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From: CMR <jackogreen.domain.name.hidden>

Date: Fri, 24 Oct 2003 11:33:15 -0700

*> Bruno,
*

*>
*

*> Thanks for the Hardy quote: it still reads well, indeed,
*

*> But I am afraid I can't agree with your reading of it or
*

*> your version of mathematical realism (and physical realism)
*

*> which strikes me as quite orthogonal to Hardy's. Nowhere does
*

*> he claim or suggest that "physical reality could be mathematical
*

*> reality seen from the inside"! What he stresses is that "mathematical
*

*> reality" is something entirely more precisely known and accessed
*

*> than "physical reality" and he is surely correct as the whole EPR
*

*> debacle clearly demonstrates. One may add that what little we
*

*> know about "physical reality" is what we manage to map to
*

*> mathematical reality! In truly Platonic terms "Reality" is
*

*> purely mathematical and what Physics is about is better
*

*> named "appearance" (or corrupted reality).
*

*>
*

Greetings,

I would tend to agree with Chaitin that your apparent confidence in the

"precise accessibility" of Mathematics as opposed to that of physics may be

misplaced; I would also agree that Leibniz's insights are probably more

useful than Plato's on the ultimate "nature" of reality:

*>From "Should Mathematics Be More Like Physics? Must Mathematical Axioms Be
*

Self-Evident?"

Gregory Chaitin, IBM Research Division

"A deep but easily understandable problem about prime numbers is used in the

following to illustrate the parallelism between the heuristic reasoning of

the mathematician and the inductive reasoning of the physicist...

[M]athematicians and physicists think alike; they are led, and sometimes

misled, by the same patterns of plausible reasoning."

-George Pólya, "Heuristic Reasoning in the Theory of Numbers", 1959,

reprinted in Alexanderson, The Random Walks of George Pólya, 2000.

"The role of heuristic arguments has not been acknowledged in the philosophy

of mathematics, despite the crucial role that they play in mathematical

discovery. The mathematical notion of proof is strikingly at variance with

the notion of proof in other areas... Proofs given by physicists do admit

degrees: of two proofs given of the same assertion of physics, one may be

judged to be more correct than the other."

-Gian-Carlo Rota, "The Phenomenology of Mathematical Proof", 1997, reprinted

in Jacquette, Philosophy of Mathematics, 2002, and in Rota, Indiscrete

Thoughts, 1997.

"There are two kinds of ways of looking at mathematics... the Babylonian

tradition and the Greek tradition... Euclid discovered that there was a way

in which all the theorems of geometry could be ordered from a set of axioms

that were particularly simple... The Babylonian attitude... is that you know

all of the various theorems and many of the connections in between, but you

have never fully realized that it could all come up from a bunch of

axioms... [E]ven in mathematics you can start in different places... In

physics we need the Babylonian method, and not the Euclidian or Greek

method."

-Richard Feynman, The Character of Physical Law, 1965, Chapter 2, "The

Relation of Mathematics to Physics".

"The physicist rightly dreads precise argument, since an argument which is

only convincing if precise loses all its force if the assumptions upon which

it is based are slightly changed, while an argument which is convincing

though imprecise may well be stable under small perturbations of its

underlying axioms."

-Jacob Schwartz, "The Pernicious Influence of Mathematics on Science", 1960,

reprinted in Kac, Rota, Schwartz, Discrete Thoughts, 1992.

"It is impossible to discuss realism in logic without drawing in the

empirical sciences... A truly realistic mathematics should be conceived, in

line with physics, as a branch of the theoretical construction of the one

real world and should adopt the same sober and cautious attitude toward

hypothetic extensions of its foundation as is exhibited by physics."

-Hermann Weyl, Philosophy of Mathematics and Natural Science, 1949, Appendix

A, "Structure of Mathematics", p. 235.

The above quotations are eloquent testimonials to the fact that although

mathematics and physics are different, maybe they are not that different!

Admittedly, math organizes our mathematical experience, which is mental or

computational, and physics organizes our physical experience. [And in

physics everything is an approximation, no equation is exact.] They are

certainly not exactly the same, but maybe it's a matter of degree, a

continuum of possibilities, and not an absolute, black and white difference.

