Why the attack on Cantor's Theory?
David Petry
Anyone who follows sci.math must be aware that there
are many people who find something to object to about
Cantor's Theory. I believe that the attack on Cantor's
theory will not end until someone has found a way to
drive a stake right through the heart of the theory.
Our whole culture has changed radically in the last twenty
years. Things that might have seemed reasonable to
the last generation will seem absurd to the new generation.
One of the things that must seem absurd to the new
generation is Cantor's Theory.
What has changed in the last twenty years is the ubiquity
of computers. Today's children are growing up surrounded
by computers, and they're growing up in families where
the parents are also comfortable with computers. They're
living in a computer culture.
So what does that have to do with Cantor's Theory?
To someone who has grown up with computers, the world
of computation must seem like a very real world. It's not
a physical world, to be sure, but it is concrete and has an
objective existence nonetheless. It's a world revealing a
rich set of phenomena. And in fact, it's a maximally rich
world, in the sense that any phenomena in any world can
be modeled in the world of computation.
So what's the point?
To someone who is comfortable with thinking about the
world of computation as a real world, "mathematics" is
most usefully defined as the study of phenomena that are
observable in the world of computation. That is, we can
think of the computer as a microscope that helps us peer
into the world of computation, and "mathematics" as the
science that studies the phenomena observed through
that microscope.
So where does Cantor's theory fit in to this scenario?
It doesn't. And that's the problem.
To someone who is comfortable with the notion of a world
of computation, it is extremely plausible to define the world
of mathematical objects that "exist" to be the universe of
objects that can be seen through the microscope. Cantor's
theory formally implies the "existence" of objects that cannot
be seen through the microscope, and hence Cantor's theory
must be viewed as a mythology.
When the classical mathematicians hear about all this, they
generally go into a tizzy and refuse to think straight about it.
They go into a defense mode. They seem to believe that
they are about to lose everything they've worked for. But in
fact, very little is lost.
For one thing, nothing of applied mathematics is lost. The
idea of applying mathematics means to create computational
models of real world phenomena in which the phenomena
observed in the model correspond to the phenomena observed
in the real world. So this notion of mathematics being the
study of phenomena observable through computation fits
applied mathematics like a glove, so to speak.
But we also don't lose most of the abstractions that the pure
mathematicians are fond of, though we are forced to understand
them from a different perspective.
For starters, we can build up a theory of the infinite, so long as
we don't allow anything into the universe of the infinite that does
not have corresponding approximations that can be seen
through our microscope. Thus, for example, we can say that
perfect circles "exist" in the sense that we can see arbitrarily
close approximations to circles. Likewise, pi "exists" in that
we can compute arbitrarily close approximations. Functions
like exp(x) "exist" likewise. Fourier Transforms "exist".
Manifolds "exist". Solutions to differential equations "exist".
What's more, infinite sets "exist", as long as we keep in mind
what it means to say that they "exist". It means that we can
see arbitrarily close approximations to them. For example, for
the set of positive integers, we can "see" sets of the form {1..N}
for arbitrarily large N, and such sets can be taken to be
approximations of the set of all positive integers. For the set of
real numbers (a continuum), an approximation must be a finite
set of finite approximations of real numbers.
So if infinite sets "exist", and power sets "exist", what's wrong
with Cantor's theory?
What I'm proposing is that mathematics should be built upon
a notion of "observability". We can say that a mathematical
object is observable if we can observe arbitrarily close
approximations to it in our microscope. We can say that
the object has certain properties, if and only if we can see that
approximations to the property hold for approximations to
the object. We can say that mathematical assertions have
observable content if they make predictions that can be
observed in our microscope.
This notion of observability leads to a unique and minimalist
theory of the infinite. And that theory is not Cantor's Theory.
Cantor's Theory implies that there "exist" things that are not
observable.
So what was Cantor's mistake?
Cantor started off asserting that infinite sets exist, without
consideration of what it means to observe those infinite sets.
He told us that power sets exist, without telling us what it
means to say that they exist. He went on talking about
properties of the objects, without telling us what it means
to say the objects have those properties (other than the
superficial notion that what it means for an object to have a
property is that a sentence in the formal language can be
interpreted to say that the object has a property). And then he
came to the conclusion that there must "exist" more objects
than can be observed. Cantor chose axioms which would
formally lead to the conclusion he wanted to reach, without
regard to the observable universe.
Emphatically, Cantor has not shown that there is any logical
inconsistency in assuming that the universe of observable
objects is the whole mathematical universe. He has not
given us a compelling reason to accept his fantasy world,
which goes well beyond the observable universe.
I should point out that by accepting this notion of observability,
we do not even lose Cantor's theory (as a formal theory). That
is, formal theories themselves are objects that exist in the
observable universe. They are objects which mathematicians
might want to study. It's just that Cantor's theory has no right
to the claim of being part of the foundation of mathematics.
What I am suggesting is that we must seek a theory in which
every object in our theory corresponds to an object in the
universe of objects observable through computation, and
likewise, every object in the observable universe should
correspond to an object in our theory. That's the goal. If we
don't set that as our goal, then the parts of the theory which
don't correspond to observation will grow into entrenched
dogma, stifling growth and creativity, and providing a
springboard from which oppressive dominance hierarchies
can grow. The most vicious battles in human affairs are the
battles that derive from such entrenched dogmas. Cantor's
Theory is not heading us in the right direction.
Those who have been on the net for a long time may recall that
I presented these ideas in sci.math many years ago. They weren't
well received. I gave up believing that there was any reason for
me to press the issue, as the likely result would be a tremendous
effort without reward.
Well, just recently we've seen a rather precocious eleven year
old boy take up the challenge of debunking Cantor's Theory. And
not surprisingly, the defenders of the faith have called him stupid,
insane and worse. I can't just ignore that.
Granted, the situation is somewhat like the story of the little boy who
said "the king has no clothes". Maybe he can't yet explain to true
believers what is wrong with their theories purporting to prove the
king is dressed in fine splendor, but he sure knows what his eyes
tell him. I'd like to encourage him to continue (while warning him of
the dangers; he'll be battling with some insanely devious minds).
I think he just might have enough spunk to win the battle. And the
rewards for winning could be quite substantial.
Cantor was a mad genius. He created a "logically consistent"
(assuming reality checks are not an essential criterion of logic)
and seductive fantasy world. His dreams of unifying theology and
mathematics were never realized, but he did succeed in seducing
the mathematicians for a hundred years, and thereby turning
mathematics into a pseudo-theology.
Cantor's Theory is a dogma. It's a mythology. It's an intellectual
fraud. It's destructive. It deserves to die.
Torkel Franzen
>Well, just recently we've seen a rather precocious eleven year
>old boy take up the challenge of debunking Cantor's Theory.
Yeah, right.
>Cantor's Theory is a dogma. It's a mythology. It's an intellectual
>fraud. It's destructive. It deserves to die.
Your tirade is quite pointless. If mathematics takes a turn towards
the computational, it won't be because of the anti-Cantor crackpot
brigade, nor because of the inherent rottenness of Cantorianism, but
because mathematicians produce a lot of computational mathematics that
other mathematicians find more interesting than non-computational
mathematics. So go for it, produce your interesting computational
mathematics!
Torkel, age 5
Joshua Scholar
<pe...@accessone.com> wrote:
Don't confuse Nathan's "theory" with your, (may I call it
intuitionist?) point of view.
Nathan's attack on Cantorian theory comes from a few naive ideas about
sets that any new student might have - that by the way are actually
contradictory. But from what little I've read, he seems more
interested in taunting adults than in understanding their
explanations.
If you could take the contradictions out of Nathan's arguments you
would end up with a theory that is more metaphysical than Cantor's,
not more concrete. One in which, for example, the set of Cardinal
numbers has a member that has infinite magnitude, or (in order to make
his arguments more consistent) an infinite number of members that have
infinite magnitude. Does this sound like the concrete, down to earth,
theory you've been looking for?
>Cantor's Theory is a dogma. It's a mythology. It's an intellectual
>fraud. It's destructive. It deserves to die.
I just noticed that line. Geeze! Calm down. It's just a
mathematical theory, don't get your underwear in a bunch.
Josh
gwen_...@cybergal.com
"David Petry" <pe...@accessone.com> wrote:
> To someone who is comfortable with the notion of a world
> of computation, it is extremely plausible to define the world
> of mathematical objects that "exist" to be the universe of
> objects that can be seen through the microscope. Cantor's
> theory formally implies the "existence" of objects that cannot
> be seen through the microscope, and hence Cantor's theory
> must be viewed as a mythology.
>
mathematics where the idea of an infinite set is removed, just think
of it:
- For any N there is a number N+1
- No problem, when you meet the number 'maxint', and add one to
it you either roll around to -1, or you get to 0 (or -0 as some
calculators used to have it).
This is even more fun than the millennium bug. With the millennium
bug you at least know when it will hit you. With the new 'non-Cantor'
maths it depends.
On a new workstation it will be when all 64 bits have filled up (or, in
a few years time when all 128 or 512 have). In a pc it will be when 32
have filled up, on DOS, when 16 have. Of course, if you are using an
embedded controller like a washing machine it will be when ony 8 bits
roll over.
Of course some of you may spot that this all happens already. Well, yes
it does, but programmers put in traps and checks to deal with the
situation. With the new 'non-Cantor' maths this will be unnecessary -
when you add one to a big number and get either 0 or -1 (depending on
1 or 2's compliment arithmetic) it will be the right answer! You can
just tell the bank that, when it transfers more than a certain amount
of money to another bank the money just disappears - poof! You can
make friends with your bank manager by pronouncing the 'poof' really
well.
Oh, yes, this is a brilliant idea!
Gwen Jones
There is almost nothing Welsh women have not done.
-----------== Posted via Deja News, The Discussion Network ==----------
http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own
Virgil
>Cantor's Theory is a dogma. It's a mythology. It's an intellectual
>fraud. It's destructive. It deserves to die.
In what computer model did you observe this?
--
Virgil
vm...@frii.com
Virgil
>What I am suggesting is that we must seek a theory in which
>every object in our theory corresponds to an object in the
>universe of objects observable through computation, and
>likewise, every object in the observable universe should
>correspond to an object in our theory. That's the goal.
If you want to dumb down to the level of computers, don't
try to limit the rest of us in the same way.
A happy Cantorian.
Dave Seaman
>world of computation as a real world, "mathematics" is
>most usefully defined as the study of phenomena that are
>observable in the world of computation. That is, we can
>think of the computer as a microscope that helps us peer
>into the world of computation, and "mathematics" as the
>science that studies the phenomena observed through
>that microscope.
Mathematica can evaluate the expression
Sum[1/k^2,{k,Infinity}]
and rather quickly print the result
Pi^2/6
which is exactly correct.
Are you suggesting that humans should be capable of any less?
--
Dave Seaman dse...@purdue.edu
Pennsylvania Supreme Court Denies Fair Trial for Mumia Abu-Jamal
<http://mojo.calyx.net/~refuse/altindex.html>
Charles H. Giffen
>
> In article <7b34gn$soc$1...@remarQ.com>,
> "David Petry" <pe...@accessone.com> wrote:
> > of computation, it is extremely plausible to define the world
> > of mathematical objects that "exist" to be the universe of
> > objects that can be seen through the microscope. Cantor's
> > theory formally implies the "existence" of objects that cannot
> > be seen through the microscope, and hence Cantor's theory
> > must be viewed as a mythology.
> >
Gwen -- what a pleasure to read your response to Petry! I especially
enjoy your comparisons with 64 bit workstations, 32 bit pc's, 16
bit DOS dinosaurs, 8 bit embedded controllers. It reminds me of
something I learned in school once (probably some middle-school
biology course) about the intelligence of various beings vis-a-vis
counting.
Of course, humans can *really* count, chimpanzees can count (ie.
recognize different numbers of objects) rather high -- compared,
say, to a horse or a dog. But for a bird, the counting system
(translated into English) would be something like
zero, one, two, many.
On the other end of the spectrum(?) a colleague once told me:
"Everyone knows what 'infinity' is -- it's almost a million."
Of course that was about twenty years ago, so perhaps by now
'infinity' is almost a trillion.
--Chuck Giffen
PS Is it a corollary to your comment on Welsh women that there
is almost nothing that a Welsh man (or, for that matter, a lowly
Scotsman) would do for good Welsh woman?
Thanks again for your wonderful post!
Joshua Scholar
necessarily naive and have enjoyed much more respect from the
mathematical community over the years than this over the top ridicule
would suggest.
If you've never heard of Intuitionism or of Luitzen Brower you might
start check any book on the history of Mathematics. You might check
out the following link:
http://werple.net.au/~gaffcam/phil/brouwer.htm
Also, formalizing numerical analysis by analysing limited precision
number systems is a useful topic. I've read articles about people
working on the subject precisely from the point of view of
understanding what can be computed by a machine. Once again the idea
deserves and is getting more respect from mathematicians than the
sniggering in this news group would suggest.
Josh Scholar
Timothy P. Keller
Right-on Gwen and Virgil!
From the moment I first read about Cantor's idea of transfinite
numbers, I thought his theory was one of the neatest, most
elegant ideas ever!
It is a widespread misconception that mathematics and
mathematicians exist in a mode untouched by the outside world.
Landau's definition of pi is the counter-example that comes to
mind. Landau defined sin( x ) as the ( unique ) solution of the
differential equation y'' + y = 0 so that y( 0 ) = 0 and
y'( 0 ) = 1 , and then demonstated sin ( x ) had exactly one
root between 3 and 4 , which he defined to be pi. Pretty routine
by now, and one wouldn't think it would ever have caused any
stir. However, Biberbach, denounced Landau's defintion as an
example of 'degenerate' Jewish mathematics. Landau lost his teaching
position, and almost his life as a consequence. The Third Reich's
'intelligensia' hated Cantor's mathematics with a passion.
Now don't get me wrong, I'm not accusing anyone in this discussion
of anti-semitism, or any such horrible thing. I just want to
point out that it is amazing what can happen ......
Best Wishes,
Tim
Jeremy Boden
>
>Anyone who follows sci.math must be aware that there
>are many people who find something to object to about
>Cantor's Theory. I believe that the attack on Cantor's
>theory will not end until someone has found a way to
>drive a stake right through the heart of the theory.
