds, we may not have been aware of
the deductive and computational commitments which our observations
have required. Thus, although there is no new information, we may
nonetheless be surprised at the form of the newly repackaged data.
Since you have carefully read the entire thread, of course, you are
well familiar with Hahn's writing on this point. Nonetheless, let me
quote a relevant passage (in which he deals with logic, not
mathematics, but Hahn believes that the same explicit analysis works
for math, too).
We see, then, that there are two totally different kinds of
statements: those which really say something about objects, and
those which do not say anything about objects but only stipulate
rules for speaking about objects. If I ask "what is the color of
Miss Ema's new dress?" and get the answer "Miss Ema's new dress
is not both red and blue (all over)," then no information about
this dress has been given to me at all. I have been made no wiser
by it. But if I get the answer "Miss Erna's new dress is red,"
then I have received some genuine information about the dress.
Let us clarify this distinction in terms of one more example. A
statement which really says something about the objects which it
mentions, is the following: "If you heat this piece of iron up to
800?, it will turn red, if you heat it up to 1300?, it will turn
white." What makes the difference between this statement and the
statements cited above, which say nothing about facts? The
application of temperature designations to objects is independent
of the application of color designations, whereas the color
designations "red" and "not red," or "red" and "blue" are applied
to objects in mutual dependence. The statements "Miss Erna's new
dress is either red or not red" and "Miss Erna's new dress is not
both red and blue" merely express this dependence, hence make no
assertion about that dress, and are for that reason absolutely
certain and irrefutable. The above statement about the piece of
iron, on the other hand, relates independently given
designations, and therefore really says something about that
piece of iron and is for just that reason not certain nor
irrefutable by observation.
Of course, in rebuttal to this view, Quine argues that the distinction
between analytic and synthetic statements can *not* be drawn as
sharply as necessary, and so the analysis fails.
> These are questions that YOU, Mr. Vibrating, need to answer.
I don't see why we would care much about *his* answer. Why not read a
source who has thought long and hard on this and related subjects, and
whose arguments are persuasive enough to be included in standard
philosophical compilations? Surely, such an author makes a more
interesting case for the claim than Mr. Vibrating could?
> Until the definitions for this discussion have been established, further
> meaningful communication is not possible (and I will not respond to this
> thread until then).
Goodness.
--
Jesse Hughes
"And a journal can beg me for the right to publish it [...] because
I'd rather see it in "People" magazine [...]"
--James Harris on his simple proof of Fermat's last theorem
James Whitehead
*You* may experience the two lines in the same way but first off we must
tip the hat to sol le witt especially but also Don Judd (in case anyone in
alt.postmodern is aware of modern art) Or even Carl Andre. Someone might
think a child of five could draw as well as Picasso - but or perhaps and -
philosophically you do experience a difference. And in ontological terms
the difference is as much - is it the same i dont know - as different parts
of the mandelebrot set or a truck from a house fly. (Something god is unable
to do)
>
> Picasso is not at issue here, I think, because in art one is
> allowed to draw in things from the "outside world" often
> through rather elaborate and subtle chains of reference,
> sometimes by omission -- for instance, Picasso could draw a
> face with one eye and it might Mean Something. The present
> questions are difficult enough, for me, anyway, without
> dragging Picasso into them.
I'm not sure why you want to delimit difference - two lines of 2 inches on a
piece of white paper are different - ontologically - if signed Sol Le Witt
they attain economic and artistic difference also. I dont think you can
have complexity without difference- what makes different parts of the
mandlebrot different is just that, Le Witt's lines allow us perhaps to
recognise more of the ontology of difference - time for instance which
plays apart - and what is not present, the mandelbrot set maybe is too
seductive to notice 'difference', i find it rather similar especially in
certain parts, i'd say i could count lines on paper easier than sections of
the mandlebrot set. Lets imagine two sheets - i could prepare - one with 100
2 inch lines on it the other with 100 images from the mandlebrot set and i
ask you to count them. I could slip in duplicate pictures for the mandlebrot
set - say nos 34 and 78 and you would probably count them as different. As
for dragging in subtle chains of reference and omissions - again shouldn't
this be OK in alt.postmodern, are not such events and contexts of *vital*
importance?
