Math as Religion
Timothy Golden BandTechnology.com
Mathematics is not a religion is it?
Why people adopt the religious attitude is baffling.
By challenging the fundamentals that have been layed down a type of
progress can be made that far rivals the usual progression of adding a
branch to a branch of the tree.
This is a statement on the human mind. The mathematician is most immune
to emotional argumentation, yet to claim an inadequacy at the base of
the tree hurts badly and well it should, for if the inadequacy is a
reality then all of the mathematicians who have been hoodwinked have
suffered from the social instinct of mimicry. The mathematician who
claims immunity will have to accept this lower accusation and therefor
should remain open to such possibilities.
Such possibilities have to be met with clear criticism from the
defenders of the existing tree.
The ease with which you do so is a measure of your own abilities.
If the truth is so clear cut there should be no problem arriving at it
with honest discussion.
Otherwise your math is just a religion:
The book has been written.
One must preserve the book.
-Tim
Randy Poe
Timothy Golden BandTechnology.com wrote:
> Mathematics is not a religion is it?
No.
> Why people adopt the religious attitude is baffling.
They don't. That's why every result presented in every
advanced math course isn't just given as doctrine, it's
given with a proof, a line of argument that the student
is expected to be able to read and critique.
Is your view of this "religious attitude" actually based on
taking a math course? Or are you just guessing?
- Randy
Dr. Doktur
"Timothy Golden BandTechnology.com" <tttp...@yahoo.com> wrote in message
news:1162995087....@e3g2000cwe.googlegroups.com...
>
> Mathematics is not a religion is it?
Math is a language.
Religious Studies is a religion.
Timothy Golden BandTechnology.com
This view is based on the responses from avid mathematicians here on
this newsgroup.
You will have to concede that mimicry is an inherent part of the
structure.
Every proffessor is forced to grade his students based on this level of
conformity.
That this structure could go awry has to be an accepted possibility;
otherwise why would you care to go through this rigorous process of
proof? Surely you are open to the possibility of a mistake, otherwise
your method is a vacant precept.
The most allowable room is in the realm of context and interpretation.
A tree has been built. It has plenty of redundancies in it that could
be ironed out. But beyond these the tree must be built with
fundamentals at its base. Its branchings are then consequences of these
fundamentals. When a fundamental is built high up in the tree there is
a conflict. This indicates that the tree is poorly formed. Correction
upsets the structure. The existent tree collapses and a new one forms
based on the fundamental constructions.
Magnitude is the simplest of these that I can point to. Most insist
that magnitude is not a fundamental concept and insist on building it
from a more complicated structure that is the real numbers. I argue
that instead the real numbers should be built from magnitude. In doing
so the complex numbers then fall out naturally and simply because the
generalization of sign is possible:
http://www.bandtechnology.com/PolySigned
Weak criticism is a sign of weakness in the poster's understanding.
Rejection of new ideas is normal human reaction.
This is a feature of the social mimicry that we as humans are subject
to.
This is a depth beneath which the mathematical mind needs to extend in
order to find the truth.
-Tim
Randy Poe
Timothy Golden BandTechnology.com wrote:
> Randy Poe wrote:
> > Timothy Golden BandTechnology.com wrote:
> > > Mathematics is not a religion is it?
> >
> > No.
> >
> > > Why people adopt the religious attitude is baffling.
> >
> > They don't. That's why every result presented in every
> > advanced math course isn't just given as doctrine, it's
> > given with a proof, a line of argument that the student
> > is expected to be able to read and critique.
> >
> > Is your view of this "religious attitude" actually based on
> > taking a math course? Or are you just guessing?
> >
> > - Randy
>
> This view is based on the responses from avid mathematicians here on
> this newsgroup.
You misunderstand what you read.
> You will have to concede that mimicry is an inherent part of the
> structure.
No I will not "have to concede" that. This statement is false.
The only "mimicry" is the use of a common language, not
what to think in that language.
> Every proffessor is forced to grade his students based on this level of
> conformity.
Absolutely not. The professor grades on a student's ability
to reason.
- Randy
kilian heckrodt
to be a science even, i.e. the scientific research &examination of
various religions.
Dik T. Winter
...
> Magnitude is the simplest of these that I can point to.
You have not even given a proper mathematical definition of that concept.
And that should form the basis?
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
T.H. Ray
As another writer pointed out, the only clear
cut fact of mathematics -- for whatever other properties
one might wish to assign the subject -- is that math
is a language. It coheres in common terms by which
its speakers communicate. Mathematics is no one's
private language, and -- unlike the case with religious
belief -- no theorem acquires truth status merely
because an individual believes it to be true.
Tom
Dr. Doktur
"kilian heckrodt" <kilianh...@yahoo.com> wrote in message
news:eisuse$i3u$03$1...@news.t-online.com...
unless it is a Buddhist University, etc.
Timothy Golden BandTechnology.com
I do not redefine magnitude.
My usage of this term is identical to traditional mathematics, as is my
usage of the term 'dimension'. I understand that a completed tree
structure will include these concepts as definitions but the point here
is to discuss the structure of that tree and your argument does not
preclude this. That magnitude may be defined as fundamental beneath the
real numbers is all that I ask. It's simplicity is beneath the real
numbers and has correspondence to natural phenomenon more accurate than
the real numbers. This natural correspondence includes a lack of
discrete basis. Because the real numbers have traditionally been
defined as constructed from the natural numbers I dispute this portion
of the tree. The concept of dedekind cuts embodies the definition of
magnitude. This is blatantly apparent to me. The consequences of this
reversal in the tree allows complex numbers to be defined via the same
fundamental rules that define the real numbers. Furthermore higher
dimension systems exist that obey the algebraic properties of the real
and complex numbers. Furthermore a singly signed entity exists which
has correspondence to time and the family of polysign numbers contains
inherent correspondence to spacetime. These features are strong
indicators of the gain of allotting magnitude as fundamental. I believe
that this concept involves more than just pulling the sign off of the
reals. The marriage of the natural numbers to magnitude implies
polysign numbers. These two distinct entities form an apropriate base
for the tree that will allow correspondence to nature.
In particular real numbers rely on a fixed unity value that is lacking
in natural phenomena.
The deletion of this concept may even go so far as to address the
treatment of finite and infinite concepts. The idea of a definite
versus a relative system is inherent here.
This concept may go beyond the definition of magnitude. A new term
having more meaning toward relative systems would apply. In this
context it is very easy to define natural numbers via superpostion of
these purely continuous values; a, a+a, a+a+a, ... but I might not rely
on this as a necessity. Anyhow far back above here Is the notion of
placing magnitude lower in the tree than the real numbers. This
requires that the continuum concept be granted without sign. Sign is
then developed atop this system with definite gains in the structure.
-Tim
Randy Poe
Timothy Golden BandTechnology.com wrote:
> Dik T. Winter wrote:
> > In article <1162999397.8...@i42g2000cwa.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > ...
> > > Magnitude is the simplest of these that I can point to.
> >
> > You have not even given a proper mathematical definition of that concept.
> > And that should form the basis?
> > --
> > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
>
> I do not redefine magnitude.
> My usage of this term is identical to traditional mathematics, as is my
> usage of the term 'dimension'. I understand that a completed tree
> structure will include these concepts as definitions but the point here
> is to discuss the structure of that tree and your argument does not
> preclude this. That magnitude may be defined as fundamental beneath the
> real numbers is all that I ask.
Nobody will dispute your right to do so. But "fundamental
beneath the real numbers" means you have to define it
in terms of something other than real numbers, then define
the real numbers later.
There is no fixed set of starting definitions or axioms in any
field. It is common for an author of a textbook to make his or
her own personal choice as to which things are fundamental,
then go from there. There is no One True Choice. All that
mathematics requires is rigor: that your definitions are rigorous
(a reader can use your definition and unambiguously decide
what does and doesn't fit it, without your help), and that what
you deduce is rigorous (i.e. follows logically).
I'm sorry if that upsets your worldview, but it's true. You are
completely free to stop talking about your personal choice,
and actually develop it. Recognize that it also won't be
the One True Choice.
- Randy
Timothy Golden BandTechnology.com
So Randy, would it be fair to say your belief is that mathematics is
perfect?
Languages such as Latin were once accepted as universal and even still
little bits lie around:
eg 'e.g.'
A student who studies differential geometry and rejects it and so
refuses to do any of the problems should fail the course shouldn't he?
This student is forced to mimic the teacher.
Your refusal to concede any room on the mimicry conjecture is weak
ground.
Your position evaporates by logic alone.
Languages evolve. I accept mathematics as a language and a universal
one in which there is not room for conflict. I identify a conflict with
the current linguistic structure. More than a mere interpretation this
conflict's resolution has clear consequences. There is not room in this
mathematical language for such confilct and so I struggle to
demonstrate this conflict to others. The inability to treat existing
mathematics as containing fallacy is a measure that equates mathematics
with religion:
The book has been written.
One must preserve the book.
A philosophy of improvement is irrefutable. What constitutes higher
quality may be an open problem but its pursuit is not. This philosophy
applies to the language of Math.
-Tim
Richard Tobin
Timothy Golden BandTechnology.com <tttp...@yahoo.com> wrote:
>A student who studies differential geometry and rejects it and so
>refuses to do any of the problems should fail the course shouldn't he?
>This student is forced to mimic the teacher.
If the student wants a qualification in differential geometry they
have to learn differential geometry. If they want to invent new
mathematics they can hardly expect to get qualifications in old
mathematics with it.
If a student refuses to learn Italian but insists on studying Polish,
they're not going to pass the Italian course, regardless of how good
their Polish is.
-- Richard
--
"Consideration shall be given to the need for as many as 32 characters
in some alphabets" - X3.4, 1963.
Randy Poe
No, it wouldn't. What did I say that could possibly justify that
conclusion?
> Languages such as Latin were once accepted as universal and even still
> little bits lie around:
> eg 'e.g.'
Languages such as Latin grow to accomodate new concepts.
There is a weekly Latin news broadcast on shortwave (Nuntii
Latini).
So it is with mathematics: it grows, and its language grows.
> A student who studies differential geometry and rejects it and so
> refuses to do any of the problems should fail the course shouldn't he?
Yes.
> This student is forced to mimic the teacher.
No. "Doing the problems" is not "mimicking the teacher".
> Your refusal to concede any room on the mimicry conjecture is weak
> ground.
> Your position evaporates by logic alone.
Oh? What logic would that be? Saying it's true?
>
> Languages evolve.
And so does the language of mathematics.
- Randy
Timothy Golden BandTechnology.com
"Mimicry is an important mechanism for knowledge acquisition
as is reinforcement through reward and punishment. "
-
http://pages.cpsc.ucalgary.ca/~gaines/reports/MFIT/FCS_IT/FCS_IT.pdf
"Following the evolution of a unique human ability for complex
imitation, Arbib proposes that language originated in a system of
manual gestures, and only later evolved into a primarily spoken form.
Finally, Arbib joins Hurford, Tomasello, and Davidson in arguing that
syntax emerged as the result of subsequent cultural evolution."
- http://www.ling.ed.ac.uk/~simon/0-19-924484-7.pdf
Mimicry and imitation are fundamental to the consistent interchange of
information. The consistent interchange and its variants form a
developing language. At this level language is arbitrary, but
mathematics contains a further stricture in which minimal forms are
preferred and that a strict construction of primitive forms lies
beneath every high level term.
This structuring is critical. The current mathematicsl language is
flawed. When a fundamental term is generated by a higher level
structure the structure must be inverted to correct this ambiguity. The
current structure is ambiguous and this ambiguity has left a portion of
mathematics veiled.
-Tim
Timothy Golden BandTechnology.com
The hierarchical structure is an explicit and meaningful component of
mathematics.
When the hierarchy is exposed as flawed the flaw needs to be corrected.
The ability of trained mathematicians to perceive a flaw or be in
denial of that flaw is the difference between true mathematics and
religion.
The inherent assumption of accuracy of current mathematics by modern
mathematicians is a religious precept.
This assumption is being exposed as invalid here. Is this a minor
point? It is not. The consequences of this invalid assumption may make
the difference between a natural basis of reality versus an arbitrary
basis.
The polysign numbers offer an answer and expose an ambiguity in the
current mathematical system:
http://www.bandtechnology.com/PolySigned/index.html
Magnitude is a fundamental principle whose marriage with the natural
numbers yields results.
-Tim
Stephen Montgomery-Smith
Hi Tim,
I actually think that you make a very valid point. I also have argued
these kinds of positions. I think that there is only so much depth to
the foundations of mathematics, and that ultimately its basis cannot be
proved.
I do accept mathematics as truth, but I also accept that my position is
ultimately an act of faith. In other words, I disagree with you that
mathematics is at heart mere mimicry, but I concede that I have no
foolproof argument against your assertion. The best argument that I can
come up with is that the rules of mathematics seem to be universally
accepted by all humans, even across cultures, and that a mathematical
argument has that "ring of truth" about it that sits comfortably with my
soul.
Best regards, Stephen
Nick
"Timothy Golden BandTechnology.com" <tttp...@yahoo.com> wrote in message
news:1163011682....@b28g2000cwb.googlegroups.com...
Have you deliberately dropped in the fact that you are a software engineer -
which I had already found out.
You are just trolling?
There is a story of a chap who discovered a theorem in the Maths Tripos at
Cambridge. He just attempted the one question, was given a Third and a
fellowship of the College.
I doubt that he was awarded it for being convention.
I recall once going to an Economics seminar at the LSE where the students
appeared to be sitting at the feet of the professor, but I don't ever recall
that from my maths degree. Far from it - they were all drinking in the bar.
Nick
Dik T. Winter
>
> Dik T. Winter wrote:
> > In article <1162999397.8...@i42g2000cwa.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > ...
> > > Magnitude is the simplest of these that I can point to.
> >
> > You have not even given a proper mathematical definition of that concept.
> > And that should form the basis?
>
> My usage of this term is identical to traditional mathematics, as is my
> usage of the term 'dimension'.
In standard mathematics it comes quite late in the development of arithmetic.
> I understand that a completed tree
> structure will include these concepts as definitions but the point here
> is to discuss the structure of that tree and your argument does not
> preclude this. That magnitude may be defined as fundamental beneath the
> real numbers is all that I ask.
Full magnitude is not defined without reference to the properties of
real numbers. And those properties are not there without the real
numbers themselves. As you want full magnitude, and also want to
retain the original definitions, you need real numbers before magnitude.
So if you want magnitude without reference to real numbers you need to
make a definition.
Nick
"Stephen Montgomery-Smith" <ste...@math.missouri.edu> wrote in message
news:nvu4h.1056171$084.954191@attbi_s22...
I see that this is your view of Maths or math
http://www.math.missouri.edu/~stephen/mathematics.html
You are an applied mathematician - but you can hardly expect that of a pure
mathematician.
But your argument is totally utilitarian. One can imagine telling that to an
engineer but how would mathematics attract the brightest of students if it
was just simply a faith.
I believe that in Soviet Russia many people chose to study mathematics and
other pure sciences for the reason that it allowed them to think
independently - I hardly imagine that they would have done that if they were
simply following an orthodoxy.
Nick
Nick
"Nick" <tulse0...@yahoo.co.uk> wrote in message
news:AJOdnadoRMUz5M_Y...@bt.com...
See http://www.math.wichita.edu/~pparker/personal/teresi.htm
"Mathematics is at the heart of the sciences. All of them require
mathematical formulas to express their various truths. As the saying goes,
the physicists defer only to the mathematicians, and the mathematicians
defer only to God. (Though one would be hard pressed to find a mathematician
so modest.)" (apparently Dr Leon Lederman - he won the Nobel Prize for
Physics in 1988).
Nick
Stephen Montgomery-Smith
Touche!!!
> You are an applied mathematician - but you can hardly expect that of a pure
> mathematician.