Why are all of these naive "anti-Cantor" types so incredibly verbose?
Surely it is they who deserve a good staking.
>
>Our whole culture has changed radically in the last twenty
>years.
So what?
Pythagorus, for example, lived in a totally different culture - but the
theorem named after him is still valid isn't it? Mathematics is not a
culturally determined exercise.
>To someone who has grown up with computers, the world
>of computation must seem like a very real world. It's not
>a physical world, to be sure, but it is concrete and has an
>objective existence nonetheless. It's a world revealing a
>rich set of phenomena. And in fact, it's a maximally rich
>world, in the sense that any phenomena in any world can
>be modeled in the world of computation.
Maximally? - Can you prove that?
Do you really think that *any* phenomena can be realistically be
modelled by a Computer?
You are aware that there exist propositions that are undecidable - these
can't be modelled in a concrete way.
Same idea for uncountable sets; computers can, not unreasonably only
produce computable numbers, and these are not uncountable.
>
>What's more, infinite sets "exist", as long as we keep in mind
>what it means to say that they "exist". It means that we can
>see arbitrarily close approximations to them. For example, for
>the set of positive integers, we can "see" sets of the form {1..N}
>for arbitrarily large N, and such sets can be taken to be
>approximations of the set of all positive integers. For the set of
>real numbers (a continuum), an approximation must be a finite
>set of finite approximations of real numbers.
You use an approximation because you cannot conceive of the real
picture. Less computing and a little more theory would allow you to
construct the real thing - without any approximations.
>
>Those who have been on the net for a long time may recall that
>I presented these ideas in sci.math many years ago. They weren't
>well received. I gave up believing that there was any reason for
>me to press the issue, as the likely result would be a tremendous
>effort without reward.
They still aren't - please don't post archive material which has clearly
been soundly refuted many years ago. This newsgroup suffers from a
surfeit of "anti-infinity" material.
>
>Well, just recently we've seen a rather precocious eleven year
>old boy take up the challenge of debunking Cantor's Theory.
His refusal to listen to reasoned argument from a number of people has
earned Nathan a well earned place in my killfile.
>
>Cantor's Theory is a dogma. It's a mythology. It's an intellectual
>fraud. It's destructive. It deserves to die.
It's none of those.
It's a sound axiomatic mathematical theory.
If you want to take issue with Cantor's theory, then "disagree" with the
axioms or attack the proofs.
--
Jeremy Boden mailto:jer...@jboden.demon.co.uk
Mike Oliver
Jeremy Boden wrote:
>
> In article <7b34gn$soc$1...@remarQ.com>, David Petry <pe...@accessone.com> writes
>>Cantor's Theory is a dogma. It's a mythology. It's an intellectual
>>fraud. It's destructive. It deserves to die.
> It's none of those.
> It's a sound axiomatic mathematical theory.
Well, no actually, it's not that either, at least if you're taking
seriously the attribution to Cantor and if you want to be historically
accurate.
Cantor's work was for the most part pre-axiomatic; he gave arguments
that made a lot of sense, but never got around to isolating a formal
collection of axioms. That was done by Zermelo, decades later (and
incidentally, remains a work in progress, as it always will be).
You can *still* understand Cantor's work without worrying about
the formal axiomatization. The latter is important for lots
of technical reasons, but it's not the soul of the theory.
Bennett Standeven
On Thu, 25 Feb 1999, David Petry wrote:
>
[...]
>
> To someone who is comfortable with thinking about the
> world of computation as a real world, "mathematics" is
> most usefully defined as the study of phenomena that are
> observable in the world of computation. That is, we can
> think of the computer as a microscope that helps us peer
> into the world of computation, and "mathematics" as the
> science that studies the phenomena observed through
> that microscope.
>
[...]
>
> For starters, we can build up a theory of the infinite, so long as
> we don't allow anything into the universe of the infinite that does
> not have corresponding approximations that can be seen
> through our microscope. Thus, for example, we can say that
> perfect circles "exist" in the sense that we can see arbitrarily
> close approximations to circles.
Arbitrarily close? Hardly. If your knowledge of computers is that poor...
Nico Benschop
>
> Anyone who follows sci.math must be aware that there
> are many people who find something to object to about
> Cantor's Theory. I believe that the attack on Cantor's
> theory will not end until someone has found a way to
> drive a stake right through the heart of the theory.
... or explain it better, with a feeling for the *real* issue:
which is not infinity N (the naturals) vs. infinity 2^N (powerset),
but a difference in TYPE : singletons (n \in N)
versus subsets of singletons {n_1, ..., n_m} \in 2^N.
I've tried to understand Cantor's diagonal argument, and came up with
a quite simple model (see http://www.iae.nl/users/benschop/cantor.htm)
which *can* be 'pulled-up from the finite by induction: the same
issues: (Sequentila-)Generation and Closure - play also in the finite!
The only requirement is that one should expand the context beyond
naturals N (generated with Peano _sequentially_ by (+1)* [star=repeat]
--> to integers Z, generated sequentially by (+1)* and (-1)*, and
consider powerset 2^Z as the fixedpoints of permutations in group Z!
By this expanded _sequential_ viewpoint, notice that all Z! permutations
can sequentially be generated by only 3 elements (permutations of Z)
namely generator set A = {+1,-1,swap2} the latter swapping any two
integers, say 0 <--> 1. Because any full ('symmetric') fiite group is
known to be generated by just 2 generators, and the infinite group Z!
needs the extra (-1) as generator [there is finite closure to provide
the inverse as g^{|G|-1} ].
Then Z! = A*/Z -- an Integer State Machine (!)
(input alphabet A, state_set Z)
which include 2^Z, and 2^N as fixpoint sets of permutations
(and their complements: "moved-point-sets")
> [...rather emotional outburst, miss-guided; probably angry
> [...because of unclear understanding of artifical Cantor proof ;-]
>
> Cantor was a mad genius. He created a "logically consistent"
> (assuming reality checks are not an essential criterion of logic)
> and seductive fantasy world. His dreams of unifying theology and
> mathematics were never realized, but he did succeed in seducing
> the mathematicians for a hundred years, and thereby turning
> mathematics into a pseudo-theology.
> Cantor's Theory is a dogma. It's a mythology.
> It's an intellectual fraud. It's destructive. It deserves to die.
^^^^^^^^^^^^^^^^^^^^^
No it is not. It treats an essential point - but with the wrong tools,
or rather with too restricted concepts: for some reason obsessed to
map everything onto a single "real" line -- While the concepts involved
can be well explained in a larger context of sequential logic (groups).
From which Peano actually started, Nota Bene: the integers N = (+1)*
as basic concept of a 1-dimensional (one seq-generator) 'closure'.
Restricting onself to sets, subsets, intersection & union operators
forces a combinational viewpoint, which is *too restricted* to notice
the essential point: N < Z < 2^Z < Z! -- be they finite or infinite...
seq_dim: 1 2 3 3
Just count the minimal number of Generators for each full Closure
(sequential dimension;-) and you see their essential difference,
don't you?
--
Ciao, Nico Benschop. | AHA: One is Always Halfway Anyway
http://www.iae.nl/users/benschop | xxxxxxxxxxxxxxx1.1xxxxxxxxxxxxxxx
http://www.iae.nl/users/benschop/cantor.htm
---- Euler's trick: If a problem is *really* hard, generalize it ----
(to be able to walk around it & find an entry)
Nico Benschop
> [...]
> Anyone who follows sci.math must be aware that there
> are many people who find something to object to about
> Cantor's Theory.
> I believe that the attack on Cantor's theory will not end
> through the heart of the theory.
Re(#):
Or until it is explained better, using Generation & Closure concepts.
It touches an essential concept, namely that of "sequential dimension",
which is the minimum number of generators of a sequential closure.
Cantor compares the infinity types of the set of naturals N, and
its powerset 2^N : the collection of all its subsets, using
combinational concepts like set, subset, intersection and union.
This is the weak point of his presentation, because a much clearer
point of view is the sequential generation aspect of the sets involved.
Namely: N is sequentially generated by +1, the succesor *function* of
Peano, in short: N = (+1)* where the star* is repetition.
So N is not just a set, but one with the sequential ordering of a
'closure' with one generator, in short: seqdim(N) = 1.
The next more complex seq.closure has seqdim=2, namely any finite
group G_m of all m! permutations of set S of m states. Two generators
can be chosen to be a full m-cycle, and any 2-cycle.
Then the concept of "subset" can be taken as the "fixpoints" of a
permutation, and its complement: the subset of "movedpoints" of that
permutation. So all 2^m subsets of set S with |S|=m can be seq_generated
in a group with just two generators: the next level beyond Peano.
Now N is not a group, since the inverse -1 of 1 is not in N.
So take the integers Z, with two generators +1 and -1, which is an
infinite group. And all its permutations Z! are generated sequentially
by three generators: A={+1, -1, sw2} where sw2 is a 2_swap, say 0<-->1.
Then Z! = A*/Z -- like an Integer State Machine IMS(Z,A) with
infinite state set Z, and input alphabet A of only 3 generators.
And 2^Z, the Boolean Lattice of all subsets of Z, is contained in a
closure of seqdim(Z!)=3. Apparently, this is again one step beyond N
and Z, forming a nice hierarchy: N < Z < 2^Z < Z!
with seq.dim: 1 2 3' 3
Here 3' is artificial, since powerset 2^Z is the only closure (lattice)
that is not sequential, but combinational (meaning its operators are
idempotent=non-sequential: intersection, and union).
This difference in *type* of object between N, Z, Z! on the one hand,
versus 2^Z on the other, makes Cantor's theory so artificial. It is not
only the cardinality of the infinities involved, but more basically the
*types* of object (combin. resp sequential), and their seq_dimension,
that should be considered and clearly distinguished.
Which, by the way, is already evident in the finite -- a great
advantage: Cantor's Theory can perfectly well be 'pulled-up' from the
finite, via induction (Peano's sequential tool, the basis of
constructive proofs, since the Greeks of old...;-)
> Cantor was a mad genius. He created a "logically consistent"
> (assuming reality checks are not an essential criterion of logic)
> and seductive fantasy world. His dreams of unifying theology and
> mathematics were never realized, but he did succeed in seducing
> the mathematicians for a hundred years, and thereby turning
> mathematics into a pseudo-theology.
>
> Cantor's Theory is a dogma. It's a mythology.
> It's an intellectual fraud. It's destructive. It deserves to die.
^^^^^^^^^^^^^^^^^^^^^?
I hope to have convinced you that, although his theory is a bit cripple,
as it were, he discovered something very essential: higher dimensional
objects beyond Peano's naturals (of seqdim=1). Unfortunately, the
presentation leaves something to be desired;-) Moreover, the obsessive
urge to map these higher dimensional objects onto the "linear" real line
is not helping very much. Rather leave the subsets of N cq Z in their
own character, with Lattice closure as their natural habitat.
And the "sequential logic" of groups (and my hobby: semigroups & State
Machines, especially the finite ones;-), with their associative algebra,
is a much more natural context for further developments than the 'old'
and a bit stuffy (last century) - and certainly artificial type of set
theory, do't you think ?-)
I made an initial attempt to describe this approach,
from the finite and sequential generative standpoint,
in: http://www.iae.nl/users/benschop/cantor.htm
Let me know what you think of it. Much needs to be further developed
and defined, cq axiomatized (especially the infinity part of it).
Nico Benschop
> [...]
> Anyone who follows sci.math must be aware that there
> are many people who find something to object to about
> Cantor's Theory.
> I believe that the attack on Cantor's theory will not end
> until someone has found a way to drive a stake right
> through the heart of the theory.
Or until it is explained it better, using Generation & Closure
It touches an essential concept, namely that of "sequential dimension",
which is the minimum number of generators of a sequential closure.
Cantor compares the infinity types of the set of naturals N, with
that of their powerset 2^N : the collection of all its subsets,
using combinational concepts like set, subset, intersection and union.
This is the weak point of his presentation, because a much clearer
point of view is the sequential genration aspect of the sets involved.
Namely: N is sequentially generated by +1, the succesor *function* of
Peano, in short: N = (+1)* where the star* is repetition. So N is not
just a set, but one with a sequential ordering, that of a 'closure'
with 1 generator, in short: seqdim(N) = 1.
Now we all know (I hope) another more complex seq.closure of seqdim=2
namely any finite group G_m of all m! permutations of set S of m states.
Two generators can be chosen to be a full m-cycle, and any 2-cycle.
Then notice that the concept of "subset" can be taken as the "fixpoints"
of a permutation, and its complement: the subset of "movedpoints" of the
state-set. So all 2^m subsets of set S with |S|=m can be seq_generated
by just two generators: the next level beyond Peano.
Now N is not a group, since the inverse -1 of 1 is not in N.
So take the integers Z, with two generators +1 and -1, which is an
infinite group. And all its permutations Z! are generated sequentially
by three generators: A={+1, -1, sw2} where sw2 is a 2-swap, say 0<-->1.
Then Z! = A*/Z -- like an Integer State Machine IMS(Z,A), with
stateset Z, and input alphabet A of only 3 generators.
And 2^Z, the Boolean Lattice of all subsets of Z, is contained in a
closure of seqdim(Z!)=3. Apparently, this is again one step beyond N
and Z, forming a nice hierarchy: N < Z < 2^Z < Z!
with seq.dim: 1 2 3' 3
Here 3' is artificial, since powerset 2^Z is the only (Lattice) closure
that is not sequential, but combinational (meaning its operators are
idempotent: intersection, and union). This difference in *type* of
object between N, Z, Z! on the one hand, versus 2^Z on the other, makes
Cantor's theory so artificial. It is not only the cardinality of the
infinities involved, but more basically the *types* of object (combin.
resp sequential), and their seq_dimension, that should be considered
and clearly distinguished. Which, by the way, is already evident in
the finite -- a great advantage: Cantor's Theory can perfectly well be
'pulled-up' from the finite, via induction (Peano's sequential tool,
the basis of constructive proofs, since the Greeks of old...;-)
> Cantor was a mad genius. He created a "logically consistent"
> (assuming reality checks are not an essential criterion of logic)
> and seductive fantasy world. His dreams of unifying theology and
> mathematics were never realized, but he did succeed in seducing
> the mathematicians for a hundred years, and thereby turning
> mathematics into a pseudo-theology.