>
>
> > > > > I realize I am arguing with a metaphor which is perhaps more
> > > > > the realm of poetry than philosophy and is certainly non-
> > > > > mathematical, but Hahn started it.
> > >
> > > > There is i think something very strange about your metaphor -
firstly i
> > > > think that one box should be an object inside the other :-)
> > >
> > >
> > > Well, that raises other problems. For the moment these two
> > > boxes are categories of experience, and I'm not privileging
> > > the ontology of one over the ontology of the other. If you
> > > wanted to do that, it could go either way.
> >
> > But youve put yourself outside and are logiaclly contructing
arguements? - i
> > think you must be talking from out of one of the boxes :-)
>
>
> No. Again, you can conceptually put me in either of the
> boxes, or both of them, one inside the other, but at this
> point I'm not considering that possibility. They're just two
> categories of experience.
Well 'No' also - your whole two box thought experiment brings with it the
question of which box it should go into itself, and i think this is both
interesting and significant. You seem to want to imagine finding
similarities between the contents of the two boxes - its a nice idea - one
explored by Wittgenstein in Investigations. You look in one box and "see"
^^
' '
= ' =
^
^^
' '
= ' =
^
In another you "see" somehow "2" (and in a third "CAT")
And so see the similarity sufficient to say "I see Two Cats"
I'm kind of paraphrasing Wittgenstein here - but is that what you do when
you enter a room?
>
>
> > > > And the
> > > > objects you take out of one will be significantly *different* to the
> > other.
> > >
> > >
> > > Yes, this is why it's remarkable to me that there seems to be
> > > a correspondence between the contents of one box and the other.
> >
> > Not to me - that there isn't one - if they are all the same kinds of
things
> > then why have two boxes. Otherwise i could worry a bit of calculus until
it
> > goes off its food and dies, i could lock up quadratic equations - or
murder
> > them in the night while they sleep. Fractions could pull back jobs and
> > triangles molest young children. Conic sections could become Anglican
> > Bishops.
> > (the later is probably true - you might have a point!)
>
>
> I didn't say they had the same kinds of things in both
> boxes, only that there are evident correspondences between
> objects in one box and objects in the other.
Mathematics and perhaps language has objects - but does the world? The
problem is you begin by saying both boxes have objects in them. A
metaphysical assumption. I'm thinking that why the objects appear to have
correspondences is such a metaphysical mistake - that is you have two and
not one box - labelled "objects." Now if we have a second labelled
non-objects i dont see how we can get correspondences.
>
>
> > > > The objects you take out of the box labelled physical universe will
be
> > > > strange fuzzy things which appear and disappear - which the audience
> > will
> > > > shout oooooo! a rabbit - naaaaa its a hare etc. And now its my
> > lunch.....
> > > > whereas the objects you take out of the maths box will be just that
> > > > "objects" - now a final question - in which box will you find
'words'?
> > >
> > >
> > > I think that may be yet a different box.
> >
> > Now you have three boxes! Lets try to see not how a "verb" is like a
> > butterfly or a fraction, but why should they be put in separate boxes.
How
> > do you if one night i (or some demon) empties all the boxes on the
living
> > room floor - how when you wake do you sort them out and put them back?
Thats
> > the help i think Wittgenstein was offering you. (he also had three
boxes,
> > Science, Logic and Nonsense)
>
>
> Well, "nonsense" is nonsense. It's not a very useful
> category. Warum man nicht sprechen kann, darueber musst er
> singen.
But its extremely interesting Nicht Probleme der Naturwissenschaft sind ja
zu losen.....Warum gibt es Sachen, die anstatt nichts sind? Die ist die
Frage.
"We feel that even when all possible scientific questions have been
answered the problems of life remain ..."
>
> In any case I am not ready to go on to a third
> box.
>
Why not?
> The physical-world box and the mathematics box are
> categories of experience. I am not claiming some deeper
> ontology for them, so the idea of a demon emptying the two
> boxes on the floor and mixing up the contents, while amusing,
> doesn't mean much -
I'm glad its amusing!
- it's possible that I could suffer some
> kind of brain damage which would make it impossible for me to
> distinguish between a quadratic equation and a cat, but that
> doesn't say anything about my previous experience where I
> could easily distinguish between them.