No. I am a pure mathematician.
http://www.math.missouri.edu/~stephen/preprints/. The only work of mine
which has found any practical use, as far as I know, is a screensaver
that I wrote for Linux:
http://www.math.missouri.edu/~stephen/software/#polyominoes - it found
an unexpected use in the design of phased array radars.
> But your argument is totally utilitarian. One can imagine telling that to an
> engineer but how would mathematics attract the brightest of students if it
> was just simply a faith.
No. I think that my argument is exactly the opposite to how you have
characterized it. Certainly what initially attracted me to mathematics
was its shear beauty. I think also for a long time it conveyed a sense
of inner truth to me, but since studying mathematical logic, and
thinking a lot about its true foundations, I lost this sense. At least
I lost the sense that it is THE truth.
But also, even though I hold mathematics to represent truth as an act of
faith, I still hold it to represent truth. Do not minimize my strong
feelings for a subject simply because I hold its basic tenants as an act
of faith, not as an act of certainty.
> I believe that in Soviet Russia many people chose to study mathematics and
> other pure sciences for the reason that it allowed them to think
> independently - I hardly imagine that they would have done that if they were
> simply following an orthodoxy.
Me neither.
Stephen Montgomery-Smith
> See http://www.math.wichita.edu/~pparker/personal/teresi.htm
>
> "Mathematics is at the heart of the sciences. All of them require
> mathematical formulas to express their various truths. As the saying goes,
> the physicists defer only to the mathematicians, and the mathematicians
> defer only to God. (Though one would be hard pressed to find a mathematician
> so modest.)" (apparently Dr Leon Lederman - he won the Nobel Prize for
> Physics in 1988).
Now it is statements like this that cause others to say that
mathematicians regard their work as a religion.
I think it more likely that we managed to find some apparently universal
abstract truths about the universe, and then lucked out because we could
state so many other apparently universal non-abstract truths in terms of
this "mathematics." The trend in this direction looks good right now,
but to assume that this will continue indefinitely is in my view an act
of faith (a reasonable act of faith, but an act of faith nevertheless).
I prefer the far more humble approach taken by Isaac Newton:
I seem to have been only like a boy playing on the seashore, and
diverting myself in now and then finding a smoother pebble or a prettier
shell than ordinary, whilst the great ocean of truth lay all
undiscovered before me.
... namely, that the great mathematical truths we discover are simply
exquisite gems that give us some small insight into how beautifully made
our universe is, but to think that we have captured the essence of truth
is self-delusion.
Stephen
Nick
Clearly I haven't gone as far in maths as you - I became a not very good
statistician - I was better at pure maths or I understood it better than
applied maths (my father is an engineer and whilst I appreciate the beauty
of a bridge I never had the skills to follow him).
I must admit that truth tables always flawed me - there was a recent post to
this group where someone had a question about that.
I do recall that at school a friend of mine tried to convince us that in
fact, that lightbulbs emitted darkness and that the natural state of things
was light! He went on to get a PhD in Chemistry.
Nick
Nick
> Nick wrote:
>
>> See http://www.math.wichita.edu/~pparker/personal/teresi.htm
>>
>> "Mathematics is at the heart of the sciences. All of them require
>> mathematical formulas to express their various truths. As the saying
>> goes, the physicists defer only to the mathematicians, and the
>> mathematicians defer only to God. (Though one would be hard pressed to
>> find a mathematician so modest.)" (apparently Dr Leon Lederman - he won
>> the Nobel Prize for Physics in 1988).
>
> Now it is statements like this that cause others to say that
> mathematicians regard their work as a religion.
I actually think that others might regard mathematics as a religion because
they don't understand it.
After all, if I told anyone that I studied mathematics it is assumed that I
am very clever (which, of course, I don't mind). And therefore it is endowed
with a mystery because they don't understand it.
Isaac Newton's quote that you gave would presumably reflect the fact that he
was a devout Christian.
Nick
Stephen Montgomery-Smith
> I do recall that at school a friend of mine tried to convince us that in
> fact, that lightbulbs emitted darkness and that the natural state of things
> was light! He went on to get a PhD in Chemistry.
That is a beautiful answer to my ramblings.
Best regards, Stephen
Stephen Montgomery-Smith
> "Stephen Montgomery-Smith" <ste...@math.missouri.edu> wrote in message
> news:b6v4h.273270$1i1.167872@attbi_s72...
>
>>Nick wrote:
>>
>>
>>>See http://www.math.wichita.edu/~pparker/personal/teresi.htm
>>>
>>>"Mathematics is at the heart of the sciences. All of them require
>>>mathematical formulas to express their various truths. As the saying
>>>goes, the physicists defer only to the mathematicians, and the
>>>mathematicians defer only to God. (Though one would be hard pressed to
>>>find a mathematician so modest.)" (apparently Dr Leon Lederman - he won
>>>the Nobel Prize for Physics in 1988).
>>
>>Now it is statements like this that cause others to say that
>>mathematicians regard their work as a religion.
>
>
> I actually think that others might regard mathematics as a religion because
> they don't understand it.
Yes, but I don't think that this was the point Timothy was trying to
make. Rather, I sensed that he was trying to say that we all engage in
mimicking one another in the pursuit of mathematics, but how do we know
whether it represents some kind of fundamental truth, and is not merely
some social custom that we have acquired? While I am sure that it is
not merely a social custom, I don't think that I can dispute in a
foolproof manner, nor do I think that his question should be regarded as
foolish.
> After all, if I told anyone that I studied mathematics it is assumed that I
> am very clever (which, of course, I don't mind). And therefore it is endowed
> with a mystery because they don't understand it.
>
> Isaac Newton's quote that you gave would presumably reflect the fact that he
> was a devout Christian.
I have heard that he spent a great deal of time studying theology, but
that in many ways he was not orthodox (with a small 'o') in his beliefs.
Stephen
ste...@nomail.com
> No. I think that my argument is exactly the opposite to how you have
> characterized it. Certainly what initially attracted me to mathematics
> was its shear beauty. I think also for a long time it conveyed a sense
> of inner truth to me, but since studying mathematical logic, and
> thinking a lot about its true foundations, I lost this sense. At least
> I lost the sense that it is THE truth.
Who has said that mathematics is THE truth? The "mainstream"
folks disagreeing with Timothy Golden have made no claims
of truth. Timothy Golden is the one who seems to think that
it is THE truth that magnitude is more fundamental than the reals.
If he wishes to define magnitude rigourously and then define
the reals based on that, he is free to.
Stephen
Stephen Montgomery-Smith
Fair enough. But my remarks were in reply to Nick, and were not related
to Tim's claims of magnitude being a more fundamental concept than the
reals. I now realize that Tim's initial claims (which as you know I
support) were meant to be a preamble to something quite different.
Stephen
T.H. Ray
Mathematics has no priesthood. That you have come up
with an idea that has not found wide acceptance, does
not logically imply that some force of authority is
conspiring to withold recognition.
I suggest you look for flaws in your argument or in your
method of presentation. That's what the referee
process is for.
Tom
T.H. Ray
Indeed, his "ocean of truth" remark relates more closely
to Newton's devotion to alchemy, than Christianity. The
believing Christian, after all, has faith that truth
lives within his or her personal belief. For a scientist,
truth is a quest for external validation; in the
alchemist's manifesto, "as above, so below."
Tom
Timothy Golden BandTechnology.com
I've read your discussion. The admission of a mathematician that the
present structure may be flawed would relieve the accusation of that
mathematician being religious in his mathematical beliefs. A physicist
would have far less trouble with this notion but the principles of
rigour in mathematics make this point more challenging for this group.
This is after all what they have spent all of their time on. The
lengths that mathematicians go to in order to accept a construction are
extraordinary so to be open to an accusation of error is difficult.
If an option or a discrepancy were exposed in a new mathematical
construction that might become a branch within the structure, but
certainly any conflict is not allowed in this language. For instance a
branch might be formed between discrete mathematics and continuous
mathematics. Now discrepancies can be resolved as one of the two and if
there be a third set which is a bit of both it will be mapped out so
that no conflicts arise. Also if a very complicated and roundabout
structure existed to describe something that could have been described
very simply the simplistic approach will be given preferential
treatment.
This revisionary process even goes on in religion, but to admit it is
another thing. Certainly math allows for building out the structure
whereas religion typically does not. The fervent belief that the work
is correct is the part that I am drumming on. Is the mathematical mind
willing to admit that a mistake may have been trained into their
knowledge? Particularly such a large clan of similarly believing people
in their collective and consistent beliefs will have formed a religion
where any new belief that conflicts will be a minority position might
be snuffed out.
In the physics community an inversion is going on. String theorists
have taken over the academic community according to some accounts. To
posit string theory as invalid introduces quite a conflict for this
community. Protectionism and a whole array of human problems are just
around the corner. The ego is tied up in all of this somewhere very
deep.
The polysign construction is a primitive thing. It operates down low on
the tree. This part of the tree is quite old yet a new shoot is forming
there. This shoot is even below the real numbers on the tree. It wil
have a branch that will replicate the real number branch of the tree.
Right next to that is another branch that will replicate the complex
numbers. Then there are even other branches that though simply defined
aren't fully understood. The paradoxically simple P1 that has time
correspondence will probably never be built any other way.
This little shoot exists because a portion of mathematics has been
overlooked for a long time. It has been overlooked because of a
fixation on the real numbers. That sign can be generalized to a natural
valued component is a grudge to the existing definitions of the reals.
These two constructions are in conflict. Is this type of conflict
permitted in the realm of mathematics?
-Tim
Timothy Golden BandTechnology.com
I understand what you are saying Dik and this is why the question
arises of mathematicains being capable of addressing such a conflict.
Your side of the argument is well established though the intuitive
simplicity of magnitude is immediately felt.
Now please allow me to present my side.
The fundamental law which I have applied is
Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is
magnitude)
This law for n = 2 (the reals) simply states that
- x + x = 0 .
We cannot dispute this as true for both systems.
The portions of the constuction to which this law applies do not
contain this law. This is merely the logic of construction. So the
constituent parts
s , x
do not contain this law. They are components to which this law is
applied.
This leaves x as a raw magnitudinal value and s a raw natural
value(sign), where the notion of magnitude is devoid of sign. Here the
concepts of sign and magnitude are distinct and married to form
polysign numbers.
Since you claim that magnitude inherently contains a notion of sign I
simply refute this as a failing of the traditional mathematical
construction. And so I acccuse you of being a mathematical religionist.
The strength of my position is furthered because the families of the
polysign system have strength and inherent properties from very few
rules. Particularly that complex numbers are three-signed numbers is
strong evidence of validity. That traditional mathematics is flawed is
the only way that this discovery could have gone veiled for so long.
Diks fixation on the real numbers is a handicap that traditional
mathematics has trained into all of us by mimicry.
-Tim
Stephen Montgomery-Smith
I think that you are being unfair both to mathematicians, and to
religion. First let me describe my personal experience of
mathematicians and scientists. I have found that if you present them
with new ideas, even ideas that go against the orthodoxy, and if it
comes clear to them that these ideas are interesting or useful, that at
least a decent proportion of them will quickly pick up your ideas.
Contrary to many people's perceptions of them, they are remarkably open
minded.
To some extent this is also true in religion. For example, the ideas of
the reformation quickly spread, and these ideas really involved trying
to think clearly about what their religion really meant. Another
example, if you read the works of Augustine, it comes clear that he is
trying to work it out for himself, and not merely wishing to follow the
orthodoxy of his time.
> In the physics community an inversion is going on. String theorists
> have taken over the academic community according to some accounts. To
> posit string theory as invalid introduces quite a conflict for this
> community. Protectionism and a whole array of human problems are just
> around the corner. The ego is tied up in all of this somewhere very
> deep.
While protectionism and protection of egos does play its role in the
development of science, these forces can only hold back progress so
much. Let me cite your example of string theory. Quantum physics as a
discipline is going through a rut right now, and so is on the wane. It
represents a very small segment of academia these days. Its golden
period of the early and mid twentieth century is coming to an end. I am
not saying that someone might be able to reinvigorate it one day, but my
sense is that all the hoopla about string theory is media hype. The
real action right now is in other areas like life science. They get by
far most of the funding these days, and life science is definitely in
its golden era.
> The polysign construction is a primitive thing. It operates down low on
> the tree. This part of the tree is quite old yet a new shoot is forming
> there. This shoot is even below the real numbers on the tree. It wil
> have a branch that will replicate the real number branch of the tree.
> Right next to that is another branch that will replicate the complex
> numbers. Then there are even other branches that though simply defined
> aren't fully understood. The paradoxically simple P1 that has time
> correspondence will probably never be built any other way.
>
> This little shoot exists because a portion of mathematics has been
> overlooked for a long time. It has been overlooked because of a
> fixation on the real numbers. That sign can be generalized to a natural
> valued component is a grudge to the existing definitions of the reals.
> These two constructions are in conflict. Is this type of conflict
> permitted in the realm of mathematics?
I looked over your polysigned numbers. You should look into the
possibility that the reason it is not finding wide acceptance is not
because of an inherent religiosity in math, but perhaps because people
are not finding your ideas interesting or useful.
Certainly, I don't think that polysigned numbers help my issue, which is
what are the proper foundations of mathematics.
Best regards, Stephen
Timothy Golden BandTechnology.com
That's fine criticism. Mathematics is a big place. I'm mostly concerned
with nature and physics and want nature's foundation; a basis. Still,
if this polysign construction is in conflict with traditional
mathematics then that also has to be adressed. There are many
references to the relationship of the mathematician and the physicist
and the way that they use the tools, physics being a more loose
pursuit. But here is a tight math that gets a physical basis from very
little. Also it gets much mathematically too. But inevitably either
community perceives the mathematical component of magnitude as "R+".
This is very wrong from the context of the polysign numbers. They build
R and so are not built from R. This contradiction has to go away and so
I suppose that I am left writing a manual that burrows down all the way
to the Dedekind level where I suppose a magnitudinal product must
exist. Yet the magnitude that we all know is the magnitude that I speak
of. The problem is not mine so much as it is the persistent effect of
the traditionally trained mind to go over to the real numbers
immediately.
In some regards I see all treatment of sign as a farce, yet it produces
results that are pertinent.In your pursuit of a mathematical foundation
perhaps you will come across a motive for sign. The product sits
nearby.
-Tim
Sean Holman
> if this polysign construction is in conflict with
> traditional
> mathematics then that also has to be adressed. There
I think what people are trying to tell you is that your polysigned numbers are not in conflict with traditional mathematics.
Timothy Golden BandTechnology.com
Well that's encouraging. They are not far off from the existing but as
soon as I claim that they build the real numbers from magnitude a
breakpoint is reached at which many turn away. Another common
insistence is that the sign rays are actually representable with real
lines.
The way that most mathematicians reflect the construction is to use R+
for the sign components. The real numbers are deeply embedded in the
common thinking and it just isn't right for this construction to be
conceived on top of the reals since the polysign construction develops
sign and in doing so develops the reals at n=2.
Thanks for weighing in. I hope you are right.
-Tim
Dik T. Winter
> Dik T. Winter wrote:
> > In article <1163004891.0...@f16g2000cwb.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > > I do not redefine magnitude.
> > Full magnitude is not defined without reference to the properties of
> > real numbers. And those properties are not there without the real
> > numbers themselves. As you want full magnitude, and also want to
> > retain the original definitions, you need real numbers before magnitude.
> > So if you want magnitude without reference to real numbers you need to
> > make a definition.
> I understand what you are saying Dik and this is why the question
> arises of mathematicains being capable of addressing such a conflict.
Apparently not. If A is normally defined based on B you want A with
the normal definition while maintaining that B should be based on A.
A being magnitude, B being the reals.
> Your side of the argument is well established though the intuitive
> simplicity of magnitude is immediately felt.
I do not feel it.