>
> Cantor's Theory is a dogma. It's a mythology.
> It's an intellectual fraud. It's destructive. It deserves to die.
^^^^^^^^^^^^^^^^^^^^^?
I hope to have convinced you that, although his theory is a bit cripple,
as it were, he discovered something very essential: higher dimensional
objects beyond Peano's naturals (of seqdim=1). Unfortunately, the
presentation leaves something to be desired. Moreover, the obsessive
urge to map these higher dimensional objects onto the "linear" real line
is not helping very much. Rather leave the subsets of N cq Z in their
own character, with Lattice closure as their natural habitat.
And the "sequential logic" of groups (and my hobby: semigroups & State
Machines, especially the finite ones;-), with their associative algebra,
is a much more natural context for further developments than the 'old'
and a bit stuffy (last century) type of set theory, don't you think ?-)
David Petry
Dave Seaman wrote in message <7b6dp2$k...@seaman.cc.purdue.edu>...
>Mathematica can evaluate the expression
>
> Sum[1/k^2,{k,Infinity}]
>
>and rather quickly print the result
>
> Pi^2/6
>
>which is exactly correct.
>
>Are you suggesting that humans should be capable of any less?
No.
I don't think you would have asked that question if you had read
the whole article.
David Petry
So far most of the responses to my article have been
the typical knee-jerk reactions of people who have not
given the subject any serious thought. I'm not sure why
people bother to post such nonsense.
If you want to think seriously about the ideas I'm proposing,
let me give some things to think about.
As I pointed out in my article, all of applied mathematics
fits easily within the paradigm I'm proposing (namely that
mathematics should be considered as the science of
phenomena observable within computation). But
furthermore, all the mathematics done before Cantor came
along, easily fits within that paradigm. What's being objected
to here is Cantor's notion that there must "exist" a world
of mathematical objects beyond the observable world,
and that idea originated with Cantor.
So here's something to think about. Is it plausible that if
someone had articulated this notion of observability before
Cantor came along, the mathematics community would
have accepted it? I think the answer is a strong "yes". In
fact, I think that if there was to have been any objection at
all to the notion, the objection would have been that it's all
so obviously true that it needn't be articulated.
So now, pretend that the mathematics community had
accepted the notion of observability. Then when Cantor
came along, could he have convinced the mathematicians
to abandon the idea? Why would the mathematicians
abandon the idea? Note that the diagonalization method
fits well within the paradigm of observability.
Here' another way to look at it. As I have explained, all of
the mathematics that is relevant to science and technology
fits within this paradigm of observability. So Cantor's Theory
cannot be essential to a technologically advanced society.
Presumably, someday we will interact with technologically
advanced alien societies (i.e. from other planets). Will those
alien cultures necessarily accept something equivalent to
Cantor's Theory? Why or why not? Will they necessarily
accept the mathematics that fits within the paradigm of
observability? Why or why not?
If you want to post your answers, may I request that you
spend more time thinking than typing?
David Petry
Joshua Scholar wrote in message <36d75445...@news.select.net>...
>On Thu, 25 Feb 1999 01:20:42 -0800, "David Petry"
><pe...@accessone.com> wrote:
>
>(may I call it intuitionist?)
I don't like that term. For one thing, "intuition" plays no special role
in the theory I'm proposing.
As I've defined it, a statement has observable content if it makes a
testable or falsifiable prediction about the results of a possible
computation. It's often the case that a statement is falsifiable
whereas its formal negation is not, hence the law of the excluded
middle cannot be consider a universally valid law. In this respect,
the logic of the observable mathematical universe has an
intuitionist flavor.
I wouldn't object to calling the theory a variant of constructivism,
although it's definitely not classical constructivism.
David Petry
Marty Fouts wrote in message ...
>
> You would deny us any
>advance in mathematics that the our current limited computational
>devices are yet incapable of dealing with.
That's utterly false.
By the way, I went so far as to read the second paragraph you
wrote, but I don't know why.
David Petry
Jeremy Boden wrote in message ...
> Mathematics is not a culturally determined exercise.
Certainly the mathematics that can be described as the
science of phenomena observable through computation
is not culturally determined. But Cantor's Theory is very
much a cultural phenomenon. It contains an element
of mysticism which would not be accepted in every
culture.
Aatu Koskensilta
David Petry <pe...@accessone.com> wrote in article
<7b8kpu$jtt$1...@remarQ.com>...
> As I pointed out in my article, all of applied mathematics
> fits easily within the paradigm I'm proposing (namely that
> mathematics should be considered as the science of
> phenomena observable within computation). But
> furthermore, all the mathematics done before Cantor came
> along, easily fits within that paradigm. What's being objected
> to here is Cantor's notion that there must "exist" a world
> of mathematical objects beyond the observable world,
> and that idea originated with Cantor.
If you wish to disagree with platonic realism in mathematics,
please do so. A number of people will get quite irritated, no
wonder considering the number of silly anti-infinity posts here
nowadays, as you start bashing Cantor *and* his theory. This
is due to the fact that you seem to be completely unable to
tell the difference between a mathematical theory and someone's
philosophy of mathematics. Yes, Cantor was a platonic realist
and so have been numerous great mathematicians both before
and after him, but that has nothing to do with whether or not
his theories are right.
I, myself, am a non-platonist, but I see
no reason to change the way we do mathematics - especially
into something as insane as your proposed "computable
universe". All our mathematics is, in a sense, finitary (due to
perfectly normal limitations - such as available number of
pages in the world and so forth), but that need not imply it
were computable/constructable/existing/etc. No matter what
is the ontological status of mathematical objects, we can
go on about doing mathematics the way we like.
---
Aatu Koskensilta (squ...@seaga.org)
"Wovon man nicht reden kann, daruber muss man schweigen"
- Ludwig Wittgenstein
Virgil
>Here' another way to look at it. As I have explained, all of
>the mathematics that is relevant to science and technology
>fits within this paradigm of observability. So Cantor's Theory
>cannot be essential to a technologically advanced society.
>Presumably, someday we will interact with technologically
>advanced alien societies (i.e. from other planets). Will those
>alien cultures necessarily accept something equivalent to
>Cantor's Theory? Why or why not? Will they necessarily
>accept the mathematics that fits within the paradigm of
>observability? Why or why not?
==================================
"paradigm of observability"?
According to your proposal, if computer programmers cannot make
a computer see, or approximately see, a thing, it shouldn't be
allowed to exist. Thus you wish to suppress Cantor's theories
because computers don't "like" them.
It appears that you want to limit mathematical thought to
standards set by non-mathematicians, and perhaps by non-humans.
This strikes me as a form of thought police. I am
strongly opposed to any such strictures on human thought,
whether of a mathematical bent or not.
There may be legitimate arguments against Cantor's theories,
but I do not see that you have brought forth any of them.
Until you have better reasons than you have so far expressed,
I reject your thesis utterly.
--
Virgil
vm...@frii.com
Mike Oliver
David Petry wrote:
> But Cantor's Theory is very
> much a cultural phenomenon. It contains an element
> of mysticism which would not be accepted in every
> culture.
It seems to me that you're the mystic here, David.
You're committed, a priori and without evidence,
to the idea that useful thought processes must
use only constructs which "look like" the objects
you ultimately wish to know about.
Whereas on the other hand, we pragmatic Cantorians
have been happily working away, producing a framework
of thought in which mathematics has been produced that
is directly relevant to computation, without imposing
on ourselves some artificial restriction that says we
can only think about states inside a computer. What
have the Brouwer fans produced in the mean time?
--
Disclaimer: I could be wrong -- but I'm not. (Eagles, "Victim of Love")
Finger for PGP public key, or visit http://www.math.ucla.edu/~oliver.
1500 bits, fingerprint AE AE 4F F8 EA EA A6 FB E9 36 5F 9E EA D0 F8 B9
Doug Norris
>I don't think you would have asked that question if you had read
>the whole article.
I think using the term "article" to describe what you wrote is a bit
generous.
Doug
----------------------------------------------------------------------------
Douglas Todd Norris (norr...@euclid.colorado.edu) "The Mad Kobold"
Hockey Goaltender Home Page:http://ucsu.colorado.edu/~norrisdt/goalie.html
----------------------------------------------------------------------------
"Maybe in order to understand mankind, we have to look at the word itself.
Mankind. Basically, it's made up of two separate words---"mank" and "ind".
What do these words mean? It's a mystery, and that's why so is mankind."
- Deep Thought, Jack Handey
Jeremy Boden
writes
>Jeremy Boden wrote in message ...
>> Mathematics is not a culturally determined exercise.
>
>Certainly the mathematics that can be described as the
>is not culturally determined. But Cantor's Theory is very
>much a cultural phenomenon. It contains an element
>of mysticism which would not be accepted in every
>culture.
>
But pure mathematics may start out in a perfectly "acceptable" way but
lead to deductions which are very much non-intuitive and possibly very
mystical.
A good example would be that of the Banach-Tarski theorem which says,
assuming the axiom of choice, that it is possible to sub-divide a sphere
into a finite number of pieces (4?) in such a way that two spheres can
be constructed, each congruent to the original one.
I find that very mystical - but it doesn't disprove Cantor or axiomatic
set theory.
Is there an example of a modern culture which finds parts of mathematics
unacceptable?
Jeremy Boden
writes
...
[49 Lines excised...]
>If you want to post your answers, may I request that you
>spend more time thinking than typing?
>
Ulrich Weigand
>As I pointed out in my article, all of applied mathematics
>fits easily within the paradigm I'm proposing (namely that
>mathematics should be considered as the science of
>phenomena observable within computation). But
>furthermore, all the mathematics done before Cantor came
>along, easily fits within that paradigm. What's being objected
>to here is Cantor's notion that there must "exist" a world
>of mathematical objects beyond the observable world,
>and that idea originated with Cantor.
This idea is certainly much older than Cantor (this is what
is usually called 'Platonism' or 'platonic realism'). What
originated with Cantor is something completely different,
namely set theory, i.e. the notion that 'sets' can be considered
as mathematical objects, in addition to the 'traditional' objects
of mathematics, like numbers, functions, geometric shapes etc.
The philosophical question of what ontological status mathematical
objects have, i.e. whether they 'really' 'exist', whether they are
just human inventions or just abstractions from observable events,
applies to the 'traditional' mathematical objects just the same as
to Cantor's notion of sets.
It is most certainly *not* necessary to adopt a Platonistic view to
be able to work in set theory. If you prefer a formalistic/
computational view, this is perfectly fine. If you take e.g.
the formal theory of ZFC, well, this is just a first-order theory
with a recursive set of axioms, so the question whether a certain
statement holds in ZFC is nothing more than the question whether
a certain computer program terminates or not. Is this question
acceptable as 'phenomenon observable within computation'?
I'd say so ... And, to return to the origins of this discussion,
the statement expressing that there is no bijection between the
sets of real and natural numbers, is easily shown to be true in ZFC
(without any need to apply some Platonistic view that sets 'really
exist' or whatever ...).
--
Ulrich Weigand,
IMMD 1, Universitaet Erlangen-Nuernberg,
Martensstr. 3, D-91058 Erlangen, Phone: +49 9131 85-7688
David Petry
Ulrich Weigand wrote in message
<7ba5jj$2...@faui11.informatik.uni-erlangen.de>...
>"David Petry" <pe...@accessone.com> writes:
>What's being objected
>>to here is Cantor's notion that there must "exist" a world
>>of mathematical objects beyond the observable world,
>>and that idea originated with Cantor.
>This idea is certainly much older than Cantor (this is what
>is usually called 'Platonism' or 'platonic realism').
Certainly philosophers, theologians and mystics have always
held that there exists a world beyond what can be seen. It
was Cantor who introduced the idea that such mysticism
could be incorporated into the framework of mathematics
itself. In order to accomodate Cantor's ideas, mathematicians
have had to abandon the idea that mathematics talks about
something (i.e. the world of computation) and have replaced
that idea with the notion that mathematics is merely a
consistent formalism with no real meaning.
> so the question whether a certain
>statement holds in ZFC is nothing more than the question whether
>a certain computer program terminates or not. Is this question
>acceptable as 'phenomenon observable within computation'?
>I'd say so
Actually, I addressed that question in my article. As I pointed
out, formal systems themselves are observable objects.
David Petry
Jeremy Boden wrote in message ...
>Is there an example of a modern culture which finds parts of mathematics
>unacceptable?
There's a sub-culture known as the constructivists.
Robin Chapman
>
> Attacks on the use of the infinite in mathematics are neither new nor
> necessarily naive and have enjoyed much more respect from the
> mathematical community over the years than this over the top ridicule
> would suggest.
Not just in mathematics, throughout the history of Western thought,
the rejection of the completed infinite has been alomost universal.
Indeed the refusal to consider completed infinities has been almost
a philosophical orthodoxy.
Cantor's great achievement was to cast aside this stale philosophical
dogma and to demonstrate how one could reason consistently with
infinite collections of objects. His theory of sets has enabled
mathematicians to introduce powerful new methods and to simplify
old ones. Of course, many people since have wished, due to their
philosphical prejudices, to turn the clock back...
--
Robin Chapman + "Going to the chemist in
Department of Mathematics, DICS - Australia can be more
Macquarie University + exciting than going to
NSW 2109, Australia - a nightclub in Wales."
rcha...@mpce.mq.edu.au + Howard Jacobson,
http://www.maths.ex.ac.uk/~rjc/rjc.html - In the Land of Oz
Nico Benschop
>As I pointed out in my article, all of applied mathematics fits
>easily within the paradigm I'm proposing (namely that mathematics
>should be considered as the science of phenomena observable within
>computation). But furthermore, all the mathematics done before
>Cantor came along, easily fits within that paradigm.
> What's being objected to here is Cantor's notion that there must
> "exist" a worldof mathematical objects beyond the observable world,
> and that idea originated with Cantor.
Ulrich Weigand <wei...@informatik.uni-erlangen.de> answered, in part:
This idea is certainly much older than Cantor (this is what
is usually called 'Platonism' or 'platonic realism'). What originated
with Cantor is something completely different, namely set theory,..(#)
i.e. the notion that 'sets' can be considered as mathematical objects,
in addition to the 'traditional' objects of mathematics, like numbers,
functions, geometric shapes etc.