But how do you! - its all very well to say they are in two separate boxes -
but you put them in there, all i'm asking is how you did it? Its a
reasonable question isnt it?
>
>
> > > > (of course you won't be able to show us any of the 'objects' from
the
> > maths
> > > > box- e.g circles - straight lines prime numbers etc. - i guess it
will
> > > > look to all intents empty! )
> > >
> > >
> > > You can "see" some mathematical objects if you want to, through
> > > well-known techniques of visualization. Others, like seven-
> > > dimensional hypercubes, may be somewhat resistant. Of
> > > course, if you _don't_ want to see them, you won't. But
> > > this is also true of physical objects.
> >
> > Try as i might the bill for the electricity i cannot make it invisible-
i
> > can using 'well-known techniques of visualization' see pixies and
Platonic
> > ideals - but i havent yet managed to see one for real.
>
>
> Heh. What do you mean by "real"?
Thats a very good question! How do we distinguish between real and false
pixies?
>
>
> > > Note that when you close your eyes, however, neither set of
> > > objects goes away -- a test of "reality".
> > >
> > Then what does go away?
>
>
> Your attention.
msuigb shjmsj sdjdjpsm,skkkdkjkj
translated with eyes open- my eyes are closed and i'm paying attention - i
cant touch type.
James Whitehead
[...]
>
> "G*rd*n" <g...@panix.com>
> > > It is not. When you remove the a's from 'a = a', by
> > > exerting a rule for '=', you still have '='. How are you
> > > going to get rid of it? But this is beside the point.
>
> "James Whitehead" <Abx4...@jjh76g7856gh.com>:
> > I dont need to get rid of it - but to say nothing equals nothing i would
> > suggest doesnt say much - and has therefore no content. One might just
as
> > well say nothing - to say this equals itself is logically tautologous or
> > needless. Not that the realisation of this is needless- that is
essential to
> > seeing how statements of logic are different from statements about
medium
> > sized dry goods - ethics et al. Which was i think Wittgenstein's point.
>
>
> Saying "x = x" doesn't tell us much about x, whether x "is"
> something or nothing. But it does tell us something about
> '='. '=', especially in concert with its little buddies,
> turns out to be very productive, in fact. Neither you nor
> Witt can get rid of it by waving it off.
Well i'm not waving it off at all and neither i think was Wittgenstein -
quite the reverse. I said above it was essential, without this why bother to
use maths in problem solving?
It's _there_.
> Saying that all that sort of thing is nothing is simply
> privileging one sort of ontology over another, and I don't
> see where you get the charter to do that when so many people
> have experiences which run directly counter to it.
>
My position is unclear - i asked the question regarding mathematics to the
effect thats its standing regarding truth of its propositions which is
guaranteed by tautology is no longer there. In other words are the
propositions of mathematics now like those of science and "provisional".
>
[..]
This is baffling? Such a model would therefore be devoid of any difference
in what we might call its surface - if we could perceive one- which of
course we could not.
If you insist on indeterminacy,
> I give you many-worlds.
Which in effect colours in our space to total opacity...
So now we can say that a sufficiently
> intelligent being could "see" the physical world as an unchanging
> mathematical object.
How - one is transparent the other opaque. Let your being try to do the eye
test or a puzzle or read a novel. The enjoyment of any of these is denied,
imagine God reading a detective! How on earth can god read - perceive a
difference, i dont know - perhaps you can help me out on this one.
Of course, we don't experience the world
> in this way, and I'm not saying it "is" an u.m.o., only that
> such a modeling appears hypothetically possible.
Time is experienced by temporal beings, would god need to parse the
u.m.o. - or would it simply be tautlogous. We have two problems telling God
a joke - first he already knows it - and second he's at the punch line
before we begin. There is a man who has a damaged brain such that he is
constantly experiencing De Ja Vu. He is extremely angry and frustrated in
this state of constantly repeating what has already happened - perhaps this
accounts for the grumpyness of the Almighty of the O.T. and those maths
chaps who were quite rude. Oh they've seen it all before! Its like trying
to give God directions - he already knows the way - worse he's already
there.
So there
> can be a correspondence between the apparently unchanging
> facts of mathematics and the changing facts of the physical
> world.
I dont see how. You may see a comparison between a changing line on a graph
and ones temperature.