Certainly, as both fields are currently practiced, there is a definite

difference in style. But that could change, and is to a certain extent a

matter of fashion, not a fundamental difference.

A good source of essays that I-but perhaps not the authors!-regard as

generally supportive of the position that math be considered a branch of

physics is Tymoczko, New Directions in the Philosophy of Mathematics, 1998.

In particular there you will find an essay by Lakatos giving the name

"quasi-empirical" to this view of the nature of the mathematical enterprise.

Why is my position on math "quasi-empirical"? Because, as far as I can see,

this is the only way to accommodate the existence of irreducible

mathematical facts gracefully. Physical postulates are never self-evident,

they are justified pragmatically, and so are close relatives of the not at

all self-evident irreducible mathematical facts that I exhibited in Section

VI.

I'm not proposing that math is a branch of physics just to be controversial.

I was forced to do this against my will! This happened in spite of the fact

that I'm a mathematician and I love mathematics, and in spite of the fact

that I started with the traditional Platonist position shared by most

working mathematicians. I'm proposing this because I want mathematics to

work better and be more productive. Proofs are fine, but if you can't find a

proof, you should go ahead using heuristic arguments and conjectures.

Wolfram's A New Kind of Science also supports an experimental,

quasi-empirical way of doing mathematics. This is partly because Wolfram is

a physicist, partly because he believes that unprovable truths are the rule,

not the exception, and partly because he believes that our current

mathematical theories are highly arbitrary and contingent. Indeed, his book

may be regarded as a very large chapter in experimental math. In fact, he

had to develop his own programming language, Mathematica, to be able to do

the massive computations that led him to his conjectures.

See also Tasic, Mathematics and the Roots of Postmodern Thought, 2001, for

an interesting perspective on intuition versus formalism. This is a key

question-indeed in my opinion it's an inescapable issue-in any discussion of

how the game of mathematics should be played. And it's a question with which

I, as a working mathematician, am passionately concerned, because, as we

discussed in Section VI, formalism has severe limitations. Only intuition

can enable us to go forward and create new ideas and more powerful

formalisms.

And what are the wellsprings of mathematical intuition and creativity? In

his important forthcoming book on creativity, Tor Nørretranders makes the

case that a peacock, an elegant, graceful woman, and a beautiful

mathematical theory, are all shaped by the same forces, namely what Darwin

referred to as "sexual selection". Hopefully this book will be available

soon in a language other than Danish! Meanwhile, see my dialogue with him in

my book Conversations with a Mathematician.

see:

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/bonn.html

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/kirchberg.html

CMR

<-- insert gratuitous quotation that implies my profundity here -->

Received on Fri Oct 24 2003 - 15:34:57 PDT

Date: Fri, 24 Oct 2003 11:33:15 -0700

Greetings,

I would tend to agree with Chaitin that your apparent confidence in the

"precise accessibility" of Mathematics as opposed to that of physics may be

misplaced; I would also agree that Leibniz's insights are probably more

useful than Plato's on the ultimate "nature" of reality:

Self-Evident?"

Gregory Chaitin, IBM Research Division

"A deep but easily understandable problem about prime numbers is used in the

following to illustrate the parallelism between the heuristic reasoning of

the mathematician and the inductive reasoning of the physicist...

[M]athematicians and physicists think alike; they are led, and sometimes

misled, by the same patterns of plausible reasoning."

-George Pólya, "Heuristic Reasoning in the Theory of Numbers", 1959,

reprinted in Alexanderson, The Random Walks of George Pólya, 2000.

"The role of heuristic arguments has not been acknowledged in the philosophy

of mathematics, despite the crucial role that they play in mathematical

discovery. The mathematical notion of proof is strikingly at variance with

the notion of proof in other areas... Proofs given by physicists do admit

degrees: of two proofs given of the same assertion of physics, one may be

judged to be more correct than the other."

-Gian-Carlo Rota, "The Phenomenology of Mathematical Proof", 1997, reprinted

in Jacquette, Philosophy of Mathematics, 2002, and in Rota, Indiscrete

Thoughts, 1997.

"There are two kinds of ways of looking at mathematics... the Babylonian

tradition and the Greek tradition... Euclid discovered that there was a way

in which all the theorems of geometry could be ordered from a set of axioms

that were particularly simple... The Babylonian attitude... is that you know

all of the various theorems and many of the connections in between, but you

have never fully realized that it could all come up from a bunch of

axioms... [E]ven in mathematics you can start in different places... In

physics we need the Babylonian method, and not the Euclidian or Greek

method."