> Now please allow me to present my side.
> The fundamental law which I have applied is
> Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is
> magnitude)
Perhaps. But you do not define magnitude. And your statement makes
no sense at all.
> This law for n = 2 (the reals) simply states that
> - x + x = 0 .
> We cannot dispute this as true for both systems.
What I dispute is that your statement simply states that at all.
For n = 2 your statement says:
sum for s = 1 to 2 (s x) = 0
which translates to
1 x + 2 x = 0
which is obviously false. You have to properly state things (and
properly define things.)
> Since you claim that magnitude inherently contains a notion of sign I
> simply refute this as a failing of the traditional mathematical
> construction. And so I acccuse you of being a mathematical religionist.
And I never have claimed that.
Gene Ward Smith
ste...@nomail.com wrote:
> Who has said that mathematics is THE truth? The "mainstream"
> folks disagreeing with Timothy Golden have made no claims
> of truth. Timothy Golden is the one who seems to think that
> it is THE truth that magnitude is more fundamental than the reals.
> If he wishes to define magnitude rigourously and then define
> the reals based on that, he is free to.
As has been pointed out on numerous times, this is in fact an old idea,
going back to the Greeks. Landau, for a modern example, develops
positive reals from second order arithmetic of
positive integers, and goes on from there. This has some advantages, in
particular that the positive rationals, as the ratios of positive
integers, may be constructed without worrying about division by zero,
and then the positive reals (or magnitude) can be constructed next.
One can use polysigned numbers, if one so chooses, for constructions.
But Tim seems unable to say why we should.
Timothy Golden BandTechnology.com
I also have to accuse you of insincerity.
Of the handful of people who have demonstrated understanding of the
polysign construction Dik ranks highly and is perhaps above myself in
his understanding especially in relation to existing mathematics. In
particular the division problem has been one that I have attempted to
work out with poor results and which he has excelled at.
But the cheap revisions that he poses here are weak and I have seen
this behavior before. For instance in my original communication there
is an explicit declaration that s is representing sign. Such
representation is a new concept and so deserves room. This type of
flexibility is what the polysign construction requires of
mathematicians. Dogma must make way for a new paradigm. The way to
understanding polysign is to dismiss old usage of sign and recover the
old usage from a general form. This general form is that of the natural
numbers. The elemental
s x
representation does need to be explicitly introduced to allow
adaptation to the new system, but let's not forget that as children we
were taught the real numbers without higher level mathematics. The same
can be done for polysign numbers. I prefer this approach since it does
not require much knowledge and so is more accessible to a less educated
person. Still, I have started to sprinkle the sx representation into my
web pages. None the less, here is a perfectly capable mathematician who
is incapable of accepting a new construction. Even after the
construction has been clearly communicated the old ways are still the
default ways and so such a primitive and simple system as polysign
numbers proves challenging. Generalization of sign as a reality is
beyond the human mind due to the ingrained real number perspective.
Let's not forget the literal word 'real' here as well. This
construction diminishes the real numbers and favors magnitude as
fundamental.
-Tim
Timothy Golden BandTechnology.com
Gene, like Dik, represents the establishment.
This is Gene on a prior thread:
"I've pointed out several times that you do not have such a
construction. I'll repeat it: you have NOT constructed the reals. This
is because your definition requires that the reals have already been
constructed. "
- http://groups.google.com/group/sci.math/msg/a340f8254714b780
The important distinction that allows this conflict is in how we
dissect the number system.
Because the polysign construction imposes the identity law
Sum for s = 1 to n ( s x ) = 0
we need not include this piece of information in the constituent
components of this law. The law builds sign and so to claim that sign
is a component of its constituents is inconsistent. This is a new
dissection and inverts a small piece of branching of the old tree. If
we adopt the new way then the old way (Gene's context) is suspect.
Though the incompatibility is minor the stricture of mathematics does
not allow even the most minor conflict. For n=2 the polysign numbers
are the real numbers and identity law above expands out to
- x + x = 0 .
This is recognizable as accurate to even a grade school child, but this
law is not a standard part of the definition of the real numbers. Yet
it is this form which allows the generalization of sign. Since this
information has been stated at this point there is no need to repeat it
anywhere else. This information principle is what makes a tight
construction. Certainly what is left of the continuous portion x is
merely a magnitude.
The benefit of this approach is that the complex numbers are barely
different than the real numbers. The reals are P2 and the complex
numbers are P3. Simply changing n by 1 gets the complex numbers from
the same laws that define the real numbers. This statement alone is
enough reason to answer why any mathematician should take interest in
the polysign construction. This is a primitive and productive
construction that poses and answers many questions:
Are the field criteria accurate?
Must a linear system obey the magnitudinal law
| A B | = | A | | B | ?
Does time correspond to P1?
Do improper transformations model electron spin?
Do n-poles exist?
Why spacetime?
That magnitude is fundamental and can be married to sign is the
foundation which allows the polysign concept to thrive. Previously in
debating this I have provided the gorilla conjecture, which poses that
since we can teach a gorilla the principle of magnitude that magnitude
is fundamental. Here we may enter into a psychological examination of
mathematical learning. Is it considered sufficient by the teaching
mathematician that a student be capable of repeatable error free
results to demonstrate understanding? Under this criteria the magnitude
is a primitive feature and the reals a sincere failing point. How many
sign errors have been generated by the human race? Yet how many
children unschooled in mathematics at all can pick out the larger of
two objects? This is how brutally simple magnitude is. The uneducated
mind is capable of percieving it, yet the highly educated mathematician
refutes the principle.
And so a farce is made of mathematicians. I do not respect religion and
I do not respect mathematicians who practice their subject as one.
-Tim
T.H. Ray
> Gene, like Dik, represents the establishment.
> This is Gene on a prior thread:
>
> "I've pointed out several times that you do not
> not have such a
> construction. I'll repeat it: you have NOT
> constructed the reals. This
> is because your definition requires that the reals
> have already been
> constructed. "
>
> -
> -
> -
> http://groups.google.com/group/sci.math/msg/a340f82547
No one has accused me of representing the establishment.
Nevertheless, the establishment as it represents itself
(as I pointed out, mathematics has no priesthood)scores
solid logical arguments which I, an outsider, can judge
to be coherent.
1. Theorems are supported by definitions. Weak
definitions cannot lead to comprehensible theorems.
2. You pose identities between empirical phenomena and
mathematical method without any reason for why this
should be so, other than that it is apparent to you.
(Isn't that what religionists claim for their side?)
Looking at your list of answered physical questions above,
one can easily counter: what field criteria are you
talking about? What do you mean by time? And spacetime
-- do you mean the Minkowski spacetime, and if so, how
do you reconcile this continuous phenomenon with your
idea of discrete magnitude?
I'm all for exiling mythology from mathematics. I am much
opposed, however, to replacing old myths with new ones.
Tom
Timothy Golden BandTechnology.com
>
> No one has accused me of representing the establishment.
> Nevertheless, the establishment as it represents itself
> (as I pointed out, mathematics has no priesthood)scores
> solid logical arguments which I, an outsider, can judge
> to be coherent.
>
> 1. Theorems are supported by definitions. Weak
> definitions cannot lead to comprehensible theorems.
>
> 2. You pose identities between empirical phenomena and
> mathematical method without any reason for why this
> should be so, other than that it is apparent to you.
> (Isn't that what religionists claim for their side?)
> Looking at your list of answered physical questions above,
> one can easily counter: what field criteria are you
> talking about? What do you mean by time? And spacetime
> -- do you mean the Minkowski spacetime, and if so, how
> do you reconcile this continuous phenomenon with your
> idea of discrete magnitude?
>
> I'm all for exiling mythology from mathematics. I am much
> opposed, however, to replacing old myths with new ones.
>
> Tom
This is an interesting perspective. I'll try to go down the paths you
propose. First, I believe that mathematics should be open to any
construction and that the measure of that construction's value is
directly related to its consequences. Every problem can be approached
as an open problem, even one that has been rigorously demonstrated in
the past.
Let's take the perspective of a thoughtful young girl in 6th grade. She
ponders time and was shown in science class how time works and how that
relates to her math. She considers that while time goes in one
direction that somehow it fits on the real line as the past and the
future yet we are always at the present on this line. But the one
directional nature of time causes her to consider that instead of a
real line with two directions that time might have a starting point and
go in one direction only. This clicked in her mind with a Nova
television show that she had watched a few months before where they
discussed the big bang.
So Jen adopted a one-signed approach to time and becomes fixated on
comparing this with the two-signed numbers that she was taught in
class. This caused her to consider that perhaps a third sign could
exist and that a set of three-signed numbers might follow the
two-signed numbers.
In eighth grade Jen was learning trigonometry in two dimensions and has
been introduced to the notion of 3D space as the space about her and
understands how it is constructed via Cartesian coordinates, and that
all of this gets put together with time to become spacetime, which she
is comfortable with from all of the Nova shows that she has watched
since she was a little girl. Her notion of time, the real line, and the
possibility of these three-signed numbers fitting in together to form
spacetime became prominent in her mind.
In eleventh grade having mastered algebra she had a very important
realization about her supposed three-signed numbers. If they are going
to work correctly they have to be symmetrical and yet fit in a
progression from the one-signed to the two-signed to the three-signed,
and possibly even onward. And it was immediately apparent that her
third sign ought to be '*' because it has three lines through it and so
will be next in a progression from '-' (one) to '+' (two) to '*'
(three) and possibly even '#' (four).
Some time passed. She developed a habit of having paper and pencil
handy and any time a thought passed through her mind of any structure
at all she would jot it down. She became quite good at this and
realized that she was training and improving her mind. Many ideas were
growing and the three sign problem danced in and out but would not
come out to make sense. She saw that the signs should mimic the real
numbers and that they would jump from one sign to the next, or preserve
sign and she settled into a pattern that would work at any sign level.
Minus would jump one sign, plus would jump two signs, and star would
jump three signs and so on. This is what the real numbers do and so she
realized the she had developed a general sign product. She knew it was
right. How else could it be? She had tried everything else twice over.
Still something was missing. The struggle that posessed her was tiring.
One summer day she was relaxing at the beach drawing her sign product
table out in the sand and considering symmetry. It struck her all at
once that the symmetry of the reals does not lie in the product, it
lies in the sum:
- x + x = 0 !!!
The extensibility was obvious and the choice was really not a choice at
all. The three-sign system must obey
- x + x * x = 0 .
And so the path was laid and she realized that she could have gotten
here back in sixth grade because it is so simple. Yet the human mind
does not work simply and her struggle was proof. Many realizations
followed but at this point the system was established and needless ot
say her struggle goes on to this day.
This is a naive approach and works very nicely, but this story is a
fabrication. I have found the question posed by others in the past but
I seem to be the first who has persisted to this point. There is a huge
amount of work to do and this is an entire and fundamental branch of
mathematics. How it went for so long without being found and whether
other fundamental constructions exist that have been overlooked are
mysterious questions. I know that this type of find is rare but it's
simplicity is such that a sixth grader with enough persistence and a
simple-minded approach could independently find this system. It's
foreign nature is merely an artifact of the educational system which
has wired us for two-signed math.
I'll try to answer your other questions later on today.
-Tim
T.H. Ray
Ray:
Sure. Any who wish to mold mathematics to suit their
philosophy do the same. From Intuitionists (Brouwer, et
al) to Constructivists (E. Bishop, et al) to Formalists
(Russell, Frege, Hilbert, et al)to Platonists (Godel,
Penrose, et al). After all, mathematics would hardly
be a meaningful language if one were unable to tie a
result to an example, and an example to a problem.
There are also, though, versions of mathematical
realism that the working researcher pursues as a matter
of principle, independent of philosophy, to support
results that would otherwise rest on philosophy alone.
Those principles are, I think, best reflected in the
work of analysts like Weierstrass, Cauchy, Dedekind,
H. Weyl, who -- in relentless pursuit of the meaning
of continuous functions (rather than imposing some
philosophical meaning on the results)-- reclaimed
arithmetic from analysis. What do you find inadequate
in these analytical constructions, in the Cauchy Integral
or Dedekind Cuts, e.g., so that your proposed philosophy
is a better alternative to comprehending and teaching
the foundations of mathematics, or the origin of numbers?
There are sound reasons why many, perhaps most,
mathematicians consider an education in mathematics to
begin with the calculus.
Golden:
Ray:
It is just such philosophical convolutions on the subject
of time that led me in a recent conference paper
to try and fix a more rigorous definition of time,
consistent with empirical knowledge while not excluding
the metaphysical realism which logically entails in
talking about the subject of time in mathematical
language. The conference was themed on complex systems
(ICCS 2006) and you can locate the paper at necsi.org or
in the online journal InterJournal, if you are interested.
Point is, our personal experience of time is not
necessarily consistent with the meaning of time.
Tom
Lester Zick
As I just picked up this thread I'm commenting variously. - LZ.
On Fri, 10 Nov 2006 13:34:14 EST, "T.H. Ray" <thra...@aol.com> wrote:
>T.H. Ray wrote:
>>
>> No one has accused me of representing the establishment.
>> Nevertheless, the establishment as it represents itself
>> (as I pointed out, mathematics has no priesthood)scores
>> solid logical arguments which I, an outsider, can judge
>> to be coherent.
To the extent mathematics does not argue truth in universal terms it
is faith based and practitioners certainly constitute a priesthood.
>> 1. Theorems are supported by definitions. Weak
>> definitions cannot lead to comprehensible theorems.
But the only relevant issue with respect to "Math as Religion" is
whether it is true in universal terms. There are many religious
arguments which are not inconsistent with religious assumptions. Which
is exactly the same rationale used in mathematical proof except the
assumptions are different and some more likely true.
>> 2. You pose identities between empirical phenomena and
>> mathematical method without any reason for why this
>> should be so, other than that it is apparent to you.
>> (Isn't that what religionists claim for their side?)
Well if mathematical assumptions turn out to be true in universal
terms one would certainly expect them to be in conformance with
empirical phenomena.
>> Looking at your list of answered physical questions above,
>> one can easily counter: what field criteria are you
>> talking about? What do you mean by time? And spacetime
>> -- do you mean the Minkowski spacetime, and if so, how
>> do you reconcile this continuous phenomenon with your
>> idea of discrete magnitude?
The idea that self contradictions of the form "A and not A" can be
true is all the rationale need for the idea of time in logical terms.
In physical terms time is defined by frequency.
>> I'm all for exiling mythology from mathematics. I am much
>> opposed, however, to replacing old myths with new ones.
Yea. Finally someone gets it.
>> Tom
>
>This is an interesting perspective. I'll try to go down the paths you
>propose. First, I believe that mathematics should be open to any
>construction and that the measure of that construction's value is
>directly related to its consequences. Every problem can be approached
>as an open problem, even one that has been rigorously demonstrated in
>the past.
The truth of mathematics is not measured by its consequences. That's a
purely empirical rationale.
>Ray:
>
>Sure. Any who wish to mold mathematics to suit their
>philosophy do the same. From Intuitionists (Brouwer, et
>al) to Constructivists (E. Bishop, et al) to Formalists
>(Russell, Frege, Hilbert, et al)to Platonists (Godel,
>Penrose, et al). After all, mathematics would hardly
>be a meaningful language if one were unable to tie a
>result to an example, and an example to a problem.
>
>There are also, though, versions of mathematical
>realism that the working researcher pursues as a matter
>of principle, independent of philosophy, to support
>results that would otherwise rest on philosophy alone.
>Those principles are, I think, best reflected in the
>work of analysts like Weierstrass, Cauchy, Dedekind,
>H. Weyl, who -- in relentless pursuit of the meaning
>of continuous functions (rather than imposing some
>philosophical meaning on the results)-- reclaimed
>arithmetic from analysis. What do you find inadequate
>in these analytical constructions, in the Cauchy Integral
>or Dedekind Cuts, e.g., so that your proposed philosophy
>is a better alternative to comprehending and teaching
>the foundations of mathematics, or the origin of numbers?
