Fermat's last theorem and Hilbert's program

Mathematical Intelligencer

Volumn 19, Issue 1, 1997, Pages 64-66

  Velleman, Daniel J ^a  

^a NONE

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EID: 0039921923     PISSN: 03436993     EISSN: None     Source Type: Journal    
DOI: 10.1007/bf03024343     Document Type: Article

References (17)

1. I would like to thank Alexander George, Faan Tone Liu, Joseph Moore, and David Velleman for helpful comments on earlier drafts of this paper.
2. In his paper "On the infinite" (in Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics: Selected Readings, 2nd ed., Cambridge: Cambridge University Press, 1983, pp. 183-201, a translation of "Über das Unendliche," Math. Annalen 95 (1926), 161-190), Hilbert makes a distinction between what he calls finitary statements and infinitary, or ideal statements. According to Hilbert, finitary statements are meaningful; infinitary statements are not meaningful, but are useful in deriving finitary statements in the same way that, for example, the imaginary number i is sometimes useful in deriving facts about the real numbers. He describes mathematics as "a stock of two kinds of formulas: first, those to which the meaningful communications of finitary statements correspond; and, secondly, other formulas which signify nothing and which are the ideal structures of our theory" (p. 196, italics original). Although Hilbert was not very precise about where the line between finitary and infinitary should be drawn, it appears that he would have considered Fermat's Last Theorem to be a finitary statement. Certainly each instance of the theorem is a finitary statement, and the theorem itself can be thought of as just a shorthand way of asserting all of its instances. Hilbert considered such a statement to be "a hypothetical judgment which asserts something for the case when a numerical symbol is given" (p. 194). In the case of Fermat's Last Theorem, four numerical symbols must be given - namely, numerals to be substituted for x, y, z, and n.
3. In his paper "On the infinite" (in Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics: Selected Readings, 2nd ed., Cambridge: Cambridge University Press, 1983, pp. 183-201, a translation of "Über das Unendliche," Math. Annalen 95 (1926), 161-190), Hilbert makes a distinction between what he calls finitary statements and infinitary, or ideal statements. According to Hilbert, finitary statements are meaningful; infinitary statements are not meaningful, but are useful in deriving finitary statements in the same way that, for example, the imaginary number i is sometimes useful in deriving facts about the real numbers. He describes mathematics as "a stock of two kinds of formulas: first, those to which the meaningful communications of finitary statements correspond; and, secondly, other formulas which signify nothing and which are the ideal structures of our theory" (p. 196, italics original). Although Hilbert was not very precise about where the line between finitary and infinitary should be drawn, it appears that he would have considered Fermat's Last Theorem to be a finitary statement. Certainly each instance of the theorem is a finitary statement, and the theorem itself can be thought of as just a shorthand way of asserting all of its instances. Hilbert considered such a statement to be "a hypothetical judgment which asserts something for the case when a numerical symbol is given" (p. 194). In the case of Fermat's Last Theorem, four numerical symbols must be given -namely, numerals to be substituted for x, y, z, and n.
4. By "proofs like Wiles's" I mean proofs in which reasoning involving infinitary statements is used to justify a finitary conclusion. (See note 2 for the meanings of these terms.)
5. See, for example, Frege's The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number (Oxford: Blackwell, 1950, a translation of Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl, Breslau: Koebner, 1884), or Russell and Whitehead's Principia Mathematica, 2nd ed. (Cambridge: Cambridge University Press, 1925-1927).
6. See, for example, Frege's The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number (Oxford: Blackwell, 1950, a translation of Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl, Breslau: Koebner, 1884), or Russell and Whitehead's Principia Mathematica, 2nd ed. (Cambridge: Cambridge University Press, 1925-1927).
7. See, for example, Frege's The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number (Oxford: Blackwell, 1950, a translation of Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl, Breslau: Koebner, 1884), or Russell and Whitehead's Principia Mathematica, 2nd ed. (Cambridge: Cambridge University Press, 1925-1927).