Therefore, if the physical world has content, the
> mathematical world can also be said to have content, whatever
> "having content" means -- I guess I would say it means supporting
> experience.
One can experience maths - do a sum - as we used to say, but the content
here is psychological. The sum is i think empty. I dont know what 7 8s means
though i know its 56. 56 chickens or cats all have contents but does 56?
Does it have 7 8s inside it like the organs of the chickens.
Neither did Wittgenstein - there is another meaning of the word which is to
mean as Christopher pointed out
"According to Merriam-Webster (http://www.m-w.com/cgi-bin/dictionary ),
tautologous is defined to be:
1 : involving or containing rhetorical tautology : REDUNDANT
2 : true by virtue of its logical form alone"
The second....
> mathematics and logic, as I said, _tautology_ appears to have
> other meanings and connotations.
Then we have no argument. Save that in Rhetoric the repetition gives a
force.... a subjective one, which is not redunant - and the doesnt the
completion of the equation make the problem redundant.
>
> I'm not ready to go on to language.
You cant avoid it :-)
>
>
> > > All this seems very different to me from the Buddhist notion
> > > of emptiness, but maybe I'm just being stubbornly attached
> > > to detachment or something.
> > >
> > Its from India that zero came.... like enlightenment.
>
>
> Well, that's been debated. I believe the zero came into use
> because of positional notation, which I seem to recall was
> an invention of the Sumerians or Babylonians. They may have
> used an empty column between drawn lines indicating the
> positions of the digits, or a point -- I don't remember now.
> Of course in India they claim it was invented there. But once
> you invent zero it's no longer "nothing" and you've blown your
> enlightenment out the window.
>
I think the Indians also used empty columns, but had the idea of zero.. i
did have that book on nothing......
James Whitehead
You miss the point - your original statement was
"If anyone claims that mathematics has no content, I dare them to provide
a (correct) proof of the Four Color Theorem."
I did a google search and found two - and provided them to you. Your
original had the proviso that it must be correct - but non of the provisos
bellow - which came afterwards. If your horse gets struck by lightening just
before the winning post you lose, if mine gets across the line first
therefore i win, unless you make specifics *before* placing the bet. I'll be
round with my Italian brother in law - and his *work colleagues* to collect!
Dont try welshing - btw they are worse at maths than i am. If they call it a
dot - its a dot OK.
>
> The exact parameters of my bet is the following. You must prove the
4-Color
> Theorem, using only the following facts:
No when you placed it it wasnt.
>
> (1) The plane is a 2-dimensional Euclidean plane. (In fact, I'll let you
use
> anything from Euclid's _Geometry_.)
> (2) The definition of a function, and continuity. All of Set Theory.
> (3) A graph G consists of a finite set of vertices (usually denoted V) and
> a set of edges (usually denoted by E), where every element of E is an
> ordered pair, where each component is an element of V.
> (4) An embedding of a graph G consists of two functions, f: V -> P, and
> g: E x [0,1] -> P, where P is the Euclidean plane. Furthermore, an
> embedding must be one-to-one (f(u) =/= f(v), if u and v are two
distinct
> vertices), g must be a continuous function, and if e is an edge (u,v),
> then g((u,v),0) = f(u) and g((u,v),1) = v.
> (5) A graph G is planar if there is an embedding so that each point p in P
> is f(v) for some vertex v, or there is at most one pair (e, t) so that
> g(e, t) = p. In other words, "the edges don't cross".
> (6) A 4-coloring of G is a function c:V -> {1, 2, 3, 4}, such that for
every
> edge (u,v) in G, c(u) =/= c(v).
>
> Now, the Four Color Theorem says: If G is a planar graph with no loops (no
> edges are of the form (v,v)), then there is a 4-coloring of G.
>
> You (anyone) are supposed to show that the Four Color Theorem is true,
using
> ONLY the items mentioned in the list above ((1)-(6)).
I'm not taking you up as you've welted on the first one - but if i did here
is how i'd do it,
You say "show that the Four Color Theorem is true" well i'd hold up a
sausage with any formulae you want on it and say to "Lucky" - good boy lucky
bark if this shows YOU LUCKY that this is a *correct* proof of the 4 colour
theorem - using only the 6 points above, he'd bark and get the sausage and
i'd be up by 2 million dollars if my sums are correct.