-----------------------------------------------------------------**]
Re(#): I beg to differ here. If I'm not mistaken it was Boole who
originated set theory concepts, in his monograph (1848, expanded in
"The Laws of Thought", 1854) on combinational logic - Aristotle - as
the idempotent & commutative branch of arithmetic: precisely the way
he introduces it there: properties x and y of objects with arithmetic
syntax x^2=x, y^2=y, x.y=y.x --- Hence x(x-1)=0 so x=0 or x=1 -->
necessarily binary logic, lacking the concept of (sequential)
iteration.
BTW: Notice that Boole (as a vicar's son, I believe;-) was applying
his compact notation to analyse the God-existence proofs of
Spinoza and Clarke, which turned out to be 'circular' (the
result is in the axioma's, as it were...) Only by 1938 Shannon
recognized, as MIT student, the usefulness of Boole's notation
to the design of combinational logic circuits (relay networks,
applied at Bell Labs soon), with (+)=union = parallel and
(*)=intersection = series connection of binary switches.
Later, in the past century, this was recognized as the essence of
set theory. And Cantor expanded this to infinity, comparing the
cardinalities of Peano's naturals N = (+1)* and their powerset 2^N,
introducing the diagonal argument -- which mixes singletons with
[sub]sets -- a rather tricky but effective concept of mixing two
distinct levels in a hierarchy of object types. The unfortunate
byproduct (probably his original motivation) was to compress all his
results onto the linear "real" line -- disconnecting the new Set
concept from its natural Lattice algebraic structure (developed later:
Birkhoff this century).
The two main algebraic ontipodes: Lattices and Groups never quite got
off the ground in an integrated fashion, although merged already by
Schushkewitch in 1928 to Semigroups - his detailed structure analysis
of simple semigroups: having no porper "ideal". --Of course, Category
Theory and (infinite;-) inverse Semigroups took off al right, but not
in the way that Boolean algebra (for logic synthesis) and Groups (for
say Chrystal structure, and Theoretical Physics) did.
Although I sympathize with David's motivation: the emphasis on Objects
and their associated operations (syntax) has slowly disappeared from
basic math. -- But freedom in exploration should be allowed in any
direction. It's fruitfulness will develop in time, no doubt. But I
agree that Cantor's thinking, although very original & strongly tied
to object representation (his "diagonal") somehow was taken, by his
followers, to an extreme that seems to miss essential balance...
(re: my recent posting with "BOOA constructor" in the title)
Balanced Object oriented Algebra)
I would appreciate some comment on my effort to analyse Cantor's
diagonal in the context of 3D infinite group Z! -- Using 'sequential
-logic' concepts to embed (in)finite sets into a sequential closure
of very low "seq.dimension" (= minimal generating set), starting with
Peano's N={+1}* via integers Z={+1,-1}* to group Z! ={+1,-1,swap2}.
See http://www.iae.nl/users/benschop/cantor.htm
Or is this too inter-disciplanry ?-)
Ciao, Nico Benschop -- http://www.iae.nl/users/benschop
C.Shannon: "It just happened that no one else was familiar with
both fields at the same time" (his info-theory, 1948) ---
Nico Benschop
Attacks on the use of the infinite in mathematics are neither new nor
necessarily naive and have enjoyed much more respect from the
mathematical community over the years than this over the top ridicule
would suggest.
If you've never heard of Intuitionism or of Luitzen Brower you might
start check any book on the history of Mathematics. You might check
out the following link:
http://werple.net.au/~gaffcam/phil/brouwer.htm
Also, formalizing numerical analysis by analysing limited precision
number systems is a useful topic. I've read articles about people
working on the subject precisely from the point of view of
understanding what can be computed by a machine. Once again the idea
deserves and is getting more respect from mathematicians than the
sniggering in this news group would suggest. -- Josh Scholar
------------------------------------------------------------------**]
Amen.
The Finite (say Gauss' residue arithmetic;-) and the Infinite
(say Cantor's diagonal argument) are complementary.
Taking only one of these as the "Real" Math is rather childish and
unbalanced. It does require a double focus though, because they are
essentially different -- as are the Discrete and the Continuous ;-)
However, for solid logic and essentially trouble-free math results,
I bet any time on the Finite ...;-)
------ The Finite as a [prefix-] projection of the Infinite --------
------ (object/semantics) (compositionrule/syntax) --------
Still a lot needs to be done there: ever seen a good general method
of attacking Diophantine eqn's? (re: Hilbert's 10-th problem, 1900).
Or a general structure theory (all details) of any finite semigroup,
as sequential closure of a Finite State Machine (=deterministic
computer model, Mealy 1956)?
So get cracking, finally. Don't be shy, the century is almost gone ;-)
Ciao, Nico Benschop -- http://www.iae.nl/users/benschop
"On Constant Rank State Machine structure"
(and the 5 basic State Machines)
http://www.iae.nl/users/benschop/c-ranksm.dvi
http://www.iae.nl/users/benschop/cantor.htm (Finite Sequential view)
http://www.iae.nl/users/benschop/fewago.htm (Fermat, Waring, Goldbach)
Arthur L. Rubin
>
> Anyone who follows sci.math must be aware that there
> are many people who find something to object to about
> Cantor's Theory. I believe that the attack on Cantor's
> theory will not end until someone has found a way to
> drive a stake right through the heart of the theory.
To begin with, it's a misnomer to call it Cantor's theory.
I think von Neumann might be a better choice, although
you seem to be talking about Zermelo-Frankael or
von Neumann-Bernays-G:odel set theory.
> To someone who has grown up with computers, the world
> of computation must seem like a very real world. It's not
> a physical world, to be sure, but it is concrete and has an
> objective existence nonetheless. It's a world revealing a
> rich set of phenomena. And in fact, it's a maximally rich
> world, in the sense that any phenomena in any world can
> be modeled in the world of computation.
Try to prove that Sum_n (1/(n log(n) log(log(n)))) diverges
in a computational manner. I realize that that does not
constitute a physical phenomenon, but it still diverges, even
though it doesn't do so in any actual evaluation. OTOH, if
you say it doesn't diverge, you reject the concept of
accellerated converage of (positive) series, which is actually
used in computer analysis.
> So what's the point?
>
> To someone who is comfortable with thinking about the
> world of computation as a real world, "mathematics" is
> most usefully defined as the study of phenomena that are
> observable in the world of computation. That is, we can
> think of the computer as a microscope that helps us peer
> into the world of computation, and "mathematics" as the
> science that studies the phenomena observed through
> that microscope.
...
>
> For one thing, nothing of applied mathematics is lost.
Unless I don't understand what you are trying to say,
you are wrong there, in more than one way.
Either,
(1) You only allow to be considered as mathematical that
which can be evaluated, which eliminates much of the
mathematics of Quantum Mechanics, as the equations cannot
actually be solved; or
(2) You allow the concept of a computer-aided proof, which
is (classically mathematically) equivalent to that of a
formal prove, which puts us back where we started.
--
Arthur L. Rubin 216-...@mcimail.com
Brian M. Scott
> Ulrich Weigand wrote
> It
> was Cantor who introduced the idea that such mysticism
> could be incorporated into the framework of mathematics
> itself. In order to accomodate Cantor's ideas, mathematicians
> have had to abandon the idea that mathematics talks about
> something (i.e. the world of computation) and have replaced
> that idea with the notion that mathematics is merely a
> consistent formalism with no real meaning.
If this is supposed to be a statement about mathematicians in general,
it is simply false.
Brian M. Scott
Ulrich Weigand
>David Petry wrote:
>> Ulrich Weigand wrote
Just to keep the attributions straight I'd like to point out that
David Petry, not me, wrote the paragraph above ;-) I agree, of
course, that that statement is false.
Furthermore, I don't quite get why Petry wants to contrast 'computation'
with 'formalism'; if there is any area of mathematics where formalism
is appropriate it is the consideration of computational processes ...
In the case of set theory, say, you might distinguish the realist view
that sets really exist out there from the formalist view that we are
just manipulating formulae of ZFC; but computational processes are
of a formalistic nature to start with!
David Petry
Mike Oliver wrote in message <36D85F28...@math.ucla.edu>...
>David Petry wrote:
>> [Cantor's Theory] contains an element
>> of mysticism which would not be accepted in every
>> culture.
>It seems to me that you're the mystic here, David.
You could say that, but in so doing you'd be giving new meaning
to the word "mysticism".
>Whereas on the other hand, we pragmatic Cantorians
>have been happily working away, producing a framework
>of thought [...]
The Cantorians are "pragmatic" in the sense that "when in Rome,
do as the Romans do" is pragmatic.
The idea that Cantor's mythology is a "framework of thought" is
a plausible view of things. In fact, I wouldn't attack Cantor's
Theory for being a mythology, if I didn't believe that I have a very
much superior (non-mythical) idea to replace it.
Cantor's "framework of thought" has become a dogma, and as
is very apparent from this thread alone, the true believers are
almost ready to take up the sword to defend the dogma. That
alone is a good reason for taking this issue seriously.
>What have the Brouwer fans produced in the mean time?
That's a cheap shot.
David Petry
Robin Chapman wrote in message <36D9C6B2...@mpce.mq.edu.au>...
>Indeed the refusal to consider completed infinities has been almost
>a philosophical orthodoxy.
>Cantor's great achievement was to cast aside this stale philosophical
>dogma and to demonstrate how one could reason consistently with
>infinite collections of objects. His theory of sets has enabled
>mathematicians to introduce powerful new methods and to simplify
>old ones. Of course, many people since have wished, due to their
>philosphical prejudices, to turn the clock back...
To go back and correct a mistake is not the same as turning the
clock back.
The desire to debunk mythology is not reasonably called a
"philosophical prejudice".
Perhaps you could say that "Cantor demonstrated how one could
reason consistently with infinite collections", but only if you choose
a convenient definition of "consistent". That is, can you consistently
maintain that you are being consistent, if what you are saying has no
meaning? Can you consistently maintain that what you are saying
has meaning, if what you are saying has no implications for the
observable universe? (Hint: if you answered "yes" to either question,
quit reading now.)
Now it's true that we have a notion of "relative consistency" which we
use when we reason about consistency in general. And we sometimes
use the word "consistent" in relation to fiction, where we allow the work
of fiction to set up a premise which needn't be consistent with the
observable universe, and then require that the remainder of the work
of fiction be consistent relative to the premise.
In order to say that Cantor's Theory is consistent, we must admit that
Cantor's Theory is a work of fiction. Its major premise (Cantor's
intuitions about how the world of the infinite should work) is not
consistent with the observable universe.
Let me take this opportunity to outline the basic ideas I've been
advocating.
We can incorporate a notion of "observability" into mathematics. We
can talk about the existence of a "world" of computation. We can think
of mathematics as the science of phenomena observable in the world
of computation. We can say that a statement is "meaningful" if it makes
a prediction about what we will see when we perform an experiment in
the world of computation.
When we do this, we can still do mathematics much as it is done today.
We do not lose the abstractions which the pure mathematicians are fond
of; in fact, we need such abstractions to reason about the observable
world of computation. We can still do set theory. We can develop a
theory of the infinite. We can even talk about a "completed infinite" if we
so desire. We can even study Cantor's Theory as a formal system which
meets the requirements of observability.
However, when we develop of theory of the infinite which is based on
the notion of observability, the theory we get is not Cantor's Theory.
In particular, any attempt to unify observability with Cantor's Theory will
formally lead to the conclusion that there "exists" a world which is not
observable.
The theory I am advocating allows us to talk consistently about infinity,
even in the strengthened sense of "consistent" which requires
meaningfulness.
This theory is unique (a *good* thing) and essential (a *good* thing)
and minimalist (another *good* thing). Cantor's Theory is none of the
above.
And furthermore, the theory of the infinite we derive from the notion of
observability is (arguably) much closer to most people's intuitions
about the infinite than Cantor's Theory.
It's my contention that Cantor's world of mathematical objects which
cannot be observed is a mythical world, and we should expunge it
from mathematics. This is not a step backwards.
Virgil
Petry is the reincarnation of Kronecker come back to continue his
character assasination of Cantor on Cantor's works!
--
Virgil
vm...@frii.com
David Petry
Arthur L. Rubin <216-...@mcimail.com> wrote in message
<36D9F0...@mcimail.com>...
>David Petry wrote:
>>
>> Anyone who follows sci.math must be aware that there
>> are many people who find something to object to about
>> Cantor's Theory
>To begin with, it's a misnomer to call it Cantor's theory.
>I think von Neumann might be a better choice, [...]
Let's look at exactly what is being objected to here.
Cantor claimed that he could use the diagonalization method
to "prove" that there must "exist" "more" real numbers than
algebraic numbers, and hence that transcendental numbers
must "exist". (I realize how silly it looks to use so many quotes,
but the point is that each of the quoted words has a different
meaning in Cantor's theory than it had before Cantor came
along.)
With a little more work, (actually I should say "different" work,
rather than "more" work) one can use the diagonalization
method to actually construct a number which is not an
algebraic number. This number is something which exists
within the world observable through computation.
The problem with Cantor's idea is that it leads to the
conclusion that their "exists" a world beyond the world that
can be seen through computation. But we are free to assert
that mathematics is the study of the phenomena observable
through computation, and to assert that the world that
"exists" is the world of observable objects. We really are
free to do that. Cantor has not shown that we will be
inconsistent if we do that. The idea that mathematical
statements must be about something observable gives
"meaning" to mathematical statements.
So it's my contention that Cantor, rather than "proving"
anything at all, gave new meaning to the word "proof"
(it no longer is a compelling argument) and to the word
"exists" (things which plainly do not "exist" in the observable
world, nevertheless "exist" in Cantor's world if a formal
conclusion of his axioms can be interpreted to assert that
something "exists").
So I claim that Cantor's world of objects that lie beyond
the world we can observe through computation, is a
mythical world.
>Try to prove that Sum_n (1/(n log(n) log(log(n)))) diverges
>in a computational manner.
Whoa! If I understand what you're getting at, you badly
misunderstand what I've been saying. What do you mean
by "prove in a computational manner"?
Mathematicians could easily deal with the above sum long
before Cantor came along.