There is no demonstrable argument of truth in universal terms.
>There are sound reasons why many, perhaps most,
>mathematicians consider an education in mathematics to
>begin with the calculus.
>
>Golden:
The following represents a socratic dialectical story telling approach
by ambiguous analogy.
And you're just playing the counter intuitive card used pretty
consistently throughout twentieth century science and math to justify
various concepts at odds with personal experience.
>Tom
>
>>
>> I'll try to answer your other questions later on
>> today.
>>
>> -Tim
>>
~v~~
Timothy Golden BandTechnology.com
> 1. Theorems are supported by definitions. Weak
> definitions cannot lead to comprehensible theorems.
>
> 2. You pose identities between empirical phenomena and
> mathematical method without any reason for why this
> should be so, other than that it is apparent to you.
> (Isn't that what religionists claim for their side?)
I am not trying to refute the principle of axioms.
If there is competition amongst equivalent axiom sets then the simpler
of them should win out so long as the results are equivalent. But
especially if the results of one set are superior to the other and the
axioms are simpler then there is strong favoritism.
I believe that mathematics has to be open to new constructions.
In this case this new construction conflicts slightly with existing
mathematics.
Because the new construction has merit the conflict is worth
addressing.
The ability of the mathematics community to accept the construction
requires addressing even the slightest conflict since this is what
mathematicians spend their time doing and how the current system comes
to be guarded with propensity. The ability of the mathematicians to
accept a flaw in their existent system is why this thread is titled as
it is. Should the supposed flaw be exposed and accepted a small
branching in the existing tree will have to be inverted. As a
consequence the polysign numbers can flourish. If the flaw is denied
then the polysign construction is in conflict and will not become
accepted. I think most see that the flaw is very minor yet this is not
a suitable argument to a mathematician.
Ultimately I would like to see magnitude accepted as axiomatic because
it is accessible to animals devoid of any mathematical education.
Magnitude is inherent. The natural numbers fall out very easily from
here without the unity problem that exists in existing Cartesian
representations. This again is a very minor point (the unity problem)
but these minor points add up to quite a large shift. To expect an
immediate change is not realistic so this problem is theoretical.
Needless to say I am not joking about this and as egotistical as it
sounds to go here I believe that I have done a good job of wiping that
ego clean off of the plate. So I hope that you and others like you can
consider this problem without taking it too personally. It's just a
theoretical problem.
This is a currently opening problem that deserves serious attention.
The simplifications that are possible by granting magnitude as
fundamental are considerable.
> Looking at your list of answered physical questions above,
> one can easily counter: what field criteria are you
> talking about?
The field problem lies in division with P4 and up. For instance in P4
( - 3 * 3 )( * 1 # 1 )
= # 3 - 3 + 3 * 3
= 0 .
Here two nonzero values are producing zero via product. There are many
such instances and so the division criteria of the field definition
cannot be met. Yet these strange zero products are a part of a
continuous and well behaved system that obeys the associative,
distributive, and commutative laws just as the real numbers do. The
polysign system begs that the field criteria be given more room in its
exclusionary range of the division operator. For points off of the
identity axis and off of the plane normal to the identity axis division
can be had. Since P4 is three dimensional a graphical representation
aids the perception:
http://www.bandtechnology.com/PolySigned/Deformation/AxisDualDeformStudy.gif
This is a product survey of P4. The product is to the left and the
source arguments are the unit sphere on the right multiplied by the
red dot on the right. The red dot travels the surface of the sphere.
See where the product forms a line? The sphere has been squashed int a
one dimensional object. Therefor the reversal back to a 3D sphere via
division cannot happen. Likewise there is another position where the
sphere is squashed into a plane and likewise the dimensionality has
been reduced such that the informational reversal cannot happen. Yet
these are merely portions of a well behaved continuous process. Other
than these two subsets of points division can be had in P4. This makes
me question whether the field laws are perfect, particularly the
division criteria. Dik Winter introduced me to the term "zero divisor"
and has given me some education on this along with some other
contributors. There are some consequences lent from topology and
associative algebra theorems that have already been established. I
don't fully understand these things. What I do understand is that
polysign numbers nearly match the existing field criteria in any
dimension and that could be called an astounding realization to someone
who this is all foreign to. These are not discrete constructions. These
are pure and continuous spaces in any dimension, all perfectly well
behaved algebraically just as the real numbers and complex numbers are.
There is the division caveat here but it has no effect on the product
and sum operations.
> What do you mean by time?
P1 has perfect correspondence to time.
P1 suffers a paradox under the identity law:
- x = 0 .
Though arithmetic can be performed the graphical results are zero
dimensional.
Hence I assert that time is zero dimensional. This coincides with the
notion of the present and the lack of freedom to travel either backward
or forward in time. So you see the cute little story about how the girl
looks at time takes on this strange twist when she applies what she has
learned up in high sign space. Yet the paradoxical behavior of P1
exactly matches the paradoxical behavior of time. This topic for its
brutal simplicity may not resolved by consensus for some time. It is
beyond most people to accept P1.
> And spacetime
> -- do you mean the Minkowski spacetime, and if so, how
> do you reconcile this continuous phenomenon with your
> idea of discrete magnitude?
No. Not at all the usual spacetime. The polysign spacetime includes
this notion of unidirectional zero dimensional time. The spacetime
model materializes when we look at the family of polysigned numbers in
its entirety. As you have already seen P4 and up take on some strange
behaviors. There is a natural breakpoint beyond P3. The simplest
observation is that
| A B | = | A | | B |
is broken for P4+ but is obeyed for P3- .
Distance under product takes on different character for the higher
signs.
P1, P2, and P3 form a sufficient representation of spacetime.
The natural representation of a coordinate under this topology is
a11
a21 a22
a31 a32 a33
...
where all of these a's are magnitudes and their position indicates
their sign and sign level.
So for instance a31 is the minus component of the P3 component.
I call this a 'tatrix' for triangular matrix. Especially because the
components are raw magnitudes the nomenclature is appropriate.
This is vastly different from a Minkowski representation.
This topology is a
0D + 1D + 2D ...
model. You can probably see that it could lend itself to
electromagnetic and sub-brane types of analysis. This sort of topolgy
is already being applied to gravity though without the 0D level. If the
spacetime model is correct then the arithmetic product lays at the
heart of physics, for this is the operation that begets spacetime.
Product operations are already a well formed part of classical physics
and I see some glimmerings of hope that electron spin might be resolved
as improper transformation such as the P2 arithmetic product. This is
quite abstract but a strong lead I think.
-Tim
Gene Ward Smith
Timothy Golden BandTechnology.com wrote:
> Gene Ward Smith wrote:
> Gene, like Dik, represents the establishment.
Gene, like Dik, knows mathematics and has helped you in your polysign
quest.
> This is Gene on a prior thread:
>
> "I've pointed out several times that you do not have such a
> construction. I'll repeat it: you have NOT constructed the reals. This
> is because your definition requires that the reals have already been
> constructed. "
That was clearly true at the time. Have you subsequently done the work
of constructing the real numbers, and if so, can you give a url by way
of a citation?
> This is a new
> dissection and inverts a small piece of branching of the old tree. If
> we adopt the new way then the old way (Gene's context) is suspect.
This represents a complete misunderstanding of the nature of the
problem. Various constructions are not in conflict.
> Though the incompatibility is minor the stricture of mathematics does
> not allow even the most minor conflict.
There isn't one.
> For n=2 the polysign numbers
> are the real numbers and identity law above expands out to
> - x + x = 0 .
> This is recognizable as accurate to even a grade school child, but this
> law is not a standard part of the definition of the real numbers.
That depends entirely on how you define them. However, if eg you say in
your definition that the real numbers form an abelian group under
addition, then this would be included.
> The benefit of this approach is that the complex numbers are barely
> different than the real numbers. The reals are P2 and the complex
> numbers are P3. Simply changing n by 1 gets the complex numbers from
> the same laws that define the real numbers. This statement alone is
> enough reason to answer why any mathematician should take interest in
> the polysign construction.
It makes it mildly interesting, but why more than that? There seems to
be nothing fundamental involved.
> This is a primitive and productive
> construction that poses and answers many questions:
>
> Are the field criteria accurate?
This is meaningless.
> Must a linear system obey the magnitudinal law
> | A B | = | A | | B | ?
If that is how you define "linear system", yes. Normally, this is taken
as a part of the definition of an absolute magnitude on a field, in
particular the reals, the complex numbers, and the p-adic numbers and
their extensions.
> Does time correspond to P1?
This is not a mathematical question and I doubt it is a physics
question either.
> And so a farce is made of mathematicians. I do not respect religion and
> I do not respect mathematicians who practice their subject as one.
Yet it is you who are going the JSH route, becoming manifestly
irrational and if you will "religious". Mathematicians don't really
need your respect, but it is clear you want the respect of
mathematicians. You need to do the required work to get it, and also
need to learn to limit your claims to what you've actually done. Then
again, Spencer Brown didn't do this so I suppose there is hope for you
if you can persuade someone you are a genius.
Timothy Golden BandTechnology.com
This is quite a question. I have some of Weyl's books and enjoy reading
his prose. I am not sharp on Dedekind and Cauchy; I took a real
analysis course once (that's when I first got the idea for three-signed
numbers) but I'm not a master of existing mathematics. Do you see the
laws of magnitude embedded in the Dedekind cut? If magnitude is taken
as fundamental these principles should simply be embedded in magnitude.
Whether we can take magnitude as purely axiomatic I am not sure. I just
see that it is very primitive. The gorilla conjecture only works to
twist someones arm into admitting this primitive nature. I'd like
magnitude to be axiomatic; but how many axioms?
Mathematicians will insist that the real line has a distinct value
unity. Yet when we go to spatial representations this unity is not
apparent. Instead it is an arbitrary choice where we put unity. Under a
magnitude definition there may be a way out of this so that we call the
element 'u' unity. An element 'a' can just as easily generate the
natural numbers as the element u; a + a = 2a; u + u = 2u . The
symbology is identical. So the natural numbers can fall out without
ever even obeying the stricture of a fixed unity. Now you can have two
apples and two oranges and they are equivalent on a symbolic level.
These natural scalars are merely shorthand notation for a, a+a, a+a+a,
... where a is now anything.This is about as far as I've gotten on
this. I really don't know how valuable it is. The natural numbers have
been treated as coming easy already but the unity distinction may be
important. At some level down here I lose interest because the only
thing that really matters is if we enter into a conflict. So long as we
don't make a self-contradiction we are free to build what we like. I
have pointed to a few small contradictions in existing mathematics. The
only reason I care is that these small contradictions endanger the
polysign construction. Today it looks like these little changes
coalesce into a radical shift but usually I just think the whole
effort of arguing with mathematicians is futile.
Do you know much about the concern over how to define magnitudinal
product? I've been told that the answer lies somewhere down in the
Cauchy/Dedekind area but I haven't really gotten into it yet. At the
point that number symbology becomes concrete numerics I have a gap and
I'm not sure how important that is to understand.
>
> There are sound reasons why many, perhaps most,
> mathematicians consider an education in mathematics to
> begin with the calculus.
This is an important note I think that you make on the calculus.
The differential is devoid of a natural value yet needs the natural
valued series construction to come to fruition. It would be really
something if there is a tie in between fundamental magnitude and the
differential, and yielding a finite system as a result. When you are
willing to break the law an awful lot opens up. I have tried to
consider the magnitudinal product as an integral but haven't gotten
farther than that. That opens up an issue of whether the product can
really remain in the same domain. So far the polysign products are
arithmetical and so they do. Perhaps since magnitude in its raw form is
a sort of universal dimensional it can too.
The polsign system can be viewed as operating on a principle of
accumulation so that component functions are nondecreasing in nature.
So long as they remain balanced the result will remain local. The
derivatives take on distinct character; the first derivative will be
nonnegative though potentially decreasing and so a real valued
representation must ensue at the second derivative. Not quite sure what
to draw from that but the second derivative is important to physics so
I keep my eye on it.
I've only made it to page 3 of Self Organization in Real and Complex
Analysis but already I can see that the polysign construction is right
up your alley; even just the title alone says it all.
The time concern fits in too (if you accept the P1 premise).
>
> Point is, our personal experience of time is not
> necessarily consistent with the meaning of time.
Yeah. Time will continue to be discussed at the same level that it has
been I think, though I do stand by the P1 correspondence. Particularly
notions of causality and determinism are still challenging. Perhaps the
zero-dimensional nature alleviates the burden of freedom but can it
answer as the source of a dynamic? I don't think so. It only answers as
the start of a progression which forms a dynamic. So perhaps the
concept of determinism will be declared an invalid context, since it
relies upon the notion of time as a one-dimensional construct.
I'll be reading your paper. Sorry I've rambled a bit.
-Tim
Dik T. Winter
...
> > > Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is
> > > magnitude)
> I also have to accuse you of insincerity.
Where? You stated a formula above. You state for s = 1 to n. The only
way I see that can be read is for s = 1, 2, 3, ..., n, not anything else.
> But the cheap revisions that he poses here are weak and I have seen
> this behavior before.
Why cheap? I indicate to you basic mathematical problems with your
model.
> For instance in my original communication there
> is an explicit declaration that s is representing sign.
In that case you should not write "for s = 1 to n".
> The elemental
> s x
> representation does need to be explicitly introduced to allow
> adaptation to the new system,
But in mathematics it is essential that you *define* your terms.
> but let's not forget that as children we
> were taught the real numbers without higher level mathematics. The same
> can be done for polysign numbers.
Right. When teaching children you skip a lot. But *not* higher level
mathematics, but lower level mathematics. When children grow in their
mathematics education they learn more and more about those lower basics.
> I prefer this approach since it does
> not require much knowledge and so is more accessible to a less educated
> person.
You may prefer that, but your preferences do not make it a mathematical
model. But you present it as such.
> None the less, here is a perfectly capable mathematician who
> is incapable of accepting a new construction.
I do not accept a new construction if the basic definitions are missing.
You state that I must forget about the reals and only consider magnitude.
But when I consider magnitude I can not ignore the reals, because
magnitude is defined in relation to the reals. So unless you provide
a proper definition of magnitude, a am at a loss to proceed in a
mathematical way.
> This
> construction diminishes the real numbers and favors magnitude as
> fundamental.
Perhaps. But that is not the case when you do not define magnitude as
something independent from the reals. As long as magnitude is defined
in terms of the reals, there is a problem.
However, you ignore completely the way I *did* solve some of your
division problems. That was by looking at algebras over the reals,
and regarding your polysign numbers as equivalence classes within
those algebras. So when solving the division problems the reals
were much more fundamental. They allowed to calculate solutions.
Dik T. Winter
> Gene Ward Smith wrote:
> > ste...@nomail.com wrote:
> >
> > > Who has said that mathematics is THE truth? The "mainstream"
> > > folks disagreeing with Timothy Golden have made no claims
> > > of truth. Timothy Golden is the one who seems to think that
> > > it is THE truth that magnitude is more fundamental than the reals.
> > > If he wishes to define magnitude rigourously and then define
> > > the reals based on that, he is free to.
> >
> > As has been pointed out on numerous times, this is in fact an old idea,
> > going back to the Greeks. Landau, for a modern example, develops
> > positive reals from second order arithmetic of
> > positive integers, and goes on from there. This has some advantages, in
> > particular that the positive rationals, as the ratios of positive
> > integers, may be constructed without worrying about division by zero,
> > and then the positive reals (or magnitude) can be constructed next.
> >
> > One can use polysigned numbers, if one so chooses, for constructions.
> > But Tim seems unable to say why we should.
>
> Gene, like Dik, represents the establishment.
Oh.
