A Diatribe on the Philosophy of Mathematics
Herman Rubin
Dave L. Renfro <dlre...@gateway.net> wrote:
>G.E. Ivey <georg...@gallaudet.edu>
>[sci.math 10 Sep 01 12:42:50 -0400 (EDT)]
><http://forum.swarthmore.edu/epigone/sci.math/nenquermgroo>
>wrote (in part):
>> philosophically, science is DEFINED by the "scientific method"
>> which mathematics does not use: science depends on inductive
>> reasoning and experimentation and mathematics depends on
>> deductive reasoning from axioms or postulates.
>I often wonder about this. Some food for thought --->>>
> *******************************************************
> *******************************************************
>1. Here's part of a math post of mine (the comment in braces {...}
> wasn't in the original) at
><http://forum.swarthmore.edu/epigone/nctm.l/zuzoucrimp/ok9s4c...@forum.mathforum.com>:
>I wonder ... if bosonic life exists (i.e. Bose-Einstein condensate
>based) somewhere and developed intelligence, would it understand
>our mathematics at all? Recall that bosons don't satisfy the
>Pauli exclusion principle (see [1]), and so the concepts of unity,
>definiteness, and separateness might not have meaning for such
>life. Hence, one might wonder if they would understand the natural
>numbers, much less the rest of mathematics {as we know it}. [I've
>always thought the idea that intelligent ET's would recognize
>prime-number-coded messages might be a bit anthropomorphic . . .]
Mathematics is mathematics. What does the Pauli exclusion
principle have to do with it? Unless there is exactly
one condensate, the condensates themselves would be separate
individuals, even if coming together caused an exchange of
material or a coalescence. I do not see why they could not
come up with the Peano Postulates.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Herman Rubin
Nobuo Saito <genki...@hotmail.com> wrote:
>>This is essentially the "formalist" philosophy of mathematics- that
>>mathematics in "formal knowledge", that axioms are true because we
>SAY
>>they are true and that theorems are proven from axioms in a purely
>>formal way. This leads to a rather obvious question: If mathematics
>>is purely formal, WHY does it work in physics, economics, biology,
>>etc.? I argue that it works precisely BECAUSE it is formal.
>This is the "formalist" approach of mathematics
>which charcterises the 20th century mathematics.
>While I agree this is important method, I think
>this doesn't explain fully why math is useful.
Of course it doesn't. I believe it was Wigner who coined
the expression, "The unreasonable effectiveness of
mathematics."
This is most apparent in the physical sciences, and works
moderately well in the biological science, and very much
less well in economics. Mathematics just provides a
language for modeling, and how well it works depends on the
accuracy of the model. In some cases, a very simple model
works well. We have seen many of these, so there is a
tendency to generalize and say that it will always happen.
This is a very big mistake, which the philosophers made
when they claimed that the model comes from the data. No,
the models are "already there", and the data just enables
us to select. If a simple model fits, we select it. If
not, we theorize and try to come up with a better one.
>If axiom can be chosen arbitrary as long as they are consistent,
>then why there are not so many axioms in math?
>For example, why is the axiom of group so useful?
>You can choose infinitely many axioms for algebraic systems on sets.
>But it seems that groups, rings, fields, Lie algebras,
>and only several other algebraic systems are useful.
>It seems that the number of truly useful axioms is not many.
For many of these, it is almost trivial that they would
come up "naturally"; Lie algebras form the only exceptional
one in the list you have given, and they arose initially
from the study of differential equations invariant under
groups of transformations.
Randy Yates
Thank you for an inspiring and well-written "diatribe." Allow me to
attempt to cut to what I believe is the core issue by responding to
the following section:
> [...]
> The most telling aspect is this: the appropriate "base philosophy"
> for science is REALISM with its insistence that the "truth" of a
> theory depend upon its correspondence with reality: i.e. experiment.
> The appropriate "base philosophy" for mathematics is, however,
> IDEALISM with "truth" of a theory depending upon its internal
> consistency- non-Euclidean geometry is as "true" as Euclidean geometry
> because, as Poincare and Klein showed, they are equally consistent.
I think you have correctly classified the bases as REALISM and IDEALISM,
however, you have neglected to make a connection between the two. Let me
propose that the reason that mathematics, which is based on IDEAS, works
in the REAL-world is that there is a connection - no, even stronger, a
relationship - between the two. To be exceedingly terse, reality results
from thought - not vice-versa. The "thought-reality" that we as mathematicians
seem to be able to use in constructing theorems and systems of mathematics
is itself more fundamental than reality. That is why the physics and other
"reality-based" systems can use mathematics.