>> For one thing, nothing of applied mathematics is lost.
>
>Unless I don't understand what you are trying to say,
>you are wrong there, in more than one way.
>Either,
>(1) You only allow to be considered as mathematical that
>which can be evaluated, which eliminates much of the
>mathematics of Quantum Mechanics, as the equations cannot
>actually be solved; or
You're invoking esoterica here. All of the equations that
physicists actually use to make quantum mechanical predictions
of phenomena observable in the real world, can actually
be solved (numerically). I am aware that you can quote
results which would *appear* to contradict what I just said.
>(2) You allow the concept of a computer-aided proof,
Sure, why not.
>which is (classically mathematically) equivalent to that of a
>formal prove [proof],
Sort of.
>which puts us back where we started.
Way wrong. I have nothing against formal proof.
What I am against is formal proof without reality checks. I
am demanding that the axioms and laws of logic we use in
our formal proofs be derived from our experience with an
objective, observable world, and not derived from someone's
hallucinations.
Virgil
<david...@mindspring.com> wrote:
>The desire to debunk mythology is not reasonably called a
>"philosophical prejudice".
It cuts two ways, a philosophical prejudice is not reasonably
called a desire to debunk mythology.
There seems to be a strong desagreement as to which is which.
--
Virgil
vm...@frii.com
Mike Oliver
David Petry wrote:
> Cantor's "framework of thought" has become a dogma, and as
> is very apparent from this thread alone, the true believers are
> almost ready to take up the sword to defend the dogma. That
> alone is a good reason for taking this issue seriously.
I would say they're ready to take up the sword against
YOU PERSONALLY, David, not for the content of your views
but for the extraordinarily arrogant and obnoxious way
in which you express them.
There are posters here whose foundational views, in their
substance, are not all that different from yours. Keith
Ramsay and Bill Taylor come to mind. They get along
fine; they're valued contributors to the group.
But they know how to sweeten their tone, and they also
give evidence of actually knowing some mathematics.
Joe Keane
--
Joe Keane, amateur mathematician
Brian M. Scott
<david...@mindspring.com> wrote:
>So it's my contention that Cantor, rather than "proving"
>anything at all, gave new meaning to the word "proof"
>(it no longer is a compelling argument)
To whom? You appear to be saying that whether an argument is
compelling may depend on something other than its logical structure,
which (to me) is absurd.
>What I am against is formal proof without reality checks. I
>am demanding that the axioms and laws of logic we use in
>our formal proofs be derived from our experience with an
>objective, observable world, [...]
In what respect do you think that they are not?
Brian M. Scott
bensc...@my-dejanews.com
>
> In what respect do you think that they are not? -- Brian M. Scott (*)
>
Indeed, there *is* a very down-to-earth interpretation of Cantor's ideas,
*if* one is willing to look, at all;-) And take the word 'diagonal' at
face value, namely implying a square table, of say w binary strings of
length w each, hence a countable w x w table ( w = 'omega' = card(N) ).
And consider ALL diagonals (and their bitwise complements: CD = Cantor's
Diagonal_set) upon permuting the w rows of that countable square table...
Copied from my 5jan99 posting, answering Dik Winter who observed
that I was 'learning something' (sic;-) :
What I learned is roughly (correct me if I'm wrong):
an UnCountable (UC) closure must have an UC set of generators,
and: a Countable set of generators cannot generate an UC closure.
The UC concept, beyond countable and proof-by-induction (-;) may be
welcome, I guess, to an abstract person (someone who tries to take as
large a distance from reality & all practical purposes as possible;-)
However, true to my simple needs of wanting something practical out of
discrete math, I look at Cantor's Diagonal not as a "closure_breaking"
divice (countable --> uncountable), but as a "generating_device".
Namely by considering not just *one* complemented diagonal (after all: w+1
= w : no big deal -- finite intuition that one extra breaks closure does not
hold here...), but *all* possible diagonals that are generated from a
countable square w x w table (say the Boolean identity matix of order w, see
below) by permutating all |N|=w rows (or equivalently: all columns), This
yields permutation group N! which, IF powerset 2^N is UC, is also UC.
Yet N! -- or rather Z! for group-reasons (Peano's {+1}* is not a
permutation, as Dave S. noticed), might be finitely generatable (?)
At first (and still) I thought by 3 basic permutations.
But then, if generation is not of interest (let alone finite- or
countable- generators), or even excluded a priori, then I will be
satisfied with approaching 2^Z from above via Z! in some more
restricted sense. To at least have a finitely generated closure
which includes powerset 2^F, where F = {fixed-sets of all permutations
in group Z! = A*/Z }. ...If you get my drift ;-)
----------------------------------------------------copy-5jan99------:
NB: The classical "UnCountability" is not so much my aim, really, as
related to "reals". Rather, my suggestion re permutations was to
generate 2^N by just two generators sequentialy, in the N! group.
The Cantor diagonal was my natural context, with a Boolean Identity
matrix of size w x w, representing the naturals by (say) its rows:
012345678... 0<->1 row_swap
0 100000000... 1 010000000...
1 010000000... 0 100000000...
2 001000000... 2 001000000...
3 000100000... 3 000100000...
4 000010000... 4 000010000...
5 000001000...
6 000000100...
7 000000010...
8 0000000010..
.
.
.
Cantor Diagonal CD:
Permuted table:
CD= (1*)' = 0* CD= (001*)' = 110* : 2-element subset in 2^N.
----------------------------------------------------copy-9jan99----:
NB:
I generated all 4! = 24 state-permutations for a 4-state ISM(Q,A)
with stateset Q={0,1,2,3}, input alphabet A={a,b,c} defined for all
integers n \in Z (although for finite Q = Z mod 4: only a,c suffice):
a: n --> n+1 (global increment)
b: n --> n-1 (global decrement)
c: n --> n, except 0<-->1 (local swap)
Indeed |A*| = 4! with generative Spectrum = {3,6,6,5,3,1} ,
the number of new permutations in A^i (i=1..6).
Your (Dave Seaman's) pairwise twist came up last (!):
--> "aacbbc" = (0<->1)(2<->3) in A^6, a good testcase.
Notice that ab = ba = cc = e: the identity permutation.
So in general, minimal length representations over A have form
(a*.c.b*.c)* where c functions as marker ("comma"). The code then
is a sequence of repetitions (naturals) of two kinds: of a and b
alternating, possibly (?) not countable (Peano +1* / Cantor's UC).
---------------------------------------------------------endcopy---
PS: What I'm after is a characterization of the known "data types" by the
algebra's they allow (associative, commutative, idempotent) and by the
minimal number of generators of a 'full' closure (Seq_Dimension):
^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Symbol Type Seq_Dimension Generators
Algebra Closure
-------+---------+-----------+----------------+-------------------+----------
N Naturals 1 {+1}* Assoc, Commut've PosInt N(+) Z Integers 2
{+1,-1} Assoc, Commut've Ring Z(+,.) 2^Z Sets 3 (not sequent) Assoc,
Cmt,Idempotent Lattice Z! Permut's 3 {+1,-1,swap2} Assoc, Const_Rank
Group Z^Z Functions 4 {+1,-1,swap2,merge2) Assoc, Reduc_Rank Semigroup
-----------------------------------------------------------------------------
Notice: Only Sets have a non-sequential (=idempotent) algebra (intersection,
union). Quite a handicap qua generative power, wouldn't you think? --> So
try to take a seq.generative view, for a change: embed sets into groups.
Comments are welcome. Ciao, Nico Benschop (bens...@iae.nl)
-- http://www.iae.nl/users/benschop/cantor.htm
-----------== Posted via Deja News, The Discussion Network ==----------
http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own
Aakash Mehendale
>
[snip]
>
> The problem with Cantor's idea is that it leads to the
> conclusion that their "exists" a world beyond the world that
> can be seen through computation. But we are free to assert
> that mathematics is the study of the phenomena observable
> through computation, and to assert that the world that
> "exists" is the world of observable objects. We really are
> free to do that. Cantor has not shown that we will be
> inconsistent if we do that. The idea that mathematical
> statements must be about something observable gives
> "meaning" to mathematical statements.
>
Yes, you are free to assert that that's what mathematics is, and
that's the world that "exists". But you can't prove the latter, and
'defining' mathematics in this way seems a trifle ... presumptious,
shall we say?
> So it's my contention that Cantor, rather than "proving"
> anything at all, gave new meaning to the word "proof"
> (it no longer is a compelling argument)
It compels me.
> and to the word
> "exists" (things which plainly do not "exist" in the observable
> world,
Things like 3, pi, and i, which we all bump into every day walking
down the street.
> nevertheless "exist" in Cantor's world if a formal
> conclusion of his axioms can be interpreted to assert that
> something "exists").
>
> So I claim that Cantor's world of objects that lie beyond
> the world we can observe through computation, is a
> mythical world.
>
So symbolic computation doesn't cut any ice with you?
The problem here is that you seem perfectly able to say that (as
before), the number pi exists, but not the transfinite cardinal
aleph0. It's a very fine distinction to make.
--
Aakash Mehendale email:aak...@hotmail.com
ICQ:26396017
Aleph-null green bottles, hanging on the wall...
Any views and opinions expressed herein are mine alone
Mike Deeth
Joe Keane wrote:
I'm the youngest.
I will die last. :-)
Nathan the Great
Age 11
David Kastrup
> Joe Keane wrote:
>
> > This is a case where you just have to wait for people to die off.
> >
> > --
> > Joe Keane, amateur mathematician
>
>
> I'm the youngest.
> I will die last. :-)
But you have the disadvantage that your mathematics will be buried
along with you once you do.
--
David Kastrup Phone: +49-234-700-5570
Email: d...@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut für Neuroinformatik, Universitätsstr. 150, 44780 Bochum, Germany
Ulrich Weigand
>Cantor claimed that he could use the diagonalization method
>to "prove" that there must "exist" "more" real numbers than
>algebraic numbers, and hence that transcendental numbers
>must "exist". (I realize how silly it looks to use so many quotes,
>but the point is that each of the quoted words has a different
>meaning in Cantor's theory than it had before Cantor came
>along.)
This is simply not true. The words you quoted above have exactly
the same meaning in Cantor's proof as they had before. The real
numbers were defined before Cantor (say, using Dedekind cuts or
equivalence classes of Cauchy sequences of rationals). The concept
of a function from real numbers to real numbers was perfectly well
defined before Cantor. The question, whether a surjective function
from the algebraic numbers to the real numbers can exist could be
easily formulated in the usual mathematical terms before Cantor.
What Cantor did was to simply *answer* that question, using a method
of proof perfectly acceptable even before him.
This is not a question of "existence". The real numbers -all of them-
are well-defined by Dedekind cuts in advance, and no new numbers
magically come into existence by Cantor's proof. The proof simply
shows that some of the already known real numbers must be transcendent.
>With a little more work, (actually I should say "different" work,
>rather than "more" work) one can use the diagonalization
>method to actually construct a number which is not an
>algebraic number. This number is something which exists
>within the world observable through computation.
But this is just the difference between saying 'one of all these numbers
must be transcendent, but I'm not sure which one' and '*this* one number
must be transcendent'. While the second statement is surely somewhat
stronger, this does not mean that the first one is somehow invalid or
illegal ... In situations where you can prove a statement of the
first kind but not of the second kind, well, that's better than nothing,
isn't it?
>The problem with Cantor's idea is that it leads to the
>conclusion that their "exists" a world beyond the world that
>can be seen through computation. But we are free to assert
>that mathematics is the study of the phenomena observable
>through computation, and to assert that the world that
>"exists" is the world of observable objects. We really are
>free to do that. Cantor has not shown that we will be
>inconsistent if we do that. The idea that mathematical
>statements must be about something observable gives
>"meaning" to mathematical statements.
Do you claim, then, that Dedekind cuts do not define the real numbers?
So what is your definition of the real numbers? From your explanations
about "observability" it seems you mean something like the recursive
reals. If you want to argue about recursive reals, you are free to do
so, but please do not claim that these are the same as *all* real numbers
in the sense of Dedekind cuts etc. which was well-known even before
Cantor.
[snip]
>Mathematicians could easily deal with the above sum long
>before Cantor came along.
Yes, because the had a notion of 'real numbers' long before
Cantor came along. And in fact this was the very same notion
of real numbers that Cantor used, and *not* some notion of
recursive or constructible reals that came up only much later.
John Savard
>Cantor's Theory is a dogma. It's a mythology. It's an intellectual
>fraud. It's destructive. It deserves to die.
There are difficulties with the transfinite - but that simply means
that, unlike some of the more pedestrian parts of algebra and
calculus, there is still the opportunity to resolve the contradictions
and achieve new understanding.
The discomfort does not mean it deserves to die, it means that it is
very much alive.
John Savard (teneerf is spelled backwards)
http://members.xoom.com/quadibloc/index.html
ull...@math.okstate.edu
I've been wondering about that myself. (Decided not to ask, given
his tendency to ignore questions. I'm still wondering whether (i) he thinks
that the Pythagorean theorem is unimportant in applied mathematics,
or (ii) he knows a way to construct the reals without involving anything
infinite, or (iii) he agrees the theorem is important, doesn't know
how to construct the reals without using infinte gizmos (or formalisms
"about" infinite gizmos), but somehow doesn't see any inconsistency
with his assertion that the infinite is of no importance in applied
mathematics. I understand he feels the questions are "bizarre", but
I don't know what his answers are.)
feldma...@my-dejanews.com
David Kastrup <d...@mailhost.neuroinformatik.ruhr-uni-bochum.de> wrote:
> Mike Deeth <m...@ashland.baysat.net> writes:
>
> > Joe Keane wrote:
> >
> > > This is a case where you just have to wait for people to die off.
> > >
> > > --
> > > Joe Keane, amateur mathematician
> >
> >
> > I'm the youngest.
> > I will die last. :-)
>
> But you have the disadvantage that your mathematics will be buried
> along with you once you do.
>
How do you bury the empty set?
> --
> David Kastrup Phone: +49-234-700-5570
> Email: d...@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
> Institut für Neuroinformatik, Universitätsstr. 150, 44780 Bochum, Germany
>
-----------== Posted via Deja News, The Discussion Network ==----------
Jeremy Boden
...