> This is Gene on a prior thread:
> "I've pointed out several times that you do not have such a
> construction. I'll repeat it: you have NOT constructed the reals.
> This is because your definition requires that the reals have
> already been constructed. "
And Gene was right.
> The important distinction that allows this conflict is in how we
> dissect the number system.
No. The important distinction is that you use terms that you define
in existing terms (the reals). So you assume the existence of the
reals. Otherwise the definition is void.
In the quote on top Gene *did* show how you could define magnitude
without any reference to the reals. But for some reason you do not
want to follow that road.
> Because the polysign construction imposes the identity law
> Sum for s = 1 to n ( s x ) = 0
Again that basic notational flaw. Proper notation would be
sum for i = 1 to n ( s_i x) = 0
> This is a primitive and productive
> construction that poses and answers many questions:
>
> Are the field criteria accurate?
Eh? What can be inaccurate in criteria?
> Must a linear system obey the magnitudinal law
> | A B | = | A | | B | ?
To me this makes no sense. In a linear system we have a linear operator
that transforms input to output. I think that with A and B you mean the
linear operators. But in that case you have to define the meaning of |.|.
In linear algebra, when we define the norm of a vector as the Euclidean
norm, and the norm of a matrix as
sup |A.x|/|x|
the above certainly does not hold. In that case we can only show:
| A B | <= | A | | B |.
> Does time correspond to P1?
> Do improper transformations model electron spin?
> Do n-poles exist?
> Why spacetime?
These are not mathematical problems.
David Marcus
> Sure. Any who wish to mold mathematics to suit their
> philosophy do the same. From Intuitionists (Brouwer, et
> al) to Constructivists (E. Bishop, et al) to Formalists
> (Russell, Frege, Hilbert, et al)to Platonists (Godel,
> Penrose, et al).
Penrose is a Platonist? I think he's a physicist, not a mathematician.
--
David Marcus
David Marcus
> But the only relevant issue with respect to "Math as Religion" is
> whether it is true in universal terms.
Isn't the only relevant issue with religion whether it is true?
--
David Marcus
Gene Ward Smith
David Marcus wrote:
> Penrose is a Platonist? I think he's a physicist, not a mathematician.
His PhD from Cambridge is in math, on the use of tensors in algebraic
geometry. In math among other things he came up with Penrose tilings
and the Moore-Penrose pseudoinverse of a matrix.
David Marcus
I knew about the tilings, but didn't realize he was the Penrose of the
pseduoinverse.
Regardless, he clearly doesn't understand mathematical logic. In fact,
if he was posting to sci.math instead of writing books, we'd call him a
crank. I think we should give him to physics.
--
David Marcus
galathaea
<5052519.11630721253...@nitrogen.mathforum.org>, "T.H.
Ray" <thra...@aol.com> wrote:
!! > T.H. Ray wrote:
!! > > >
!! > > > Mathematics is not a religion is it?
!! > > > Why people adopt the religious attitude is
!! > > > baffling.
||
!! > > > By challenging the fundamentals that have been
!! > > > layed down a type of progress can be made that
!! > > > far rivals the usual progression of adding a
!! > > > branch to a branch of the tree.
take any mathematical theorem
here are some properties of the theorem:
- it can be _expressed_, ie. it has a symbolic representation
- the symbology has an _interpretation_ in terms of:
+ objects
+ transformations of objects
what that theorem "means"
depends on which interpretation is chosen
!! > > > This is a statement on the human mind. The
!! > > > mathematician is most immune to emotional
!! > > > argumentation, yet to claim an inadequacy at
!! > > > the base of the tree hurts badly and well it
!! > > > should, for if the inadequacy is a reality
!! > > > then all of the mathematicians who have been
!! > > > hoodwinked have suffered from the social
!! > > > instinct of mimicry.
that social instinct
is how we learn to play the games
no one suffers because of that
they suffer because of an opposing
social punishment at mistakes
this is more rigorous in mathematics
!! > > > The mathematician who claims immunity will
!! > > > have to accept this lower accusation and
!! > > > therefor should remain open to such
!! > > > possibilities.
but interpretations are not possibilities
and possibilities are not religious
religion is a particular choice
a decision in faith that is not influenced
by the scientific verificationism
of possibility
!! > > > Such possibilities have to be met with clear
!! > > > criticism from the defenders of the existing
!! > > > tree.
the cause of the suffering above
!! > > > The ease with which you do so is a measure of
!! > > > your own abilities.
clear critique
is only possible where verifiable criteria apply
interpretation cannot be criticised
except in muddy terms that
don't quite understand the argument
!! > > > If the truth is so clear cut there should be no
!! > > > problem arriving at it with honest discussion.
which truth?
it is all the same argument
the game of philosophy
truth needs to be interpreted too
!! > > > Otherwise your math is just a religion:
there has always been a close relationship
between mathematics and religion
all students of mathematical history
know the strong connections between these categories
the pythagorean legacies
< the embarrassed acknowledgements of godel >
!! > > > The book has been written.
!! > > > One must preserve the book.
one book
for one interpretation
for one truth
for one
and all
!! > >
!! > >
!! > > As another writer pointed out, the only clear
!! > > cut fact of mathematics -- for whatever other
!! > > properties one might wish to assign the subject
!! > > -- is that math is a language.
no
math is all languages
it is the lingua in all forms
the abstraction of the string of symbols
and the games they can play
!! > > It coheres in common terms by which its speakers
!! > > communicate.
it does not cohere
even on whether it should cohere
!! > > Mathematics is no one's private language, and
!! > > -- unlike the case with religious belief --
!! > > no theorem acquires truth status merely
!! > > because an individual believes it to be true.
interpretation
is a part of the game itself
it is learned through mimicry and creative expression
the thesis and antithesis
individuals make choices
with respect to others
but individuals make choices
that is the ego center in agent theory
!! > The hierarchical structure is an explicit and
!! > meaningful component of mathematics.
||
!! > When the hierarchy is exposed as flawed the flaw
!! > needs to be corrected.
||
!! > The ability of trained mathematicians to perceive a
!! > flaw or be in denial of that flaw is the difference
!! > between true mathematics and religion.
what is the ontology of truth used?
except in respect to a chosen interpretation
there is no notion of flaws and truth
!! > The inherent assumption of accuracy of current
!! > mathematics by modern mathematicians is a religious
!! > precept.
no
the accuracy of mathematics
is due to its verificationist studies
of scientific ontologies
ontologies with perceptual abstractions
shared by a communicating group
but mathematical ontologies
extend beyond the accurate
!! > This assumption is being exposed as invalid here.
||
!! > Is this a minor point?
||
!! > It is not.
||
!! > The consequences of this invalid assumption may make
!! > the difference between a natural basis of reality
!! > versus an arbitrary basis.
mathematicians developed euclidean geometry
mathematicians developed boolean logic
neither of these
are the most likely language of propositions
in the models of our reality
most modern theories
are within noneuclidean notions of space
and which have noncommuting observables
causing failures in distributive associations of propositions
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
!! > The PolySign Numbers (tm)
!! > offer an Answer
!! > and expose an Ambiguity
!! > in the current mathematical system:
!! >
!! > http://www.bandtechnology.com/PolySigned/index.html
||
!! > Magnitude is a fundamental principle whose marriage
!! > with the natural numbers yields Results(C).
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
om mani padme hum...
i prettied up your advertisement
!! Mathematics has no priesthood.
sure it does
and it should
they are the respected
because of accomplishment and skill
in an interpretation
the teachers
of the gameplay in that interpretation
the referrees and speakers for the field
whatever field
!! That you have come up with an idea that has not found
!! wide acceptance, does not logically imply that some
!! force of authority is conspiring to withold recognition.
often there is confusion
between lack of acceptance
and lack of acknowledgement
some interpretations are "uninteresting"
to other interpretations
they lack a communication channel
!! I suggest you look for flaws in your argument or in your
!! method of presentation.
||
!! That's what the referee process is for.
flaws
in which interpretation?
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
Timothy Golden BandTechnology.com
Dik T. Winter wrote:
> In article <1163159563....@i42g2000cwa.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> ...
> > > > The fundamental law which I have applied is
> > > > Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is
> > > > magnitude)
>
> > I also have to accuse you of insincerity.
>
> Where? You stated a formula above. You state for s = 1 to n. The only
> way I see that can be read is for s = 1, 2, 3, ..., n, not anything else.
>
> > But the cheap revisions that he poses here are weak and I have seen
> > this behavior before.
>
> Why cheap? I indicate to you basic mathematical problems with your
> model.
It is cheap because you ignored the specification that s is sign.
I see in some other post you suggest putting an underscore.
The point is that you choose to ignore parts of the communication.
I can go back to past threads and find the same poor style of debate.
It only weakens your position.
For you to criticize my communication by this method is paradoxical.
I concede that I can do a better job of communicating the properties of
polysign numbers.
In particular there are places where the notation is seriously
conflicted due to the usage in traditional mathematics of the '+' sign
as summation whereas under the polysign approach this does not hold
true. So to do algebra with a '+' sign as summation has to be explicity
stated and cannot mix with concrete instances.
You are a wonderful antagonist Dik.
Rather than chase about in a circle it would be preferable to simply
acknowledge the conflict with existing mathematics. If there is
something wrong with the polysign construction then it should be
pointed out. What consequence makes it fail? We do have the division
problem in P4+, but without it there would be no claim for spacetime
support.
I believe you are disputing my form of communication rather than the
polysign numbers. You have not granted that some new room must be made
if we are to generalize sign. To generalize sign is a new concept. New
concepts need room for representation. Most of the old ways can be
applied but the meaning is not the same. We can use ordered
coordinates:
( 3.4, 1.2, 5.6, 9.0 )
to represent a value in P4.
Alternatively we can use symbolic signs:
# 3.4 - 1.2 + 5.6 * 9.0 .
We could even use unit vectors:
3.4 + 1.2 i + 5.6 j + 9.0 k .
All of these work. Their meaning is the same. There is no one right
way. Beneath all of these ways is a new concept that is sign in its
general form. The generalized sign is inherently a natural number. Each
one of these instances above does this implicitly. In order to generate
these forms compactly we have:
Sum from s = 1 to n ( s x_s )
where s is sign and x_s is the sth magnitudinal component.
Do you believe that I have generalized sign?
-Tim
David Marcus
> Dik T. Winter wrote:
> > In article <1163159563....@i42g2000cwa.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > > > > The fundamental law which I have applied is
> > > > > Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is
> > > > > magnitude)
> >
> > > I also have to accuse you of insincerity.
> >
> > Where? You stated a formula above. You state for s = 1 to n. The only
> > way I see that can be read is for s = 1, 2, 3, ..., n, not anything else.
> >
> > > But the cheap revisions that he poses here are weak and I have seen
> > > this behavior before.
> >
> > Why cheap? I indicate to you basic mathematical problems with your
> > model.
>
> It is cheap because you ignored the specification that s is sign.
> I see in some other post you suggest putting an underscore.
> The point is that you choose to ignore parts of the communication.
> I can go back to past threads and find the same poor style of debate.
> It only weakens your position.
> For you to criticize my communication by this method is paradoxical.
If you write "sum s=1 to n", then it means s is an integer. If that's
not what you mean, then don't write that. You can't say, "s is a sign"
and then say "s=1 to n". The reason is that 1 is not a sign. You can
have a first sign, but then you need to give it a name, e.g., s_1 (where
the underscore means subscript). If you say that s_i is the i-th sign,
then you can write "sum_{i=1}^n s_i x".
> I concede that I can do a better job of communicating the properties of
> polysign numbers.
> In particular there are places where the notation is seriously
> conflicted due to the usage in traditional mathematics of the '+' sign
> as summation whereas under the polysign approach this does not hold
> true. So to do algebra with a '+' sign as summation has to be explicity
> stated and cannot mix with concrete instances.
If you are defining a new operation, then state that "+" means your new
operation, not the usual addition. If you need to use both operations,
then make up a new symbol for your new operation.
--
David Marcus
Timothy Golden BandTechnology.com
The simplest and most intuitive sign symbology uses
- + * # ...
Here the first symbol in one line, the second symbol two lines, etc.
These symbols represent natural numbers that will be married to a
magnitude to achieve
-1.2 , +35.6 , *5 , etc.
if the sign in these elemental values is regarded as a natural number
then the generic form is
s x
where s is sign and x is magnitude.
These are two different data types; they combine but they do not
evaluate to a singular type.
They retain structure.
That s is a natural number does not deny it this possibility.
In particular operations on the sx form go like:
( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2
s1 x1 + s1 x2 = s1( x1 + x2 )
These form the basic operations of polysign numbers, the first being
product and the second being superpostion or summation where the '+'
operator means superposition.
The product causes a sign sum s1+s2 which is actually a mod n type of
sum.
Really these rules are trying to explain something that is more
practically learned from my website using the second grader approach.
But since mathematicians like things like this sx terminology there it
is. Really there is just one more statement and the polysign
construction is completed:
Sum for s = 1 to n ( s x ) = 0
where s is sign and x is magnitude.
Because noone has gneralized sign before this notation may be
uncomfortable.
Does the s_i symbol mean anything different?
I suppose it is like a unit vector approach where we might write
1.2 + 3.4 i + 4.5 j .
The most compact form and the least cumbersome is the
- 1 + 3 # 2.5
type of notation which for the sx notation turns into
s1 x1 + s2 x2 + s3 x3
where s1=1,x1=1,s2=2,x2=3,s3=4,x3=2.5
and likewise in your notation
s_1 x1 + s_2 x2 + s_4 x3
where the sign is carried around inherently so we only need specify the
x components.
However these signs are not generic and so when we go over to the
generic form:
s_i x1 + s_j x2 + s_k x3
we have now lost the correspondence of the numerical indices.
Based on this analysis I would like to remain with the sx form.
If you see something wrong with this analysis I hope you will share it.
Your notation works, but it may cause some difficulty.
Some room must be granted for the generalization of sign.
It does not fit into existing mathematics.
-Tim
Lester Zick
Of course. I took it for granted that was understood. Mathematics is
in a different category because most assume it's true but can't prove
it. In fact apart from certain categorical doctrines I as well assume
it's true. The problem is demonstrating that truth. Otherwise you're
reduced to name calling and ad hominem nonsense such as we've seen
recently. The search for "truth" in terms of syllogistic inference is
Aristotle's intellectual legacy both to religion and science. Religion
just uses different axioms which are certainly less likely true than
mathematical axioms and science in general. But religious arguments
based on their axiomatic assumptions can be quite rigorous although
obviously they're often meretricious. The difficulty is that we can't
judge these issues critically without a rigorous definition for truth
in universal terms.
(Actually looking back at your question I notice the qualification
"only" so I suppose I should qualify my comments because religion can
have other relevant issues than truth alone in terms of empirical
social, emotional, and personal utility which I don't see applying to
mathematics in general. Either mathematics is demonstrably true or it
becomes a more reasonable variant of belief systems in general.)
~v~~
David Marcus
You are free to use whatever symbols you wish, as long as you state
clearly what you are doing.
> Here the first symbol in one line, the second symbol two lines, etc.
> These symbols represent natural numbers that will be married to a
> magnitude to achieve
> -1.2 , +35.6 , *5 , etc.
> if the sign in these elemental values is regarded as a natural number
> then the generic form is
> s x
> where s is sign and x is magnitude.
I don't know what "these symbols represent natural numbers" or "is
regarded as a natural number" mean. It looks like you've got a sequence
of "signs". The normal notation for a sequence is something like s_i,
where s_1 is the first element in the sequence, s_2 is the second, etc.
So, you could define s_1 to be -, s_2 to be +, and, in general, s_i to
be your i-th sign. This would let you refer to the i-th sign as s_i. It
is just another name for the same thing.
> These are two different data types; they combine but they do not
> evaluate to a singular type.
So, you have orderd pairs. s x means the ordered pair (s,x) where s is a
sign and x is a natural number.