8. Because of the use of concepts that go beyond the positive integers, it appears unlikely that Wiles's proof can be formalized using Peano's axiom system for arithmetic, usually abbreviated PA. It certainly can be formalized in ZF, but the full strength of ZF is not needed for the proof. Most likely the proof can be carried out in a formal system intermediate in strength between PA and ZF. A number of such systems have been studied; for example, see Solomon Feferman's paper "Theories of finite type related to mathematical practice" (in Jon Barwise, ed., Handbook of Mathematical Logic, Amsterdam: North-Holland, 1977, pp. 913-971). It would be an interesting project to determine exactly what axioms are needed for Wiles's proof. However, as far as I know this project has not yet been carried out. Furthermore, my concern in this paper is not so much with Wiles's proof itself, but rather with the use of infinitary reasoning in mathematics in general, Wiles's proof being a particularly striking and interesting example. Thus, for the purposes of this paper I will continue to refer to ZF.
9. Brouwer was the founder of intuitionism. In his paper "Intuitionism and Formalism" (Bulletin of the AMS 20 (1913), pp. 81-96, reprinted in Benacerraf and Putnam, pp. 77-89) he writes: "In the domain of finite sets in which the formalist axioms have an interpretation perfectly clear to the intuitionists, unreservedly agreed to by them, the two tendencies differ solely in their method, not in their results; this becomes quite different however in the domain of infinite or transfinite sets, where . . . the formalist introduces various concepts, entirely meaningless to the intuitionist." As a result, he concludes that "extended fields of research, which are without significance for the intuitionist are still of considerable interest to the formalist."
10. See Weyl's The Continuum: A Critical Examination of the Foundation of Analysis (New York: Dover, 1987, a translation of Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis, Leipzig: Veit, 1918). In the preface Weyl writes that "the house of analysis . . . is to a large degree built on sand. I believe that I can replace this shifting foundation with pillars of enduring strength. They will not, however, support everything which today is generally considered to be securely grounded. I give up the rest, since I see no other possibility" (p. 1).
11. See Weyl's The Continuum: A Critical Examination of the Foundation of Analysis (New York: Dover, 1987, a translation of Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis, Leipzig: Veit, 1918). In the preface Weyl writes that "the house of analysis . . . is to a large degree built on sand. I believe that I can replace this shifting foundation with pillars of enduring strength. They will not, however, support everything which today is generally considered to be securely grounded. I give up the rest, since I see no other possibility" (p. 1).
12. For example Lebesgue, in a letter to Borel, writes: "I believe that we can only build solidly by granting that it is impossible to demonstrate the existence of an object without defining it" (italics original). This letter is part of an exchange of letters among Lebesgue, Borel, Baire, and Hadamard that was published in Bulletin de la Société Mathématique de France, vol. 33 (1905), pp. 261-273. The quotation above is from an English translation that can be found in Appendix 1 of Gregory Moore's book Zermelo's Axiom of Choice: Its Origins, Development, and Influence, New York: Springer-Verlag, 1982.
13. For example Lebesgue, in a letter to Borel, writes: "I believe that we can only build solidly by granting that it is impossible to demonstrate the existence of an object without defining it" (italics original). This letter is part of an exchange of letters among Lebesgue, Borel, Baire, and Hadamard that was published in Bulletin de la Société Mathématique de France, vol. 33 (1905), pp. 261-273. The quotation above is from an English translation that can be found in Appendix 1 of Gregory Moore's book Zermelo's Axiom of Choice: Its Origins, Development, and Influence, New York: Springer-Verlag, 1982.
14. In "On the infinite", Hilbert lists two goals for his program. The first is very famous: "Wherever there is any hope of salvage, we will carefully investigate fruitful definitions and deductive methods. . . . No one shall drive us out of the paradise which Cantor has created for us." However, the second gives a more specific description of what he hoped to accomplish: "We must establish throughout mathematics the same certitude for our deductions as exists in ordinary elementary number theory, which no one doubts and where contradictions and paradoxes arise only through our own carelessness" (p. 191). Hilbert's mention of "contradictions and paradoxes" is of course a reference to the paradoxes in set theory, such as Russell's paradox, that arose near the turn of the century.
15. The axiom system Hilbert had in mind is not exactly the same as ZF. However, a proof of the consistency of ZF would certainly fulfill the goals of Hilbert's program.
16. 1 statements of number theory.
17. Of course, if it could be shown that Wiles's proof could be carried out in some axiomatic system weaker than ZF, then only the consistency of this weaker system would be needed as a hypothesis.