>Perhaps you could say that "Cantor demonstrated how one could
>reason consistently with infinite collections", but only if you choose
>a convenient definition of "consistent". That is, can you consistently
>maintain that you are being consistent, if what you are saying has no
>meaning? Can you consistently maintain that what you are saying
>has meaning, if what you are saying has no implications for the
>observable universe? (Hint: if you answered "yes" to either question,
>quit reading now.)
Yes. (I took your hint and stopped reading).
You don't appear to have much idea what *consistent* means.
In mathematics, consistency means that no contradiction ever appears. It
is up to you to interpret any results which you have deduced. It would
appear that because you find the answers unpleasant, but can't fault the
deductions that got you there, that you feel forced into a denial of a
simple concept like the infinite.
[Nihilistic viewpoint!] You say that you have problems with "meaning". I
could ask you the meaning of anything and ultimately all you could do
would be to break down into simpler and simpler building blocks. But you
would never be able to give a real "meaning" to anything.
e.g. What is the meaning of zero?
Does the concept of zero (or infinity) have any "meaning" for the
observable universe? A good case could be made for the absence of zero
from the observable universe, but perhaps not for the absence of
infinite quantities.
--
Jeremy Boden mailto:jer...@jboden.demon.co.uk
Jeremy Boden
<david...@mindspring.com> writes
...
>
>The idea that Cantor's mythology is a "framework of thought" is
>a plausible view of things. In fact, I wouldn't attack Cantor's
>Theory for being a mythology, if I didn't believe that I have a very
>much superior (non-mythical) idea to replace it.
>
vastly superior replacement theory?
Daryl McCullough
[About Nathan Deeth]
>But you have the disadvantage that your mathematics will be buried
>along with you once you do.
Let's not be too hasty. If Nathan is right that he is only 11
years old, then he has plenty of time. He can play at being a
crackpot for ten years or so and still have time to do real
mathematics.
Actually, if Nathan is really 11 (and not a 30 year old pretending)
then he is obviously quite bright. Not many 11 year olds would even
make a convincing crackpot. Maybe there is some hope for the little
guy. My guess is that all you people arguing with him is just fanning
his ego and prolonging his crackpot period. Please just ignore him
unless he is saying something intelligent or legitimately asking
for help. You're not going to convert him through arguments.
Daryl McCullough
CoGenTex, Inc.
Ithaca, NY
David Petry
Jeremy Boden wrote in message ...
>e.g. What is the meaning of zero?
>
>Does the concept of zero (or infinity) have any "meaning" for the
>observable universe?
Usually we say that statements have meaning, not objects.
>A good case could be made for the absence of zero
>from the observable universe, but perhaps not for the absence of
>infinite quantities.
If you're interested in such issues, you should perhaps check out
the philosophy newsgroups.
David Petry
> wei...@informatik.uni-erlangen.de (Ulrich Weigand) wrote:
>> Furthermore, I don't quite get why Petry wants to contrast 'computation'
>> with 'formalism';
I have made no such contrast. I have made a contrast between
objects that exist in the world observable through computation,
with objects that "exist" only in the sense that a valid statement
in some formalism can be interpreted to assert that the objects
exist.
>> if there is any area of mathematics where formalism
>> is appropriate it is the consideration of computational processes ...
That's quite true, but in no way conflicts with anything I have said.
>>but computational processes are
>> of a formalistic nature to start with!
I think you're just playing word games. It's a little like saying that
science uses words to describe truth, and religion and philosophy
use words to describe truth, and therefore they are all fundamentally
the same thing.
The difference is that science recognizes the existence of an
objective, observable universe, and then uses formalisms as a
tool for understanding that universe. Religion and philosophy
play games with formalisms, but the formalisms they use are not
typically derived from the concrete, observable, objective universe.
What I'm saying is that we can separate the science from the
philosophy in mathematics, and furthermore, there are some very
good reasons for doing so.
ull...@math.okstate.edu wrote in message
<7bh8u0$66v$1...@nnrp1.dejanews.com>...
>In article <7bejii$e...@faui11.informatik.uni-erlangen.de>,
> I've been wondering about that myself. (Decided not to ask, given
>his tendency to ignore questions. I'm still wondering whether (i) he thinks
>that the Pythagorean theorem is unimportant in applied mathematics,
We've discussed this before, and you didn't like my answer. For
applied mathematics, the finite precision version of the Pythagorean
Theorem is sufficient. This version goes something like: At every
level of precision, the sides of a right triange satisfy the relation
a^2 + b^2 = c^2 .
>or (ii) he knows a way to construct the reals without involving anything
>infinite, or
First of all, it's rather dishonest of you to use such a vague term as
"anything infinite".
I'm not going to answer a question just to "prove" I know something.
If you're interested in the topic, follow some of the never ending
threads on constructivism in this newsgroup and in sci.logic, or
get a book.
(iii) he agrees the theorem is important, doesn't know
>how to construct the reals without using infinte gizmos (or formalisms
>"about" infinite gizmos), but somehow doesn't see any inconsistency
>with his assertion that the infinite is of no importance in applied
>mathematics. I understand he feels the questions are "bizarre", but
>I don't know what his answers are.)
Yes, I think your question here is bizarre.
David Petry
Brian M. Scott wrote in message <36db6baf...@news.csuohio.edu>...
> You appear to be saying that whether an argument is
>compelling may depend on something other than its logical structure,
Not really. I'm saying that the conclusion of the argument
must have a meaningful interpretation in an objective world. If
it doesn't, I take that as proof that there is something wrong with
the logical structure. That is, I demand that "logic" produce
meaninful conclusions.
In particular, Cantor's Theory leads to conclusions which have
no meaningful interpretation in the concrete, objective, observable
world of computation. Hence my conclusion that there is something
wrong with the logical structure of Cantor's Theory.
>which (to me) is absurd.
In contexts other than mathematics you would probably quickly
agree with me that logical structure is not sufficient For example,
in politics and religion. Political and theological theories can
be very consistent internally, but still be very distorted models
of the real world. And I think anyone can see that these theories
can be enormously destructive in the real world, even if the
arguments are highly respected in the academic or religious
worlds.
In other words, it's very wrong to teach people to look only at the
logical structure of an argument, and to not closely examine the
connection of the objects in the formal argument with the objects
and phenomena in the real world.
In other words, destructive dogmas can be internally logically
consistent. It's this logical consistency that makes these
dogmas so pernicious and persistent. The way to see past
the dogma is to carefully examine the connection between
the conclusion of the dogmatic formalism and the phenomena
in the real world.
Granted, Cantor's Theory cannot be regarded as a destructive
dogma, but only because pure mathematics is so utterly
irrelevant to the real world. But by forcing students to accept
the theory without even allowing them to ask the question of
what reality the theory is intended to be a model of, the
mathematicians are indoctrinating students into a world view
in which destructive dogmas can thrive. This is not a good
thing.
>>What I am against is formal proof without reality checks.
>In what respect do you think that they are not?
Does anything I am saying make sense to you?
Neil Rickert
>In particular, Cantor's Theory leads to conclusions which have
>no meaningful interpretation in the concrete, objective, observable
>world of computation.
Perhaps. But it also leads to many conclusions which are highly
meaningful and useful, and that seems to be good enough reason to
retain Cantor.
> Hence my conclusion that there is something
>wrong with the logical structure of Cantor's Theory.
Some of us would locate the "something wrong" elsewhere.
>In other words, it's very wrong to teach people to look only at the
>logical structure of an argument, and to not closely examine the
>connection of the objects in the formal argument with the objects
>and phenomena in the real world.
Why? What is wrong with that, in the context of mathematics?
>Granted, Cantor's Theory cannot be regarded as a destructive
>dogma, but only because pure mathematics is so utterly
>irrelevant to the real world.
A strange claim. Given the usefulness of mathematics to the
sciences, it is surely not irrelevant.
> But by forcing students to accept
>the theory without even allowing them to ask the question of
>what reality the theory is intended to be a model of, the
>mathematicians are indoctrinating students into a world view
>in which destructive dogmas can thrive. This is not a good
>thing.
Personally, I encourage students to ask all kinds of questions.
Perhaps that way, they can avoid your kind of confusion, such as your
silly idea that mathematics should be be a model of anything.
Instead of thinking of mathematics as a model of reality, you should
think of it as a powerful modelling medium in which objective reality
is only one of the things that can be modelled. Instead of trying to
constrain mathematics, learn to value it for its versatility.
Virgil
<david...@mindspring.com> wrote:
>In contexts other than mathematics you would probably quickly
>agree with me that logical structure is not sufficient For example,
>in politics and religion. Political and theological theories can
>be very consistent internally, but still be very distorted models
>of the real world. And I think anyone can see that these theories
>can be enormously destructive in the real world, even if the
>arguments are highly respected in the academic or religious
>worlds.
In politics and in religion, one person's orthodoxy is another's
heterodoxy. Who belives that what is to him/her herterodox is logically
consistent? You have chosen here an exraordinarily bad example
which which to make your point.
I would say that, outside of mathematics, logical structure is
neither necessary nor sufficient. Outside of mathematics and
the "hard" sciences, it is mostly irrelevant.
Also, is it your younger generation, which you say will only be satisfied
with reality checks, the same generation which watches "the X Files" and
"Buffy, the vampire Slayer" in droves, worships UFOs and swears by the
truth according to the supermarket tabloids?
--
Virgil
vm...@frii.com
Virgil
<david...@mindspring.com> wrote:
>We've discussed this before, and you didn't like my answer. For
>applied mathematics, the finite precision version of the Pythagorean
>Theorem is sufficient. This version goes something like: At every
>level of precision, the sides of a right triangle satisfy the relation
>a^2 + b^2 = c^2 .
The trouble with this example is that we cannot verify it in the real world.
We know that it is true on a scale of human sizes to the limits of our
ability to test it.
We don't know whether it is true on a microscopic scale.
We don't know whether it is true on an astronomic scale,
and many cosmologists suspect that it is not.
--
Virgil
vm...@frii.com
Arthur L. Rubin
> Cantor claimed that he could use the diagonalization method
> to "prove" that there must "exist" "more" real numbers than
> algebraic numbers, and hence that transcendental numbers
> must "exist". (I realize how silly it looks to use so many quotes,
> but the point is that each of the quoted words has a different
> meaning in Cantor's theory than it had before Cantor came
> along.)
Cantor was among the first to use formal (or quasi-formal) proofs,
so I suppose he did change the definition of "proof" -- he
NARROWED it. He observed the diagonal arguement, and it works
in many systems, to show that:
There is a recursive function which is not primative recursive:
(The index function pr_i(j) is recursive)
There is a function which is not recursive
There is no function which enumerates the reals
etc.
--
Arthur L. Rubin 216-...@mcimail.com
David Kastrup
> David says...
>
> [About Nathan Deeth]
>
> >But you have the disadvantage that your mathematics will be buried
> >along with you once you do.
>
> Let's not be too hasty. If Nathan is right that he is only 11
> years old, then he has plenty of time. He can play at being a
> crackpot for ten years or so and still have time to do real
> mathematics.
>
> Actually, if Nathan is really 11 (and not a 30 year old pretending)
> then he is obviously quite bright.
He can spell ok. He can post. But the mathematical content of his is
negligible. It comes down to "if two sets match up, and I take away
one element from one set, then the two resulting sets will not match
up". Which obviously works only for finite sets. He has not
understood this simple principle even after having it had explained
for a month. He then resorted to sulking and ridicule. Up to now I
think he qualifies more as belligerent than as bright.
Of course, one might not necessarily expect understanding of basic
concepts of infinity by an 11 year-old, so he may be bright for his
age, after all. The topic he has chosen for his display, however,
leave us in the dark.
> Not many 11 year olds would even make a convincing crackpot.
Sorry, but being overly childish is exactly the mark of a crackpot.
He might not be overly childish for his age, but anyhow he manages to
get on people's nerves just like a crackpot and obnoxious children
would.
Brian M. Scott
> Brian M. Scott wrote in message <36db6baf...@news.csuohio.edu>...
> >On Mon, 1 Mar 1999 17:43:11 -0800, "David Petry"
> ><david...@mindspring.com> wrote:
> > You appear to be saying that whether an argument is
> >compelling may depend on something other than its logical structure,
> Not really. I'm saying that the conclusion of the argument
> must have a meaningful interpretation in an objective world. If
> it doesn't, I take that as proof that there is something wrong with
> the logical structure.
And not with its hypotheses?
> That is, I demand that "logic" produce
> meaninful conclusions.
It does. They are typically of the form 'if A, then B'.
> In particular, Cantor's Theory leads to conclusions which have
> no meaningful interpretation in the concrete, objective, observable
> world of computation. Hence my conclusion that there is something
> wrong with the logical structure of Cantor's Theory.
Your conclusion *should* be that there is something wrong with its
premises. But in fact it appears that at bottom your objection is
simply to the subject matter, a prejudice masquerading as a principled
objection.
> >which (to me) is absurd.
> In contexts other than mathematics you would probably quickly
> agree with me that logical structure is not sufficient.
Of course I wouldn't.
> For example,
> in politics and religion. Political and theological theories can
> be very consistent internally, but still be very distorted models
> of the real world.
This has nothing to do with the logical structure of arguments. You're
confusing inadequate or simply incorrect premises with faulty logic.
> >>What I am against is formal proof without reality checks.
> >In what respect do you think that they are not?
> Does anything I am saying make sense to you?
Do I understand what you're saying? I think so, yes. Do I think that
you have a point? No: your objections are confused. Possibly someone
else could present them more cleanly and convincingly, but your notion
of reality is still clearly very different from mine.
Brian M. Scott
ull...@math.okstate.edu
"David Petry" <david...@mindspring.com> wrote:
>
> ull...@math.okstate.edu wrote in message
> <7bh8u0$66v$1...@nnrp1.dejanews.com>...
> >In article <7bejii$e...@faui11.informatik.uni-erlangen.de>,
>
> > I've been wondering about that myself. (Decided not to ask, given
> >his tendency to ignore questions. I'm still wondering whether (i) he thinks
> >that the Pythagorean theorem is unimportant in applied mathematics,
>
> applied mathematics, the finite precision version of the Pythagorean
> Theorem is sufficient. This version goes something like: At every
> a^2 + b^2 = c^2 .