For a share in the winnings - (1/2 million?) i bet i could get a maths PhD
to say i'd shown them the proof. I'll make that offer to anyone here - would
anyone with a maths PhD be prepared to state that i *had* done this - i'll
split the winnings.
James Whitehead
>
> Of course, in rebuttal to this view, Quine argues that the distinction
> between analytic and synthetic statements can *not* be drawn as
> sharply as necessary, and so the analysis fails.
>
If i can ask an question here? - I've snipped the repeated quotes- but they
more or less were from similar sources where i got the idea- of tautology
and maths - but thanks- now remembering that i'm not a maths PhD, the Quine
argument you seem to say succeeded? What is the nature of its proof?
Jesse F. Hughes
It's not a mathematical, but a philosophical, argument.
It's found most famously in "Two Dogmas of Empiricism", but I believe
it also comes up in "Truth by Convention".
I would try to summarize it for you, but I confess that I can't recall
how it goes right now, and my copy of these articles is unavailable
until Monday. I hope someone else can give a brief outline.
If not, drop me an email on Monday and I'll try to have a go at it. I
read the latter article about two months ago, so I think it won't take
long to refresh my addled head.
James Whitehead
OK - forget any PhDs out there I've an even better* idea - I'll prove it to
you
You stated:
"Kurt G\"odel proved that if you make any given set of assumptions
(axioms),
then one of two things happens: Either you can prove something which is
actually not true,
or there is some statement which is true, but you can't prove it (only given
your assumptions)."
-- Christopher Heckman
And then went on to show how you could...
"I've thus managed to prove a false statement."
So all i need to do is provide you with any "incorrect proof" of the 4
colour theory and *you* can *prove* this false statement to be true!
I want my money!
* it saves me 1/2 a million.
Pickled Cucumber
> So evidently some authoritative philosophers and mathematicians claim that
> (1) all of mathematics is "tautology" and (2) makes no claims about the real
> world.
Well, maybe some do, but this account seems a bit sweeping and
overhasty to me.
As to (1), mathematics proceeds from things that it has accepted as
known to
things that it doesn't know, and this "proceeding" - if its fruits are
to be
accepted as yet another "known" - has to be presented formally in
terms of
procedures that have been accepted as standard in that field of
endeavour.
Don't all systematized fields of knowledge proceed in this manner? In
the
case of mathematics, the accepted formal presentation is, most
commonly,
a logical deduction of the "unknown" from the "known". But to say
that all of
mathematics is "tautology" seems to be founded on the assumption that
mathematical objects are identical to their linguistic descriptions --
i.e. to the axion systems that define them. They are not. The Greeks
have
tried to trisect the angle with ruler and compass long before geometry
and algebra advanced to the stage where it was possible to supply a
systematic
proof that the trisection was impossible. Those Greeks were doing
real math,
and they were certainly not playing with linguistic definitions. One
might say
that a substantial part of math has to do with trying to find adequate
definitions for mathematical objects -- i.e. a minimal selection of
properties
that are accepted as true about the object and that _would_ enable one
to derive
all other interesting properties of the object. And one might also
say that
the development of the way mathematical discoveries are formally
presented
has been towards disguising them as complicated tautologies.
As for (2), math sure makes claims about _my_ real world. If I have
to be
somewhere by a certain time, math tells me how fast I should be
moving.
If I have to build a fence, math tells me how much material I need.
Math tells me if it's likely to rain tomorrow, whether or not that
rocket will
hit that moon, whether a bridge thusly designed will be likely to
collapse.
It whispers to me things about the real world all the time.
PC
G*rd*n
"G*rd*n" <g...@panix.com>
>>>> It is not. When you remove the a's from 'a = a', by
>>>> exerting a rule for '=', you still have '='. How are you
>>>> going to get rid of it? But this is beside the point.
"James Whitehead" <Abx4...@jjh76g7856gh.com>:
>>> I dont need to get rid of it - but to say nothing equals nothing i would
>>> suggest doesnt say much - and has therefore no content. One might just as
>>> well say nothing - to say this equals itself is logically tautologous or
>>> needless. Not that the realisation of this is needless- that is essential to
>>> seeing how statements of logic are different from statements about medium
>>> sized dry goods - ethics et al. Which was i think Wittgenstein's point.