> They retain structure.
> That s is a natural number does not deny it this possibility.
> In particular operations on the sx form go like:
> ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2
> s1 x1 + s1 x2 = s1( x1 + x2 )
> These form the basic operations of polysign numbers, the first being
> product and the second being superpostion or summation where the '+'
> operator means superposition.
Hold on. You are using "+" as one of your "signs". You shouldn't now use
it for something else. If you do, then you'll get formulas like
( + 5 )( + 6 ) = ( + + + ) 5 6
What is that supposed to mean? It is very confusing.
> The product causes a sign sum s1+s2 which is actually a mod n type of
> sum.
> Really these rules are trying to explain something that is more
> practically learned from my website using the second grader approach.
> But since mathematicians like things like this sx terminology there it
> is. Really there is just one more statement and the polysign
> construction is completed:
> Sum for s = 1 to n ( s x ) = 0
> where s is sign and x is magnitude.
>
> Because noone has gneralized sign before this notation may be
> uncomfortable.
> Does the s_i symbol mean anything different?
If we define what s_i is, then it is just another name for the thing. If
I say x_22 = 55, then "x_22" and "55" are both names for the number 55
(note the use of double quotes to refer to the string of characters).
--
David Marcus
Lester Zick
I take "platonist" in general to mean mystic. And there are certainly
plenty of those among professionals in both camps.
~v~~
Lester Zick
Hey the lot of you belong in faith based pseudologic.
~v~~
Gene Ward Smith
David Marcus wrote:
> I knew about the tilings, but didn't realize he was the Penrose of the
> pseduoinverse.
>
> Regardless, he clearly doesn't understand mathematical logic.
Happily, mathematicians are not required to have a high-level
understanding of mathematical logic to count as real mathematicians.
> In fact, if he was posting to sci.math instead of writing books, we'd call him a
> crank. I think we should give him to physics.
Baloney. He wouldn't be a *mathematical* crank, so you'd be calling him
a philosophical crank, like Ayn Rand or Alfred Korzybski. That is at
best a dodgy claim; I think some people would call *you* a crank for
making it. Is there a word for notions such as that the nature of
consciousness probably depends on unknown features of gravitation? How
about "goofball"?
Timothy Golden BandTechnology.com
You have jumped in on the middle of a discussion that regards a
construction that is new. I have written out the description many
times and will do so again here for you. The proper tutorial is my
website:
http://www.bandtechnology.com/PolySigned
but let's just start from scratch. It really won't take that long to
describe the entire system.
Just suppose that there is a new domain of numbers called three-signed
numbers.
Now we need a third sign.
The existing signs are '-' and '+' and so we choose '*' for the third
sign since when we write it down on a piece of paper it has three
lines. So these symboles are numerical mnemonics.
Now, we hunt for a symmetrical property in the reals that can extend to
this three-signed system and find that
- x + x = 0
can be expanded to
- x + x * x = 0 (the identity law).
Under the reals when we do superposition if the values have the same
sign we simply add the values and preserve the sign. The same will
happen here so that:
- 1.2 - 2.3 = - 3.5 .
* 5.6 * 1.1 = * 6.7 .
The way that the identity law gets applied is when we have a value like
- 2.3 + 4.5 * 1.1
This value can be broken down to
- 1.2 + 3.4 - 1.1 + 1.1 * 1.1
where the last three values are equivalent to zero.
So really
-2.3 + 4.5 * 1.1 = - 1.2 + 3.4 .
Now you will be capable of performing any concrete summation in
three-signed math.
The product has a rotational character that matches the real numbers.
Let's just look at the sign rules of the real product as if a plus sign
jumps twice and a minus sign jumps once. In effect this is just a count
that keeps wrapping at two where the amount to count is represented by
the sign mnemonic. This sign product rule extends to three-signed
numbers so that:
- - = +
- + = *
- * = -
+ * = +
+ + = -
* * = * .
If you want to insert ones to make these concrete products that is
fine.
Really you won't need the table. its just addition.
So for example the third line of this table can be used to do:
( - 3 )( * 4 ) = - 12 .
This is no different than what we might write in the reals:
( - 2 )( - 3 ) = + 6 .
The distributive, commutative, and associative properties work so that
( - 2 + 3 )( - 1 * 2 )
= + 2 - 4 * 3 + 6
= - 4 + 8 * 3
= - 1 + 5 .
The last line is the reduced form but it really doesn't matter if you
reduce.
Perhaps you already see that these three signed numbers are
two-dimensional.
Upon graphing a value the reduction takes place automatically. Please
see my website for a drawing. It turns out that they are the complex
numbers in a natural form that extends directly from the real numbers
by generalizing sign. The proof is on my website.
The rules you have hopefully just learned are extensible and allow
algebraic geometry in any dimension. In order to discuss the entire
family in general we need a generic representation. It has taken me a
long time to arrive at the sx notation. I am still integrating it into
my website. The use of modulo sum is clunky until the zero sign is
introduced and I don't bother with that yet since it adds another
element of confusion.
Your concern over the '+' symbol is exposed here in three-sign (P3)
math.
In the reals (P2) we can write:
+ 5 - ( - 1 ) = + 6 .
and we are used to + preserving sign since it jumps two places:
- 4 + ( - 3 ) = - 7 .
But in P3 '*' takes this role so in the FOIL expansion that I wrote out
above I could have done something more like:
( - a + b )( - c * d )
= (-a)(-c) * (-a)(*d) * (+b)(-c) * (+b)(*d)
where all letters are magnitudes. So usually I am careful to say for
example
"In P9
( z1 + z2 ) z3 = z1 z3 + z2 z3 = z3 z1 + z2 z3
where '+' means superposition. This is also true for any sign level
Pn."
Otherwise someone might substitute in values and forget to change the
plus's to s_9's (your underscore notation).
This is a slightly different description than I usually write; I try to
shake it up to see how it works so I would appreciate your feedback on
what is confusing and of course feel free to ask questions. Dik and I
have already been through this level months ago. His claim of
misunderstanding the sign representation is not a real position. He
completely understands the construction. He's just giving me a hard
time about notation. His suggestion I guess is what you are pushing but
I'll stand by my earlier critique. The underscore notation will not be
as clean. Anyhow notational variation is allowed. If you want to use
the underscore notation I can follow it while communicating with you.
It does not alter the underlying construction.
-Tim
David Marcus
> David Marcus wrote:
>
> > crank. I think we should give him to physics.
>
> Baloney. He wouldn't be a *mathematical* crank, so you'd be calling him
> a philosophical crank, like Ayn Rand or Alfred Korzybski. That is at
> best a dodgy claim; I think some people would call *you* a crank for
> making it. Is there a word for notions such as that the nature of
> consciousness probably depends on unknown features of gravitation? How
> about "goofball"?
"Goofball" certainly applies. However, I think the fact that his
argument is based on a mathematical error, and he denies that it is an
error, qualifies him as a mathematical crank. He is using an incorrect
mathematical argument to support his other goofball notions. Here is
something I wrote a while ago on the topic:
Penrose's books have often been reviewed in mathematical journals,
e.g., [Barr], [Faris], [McCarthy], [Putnam]. All such reviews point
out that Penrose makes a mathematical error. Specifically, Penrose
misunderstands Gödel's Incompleteness Theorem, and so erroneously
concludes that human mathematicians can do something that a machine
cannot. Chapter 6, Section 2, of the recent book [Franzén] also
provides a clear discussion of Penrose's error.
Unfortunately, Penrose's misunderstanding of Gödel's work is the
primary basis for his argument that new physics is needed to explain
consciousness. Once we correct the mathematical error, little remains.
As [Barr] says in the final sentence of his review, "The only real
criticism [of the book] is that there is no valid evidence brought to
bear on its main thesis."
Penrose's misunderstanding is on a point that is often confusing to
students of mathematical logic. However, anyone who has taken a course
in mathematical logic should be able to find the error in Penrose's
argument. As such, it is rather surprising that Penrose should write
two books based on this error. As [Putnam] says of "Shadows of the
Mind":
And yet this reviewer regards its appearance as a sad episode in
our current intellectual life. Roger Penrose is the Rouse Ball
Professor of Mathematics at Oxford University and has shared the
prestigious Wolf Price in physics with Stephen Hawking, but he is
convinced by--and has produced this book as well as the earlier
"The emperor's new mind" to defend--an argument that all experts in
mathematical logic have long rejected as fallacious. The fact that
the experts all reject Lucas's infamous argument counts for nothing
in Penrose's eyes. He mistakenly believes that he has a
philosophical disagreement with the logical community, when in fact
this is a straightforward case of a mathematical fallacy.
Penrose's books are of course also reviewed and discussed in the
non-mathematical literature, e.g., [Dewdney], [Landauer], [Smith],
[Zurek]. Such reviewers are typically unaware of the mathematical
error that invalidates Penrose's argument. Despite this, they often
disagree with Penrose's conclusions.
It is ironic that while Penrose sees Gödel's work as demonstrating the
need for new physics to understand consciousness, Douglas Hofstadter
(in the Pulitzer-Prize winning [Hofstadter]) argued that by studying
the self-reference displayed in the works of Gödel, Escher, and Bach,
we could better understand the analogous self-reference displayed in
consciousness.
Barr, Michael, review of "The Emperor's New Mind", American
Mathematical Monthly, Vol. 97, No. 12, Dec. 1990, pp. 938-942.
Dewdney, A.K., "A Pandora's Box of Minds, Machines and Metaphysics",
Computer Recreations, Scientific American, Dec. 1989, pp. 140-142.
Faris, William, review of "Shadows of the Mind", Notices of the
American Mathematical Society, Vol. 43, No. 2, Feb. 1996, pp. 203-208.
Franzén, Torkey, "Gödel's Theorem, an Incomplete Guide to Its Use and
Abuse", A.K. Peters, Ltd., Wellesley, Massachusetts, 2005.
Hofstadter, Douglas R., "Gödel, Escher, Bach: An Eternal Golden
Braid", Basic Books, 20th Anniversary Ed., Jan. 1999.
Landauer, Rolf, "Is the Mind More than an Analytical Engine?", review of
"The Emperor's New Mind", Physics Today, June 1990, pp. 73-75.
McCarthy, John, review of "The Emperor's New Mind", Bulletin of
the American Mathematical Society, Vol. 23, No. 2, Oct. 1990, pp.
606-616.
Putnam, Hilary, review of "Shadows of the Mind", Bulletin of the
American Mathematical Society, Vol. 32, No. 3, July 1995, pp. 370-373.
Smith, John Maynard, "What Can't the Computer Do?", review of "The
Emperor's New Mind", The New York Review, March 15, 1990, pp. 21-25.
Zurek, Wojciech H., "Physics, Mathematics, and Minds", review of "The
Emperor's New Mind", Science, Vol. 248, May 18, 1990, pp. 880-881.
--
David Marcus
David Marcus
>
> You have jumped in on the middle of a discussion that regards a
> construction that is new. I have written out the description many
> times and will do so again here for you. The proper tutorial is my
> website:
> http://www.bandtechnology.com/PolySigned
> but let's just start from scratch. It really won't take that long to
> describe the entire system.
> Just suppose that there is a new domain of numbers called three-signed
> numbers.
> Now we need a third sign.
> The existing signs are '-' and '+' and so we choose '*' for the third
> sign since when we write it down on a piece of paper it has three
> lines. So these symboles are numerical mnemonics.
> Now, we hunt for a symmetrical property in the reals that can extend to
> this three-signed system and find that
> - x + x = 0
> can be expanded to
> - x + x * x = 0 (the identity law).
Are you defining a new type of number or are you defining a new binary
operation on an existing type of number, both, or neither?
--
David Marcus
Timothy Golden BandTechnology.com
The construction defines a family of number systems:
P1, P2, P3, P4, ...
where P1 is one-signed numbers, P2 are two-signed numbers, etc.
P2 are consistent with the real numbers by design.
P3 are equivalent to the complex numbers.
Pn are n-1 dimensional spaces that obey the geometry of a simplex as
their coordinate system structure. They have a product and sum defined
and they obey the algebraic principles just as the real and complex
numbers do with the exception that P4 and above fail the field division
criteria under special circumstances. Still the higher sign systems are
algebraically well behaved under product and sum. P1 has exact time
correspondence and due to the break at P4 the system supports
spacetime.
-Tim
David Marcus
>
> David Marcus wrote:
> > Timothy Golden BandTechnology.com wrote:
> > >
> > > You have jumped in on the middle of a discussion that regards a
> > > construction that is new. I have written out the description many
> > > times and will do so again here for you. The proper tutorial is my
> > > website:
> > > http://www.bandtechnology.com/PolySigned
> > > but let's just start from scratch. It really won't take that long to
> > > describe the entire system.
> > > Just suppose that there is a new domain of numbers called three-signed
> > > numbers.
> > > Now we need a third sign.
> > > The existing signs are '-' and '+' and so we choose '*' for the third
> > > sign since when we write it down on a piece of paper it has three
> > > lines. So these symboles are numerical mnemonics.
> > > Now, we hunt for a symmetrical property in the reals that can extend to
> > > this three-signed system and find that
> > > - x + x = 0
> > > can be expanded to
> > > - x + x * x = 0 (the identity law).
> >
> > Are you defining a new type of number or are you defining a new binary
> > operation on an existing type of number, both, or neither?
>
> P1, P2, P3, P4, ...
> where P1 is one-signed numbers, P2 are two-signed numbers, etc.
> P2 are consistent with the real numbers by design.
> P3 are equivalent to the complex numbers.
> Pn are n-1 dimensional spaces that obey the geometry of a simplex as
> their coordinate system structure. They have a product and sum defined
> and they obey the algebraic principles just as the real and complex
> numbers do with the exception that P4 and above fail the field division
> criteria under special circumstances. Still the higher sign systems are
> algebraically well behaved under product and sum. P1 has exact time
> correspondence and due to the break at P4 the system supports
> spacetime.
I was asking what the lines above were doing, not what is done later.
Would you first state just the definitions without including the
discussion/motivation?
--
David Marcus
Timothy Golden BandTechnology.com
The identity law...
The dimensionality of the system can be broached by this law.
The operation of rendering or graphing a value invokes this law
implicitly.
Otherwise its usage is completely optional to the arithmetic system.
Because it can be applied to a value I do believe that it deserves
operator status.
Perhaps it deserves a bit more than that.
Your question is challenging.
-Tim
Gene Ward Smith
David Marcus wrote:
> Are you defining a new type of number or are you defining a new binary
> operation on an existing type of number, both, or neither?
He's defining real commutative algebras, I think.
Timothy Golden BandTechnology.com
Here is the ultra-compact form:
Sum for s = 1 to n ( s x ) = 0 .
s1 x1 + s1 x2 = s1( x1 + x2 ) .
( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 .
The sum (s1 + s2) is a modulo n sum as described already.
-Tim
Gene Ward Smith
David Marcus wrote:
> "Goofball" certainly applies. However, I think the fact that his
> argument is based on a mathematical error, and he denies that it is an
> error, qualifies him as a mathematical crank.
No, because it isn't mathematics, it's philosophy. Calling it a
"mathematical error" is a mischaractization.
> Penrose's books have often been reviewed in mathematical journals,
> e.g., [Barr], [Faris], [McCarthy], [Putnam]. All such reviews point
> out that Penrose makes a mathematical error.
Citations saying specificially that he makes a *mathematical* error?
> Specifically, Penrose
> misunderstands Gödel's Incompleteness Theorem, and so erroneously
> concludes that human mathematicians can do something that a machine
> cannot.
Arguing on the basis of Gödel that humans can do things machines
cannot is not mathematics.
> He mistakenly believes that he has a
> philosophical disagreement with the logical community, when in fact
> this is a straightforward case of a mathematical fallacy.