-Richard Feynman, The Character of Physical Law, 1965, Chapter 2, "The

Relation of Mathematics to Physics".

"The physicist rightly dreads precise argument, since an argument which is

only convincing if precise loses all its force if the assumptions upon which

it is based are slightly changed, while an argument which is convincing

though imprecise may well be stable under small perturbations of its

underlying axioms."

-Jacob Schwartz, "The Pernicious Influence of Mathematics on Science", 1960,

reprinted in Kac, Rota, Schwartz, Discrete Thoughts, 1992.

"It is impossible to discuss realism in logic without drawing in the

empirical sciences... A truly realistic mathematics should be conceived, in

line with physics, as a branch of the theoretical construction of the one

real world and should adopt the same sober and cautious attitude toward

hypothetic extensions of its foundation as is exhibited by physics."

-Hermann Weyl, Philosophy of Mathematics and Natural Science, 1949, Appendix

A, "Structure of Mathematics", p. 235.

The above quotations are eloquent testimonials to the fact that although

mathematics and physics are different, maybe they are not that different!

Admittedly, math organizes our mathematical experience, which is mental or

computational, and physics organizes our physical experience. [And in

physics everything is an approximation, no equation is exact.] They are

certainly not exactly the same, but maybe it's a matter of degree, a

continuum of possibilities, and not an absolute, black and white difference.

Certainly, as both fields are currently practiced, there is a definite

difference in style. But that could change, and is to a certain extent a

matter of fashion, not a fundamental difference.

A good source of essays that I-but perhaps not the authors!-regard as

generally supportive of the position that math be considered a branch of

physics is Tymoczko, New Directions in the Philosophy of Mathematics, 1998.

In particular there you will find an essay by Lakatos giving the name

"quasi-empirical" to this view of the nature of the mathematical enterprise.

Why is my position on math "quasi-empirical"? Because, as far as I can see,

this is the only way to accommodate the existence of irreducible

mathematical facts gracefully. Physical postulates are never self-evident,

they are justified pragmatically, and so are close relatives of the not at

all self-evident irreducible mathematical facts that I exhibited in Section

VI.

I'm not proposing that math is a branch of physics just to be controversial.

I was forced to do this against my will! This happened in spite of the fact

that I'm a mathematician and I love mathematics, and in spite of the fact

that I started with the traditional Platonist position shared by most

working mathematicians. I'm proposing this because I want mathematics to

work better and be more productive. Proofs are fine, but if you can't find a

proof, you should go ahead using heuristic arguments and conjectures.

Wolfram's A New Kind of Science also supports an experimental,

quasi-empirical way of doing mathematics. This is partly because Wolfram is

a physicist, partly because he believes that unprovable truths are the rule,

not the exception, and partly because he believes that our current

mathematical theories are highly arbitrary and contingent. Indeed, his book

may be regarded as a very large chapter in experimental math. In fact, he

had to develop his own programming language, Mathematica, to be able to do

the massive computations that led him to his conjectures.

See also Tasic, Mathematics and the Roots of Postmodern Thought, 2001, for

an interesting perspective on intuition versus formalism. This is a key

question-indeed in my opinion it's an inescapable issue-in any discussion of

how the game of mathematics should be played. And it's a question with which

I, as a working mathematician, am passionately concerned, because, as we

discussed in Section VI, formalism has severe limitations. Only intuition

can enable us to go forward and create new ideas and more powerful

formalisms.

And what are the wellsprings of mathematical intuition and creativity? In

his important forthcoming book on creativity, Tor Nørretranders makes the

case that a peacock, an elegant, graceful woman, and a beautiful

mathematical theory, are all shaped by the same forces, namely what Darwin

referred to as "sexual selection". Hopefully this book will be available

soon in a language other than Danish! Meanwhile, see my dialogue with him in

my book Conversations with a Mathematician.

see:

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/bonn.html

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/kirchberg.html

CMR

<-- insert gratuitous quotation that implies my profundity here -->

Received on Fri Oct 24 2003 - 15:34:57 PDT

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