"G*rd*n" <g...@panix.com> wrote in message
>> Saying "x = x" doesn't tell us much about x, whether x "is"
>> something or nothing. But it does tell us something about
>> '='. '=', especially in concert with its little buddies,
>> turns out to be very productive, in fact. Neither you nor
>> Witt can get rid of it by waving it off.
"James Whitehead" <Abx4...@jjh76g7856gh.com>:
> Well i'm not waving it off at all and neither i think was Wittgenstein -
> quite the reverse. I said above it was essential, without this why bother to
> use maths in problem solving?
Above you say, "but to say nothing equals nothing i would
suggest doesnt say much - and has therefore no content." In
natural language and common sense this utterance would be
taken by many to denote the insignificance of '=' or
tautlogies. But because natural language and common sense
are necessarily full of ambiguities, it's pretty easy to
mix them up so that they don't work. Thus, one can say
"x has no content" and later "x is very important" which
are an implied contradiction in natural language and common
sense, and if someone takes exception to your utterance,
explain that you have given _content_ or _important_ special
unexpected meanings, perhaps derived from arcane branches of
Buddhist philosophy. I don't really care to play this game
because I don't think it's challenging enough. There's no
payoff: it's like passing your whole life smoking dope and
watching television.
To make one more try at playing the game my way, I'll note
that while logic and mathematics may be tautologous in the very
limited sense given in the language of those fields, they are
not tautolgous in the rhetorical sense: experientially, we
get more out of the math box than we put into it. This seems
odd because we believe nothing went into the box except what
we put consciously and knowingly into it (whereas in the case
of the physical-world box, we assume someone or something
other than ourselves put the objects in the box). The other
odd thing is that sets of objects from the math box often line
up with sets of objects from the physical-world box even though
it would seem they have very different genealogies and
constitutions.
Of course, you can quibble about "odd". I'm using metaphors
from the physical world; maybe nothing is really odd, that is,
unaccounted for, or everything is odd, and I just haven't
smoked enough dope to understand that.
> It's _there_.
>> Saying that all that sort of thing is nothing is simply
>> privileging one sort of ontology over another, and I don't
>> see where you get the charter to do that when so many people
>> have experiences which run directly counter to it.
>>
> My position is unclear - i asked the question regarding mathematics to the
> effect thats its standing regarding truth of its propositions which is
> guaranteed by tautology is no longer there. In other words are the
> propositions of mathematics now like those of science and "provisional".
Some of them may be _experientially_ provisional because we
don't already know everything. That is true both of most
physical-world objects and maybe most mathematical objects.
(The are infinitely many mathematical objects whose shape I
won't know until I actually and specifically compute them,
just as there are many physical objects I won't know until I
find and observe them.)
However, as I pointed out before, the physical world
could apparently "be", or correspond to, a hypothetical
mathematical object. (I say "apparently" because we haven't
gotten to the bottom turtle -- it may be there is something
at the ground of physical being which is unmathematizable.)
Introducing the omniscient, omnipotent God into a hitherto
(semi-)rational discussion is something like introducing
infinity into arithmetic -- many normally sensible and useful
statements and rules become senseless and intractable. For
instance, if one knows something one has a model of the known
thing in one's mind. So, if one is omniscient, if one has
total and perfect knowledge of everything, one has in one's
mind a model of everything which is necessarily identical to
this same everything, that is, the model and the thing modeled
must be one and the same. Moreover, the model must necessarily
include a complete, identical model of oneself modeling
everything -- a set which contains itself as a proper subset.
In short, we have blown our logical fuses. The idea of
knowledge as we use the term in mundane discourse is a gestalt
which assumes, in the background, a resistance to knowledge
which must be overcome by some sort of action, some transfer
of energy (see Information Theory) and which can never be
totally overcome because one doesn't possess infinite time
and energy. Omniscience seems to be a contradiction in terms
for us non-Gods. (Similar problems attend omnipotence.)
Of course, God can get out of this by pretending to herself
that she's finite and ignorant, and thus can read a
detective novel with enjoyment, both knowing and not knowing
who done it.