OK, here is a claim that Penrose commits a "mathematical fallacy". How
is it even *possible* to commit a mathematical fallacy on the question
of what humans can or cannot do? This makes no sense to me.
galathaea
Gene Ward Smith wrote:
> David Marcus wrote:
>
> > I knew about the tilings, but didn't realize he was the Penrose of the
> > pseduoinverse.
> >
> > Regardless, he clearly doesn't understand mathematical logic.
>
> Happily, mathematicians are not required to have a high-level
> understanding of mathematical logic to count as real mathematicians.
so what is required for proof?
a fuzzy feeling?
a social contract?
you do realise this one quotable quote
underscores quite clearly
where the feeling of religious application in mathematics comes from...
> > In fact, if he was posting to sci.math instead of writing books, we'd call him a
> > crank. I think we should give him to physics.
>
> Baloney. He wouldn't be a *mathematical* crank, so you'd be calling him
> a philosophical crank, like Ayn Rand or Alfred Korzybski. That is at
> best a dodgy claim; I think some people would call *you* a crank for
> making it. Is there a word for notions such as that the nature of
> consciousness probably depends on unknown features of gravitation? How
> about "goofball"?
plus the ability to disparage others
without need for explanation
_or_proof_
just one's word...
here is an axiom for living:
BEWARE THE MATHEMATICIAN
WHO DISTINGUISHES HERSELF FROM PHILOSOPHY
FOR THEY ARE RUNNING AWAY FROM SOMETHING IMPORTANT
Gene Ward Smith
galathaea wrote:
> so what is required for proof?
>
> a fuzzy feeling?
> a social contract?
(1) Find a math journal
(2) Open it to some article
(3) Read
The sort of thing you see is what is required for a proof.
galathaea
David Marcus
> David Marcus wrote:
>
> > "Goofball" certainly applies. However, I think the fact that his
> > argument is based on a mathematical error, and he denies that it is an
> > error, qualifies him as a mathematical crank.
>
> No, because it isn't mathematics, it's philosophy. Calling it a
> "mathematical error" is a mischaractization.
If I say that I am basing my philosophy on the fact that 1 = 0, that I
am doing this because I can prove that 1 = 0 is true, and you tell me
why my proof is wrong, but I insist that it is correct, am I making a
mathematical error or are we arguing philosophy?
> > Penrose's books have often been reviewed in mathematical journals,
> > e.g., [Barr], [Faris], [McCarthy], [Putnam]. All such reviews point
> > out that Penrose makes a mathematical error.
>
> Citations saying specificially that he makes a *mathematical* error?
They all say that Penrose thinks the proof of Godel's Theorem proves
something that it doesn't prove. I would call that a mathematical error.
I don't have Penrose's book anymore, but when I read it, I agreed that
Penrose says he proves something that he doesn't prove. (See below for
more on this.)
> > Specifically, Penrose
> > misunderstands G=F6del's Incompleteness Theorem, and so erroneously
> > concludes that human mathematicians can do something that a machine
> > cannot.
>
> Arguing on the basis of G=F6del that humans can do things machines
> cannot is not mathematics.
If you base your physics/biology/whatever on incorrect mathematics,
aren't you still making a mathematical error?
> > He mistakenly believes that he has a
> > philosophical disagreement with the logical community, when in fact
> > this is a straightforward case of a mathematical fallacy.
>
> OK, here is a claim that Penrose commits a "mathematical fallacy". How
> is it even *possible* to commit a mathematical fallacy on the question
> of what humans can or cannot do? This makes no sense to me.
Penrose says that we (mathematicians/logicians) prove that the Godel
sentence (say for ZFC) is true. He agrees that the proof shows that ZFC
can't prove the Godel sentence, but insists that the proof is a valid
mathematical proof showing that the Godel sentence is true. He concludes
that we have proven something that ZFC can't prove.
On the other hand, all the book reviews written by mathematicians say
that we don't prove the Godel sentence is true. All we do is prove that
if ZFC is consistent, the Godel sentence is true. Which is precisely
what ZFC proves.
Admittedly, as the logicians kept trying to tell Penrose that he was
wrong, his counter arguments got more involved, but I think the
preceding is a fair description of his basic error.
Is that a mathematical error or a philosophical error?
--
David Marcus
David Marcus
Oh. OK. Thanks.
--
David Marcus
Gene Ward Smith
David Marcus wrote:
> Gene Ward Smith wrote:
> > Arguing on the basis of G=F6del that humans can do things machines
> > cannot is not mathematics.
>
> If you base your physics/biology/whatever on incorrect mathematics,
> aren't you still making a mathematical error?
But it isn't incorrect as mathematics.
> Penrose says that we (mathematicians/logicians) prove that the Godel
> sentence (say for ZFC) is true. He agrees that the proof shows that ZFC
> can't prove the Godel sentence, but insists that the proof is a valid
> mathematical proof showing that the Godel sentence is true. He concludes
> that we have proven something that ZFC can't prove.
>
> On the other hand, all the book reviews written by mathematicians say
> that we don't prove the Godel sentence is true. All we do is prove that
> if ZFC is consistent, the Godel sentence is true. Which is precisely
> what ZFC proves.
You are simply ignoring his point here, which is that proving something
on the assumption that ZFC is consistent is regarded in the wider world
of mathematics as a proof of its truth.
Now, I think Penrose is trying to milk this for a conclusion he can't
get, but that's a philosophy argument. There's nothing wrong
mathematically with making a philosophical mountain out of a
mathematical molehill, which I think is the deal here.
> Is that a mathematical error or a philosophical error?
It is entirely a philosophical argument.
David Marcus
> But you asked for no text.
No, I asked for no discussion/remarks. Words are fine (and a good idea).
> Here is the ultra-compact form:
>
> Sum for s = 1 to n ( s x ) = 0 .
> s1 x1 + s1 x2 = s1( x1 + x2 ) .
> ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 .
>
> The sum (s1 + s2) is a modulo n sum as described already.
I asked for a definition of the object (commutative algebra?) that you
are defining. You can either construct the object or tell us the
properties that characterize it. For example, we can give a construction
of the real numbers via Cauchy sequences or Dedekind cuts or we can
write down the properties that characterize a complete ordered field and
prove that there is only one object that satisfies all the properties.
--
David Marcus
Timothy Golden BandTechnology.com
I don't really know how to answer you.
If you want to know what area of mathematics it is this then I would
have to say that it fits beneath the real numbers if we were to view
mathematics as a branching tree.
It derives dimensionality without useage of a Cartesian product.
There is implicit geometry.
I think you could tell me the answer better than I can tell you.
I am not versed in mathematics at the level that you are trying to
discuss this at.
It's polysign numbers. Sorry that this is such poor communication.
Do you understand the polysign numbers?
-Tim
Gene Ward Smith
David Marcus wrote:
> Timothy Golden BandTechnology.com wrote:
> > But you asked for no text.
>
> No, I asked for no discussion/remarks. Words are fine (and a good idea).
>
> > Here is the ultra-compact form:
> >
> > Sum for s = 1 to n ( s x ) = 0 .
> > s1 x1 + s1 x2 = s1( x1 + x2 ) .
> > ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 .
> >
> > The sum (s1 + s2) is a modulo n sum as described already.
>
> I asked for a definition of the object (commutative algebra?) that you
> are defining. You can either construct the object or tell us the
> properties that characterize it.
I think it might help to take s to be a primitive nth root of unity, in
the sense that s^n=1, but
s^i is not 1 for 0<i<n, and 1+s+s^2 +...+s^(n-1)=0, but we are not
assuming s is a complex number. Then x1, x2, etc are nonnegative real
numbers. So you have things like x1 s + x2 s^2 + ... + xn s^n. Now the
question is modulo what--when are two such expressions equated?
David Marcus
> David Marcus wrote:
> > Gene Ward Smith wrote:
>
> > > Arguing on the basis of G=F6del that humans can do things machines
> > > cannot is not mathematics.
> >
> > If you base your physics/biology/whatever on incorrect mathematics,
> > aren't you still making a mathematical error?
>
> But it isn't incorrect as mathematics.
If you mean that physics/biology/whatever isn't mathematics, then I
agree. But, if a person insists that incorrect mathematics is correct
mathematics, isn't that incorrect as mathematics?
> > Penrose says that we (mathematicians/logicians) prove that the Godel
> > sentence (say for ZFC) is true. He agrees that the proof shows that ZFC
> > can't prove the Godel sentence, but insists that the proof is a valid
> > mathematical proof showing that the Godel sentence is true. He concludes
> > that we have proven something that ZFC can't prove.
> >
> > On the other hand, all the book reviews written by mathematicians say
> > that we don't prove the Godel sentence is true. All we do is prove that
> > if ZFC is consistent, the Godel sentence is true. Which is precisely
> > what ZFC proves.
>
> You are simply ignoring his point here, which is that proving something
> on the assumption that ZFC is consistent is regarded in the wider world
> of mathematics as a proof of its truth.
That's not what he says. And, I don't think he means that. If he really
does mean that, he has a very strange way of saying it. And, if he did
say/mean that, I don't think we would have logicians complaining in math
journals that Penrose just doesn't get it. I think you are giving
Penrose too much credit.
> Now, I think Penrose is trying to milk this for a conclusion he can't
> get, but that's a philosophy argument. There's nothing wrong
> mathematically with making a philosophical mountain out of a
> mathematical molehill, which I think is the deal here.
>
> > Is that a mathematical error or a philosophical error?
>
> It is entirely a philosophical argument.
If you are right that he is merely saying that mathematicians believe
that ZFC is consistent and so believe the Godel sentence is true, then
OK--anything from then on is not mathematics. But, suppose that is not
what he is saying. Would it still be just a philosophical argument?
And, if a person claims that something is proven mathematically when it
isn't and uses this to argue that this mathematical fact supports their
physics/biology views in writings that are read by people in those
fields who do not have the mathematical training to detect the
mathematical error, what is that person?
Have you read the relevant sections in The Emperor's New Mind? If not, I
suggest you do.
--
David Marcus
Gene Ward Smith
David Marcus wrote:
> If you are right that he is merely saying that mathematicians believe
> that ZFC is consistent and so believe the Godel sentence is true, then
> OK--anything from then on is not mathematics.
I think he is saying we know it in our inner gizzard, whereas a robot
doesn't have an inner gizzard and just has to follow the rules.
> Have you read the relevant sections in The Emperor's New Mind? If not, I
> suggest you do.
Some time ago. What you see me say is what I thought he was saying.
David Marcus
> David Marcus wrote:
> > Timothy Golden BandTechnology.com wrote:
> > > But you asked for no text.
> >
> > No, I asked for no discussion/remarks. Words are fine (and a good idea).
> >
> > > Here is the ultra-compact form:
> > >
> > > Sum for s = 1 to n ( s x ) = 0 .
> > > s1 x1 + s1 x2 = s1( x1 + x2 ) .
> > > ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 .
> > >
> > > The sum (s1 + s2) is a modulo n sum as described already.
> >
> > I asked for a definition of the object (commutative algebra?) that you
> > are defining. You can either construct the object or tell us the
> > properties that characterize it. For example, we can give a construction
> > of the real numbers via Cauchy sequences or Dedekind cuts or we can
> > write down the properties that characterize a complete ordered field and
> > prove that there is only one object that satisfies all the properties.
> >
> > --
> > David Marcus
>
> I don't really know how to answer you.
>
> If you want to know what area of mathematics it is this then I would
> have to say that it fits beneath the real numbers if we were to view
> mathematics as a branching tree.
No, that wasn't what I asked.
> It derives dimensionality without useage of a Cartesian product.
> There is implicit geometry.
>
> I think you could tell me the answer better than I can tell you.
Perhaps.
> I am not versed in mathematics at the level that you are trying to
> discuss this at.
How much mathematics have you had?
> It's polysign numbers. Sorry that this is such poor communication.
> Do you understand the polysign numbers?
Nope.
--
David Marcus
Timothy Golden BandTechnology.com
So in Pn you want to know when z1=z2 ?
Your construction looks fine. There is the hair-splitting issue of
constructing it out of the reals but this will work. The sign vectors
are nonorthogonal according to the identity law. In running the numbers
through the identity law they will be reduced and so if all of their
reduced components are equal they will be equal. You say 'modulo what'
but I don't think you need to worry about any modulo math to test for
equality. The modulo math is just for sign products. With your
construction s^n = s^0 which is good. If you were to multiply two
values z1 and z2 you'd distribute the terms out and the signs would
combine by addition modulo n, and since you've implemented the zero
sign you'll be fine.
Is this a branch of existing mathematics?
-Tim
Timothy Golden BandTechnology.com
I've taken a topology course though I didn't appreciate it.
Real analysis, complex analysis, and engineering math beneath that.
You should probably just spend some time on my website to get familiar
with the polysign construction. I've found some neat stuff. The lattice
is pretty interesting and the P4 product also. There is quite a lot of
work still to do. I mostly look for physics models. Since they are
capable of generating spacetime they could make a very tight and
natural physical model.
Do you see a conflict with taking magnitude as fundamental? This has
been a sticky point for mathematicians since they build the reals then
get magnitude. This system takes magnitude and builds the reals.
-Tim
David Marcus
> David Marcus wrote:
>
> > If you are right that he is merely saying that mathematicians believe
> > that ZFC is consistent and so believe the Godel sentence is true, then
> > OK--anything from then on is not mathematics.
>
> I think he is saying we know it in our inner gizzard, whereas a robot
> doesn't have an inner gizzard and just has to follow the rules.
If that's what he's saying, then it is just philosophy. But, then why
give the Godel sentence as an example of this? I would think there are
lots simpler examples of things our inner gizzard says are true that we
can't prove.
For that matter, one of my logic professors said that the more you work
with the ZFC axioms, the less you feel you understand them. Having taken
several logic courses in graduate school, I feel the same way. I
wouldn't place a bet that ZFC is consistent. I might be willing to bet
that people will be able to fix any problems that are found, but that's
a rather different bet.
> > Have you read the relevant sections in The Emperor's New Mind? If not, I
> > suggest you do.
>
> Some time ago. What you see me say is what I thought he was saying.
OK. Does my thinking Penrose is a crank still make me a crank?
--
David Marcus
David Marcus
>
> I've taken a topology course though I didn't appreciate it.
> Real analysis, complex analysis, and engineering math beneath that.
> You should probably just spend some time on my website to get familiar
> with the polysign construction. I've found some neat stuff. The lattice
> is pretty interesting and the P4 product also. There is quite a lot of
> work still to do. I mostly look for physics models. Since they are
> capable of generating spacetime they could make a very tight and
> natural physical model.
>
> Do you see a conflict with taking magnitude as fundamental? This has
> been a sticky point for mathematicians since they build the reals then
> get magnitude. This system takes magnitude and builds the reals.
If you can really do it, then there is no conflict. I don't think you
understand enough mathematics to even know if you are really doing it. I
think what other people have been trying to tell you is that your
construction doesn't do quite what you think it does. Although, if you
are mainly interested in applications, I'm not sure why you would care
whether you use the reals to build what you want or build it from other
things.
--
David Marcus
galathaea
Ward Smith" <genewa...@gmail.com> wrote:
!! galathaea wrote:
!!
!! > so what is required for proof?
!! >
!! > a fuzzy feeling?
!! > a social contract?
!!
!! (1) Find a math journal
!! (2) Open it to some article
!! (3) Read
!!
!! The sort of thing you see is what is required for a proof.
what journals classify as mathematical?
does any symbolic manipulation count?
is this an appeal to authority
or a call for laissez-faire?
does this article
http://tinyurl.com/yxyg2s
reside in a mathematical journal
because i see symbols there
and i see them getting manipulated
to give them different meanings
in what way
are objects in reality allowed symbolic content?