Of course I cannot prove this, but from my experience and viewpoint,
it appears to be so. A corollary of this view is that mathematical
truths are discovered, not invented.
--
Randy Yates
DSP Engineer
Ericsson Inc.
Research Triangle Park, NC, USA
randy...@ericsson.com, 919-472-1124
Chan-Ho Suh
"Randy Yates" <eus...@rtp.ericsson.com> wrote in message
news:3BAA7D20...@rtp.ericsson.com...
This reminds me of a story: The Zen Master Baso became ill one day. A
student came by and asked him, "Are you feeling well today?" Baso
responded, "Sun-faced Buddha. Moon-faced Buddha." [according to Buddhists,
the Sun-faced Buddha lives 1800 years, the Moon-faced Buddha for 1 day]
Am I a Platonist or Formalist? Sun-faced Buddha. Moon-faced Buddha.
Randy Yates
Huh?
--
% Randy Yates % "...the answer lies within your soul
%% DIGITAL SOUND LABS % 'cause no one knows which side
%%% Digital Audio Sig. Proc. % the coin will fall."
%%%% <ya...@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO
http://personal.lig.bellsouth.net/~yatesc
James Harris
Perspective.
James Harris
Randy Yates
>
> Perspective.
I suppose we will never know unless a perfectly objective intelligence
reveals it to us.
Chan-Ho Suh
"This reminds me of a story: The Zen Master Baso became ill one day. A
student came by and asked him, "Are you feeling well today?" Baso
responded, "Sun-faced Buddha. Moon-faced Buddha." [according to Buddhists,
the Sun-faced Buddha lives 1800 years, the Moon-faced Buddha for 1 day]
Am I a Platonist or Formalist? Sun-faced Buddha. Moon-faced Buddha."
To which Randy Yates responded, "Huh?"
To which James Harris thoughtfully responded, "Perspective".
"Randy Yates" <ya...@ieee.org> wrote in message
news:3BAC92FC...@ieee.org...
> James Harris wrote:
> >
> > Perspective.
>
> I suppose we will never know unless a perfectly objective intelligence
> reveals it to us.
> --
Every mathematician *does* know. Some don't know they know.
James is of course correct in saying (and this was kinda part of what I was
saying) that things depend on your perspective. But that was not the entire
content of my first message. "Sun-faced Buddha" is seen as positive, while
"moon-faced Buddha" is seen as negative. But just as a coin has two sides
and is neither one or the other, this duality is only a delusion.
Saying 'perspective' implies other perspectives. Saying 'objective' implies
subjectivity. Saying "heads" implies "tails." Saying "Sun-faced Buddha"
implies "Moon-faced Buddha". It is easy to say neither in fear of this
duality. But only when one affirms both, does one know the totality.
Every mathematician knows what mathematics is. It is what he/she does.
That is what it is and nothing more or less. So it is easy to think that
anything anyone does is mathematics, as long as it is mathematics for that
person. But that is incorrect. But to deny the differences in practice is
just as incorrect. The mathematics you do is the same as mine. But it is
not the same. It is both same and not the same.
To only push symbols around as a formalist is not doing mathematics. But to
indulge in mysticism is not either. Doing mathematics is about doing
something that is at the same time one and the other. But probably I am
mistaken if I say it is about doing both.
So I pose to you the question, "Was Baso really ill?" If you respond
"Sun-faced Buddha, Moon-faced Buddha" then you've learned nothing at all!
James Hunter
Chan-Ho Suh wrote:
> Chan-Ho originally wrote:
>
> Saying 'perspective' implies other perspectives. Saying 'objective' implies
> subjectivity. Saying "heads" implies "tails." Saying "Sun-faced Buddha"
> implies "Moon-faced Buddha". It is easy to say neither in fear of this
> duality. But only when one affirms both, does one know the totality.
>
> Every mathematician knows what mathematics is. It is what he/she does.
> That is what it is and nothing more or less. So it is easy to think that
> anything anyone does is mathematics, as long as it is mathematics for that
> person. But that is incorrect. But to deny the differences in practice is
> just as incorrect. The mathematics you do is the same as mine. But it is
> not the same. It is both same and not the same.
The only difference is that one person thinks it's a religion
and the other one doesn't, so it is NOT "both the same and not the same",
since:
"That statement is not only false, -> GIBBERISH".