Notes on
Philosophy of Mathematics
An Introduction to the World of Proofs and Pictures
London & New York, Routledge, 1999
A second edition appeared in January 2008
Corrections (1st edition)
The following mistakes are in the first and second impression. They have been corrected in the third printing.
p(age) 8, l(ine) 7: "principle" should be "principal"
p 37: in the theorem the first "+" should be "="
p 44, diagram 3.10: a "b" should be inserted in the bottom line, similar to the "a".
p 52, l 14: "Cloths" should be "Clothes"
p 57, l 4: "principle" should be "principal"
p 66, l 13 "5>3" should be "5<3"
p 72, l 16, "(y=sy)" should be "(x=sy)"
p 76, near bottom: "g" should have no prime mark
p 77, l 6, the quantifier should have no quote mark
p 136, l 12: Should be "Platonism of Frege"
p 153, l 10: "aide" should be "aid"
p 170, l 27: "plant" should be "planet"
Some of the entries in the bibliography are slightly out of alphabetical order.
The title of Kitcher's 1983 book should be The Nature of Mathematical Knowledge.
Thanks to Louis Levine, Ken Manders and others for these corrections.
Corrections (2nd edition)
p 50: The diagram should have a dark line at the top left between the first two units, indicating 1 + 1. The discussion above the diagram should be “so the third is 1 + 2 = 3, the forth is 2 + 3 = 5, and so on.”
P 174, line 8: should be 6 = 3 + 3.
I would be glad to learn of any more.
Reviews/Discussions
• W. Faris in Notices of the American Mathematical Society (Vol 47, No 10, 2000, 1276-1280)
• Gila Hanna, Preuve, available at: http://www-cabri.imag.fr/Preuve/Newsletter/000102.html
• Ulrich Majer, Vienna Circle Yearbook #8, (2000) 344-349
• Marco Ruffino, Erkenntnis, Vol. 54, Issue: 3 (May 2001) 403-407
• Thomas Hofweber, British Journal for the Philosophy of Science, 52, (2001) 413-416
• Stephen Read, Studia Logica (forthcoming)
• Eduard Glas, Mathematical Reviews, 2001
• R. Cook, Mind, (2004), 154-57
• D. Boersema, Teaching Philosophy, Sept. 2002
• O. Bueno, "What does a picture prove?", Metascience, vol 11, no 1, March 2002, 61-65
• M. Colyvan, Philosophy in Review, Feb. 2002
• M. Resnik, Philosophia Matematica, Feb. 2003
• J. Folina, "Pictures, Proofs, and 'Mathematical Practice': A Reply to James Robert Brown", Brit. J. Phil. Sci. 1999, 425-429. This is a reply to my "Proofs and Pictures", Brit. J. Phil. Sci. 1997, which became (with modifications) chapter three of the book.
• J. Folina, Bulletin of Symbolic Logic, vol. 9, (2003)
I would be glad to hear of any more.
Comments
In the chapter on constructive mathematics (ch 8) I cited some work of Geoffrey Hellman, a prominent philosopher of science and mathematics (Minnesota), to the effect that constructive math couldn't provide the mathematics needed for science. Perhaps I was not sufficiently critical here, or at least failed to note that this is controversial. Douglas Bridges, a prominent constructive mathematician (Canterbury, New Zealand), rejects Hellman's claims. He sent me an email from which I quote:
First, Hellman's argument against Gleason's Theorem: this is based on a brouwerian counterexample to a version of Gleason's Theorem that is classically, but not constructively, equivalent to Gleason's original result. (In fact, his counterexample shows that the principal axes theorem can't be proved, in its exact form., constructively, a fact that had already been shown, probably on several occasions but certainly in a paper of mine, many years beforehand.) However, as Fred Richman and I showed in the paper "A constructive proof of Gleason's Theorem" (J. Functional Anal. 162, 287-312, 1999), Gleason's Theorem, in its original form dealing with measures on the projections, can be proved constructively.
Secondly, Hellman's analysis and interpretation of the Pour El & Richards result: quite
simply, he got this wrong. See my paper "Constructive mathematics and unbounded
operators-a reply to Hellman" (J. Philosophical Logic 24, 549-561, 1995). Hellman wrote
a rejoinder to my paper-I can't recall the exact reference-which was also published in J.
Phil. Logic, but which still got things wrong. Since then, I have written another paper,
"Can constructive mathematics be applied in physics", which should appear in J. Phil.
Logic any day now.