I don't recall you saying this before - don't recall you saying
anything at all about the Pythagorean theorem except calling my
mentioning it bizarre.
The version that you suggest is "sufficient" has the unfortunate
disadvantage of being false.
> >or (ii) he knows a way to construct the reals without involving anything
> >infinite, or
>
> First of all, it's rather dishonest of you to use such a vague term as
> "anything infinite".
Yes, the term "anything infinite" is a little vague. But there's
nothing vague about what it refers to in the constructions of the
reals that I'm aware of: a Dedekind cut is an infinite set, for
example.
IF I'd asserted that it was impossible to define the reals
without using anything infinite then you could rightly ask exactly
what I meant by the phrase. But I've made no such assertion.
You've stated explicitly that the infinite is not needed in
applied mathematics. I've asked you repeatedly how you define the
reals without using anything infinite - you haven't answered the
question yet, you haven't given anything like a hint of what the
answer would be. If you tell me what construction of the reals
you have in mind then we can worry about whether there is or
is not anything "infinite" involved in that construction.
Until you say what construction you intend it's hard to do that.
> I'm not going to answer a question just to "prove" I know something.
Excellent reply.
> If you're interested in the topic, follow some of the never ending
> threads on constructivism in this newsgroup and in sci.logic, or
> get a book.
Um. I've followed lots of threads on this newgroup and on
sci.logic. I own lots of books. I know _several_ ways to construct
the reals, all of which involve infinite sets. I have never seen
a construction of the reals without involving anthing infinite in
any of those books or threads.
So by all means don't answer the question - if I were you
I wouldn't try to prove I knew something either. It's a very
convenient response.
Like, I say the world is flat, and this business about
a round earth is just a fiction. Someone asks how I know the
earth is flat. I reply that I'm not going to answer this
question just to "prove" I know something - you should just
check innumerable threads in sci.geology or buy a book.
Very convenient. But I wonder if anyone would be convinced
that the earth was flat...
> (iii) he agrees the theorem is important, doesn't know
> >how to construct the reals without using infinte gizmos (or formalisms
> >"about" infinite gizmos), but somehow doesn't see any inconsistency
> >with his assertion that the infinite is of no importance in applied
> >mathematics. I understand he feels the questions are "bizarre", but
> >I don't know what his answers are.)
>
> Yes, I think your question here is bizarre.
You've explained that _several_ times now. Calling a question
bizarre doesn't quite say what the answer is.
Virgil
>>Granted, Cantor's Theory cannot be regarded as a destructive
>>dogma, but only because pure mathematics is so utterly
>>irrelevant to the real world.
Group theory was originally considered pure mathematics, but now appears
applied to many real world situations.
You, no doubt, would have objected to exposing innocent applied
mathematics students to the horrors of group theory before its
applications became so apparent.
Those who are committed to applying mathematics have been known take the
position that anyone who does "pure" mathematics is somehow cheating,
because it doesn't apply to the "real" world.
--
Virgil
vm...@frii.com
Neil Rickert
>In article <7big7k$m...@ux.cs.niu.edu>, ric...@cs.niu.edu (Neil Rickert) wrote:
>>>Granted, Cantor's Theory cannot be regarded as a destructive
>>>dogma, but only because pure mathematics is so utterly
>>>irrelevant to the real world.
>Group theory was originally considered pure mathematics, but now appears
>applied to many real world situations.
>You, no doubt, would have objected to exposing innocent applied
>mathematics students to the horrors of group theory before its
>applications became so apparent.
Please get those attributions right. You have quoted nothing that I
said. I had also disagreed with the above quoted comment.
David Petry
Neil Rickert wrote in message <7big7k$m...@ux.cs.niu.edu>...
>"David Petry" <david...@mindspring.com> writes:
>
>>Granted, Cantor's Theory cannot be regarded as a destructive
>>dogma, but only because pure mathematics is so utterly
>>irrelevant to the real world.
>A strange claim. Given the usefulness of mathematics to the
>sciences, it is surely not irrelevant.
Should I apologize for being a little sloppy?
As I have pointed out before, the mathematics which can be
described as the science of phenomena observable through
computation is indeed very relevant and extraordinarily useful
in the sciences and in technology. But the mathematics which
relies in an essential way on Cantor's Theory is nevertheless
utterly irrelevant.
If you think you have a counterexample to my claim, by all
means, share it, and we can discuss it.
David Petry
Brian M. Scott wrote in message <36DCDF...@stratos.net>...
>David Petry wrote:
>> In particular, Cantor's Theory leads to conclusions which have
>> no meaningful interpretation in the concrete, objective, observable
>> world of computation. Hence my conclusion that there is something
>> wrong with the logical structure of Cantor's Theory.
>Your conclusion *should* be that there is something wrong with its
>premises
Remember that the world of the infinite is a ficticious world. It is
not _a priori_ clear what laws of logic apply in that world. So if we
make a wrong choice about which laws of logic should be included
in that world, is there something wrong with a premise or is there
something wrong with the logical structure? Does it really matter
which way we say it?
Neil Rickert
>Neil Rickert wrote in message <7big7k$m...@ux.cs.niu.edu>...
>>"David Petry" <david...@mindspring.com> writes:
>>
>>>Granted, Cantor's Theory cannot be regarded as a destructive
>>>dogma, but only because pure mathematics is so utterly
>>>irrelevant to the real world.
>>A strange claim. Given the usefulness of mathematics to the
>>sciences, it is surely not irrelevant.
>Should I apologize for being a little sloppy?
>As I have pointed out before, the mathematics which can be
>described as the science of phenomena observable through
>computation is indeed very relevant and extraordinarily useful
>in the sciences and in technology. But the mathematics which
>relies in an essential way on Cantor's Theory is nevertheless
>utterly irrelevant.
If our forebears, throughout all history, had rejected all
mathematics that was not relevant, we would still be counting on our
fingers.
How about coming up with a complete reworking of quantum physics in
such a way as to remove all reliance on infinite sets.
Virgil
>vm...@frii.com (Virgil) writes:
>
>>In article <7big7k$m...@ux.cs.niu.edu>, ric...@cs.niu.edu (Neil Rickert)
wrote:
>Please get those attributions right. You have quoted nothing that I
>said. I had also disagreed with the above quoted comment.
Sorry, forgot to proof read before posting.
--
Virgil
vm...@frii.com
Virgil
<david...@mindspring.com> wrote:
>Remember that the world of the infinite is a ficticious world. It is
>not _a priori_ clear what laws of logic apply in that world. So if we
>make a wrong choice about which laws of logic should be included
>in that world, is there something wrong with a premise or is there
>something wrong with the logical structure? Does it really matter
>which way we say it?
The entire world of mathematics, infinite or finite, is a glorious fiction.
Brouwer's fiction is no more physical fact than Cantor's.
Why should one part be acceptable fiction while another part is
unacceptable fiction? What reality distinguishes between the fictions?
--
Virgil
vm...@frii.com
Brian M. Scott
<david...@mindspring.com> wrote:
>Remember that the world of the infinite is a ficticious world.
No more so than any other part of mathematics.
> It is
>not _a priori_ clear what laws of logic apply in that world.
It is to me.
Brian M. Scott
Ingrid Voigt
> Neil Rickert wrote in message <7big7k$m...@ux.cs.niu.edu>...
> >"David Petry" <david...@mindspring.com> writes:
> >
>
> >>Granted, Cantor's Theory cannot be regarded as a destructive
> >>dogma, but only because pure mathematics is so utterly
> >>irrelevant to the real world.
>
> >A strange claim. Given the usefulness of mathematics to the
> >sciences, it is surely not irrelevant.
>
> Should I apologize for being a little sloppy?
>
> As I have pointed out before, the mathematics which can be
> described as the science of phenomena observable through
> computation is indeed very relevant and extraordinarily useful
> in the sciences and in technology. But the mathematics which
> relies in an essential way on Cantor's Theory is nevertheless
> utterly irrelevant.
>
> If you think you have a counterexample to my claim, by all
> means, share it, and we can discuss it.
Theoretical computer science considers the question which functions
are computable. The proof that some are not uses Cantor-like
arguments.
Ingrid
Mike Deeth
Brian M. Scott wrote:
Mr. BS, does the principle of identity apply in your infinite
world?
Principle of Identity: A is A. A thing is equal to itself.
Nathan the Great
Age 11
Paul Hammond
> In article <m2g17o9...@mailhost.neuroinformatik.ruhr-uni-bochum.de>,
> David Kastrup <d...@mailhost.neuroinformatik.ruhr-uni-bochum.de> wrote:
> > Mike Deeth <m...@ashland.baysat.net> writes:
> >
> > > Joe Keane wrote:
> > >
> > > > This is a case where you just have to wait for people to die off.
> > > >
> > > > --
> > > > Joe Keane, amateur mathematician
> > >
> > >
> > > I'm the youngest.
> > > I will die last. :-)
> >
> > But you have the disadvantage that your mathematics will be buried
> > along with you once you do.
> >
>
> How do you bury the empty set?
>
Well, you don't need to as it's already a subset of the set of buried
things.
>
>
> > --
> > David Kastrup Phone: +49-234-700-5570
> > Email: d...@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
> > Institut für Neuroinformatik, Universitätsstr. 150, 44780 Bochum, Germany
> >
>
> -----------== Posted via Deja News, The Discussion Network ==----------
> http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own
>
>
----
p...@maths.nott.ac.uk
Paul Hammond | "Only teenagers could be
Maths Dept | that incoherent"
University of Nottingham |
Nottingham NG7 2RD | - Lisa Simpson
torqu...@my-dejanews.com
Paul Hammond <p...@maths.nott.ac.uk> wrote:
> > How do you bury the empty set?
>
> Well, you don't need to as it's already a subset of the set of buried
> things.
It may be a subset of the set of buried things but its not necessarily buried
itself. It's also a subset of the set of integers bigger than 11 but it is not
itself an integer bigger than 11.
--
Torque
http://www.tanelorn.demon.co.uk
arinch...@my-dejanews.com
> It may be a subset of the set of buried things but its not necessarily buried
> itself. It's also a subset of the set of integers bigger than 11 but it is not
> itself an integer bigger than 11.
> --
the point is I think that any element(???) of the empty set is
bigger than 11 , the empty set is a subset not an element of the
original set.
Ralph Frost
>
> "David Petry" <david...@mindspring.com> writes:
> >Neil Rickert wrote in message <7big7k$m...@ux.cs.niu.edu>...
> >>"David Petry" <david...@mindspring.com> writes:
> >>
> If our forebears, throughout all history, had rejected all
> mathematics that was not relevant, we would still be counting on our
> fingers.
>
What? What? Hey, what's the option? I still am counting on my fingers,
in more ways than one. My boys are learning to associate digits with
digits, too. What'd we miss?
> How about coming up with a complete reworking of quantum physics in
> such a way as to remove all reliance on infinite sets.
Use a macrophysical quantum state (analog) model.
PMI, but what does Cantor's Theory say, in English, if possible please?
Is that in a FAQ, somewhere? Also, Fibonacci??
TIA.
Ralph Frost
Ralph Frost
TIA.
Ralph Frost
David Petry
Brian M. Scott wrote in message <36de2900...@news.csuohio.edu>...
>>Remember that the world of the infinite is a ficticious world.
>No more so than any other part of mathematics.
The world observable through computation is something that
people were studying and applying for thousands of years
before mathematicians invented a formal model of that world.
Cantor's world of the infinite is something that sprang into
existence through its formalization. And furthermore, numerous
alternative and incompatible formalizations of the infinite are
possible.
As Nathan the Great says, if you close your eyes, you can
clearly see that there's no difference between the two worlds.
>> It is not _a priori_ clear what laws of logic apply in that world.
>It is to me.
OK, so you believe that you have some source of knowledge that
a computer can never access. I think such an idea is absurd, but
I acknowledge that there are some very bright people who agree
with you. Penrose, for example, thinks that we have access to
such knowledge through some sort of connection with quantum
gravity, though he is a little fuzzy on the details. Likewise, as I
understand it, Cantor thought he had access to such knowledge
by virtue of his ability to talk directly to God.
Do you have a theory as to what gives you access to knowledge
that is unavailable to a digital computer?
David Petry
Neil Rickert wrote in message <7bkji1$q...@ux.cs.niu.edu>...
>"David Petry" <david...@mindspring.com> writes:
>> But the mathematics which
>>relies in an essential way on Cantor's Theory is nevertheless
>>utterly irrelevant.
>If our forebears, throughout all history, had rejected all
>mathematics that was not relevant, we would still be counting on our
>fingers.
A truer irrelevant truism was never spoken.
Note that I have not been criticizing Cantor's Theory on the empirical
grounds that his theory is irrelevant. Rather I have been advocating
the notion of observability as a plausible and intuitively appealing
test to distinguish reality from make believe. Then I have been arguing
that any mathematical theory that does not pass this test of observability
cannot be relevant in any way to our understanding of the world around
us, and then I point out that the mathematics which relies on Cantor's
Theory in an essential way does not pass this test, and hence must be
irrelevant, and in fact is empirically irrelevant.
If people would pay more attention to the connection of theory to
reality, instead of seeing the creation of clever theories as a goal in
itself, then people might be less likely to be mislead by the seductive
ideologies which have been enormously destructive in our world.
The idea of observability which I am advocating is a *new* idea. If
people rejected every new idea simply because it conflicted with the
old dogma, the world would get bogged down in dogma and not
progress.
David Petry
Jeremy Boden wrote in message <3hiIUJAi...@jboden.demon.co.uk>...
>In article <7bf9eb$eia$1...@camel0.mindspring.com>, David Petry
><david...@mindspring.com> writes
>>The idea that Cantor's mythology is a "framework of thought" is
>>a plausible view of things. In fact, I wouldn't attack Cantor's
>>Theory for being a mythology, if I didn't believe that I have a very
>>much superior (non-mythical) idea to replace it.
>Well why not provide a brief, but short, summarised, abstract of this
>vastly superior replacement theory?
OK, here's a brief, but short, summarized, abstract:
It's not nice to ignore observability.
David Petry
Virgil wrote in message ...
