> 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 -->