-+-+-
certainty is religion
in whatever symbolic belief it occurs in
the best we can do with actual learning
is repeatability
and correlation of models
bisimulation
never guarantees against failures
Gene Ward Smith
galathaea wrote:
> In article <1163288725....@h54g2000cwb.googlegroups.com>, "Gene
> Ward Smith" <genewa...@gmail.com> wrote:
>
> !! galathaea wrote:
> !!
> !! > so what is required for proof?
> !! >
> !! > a fuzzy feeling?
> !! > a social contract?
> !!
> !! (1) Find a math journal
> !! (2) Open it to some article
> !! (3) Read
> !!
> !! The sort of thing you see is what is required for a proof.
>
>
> what journals classify as mathematical?
Apparently my reply was too complicated for you. I'll make it simpler:
get hold of an issue of Inventiones Mathematica and do the above.
galathaea
entity "Gene Ward Smith" <genewa...@gmail.com> selfaugmented:
!! galathaea wrote:
!! > In article <1163288725....@h54g2000cwb.googlegroups.com>, "Gene
!! > Ward Smith" <genewa...@gmail.com> wrote:
!! >
!! > !! galathaea wrote:
!! > !!
!! > !! > so what is required for proof?
!! > !! >
!! > !! > a fuzzy feeling?
!! > !! > a social contract?
!! > !!
!! > !! (1) Find a math journal
!! > !! (2) Open it to some article
!! > !! (3) Read
!! > !!
!! > !! The sort of thing you see is what is required for a proof.
!! >
!! >
!! > what journals classify as mathematical?
!!
!! Apparently my reply was too complicated for you. I'll make it simpler:
!! get hold of an issue of Inventiones Mathematica and do the above.
you must have been reading my mind
as i have had my eye on an article in their
"december 2006" future edition
it is called
"mirror symmetry for del pezzo surfaces:
vanishing cycles and coherent sheaves"
by auroux, katzarkov, orlov
this will be my bible
and i shall learn from it
what is required for proof in mathematics
and why mathematics is not a religion
i hope if you do not mind
i may ask some questions along the way
first
look at the ontology of definition 2.3
it assigns the adjective name "full"
to objects of type "exceptional collection" in a category
when they satisfy the property
of "generating" the entire category
questiongroup 1)
what is the scope of this naming?
does the term "full" mean this in other papers?
am i supposed to use the term "full" myself for this
or am i allowed to use any term as long as i define it?
now
i'm somewhat familiar with C-linear triangulated categories
and i have been trying to learn more about them
because i suspect they have something to do with
my multisection projection theorems
over hypergeometrics and q-hypergeometrics
but it is not entirely clear to me immediately
whether or not we can always build a full exceptional collection
in every bounded derived category of coherent sheaves on a del pezzo surface
it only states that it is known that they "have" one
2.5 constructs them from existing ones
proposition 2.7 gives a construction of a full exceptional set
but the surface is not dell pezzo
questiongroup2)
what does this mean to "know" del pezzo surfaces
"have" full exceptional collections?
do we know how to create this collection
using only the data of the del pezzo surface?
unfortunately
my local university does not have online access
to references 14 or 16
so i can only read the abstracts
which say things like
"conditions on the existence of complete exceptional sets
in these categories are derived"
and
"all such collections can be obtained from each other"
but i have found "exceptional collections and del pezzo gauge theories"
by herzog
which asserts only that there exist exceptional collections
no mention of construction or of fullness
so i feel i have reason to question further
questiongroup2')
again, when these mathematicians speak of existence
what is being proved?
proposition 4.7 is existential in the chosen paper as well
but this appears to be an existence of a different kind
the proof of proposition 4.7 actually constructs the alpha_ij's
it asserts the existence of
it actually shows how i can calculate the alpha_ij's
so that i may actually use them
questiongroup 3)
is it proper to give different importance
to this constructive kind of existence?
the same kind of existence is seen in
proposition 4.10
proposition 4.12
but then we get to remark 5.2
where we are cautioned that there is actual ambiguity
in two homology classes used in 5.1
here we "know" they exist
but we only construct them up to translation
questiongroup 4)
why does a mathematical paper contain material
that the same paper should "legitimately concern" the reader?
when ambiguity is allowed in a construction
what do the ambiguous symbols mean?
and finally
in reference to http://tinyurl.com/yxyg2s
you suggest i am being naive by assuming it is a mathematics journal
i clsim it has:
- objects
- transformation of objects
- the symbols for the objects are given meaning
through the transformations they undergo
- there are premises stated
- and conclusions given
questiongroup 5)
what distinguishes
the article http://tinyurl.com/yxyg2s
with the article by auroux, katzarkov, orlov
in mathematical content?
Timothy Golden BandTechnology.com
galathaea wrote:
> In article <1163288725....@h54g2000cwb.googlegroups.com>, "Gene
> Ward Smith" <genewa...@gmail.com> wrote:
>
> !! galathaea wrote:
> !!
> !! > so what is required for proof?
> !! >
> !! > a fuzzy feeling?
> !! > a social contract?
> !!
> !! (1) Find a math journal
> !! (2) Open it to some article
> !! (3) Read
> !!
> !! The sort of thing you see is what is required for a proof.
>
>
> what journals classify as mathematical?
> does any symbolic manipulation count?
>
> is this an appeal to authority
> or a call for laissez-faire?
>
> does this article
> http://tinyurl.com/yxyg2s
Nice Link.
You are good at critiquing mathematical constructions.
Could you please criticize the polysign construction?
http://www.bandtechnology.com/PolySigned
Any advice is appreciated.
-Tim
Timothy Golden BandTechnology.com
Dik T. Winter wrote:
> In article <1163162640.0...@k70g2000cwa.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > Gene Ward Smith wrote:
> > > ste...@nomail.com wrote:
> > >
> > > > Who has said that mathematics is THE truth? The "mainstream"
> > > > folks disagreeing with Timothy Golden have made no claims
> > > > of truth. Timothy Golden is the one who seems to think that
> > > > it is THE truth that magnitude is more fundamental than the reals.
> > > > If he wishes to define magnitude rigourously and then define
> > > > the reals based on that, he is free to.
> > >
> > > As has been pointed out on numerous times, this is in fact an old idea,
> > > going back to the Greeks. Landau, for a modern example, develops
> > > positive reals from second order arithmetic of
> > > positive integers, and goes on from there. This has some advantages, in
> > > particular that the positive rationals, as the ratios of positive
> > > integers, may be constructed without worrying about division by zero,
> > > and then the positive reals (or magnitude) can be constructed next.
> > >
> > > One can use polysigned numbers, if one so chooses, for constructions.
> > > But Tim seems unable to say why we should.
> >
> > Gene, like Dik, represents the establishment.
>
> Oh.
>
> > This is Gene on a prior thread:
> > "I've pointed out several times that you do not have such a
> > construction. I'll repeat it: you have NOT constructed the reals.
> > This is because your definition requires that the reals have
> > already been constructed. "
>
> And Gene was right.
>
> > The important distinction that allows this conflict is in how we
> > dissect the number system.
>
> No. The important distinction is that you use terms that you define
> in existing terms (the reals). So you assume the existence of the
> reals. Otherwise the definition is void.
>
> In the quote on top Gene *did* show how you could define magnitude
> without any reference to the reals. But for some reason you do not
> want to follow that road.
>
> > Because the polysign construction imposes the identity law
> > Sum for s = 1 to n ( s x ) = 0
>
> sum for i = 1 to n ( s_i x) = 0
>
> > This is a primitive and productive
> > construction that poses and answers many questions:
> >
> > Are the field criteria accurate?
>
> Eh? What can be inaccurate in criteria?
>
> > Must a linear system obey the magnitudinal law
> > | A B | = | A | | B | ?
>
> To me this makes no sense. In a linear system we have a linear operator
> that transforms input to output. I think that with A and B you mean the
> linear operators. But in that case you have to define the meaning of |.|.
> In linear algebra, when we define the norm of a vector as the Euclidean
> norm, and the norm of a matrix as
> sup |A.x|/|x|
> the above certainly does not hold. In that case we can only show:
> | A B | <= | A | | B |.
>
> > Does time correspond to P1?
> > Do improper transformations model electron spin?
> > Do n-poles exist?
> > Why spacetime?
>
> These are not mathematical problems.
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
I am willing to be flexible with notation and I don't believe that
there is a singular way. Above in this thread I have criticized your
underscore notation as lacking generality and I would suggest on that
issue if you want to take it up to take it up there:
http://groups.google.com/group/sci.math/msg/db97b7686861d8c0
Your refutation of the polysign construction as relying on the reals is
exactly what this thread is trying to address. I am looking squarely at
this from one side and you from the other.
So this is fine. This is the debate that I wish to have. And so this
debate leads to the question of whether a fine mathematician such as
yourself is capable of breaking such a rule in the interest of
improving mathematics. Hindsight tells me that breaking this rule is
not a conflict ad so this rule is no law. It is merely a tradition
which has cost mathematics the deficit of the polysign numbers.
Your ability to go here is the difference between scientific
mathematics and religious mathematics. Currently you are practicing
religious mathematics. Breaking rules is a lot of fun; especially false
rules, for these shroud the truth and should be broken. It is a
scientific mathematicians duty to tread here.
-Tim
David Marcus
> I am willing to be flexible with notation and I don't believe that
> there is a singular way. Above in this thread I have criticized your
> underscore notation as lacking generality and I would suggest on that
> issue if you want to take it up to take it up there:
> http://groups.google.com/group/sci.math/msg/db97b7686861d8c0
>
> Your refutation of the polysign construction as relying on the reals is
> exactly what this thread is trying to address. I am looking squarely at
> this from one side and you from the other.
> So this is fine. This is the debate that I wish to have. And so this
> debate leads to the question of whether a fine mathematician such as
> yourself is capable of breaking such a rule in the interest of
> improving mathematics. Hindsight tells me that breaking this rule is
> not a conflict ad so this rule is no law. It is merely a tradition
> which has cost mathematics the deficit of the polysign numbers.
>
> Your ability to go here is the difference between scientific
> mathematics and religious mathematics. Currently you are practicing
> religious mathematics. Breaking rules is a lot of fun; especially false
> rules, for these shroud the truth and should be broken. It is a
> scientific mathematicians duty to tread here.
Which rule do you want to break?
--
David Marcus
Timothy Golden BandTechnology.com
Formal mathematics seems to deny magnitude as having fundamental
status. With the polysign numbers the identity law:
Sum for s = 1 to n ( s x ) = 0
takes care of all sign information. So this information need not be
defined within either of the components s or x. This leaves x a raw
magnitude. Traditionally magnitude seems to be defined as a real number
and so people gripe. I don't personally care but if the polysign
construction is to be palatable to others this small conflict needs to
be adressed. Otherwise the strict mathematician will claim a flaw and
the system is refuted. The language of mathematics leaves no room for
conflict whatsoever.
There is another benefit in allowing magnitude to be fundamental: the
unity problem. Traditionally when mathematicians define the real line
they do so via the natural numbers. This description is inherently
carried about when we study Cartesian spaces such as 3D physical space.
Where is unity? It is an arbitrary decision, not a fixed choice. In
other words the correspondence of Cartesian coordinates to 3D space is
imperfect. By granting magnitude as fundamental the unit value can take
on a nondefinite position 'u' and yet the natural numbers will now be
recoverable under magnitude via u, u+u, u+u+u, ... which is
symbolically identical to a, a+a, a+a+a. Because the natural numbers
are so easily recovered from magnitude and this definition does not
suffer the unity problem I see this as another reason to grant
magnitude as fundamental. These natural numbers are somewhat more like
scalars but that is fine; when we talk about having three apples and
two oranges the scalar product is apparent.
The most convincing argument I call the gorilla conjecture: magnitude
is a primitive feature of existence that we needn't even be tought to
understand. A gorilla can be tought to sort sticks by length. All of
mathematics as a human construct is a priori subject to this feature.
Is this minor? In my opinion it is, but the consequences are actually
quite large if magnitude is granted as a primitive. Existing
mathematics will be vastly simplified, particularly the definition(s)
of real numbers.
-Tim
Timothy Golden BandTechnology.com
Formal mathematics seems to deny magnitude as having fundamental
status. With the polysign numbers the identity law:
Sum for s = 1 to n ( s x ) = 0
Lester Zick
I find it incredibly pretentious and totally disingenuous that you
would use the term "true" several times in the context of this post
without being able to define the term while demanding I define it for
you in other posts. Is this an example of the exact and precise
meanings you claim mathematicians are trained to employ?
~v~~
Lester Zick
<David...@alumdotmit.edu> wrote:
>Gene Ward Smith wrote:
>> David Marcus wrote:
>> > Gene Ward Smith wrote:
>>
>> > > Arguing on the basis of G=F6del that humans can do things machines
>> > > cannot is not mathematics.
>> >
>> > If you base your physics/biology/whatever on incorrect mathematics,
>> > aren't you still making a mathematical error?
>>
>> But it isn't incorrect as mathematics.
>
>If you mean that physics/biology/whatever isn't mathematics, then I
>agree. But, if a person insists that incorrect mathematics is correct
>mathematics, isn't that incorrect as mathematics?
And perhaps you'd be so good as to define "correct" and "incorrect"
mathematics for the heathen? Or even just the term "mathematics" might
do for a starter. I mean purely in the interests of clear and precise
meanings modern mathematicians are trained to employ.
~v~~
Lester Zick
No but your thinking you know what you're talking about does.
~v~~
galathaea
"Timothy Golden BandTechnology.com" <tttp...@yahoo.com> wrote:
!! galathaea wrote:
!! > In article <1163288725....@h54g2000cwb.googlegroups.com>, "Gene
!! > Ward Smith" <genewa...@gmail.com> wrote:
!! >
!! > !! galathaea wrote:
!! > !!
!! > !! > so what is required for proof?
!! > !! >
!! > !! > a fuzzy feeling?
!! > !! > a social contract?
!! > !!
!! > !! (1) Find a math journal
!! > !! (2) Open it to some article
!! > !! (3) Read
!! > !!
!! > !! The sort of thing you see is what is required for a proof.
!! >
!! >
!! > what journals classify as mathematical?
!! > does any symbolic manipulation count?
!! >
!! > is this an appeal to authority
!! > or a call for laissez-faire?
!! >
!! > does this article
!! > http://tinyurl.com/yxyg2s
!! > reside in a mathematical journal
!! >
!! > because i see symbols there
!! > and i see them getting manipulated
!! > to give them different meanings
!! >
!! > in what way
!! > are objects in reality allowed symbolic content?!!
||
!! Nice Link.
!!
!! You are good at critiquing mathematical constructions.
!! Could you please criticize the polysign construction?
!! http://www.bandtechnology.com/PolySigned
!! Any advice is appreciated.
here is what i see:
i see a theory
that has made it through the first stages of selection
in other words
it has established an apparently consistent worldview
that can claim repeatability of its derivations
mathematicians should hunger for these moments
as they mark the transition into
the time of connections
where a theory establishes its context
in terms of related theories and theorems
you have already made some good progress here
and it is apparent that you have the drive
to develop many more such relationships
you may even have a "vision"
an unstated view of goals and structure
you are still trying to give formalism to
i see with works like your
"proof of P3's equivalence to C"
and
"deformation study of the P4 product"
that you are developing connections with roots of unity
and exploring the product structure
but i think
as is clear from newsgroup reactions
that you will likely continue this
as a mostly solitary endeavor
you feel there is a great simplification here
there is still a lot that must be developed
in order to show this to others
i might suggest checking out cyclotomic fields
but follow what drives you
as drive is ultimately what gets you there
!! > -+-+-
!! >
!! > certainty is religion
!! > in whatever symbolic belief it occurs in
!! >
!! > the best we can do with actual learning
!! > is repeatability
!! > and correlation of models
!! >
!! > bisimulation
!! > never guarantees against failures
always be open to critique of your argument
expect that you will sometimes be wrong