Thirdly, Hellman's interpretation of Dummett: again, this is totally off-beam. See my two
J. Phil.. Logic papers, and if you are still unconvinced, contact Michael Dummett himself.
(He has already sent me a draft of something about this issue that he will include in the
revised edition of his book on intuitionism, which is due out later this year.)
Fourthly: Hellman's more recent paper about constructive mathematics and general relativity. It is harder to counter this one precisely, since there is, as yet, no decent constructive theory of manifolds.... One person, Helen Billinge, has a preprint rebutting Hellman's claims on this topic; she is at King's College, London, and would probably let you have a copy.
... It is hard to get people to realise what is actually going on. In this regard, I would recommend you to look at papers like"Interview with a constructive mathematician" and "Intuitionism as generalization" on Fred Richman's home pages (http://www.math.fau.edu/Richman/). You will see there that some of us working in constructive mathematics are not actually hidebound by philosophical dogma, and regard the subject as something more akin to investigating mathematics with intuitionistic logic.
-Douglas Bridges (October 1999)
Geoffrey Hellman replies:
A theorem is a theorem and it seems that Gleason's theorem, stripped of any info about principal axes is constructively provable. A graduate student, Matthew Frank, at the Univ. of Chicago (math dep't) claims to have independently arrived at the Richman/Bridges result. Assuming these results are correct (and I have no reason to doubt them), I am pleased that I helped stimulate a search for them. It remains open whether the non-constructive content of the version I dealt with (as stated in Cooke-Keane-Moran, appendix to RIG Hughes' book on QM) has any scientific applications.
Bridges claims that I got Pour-El and Richards wrong, but I don't think so. I granted that what I showed goes beyond the constructivist framework--my argument is only intelligible to the classicist. But I believe it is correct and no less important for that. The unbounded operators that the constructivist may recognize are provably different from the ones used in QM even on the restricted domain of constructive inputs (vectors in Hilbert space). And because of discontinuities, the constructivist cannot "come arbitrarily close" to the classicists values for inputs on which the constructive substitute operators cannot be defined. So I believe that my points in my published (JPL, '97) rejoinder to Bridges still stand.
Similarly, I don't think Bridges' interpretation of Dummett comes to grips with the radical views he espouses, nor with Dummett's attempt to ground intuitionism in Wittgensteinian "theory of meaning". Indeed, Dummett has been heard to say, at conferences, that if physics must believe non-constructive mathematics, then "so much the worse for physics". Bridges doesn't seem to realize how different his own position actually is.
Helen Billinge has written a paper dealing with my more recent BJPS paper on space-time physics and constructive math. You would find it revealing, I think. She concedes that there is little reason to think the singularity theorems, for example, really can be "constructivized", but she also thinks that the radical constructivist will not care: philosophical a priorism here is simply an integral part of the radical's view, and there is no hope of rational debate about it. I hope this appears in print: I couldn't ask for stronger confirmation of my position!
- Geoffrey Hellman (October 1999)
Matthew Frank (a mathematician at Chicago) adds the following to the above discussion.
I suggest the following example to clarify point 2 in the above comments; I think it shows all the relevant features of the Pour-El/Richards theorem. This is acceptable in both constructive and classical mathematics: Consider the Hilbert space H of all sequences (a_n) of complex numbers for which the sum of |a_n|^2 converges. Consider the operator T sending (a_n) to (n a_n) ("multiply the nth coordinate by n",) defined on the domain where the sum of n^2 |a_n|^2 converges. Let b_n = 1/n if 2n is the first even number >2 not the sum of two primes, 0 otherwise. Then (b_n) is a vector in H, and computable. One can prove classically that (b_n) is in the domain of T, and that T(b_n) = (n b_n) is not a computable vector. However, one can not (at the moment) prove constructively that (b_n) is in the domain of T. If one could prove this constructively, one would also prove that the sum of n^2 |a_n|^2 converged, and by seeing whether this sum was near 0 or near 1 one could settle the Goldbach conjecture.
The Pour-El/Richards result shows that this happens generically: (roughly) any closed unbounded operator on any Hilbert space takes some computable input to some uncomputable output. Hence, for any unbounded operator defined in constructive mathematics, there are computable vectors which one can prove classically, but not constructively, to be in the domain of the operator.
- Matthew Frank (February 2000)
Douglas Bridges adds:
The point made by Matt Frank in the last paragraph of his contribution was made by me
as well, on page 445 of my article "Can constructive mathematics be applied in physics?"
(J. Phil. Logic 28, 39-453, 1999).
I would be glad to include further comments on any aspect of the book.