> Of course, we don't experience the world
>> in this way, and I'm not saying it "is" an u.m.o., only that
>> such a modeling appears hypothetically possible.
>
> Time is experienced by temporal beings, would god need to parse the
> u.m.o. - or would it simply be tautlogous. We have two problems telling God
> a joke - first he already knows it - and second he's at the punch line
> before we begin. There is a man who has a damaged brain such that he is
> constantly experiencing De Ja Vu. He is extremely angry and frustrated in
> this state of constantly repeating what has already happened - perhaps this
> accounts for the grumpyness of the Almighty of the O.T. and those maths
> chaps who were quite rude. Oh they've seen it all before! Its like trying
> to give God directions - he already knows the way - worse he's already
> there.
You're talking about the Demiurge now. If we have a grumpy
God we're in Gnostic territory.
> So there
>> can be a correspondence between the apparently unchanging
>> facts of mathematics and the changing facts of the physical
>> world.
>
> I dont see how. You may see a comparison between a changing line on a graph
> and ones temperature.
I don't see your problem. One's temperature can be considered
to be an object in some sort of n-dimensional manifold by
making the time dimension spatial. If the universe of one's
temperature has to be non-deterministic, we can still do this
by postulating a many-worlds model in which all the possibilities
of one's temperature's history exist "simultaneously".
> Therefore, if the physical world has content, the
>> mathematical world can also be said to have content, whatever
>> "having content" means -- I guess I would say it means supporting
>> experience.
>
> One can experience maths - do a sum - as we used to say, but the content
> here is psychological. The sum is i think empty. I dont know what 7 8s means
> though i know its 56. 56 chickens or cats all have contents but does 56?
> Does it have 7 8s inside it like the organs of the chickens.
8 x 7 = 56 seems like a very frail, tenuous thing, the merest
wisp of an idea, almost not there at all, but then it keeps
popping up in the physical world, and seems to possess
remarkable powers of endurance. You can call it "empty" if
you like but it's still there.
So why is it "empty"? What does "empty" mean?
>> mathematics and logic, as I said, _tautology_ appears to have
>> other meanings and connotations.
>
> Then we have no argument. Save that in Rhetoric the repetition gives a
> force.... a subjective one, which is not redunant - and the doesnt the
> completion of the equation make the problem redundant.
Often it doesn't, because the two sides of an equation have
significantly different apparent properties. In the case of
physics, for example, suppose 'e = mc^2' is true for all the
phenomena we can observe. Yet on the two sides of the
equation we have very different sets of phenomena which '='
links. Far from being empty, '=' marks a tremendous leap
of intelligence.
> ....
Proginoskes
>
> Why ask what he means? Wiser men than he is have made a claim such as
> this, so let's try to deduce what *they* meant in order to evaluate
> the claim.
You can't play checkers according to the rules of chess. Also, his idea of
tautology might be something other than mine.
> > * What do you mean by "tautology"?
>
> The only reasonable meaning is the same one I've given repeatedly.
> Namely, a theorem of first order logic, i.e., a statement provable via
> pure logic alone, the strongest reasonable form of necessary truth in
> this context.
Okay. Then not all mathematical theorems are tautologies. For instance, it
is impossible to trisect an angle using just a compass and straightedge.
To prove this, you have to introduce the idea of fields and field extensions,
neither of which is part of the original statement. If the proof of this
statement were tautological, you wouldn't need to do this. Q.E.D., end of
thread.
> > Until the definitions for this discussion have been established, further
> > meaningful communication is not possible (and I will not respond to this
> > thread until then).
>
> Goodness.
If you are thinking me daft or aloof for my answer, you haven't read all of
my responses to this post. I will then put the following curse on you: May
you have to reply to a thread with the same information ten times.
-- Christopher Heckman
Proginoskes
>
> You miss the point - your original statement was
>
> "If anyone claims that mathematics has no content, I dare them to provide
> a (correct) proof of the Four Color Theorem."
You miss the point - the statement was put in a certain context.
-- Christopher Heckman
P.S. Centuries ago, delivering letters took weeks; there was no such thing
as sending a letter off in a fury. Usually, if you wrote a hateful letter
in the evening, by the time you'd gotten some sleep and could drop off the
letter, you'd decide that you didn't really want to send it after all. Usenet
should come with this feature.