>In article <7bq1p0$j9g$1...@camel25.mindspring.com>, "David Petry"
><david...@mindspring.com> wrote:
>
>>Do you have a theory as to what gives you access to knowledge
>>that is unavailable to a digital computer?
>
Perhaps I should have said "... unavailable to an artificial intelligence
running on a digital computer".
David Petry
Neil Rickert wrote in message <7bqb59$8...@ux.cs.niu.edu>...
>We don't observe complex numbers, so they must be irrelevant. We
>don't observe Euclid's perfect lines and circles, [...]
Evidently you have not even read the article I wrote which started
this thread. So you really have no business participating in this
discussion.
Ralph Frost
> ><david...@mindspring.com> wrote:
>
>
> Do you have a theory as to what gives you access to knowledge
> that is unavailable to a digital computer?
Different materials of construction
Greater mobility
Autonomy
Bio-energetic
Able to resonantly process tacit knowledge into conjectural knowledge
On a good day, at least partly conscious of some of the surroundings
Able to decide to talk to God -- or not
Basically, it boils down to existing at a different base level resonance
state.
ull...@math.okstate.edu
"David Petry" <david...@mindspring.com> wrote:
>
> >> It is not _a priori_ clear what laws of logic apply in that world.
{Scott]
> >It is to me.
[Petry]
> OK, so you believe that you have some source of knowledge that
> a computer can never access. I think such an idea is absurd, but
> I acknowledge that there are some very bright people who agree
> with you. Penrose, for example, thinks that we have access to
> such knowledge through some sort of connection with quantum
> gravity, though he is a little fuzzy on the details. Likewise, as I
> understand it, Cantor thought he had access to such knowledge
> by virtue of his ability to talk directly to God.
>
> Do you have a theory as to what gives you access to knowledge
> that is unavailable to a digital computer?
What??? He didn't say anything about having access to information
unavailable to a computer!
You seem to have missed a point or two. You accuse us of
"mysticism", but you're the guy who's taking a mystical attitude
to all this. The point you seem to have missed: ZF is a _formal_
theory. We can say exactly what the axioms are, we can define
exactly what a proof is, in such a way that it's _trivial_
to write a computer program that will check whether a given
sequence of symbols is or is not a valid proof of a theorem
of ZF. You go through the purported proof one line at a time;
for each line you ask whether it's an axiom of ZF (you can
check that mechanically) or follows from previous lines by
modus ponens. If each line passes one of these tests it's
a valid proof, otherwise not.
That's proofs of the theorems of ZF involving infinite
sets. A computer can verify these once they're formalized.
The idea that reasoning about infinite sets requires some
sort of transcendental understanding is simply wrong.
Of course Penrose is babbling nonsense (well, I'm not
really qualified to state an opinion there, that's the way
it seems to me, as well as to many people eminently qualified
to have an opinion.) I wasn't aware that Cantor thought
he could talk to God. If so, so what? Lots of people have
explained a lot of problems with a lot of the things you've
said - the fact that Cantor could talk to God has not been
part of any of the explpanations, has it?
If you _really_ think that Brian implied or meant to
imply that humans had "some source of knowledge that a
computer can never access" that simply indicates you're
totally misunderstanding the issues. (Or it _could_ be that
I totally misunderstood what Brian meant and your interpretation
is what he intended. I doubt it...)
Neil Rickert
Yes, I read it. I'm not impressed by your dismissal. In the early
days there were challenges against complex numbers, on the same sort
of basis that you use in you opposition to Cantor's innovations.
My accusation is that you are setting yourself up as God, declaring
that you have perfect knowledge of what parts of mathematics will be
useful in the future. But mathematics has always been ahead of its
uses. The parts of mathematics that you want to discard have already
proved their usefulness, although with your head buried in the sand
you are incapable of seeing this.
Arthur L. Rubin
>
> Brian M. Scott wrote in message <36de2900...@news.csuohio.edu>...
> >On Wed, 3 Mar 1999 15:44:51 -0800, "David Petry"
> ><david...@mindspring.com> wrote:
>
>
> >No more so than any other part of mathematics.
>
> The world observable through computation is something that
> people were studying and applying for thousands of years
> before mathematicians invented a formal model of that world.
...and produced somewhat limite results until...
> Cantor's world of the infinite is something that sprang into
> existence through its formalization. And furthermore, numerous
> alternative and incompatible formalizations of the infinite are
> possible.
proper handling of the infinite was formalized.
--
Arthur L. Rubin 216-...@mcimail.com
Brian M. Scott
> If you _really_ think that Brian implied or meant to
>imply that humans had "some source of knowledge that a
>computer can never access" that simply indicates you're
>totally misunderstanding the issues. (Or it _could_ be that
>I totally misunderstood what Brian meant and your interpretation
>is what he intended. I doubt it...)
You're quite right, and you've saved me the trouble of answering;
thanks. (On the matter of human vs. artificial intelligence I'm
agnostic.)
Brian M. Scott
Brian M. Scott
<david...@mindspring.com> wrote:
>Brian M. Scott wrote in message <36de2900...@news.csuohio.edu>...
>>On Wed, 3 Mar 1999 15:44:51 -0800, "David Petry"
>><david...@mindspring.com> wrote:
>>>Remember that the world of the infinite is a ficticious world.
>>No more so than any other part of mathematics.
>The world observable through computation is something that
>people were studying and applying for thousands of years
>before mathematicians invented a formal model of that world.
The world isn't observable through computation. If you're computing
(beyond simple counting, anyway), you're applying some mathematical
model of the world.
>Cantor's world of the infinite is something that sprang into
>existence through its formalization. And furthermore, numerous
>alternative and incompatible formalizations of the infinite are
>possible.
So? Which flavour of geometry would you like with your tea this
morning? In any case, the diagonalization argument, which at least
indirectly started all this, isn't affected by any of these
variations.
Brian M. Scott
LordBeotian
David Petry wrote
>I don't like that term. For one thing, "intuition" plays no special role
>in the theory I'm proposing.
>
>As I've defined it, a statement has observable content if it makes a
>testable or falsifiable prediction about the results of a possible
>computation.
There is not a single terminating algorithm to decide if a statement can be
tested or falsified.
So the observability is in general not observable.
So if statements that are not observable are meaningless also your
distinction between observable and not observable is meaningless.
>It's often the case that a statement is falsifiable
>whereas its formal negation is not, hence the law of the excluded
>middle cannot be consider a universally valid law.
Formally, what does "falsifiable" mean?
Greetings
LordBeotian
Jeremy Boden
<david...@mindspring.com> writes
>
>Jeremy Boden wrote in message <3hiIUJAi...@jboden.demon.co.uk>...
>>In article <7bf9eb$eia$1...@camel0.mindspring.com>, David Petry
>><david...@mindspring.com> writes
>
>>>The idea that Cantor's mythology is a "framework of thought" is
>>>a plausible view of things. In fact, I wouldn't attack Cantor's
>>>Theory for being a mythology, if I didn't believe that I have a very
>>>much superior (non-mythical) idea to replace it.
>
>>Well why not provide a brief, but short, summarised, abstract of this
>>vastly superior replacement theory?
>
>OK, here's a brief, but short, summarized, abstract:
>
>It's not nice to ignore observability.
>
convenient to use mathematics as a language for expressing scientific
ideas. However, it isn't necessary to do so - it's purely a matter of
convenience.
More importantly, there is no necessity to place restrictions on a
purely mathematical theory as to whether it meets some criterion of
"reality". My interpretation of what is truly reality might not be the
same as yours - in which case a mathematical proof becomes simply a
matter of opinion.
For example, is the (local) curvature of space an observable? A modern
physicist would certainly say that it is. But a classical physicist
would probably say the question is meaningless, i.e. that the curvature
of space is *not* an observable. So choose your reality.
"Reality" comes down to a value judgement - unfortunately reality is not
an objective concept.
--
Jeremy Boden mailto:jer...@jboden.demon.co.uk
Alehandro Taptaptaptaptap
wrote:
>Formally, what does "falsifiable" mean?
"My dousing rod can detect water."
But your dousing rod didn't detect water!
"Well, today the magnetic fields were off."
The hypothesis "My dousing rod can detect water." is not falsifiable
in this case, because any failure will be rationalized. A falsifiable
statement is one that can be proved false (whether or not it is. 2+2 =
4 is a falsifiable statement that's generally accepted as true.)
"My dousing rod can detect water every Tuesday at 10 PM, regardless of
the surrounding conditions." is falsifiable.
This sort of thing usually pops up in the "scientific method" section
of your typical chemistry or psychology textbook.
--
sqrt email
Fermat's dead, Jim.
LordBeotian
David Petry wrote:
>OK, so you believe that you have some source of knowledge that
>a computer can never access. I think such an idea is absurd,
What about the formula:
"for any positive integer N, N=N"
No computer can test it for any N, but we know that it is true.
>Do you have a theory as to what gives you access to knowledge
>that is unavailable to a digital computer?
Logic and intuition.
Greetings
LordBeotian
David Petry
Neil Rickert wrote in message <7brq0i$a...@ux.cs.niu.edu>...
>In the early
>days there were challenges against complex numbers, on the same sort
>of basis that you use in you opposition to Cantor's innovations.
There are similarities in a touchy-feely sort of sense, I suppose.
>My accusation is that you are setting yourself up as God, declaring
>that you have perfect knowledge of what parts of mathematics will be
>useful in the future.
Well, uh, I see things differently.
I'm claiming that mathematics is part of mankind's quest for
understanding. We want to understand ourselves, and we want
to understand the world around us.
This quest for understanding requires us to attempt to distinguish
fact from fiction, truth from myth, reality from fantasy. What it means
to "understand" some aspect of the world is to have a formal model
which gives us the power to make reliable predictions about the
results of experiments and tests relating to that aspect of the world.
So indeed, formalisms play a key role in our quest for understanding.
However, the pursuit of formalism is not identical to the pursuit of
understanding. That is, if we use formalism to create new worlds,
we are creating fantasies; when formalisms lose their connection
to the observable world, they must be regard as myths, regardless
of how clever and precise they may be.
I'm claiming that the concept of observability gives us a criterion
for distinguishing fantasy from reality in mathematics. I'm claiming
that observability can serve as a guide for the creation of a new
foundation for mathematics which is free from the fantasies
inherent in Cantor's theory.
> The parts of mathematics that you want to discard have already
>proved their usefulness, although with your head buried in the sand
>you are incapable of seeing this.
That conclusion is based on your own very incomplete and inaccurate
understanding of what I am saying.
David Petry
LordBeotian wrote in message <7bs7jt$j4i$4...@nslave1.tin.it>...
>
>David Petry wrote
>>As I've defined it, a statement has observable content if it makes a
>>testable or falsifiable prediction about the results of a possible
>>computation.
>There is not a single terminating algorithm to decide if a statement can
be
>tested or falsified.
Why do you say that?
>Formally, what does "falsifiable" mean?
"Falsifiable" means that it is possible to search for a counterexample.
For example, the statement "For all n and m, n^2-m^2 = (n+m)(n-m)"
is falsifiable, in that we can search for counterexamples by picking values
for n and m, and evaluating the expressions on each side of the equality,
and then verifying the equality of the two expressions. If we were to find
that equality didn't hold for some n and m, we would have a counterexample
to the general statement.
David Petry
ull...@math.okstate.edu wrote in message
<7brsuv$9dd$1...@nnrp1.dejanews.com>...
> That's proofs of the theorems of ZF involving infinite
>sets. A computer can verify these once they're formalized.
Right. The key words here are "once they're formalized".
I've already pointed out that formalisms are objects that exist
in the world observable through computation.
>The idea that reasoning about infinite sets requires some
>sort of transcendental understanding is simply wrong.
Sure, "ONCE THEY'VE BEEN FORMALIZED". (pardon
my shouting).
The belief that humans can somehow "see" what rules apply to
the world of the infinite is a transcendental notion. That's what
Brian Scott was claiming he could see.
Neil Rickert
>Neil Rickert wrote in message <7brq0i$a...@ux.cs.niu.edu>...
>>My accusation is that you are setting yourself up as God, declaring
>>that you have perfect knowledge of what parts of mathematics will be
>>useful in the future.
>Well, uh, I see things differently.
>I'm claiming that mathematics is part of mankind's quest for
>understanding. We want to understand ourselves, and we want
>to understand the world around us.
So far, I can agree.
>This quest for understanding requires us to attempt to distinguish
>fact from fiction, truth from myth, reality from fantasy. What it means
>to "understand" some aspect of the world is to have a formal model
>which gives us the power to make reliable predictions about the
>results of experiments and tests relating to that aspect of the world.
I agree with that too. But it is the role of science to do that.
Mathematics has a different role, as a tool to be used within
science.
>So indeed, formalisms play a key role in our quest for understanding.
Right.
>However, the pursuit of formalism is not identical to the pursuit of
>understanding. That is, if we use formalism to create new worlds,
>we are creating fantasies; when formalisms lose their connection
>to the observable world, they must be regard as myths, regardless
>of how clever and precise they may be.
And that is where I see your view as nonsensical. The heart of
mathematics is in mathematical technique. We invent fictitious
worlds to test this technique and to find the limitations on its
capabilities. By this means we hone our mathematical tools so as to
make them useful in real life applications.
It matters not one iota whether mathematical systems are fictions, as
long as they serve their roles for testing our mathematical
technique. If we limited our testing of mathematical technique to
real worlds, then our tests would not be sufficiently thorough and we
could not rely on the results of those tests.
>I'm claiming that the concept of observability gives us a criterion
>for distinguishing fantasy from reality in mathematics. I'm claiming
>that observability can serve as a guide for the creation of a new
>foundation for mathematics which is free from the fantasies
>inherent in Cantor's theory.
Cantor world is no more a fantasy than Euclid's points lines and
circles.
>> The parts of mathematics that you want to discard have already
>>proved their usefulness, although with your head buried in the sand
>>you are incapable of seeing this.
>That conclusion is based on your own very incomplete and inaccurate
>understanding of what I am saying.
You want to discard mathematics that has proved useful to quantum
physicists.
Brian M. Scott
> The belief that humans can somehow "see" what rules apply to
> the world of the infinite is a transcendental notion. That's what
> Brian Scott was claiming he could see.
It was not. My claim was (and is) that the rules of logic don't change
when one considers the infinite.
Brian M. Scott