Abstract
What do mathematicians do? There are many ways of approaching this question, but one kind of answer is easy to give. Mathematicians study structures of different kinds: for instance, natural numbers, real numbers (the continuum), geometrical structures, groups, lattices, topological spaces, etc. For these structures, they develop corresponding theories, such as number theory, real analysis (theories of measure and integration), Euclidean geometry, group theory, lattice theory, topology, etc. In the late nineteenth century, a large and central class of foundational problems came up concerning the nature, the basic assumptions, and the presuppositions of theories of this general kind. These problems included questions concerning the axiomatization of important mathematical theories, the definitions of their basic concepts (e.g., natural numbers, real numbers, different geometrical objects), the relation of these mathematical theories to logic, etc. These questions are clearly among the foundational problems which working mathematicians are likely to find relevant. Hilbert’s efforts in his axiomatization of elementary geometry constitute a representative example of this early foundational work.1
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Notes
David Hilbert, Foundations of Geometry, translated by Leo Unger, tenth edition, revised and enlarged by Paul Bernays (La Salle, Illinois: Open Court, 1971) German original (first edition), 1899.
See here Jaakko Hintikka. “On the Development of the Model-Theoretic Viewpoint in Logical Theory,” Synthese 77 (1988): 1–36.
Jean van Heijenoort, “Logic as Language and Logic as Calculus.” Synthese 17 (1967): 324–330; Warren Goldfarb, “Logic in the Twenties,” Journal of Symbolic Logic 44 (1979): 351–368.
For a survey, see J. Barwise and S. Feferman (eds.), Model-Theoretic Logics. (New York: Springer-Verlag, 1985).
See Kurt Gödel, “On Formally Undecidable Propositions of Principia Mathematica and Related Systems,” ed. Jean van Heijernoort, From Frege to Gode! Source Book in Mathematical Logic 1879–1931 (Cambridge: Harvard University Press, 1967) 592–616.
Kurt Gödel “The Completeness of the Axioms of the Functional Calculus of Logic”. van Heijenoort, 582–591.
C. P. Snow, “Foreword” to G. H. Hardy, A Mathematician’s Apology (Cambridge: Cambridge University Press, 1967) 36.
These definitions go back to Alfred Tarski, “The Concept of Truth in Formalized Languages,” Logic, Semantics, Metamathematics (Oxford: Clarendon Press, 1952) 152–278.
How unruly the construction process may have to be is shown by the fact that in nonstandard models of elementary arithmetic the basic arithmetical relations are nonrecursive. See Solomon Fefermann, “Arithmetically Definable Models of Formalized Arithmetic,” Notices of the American Mathematical Society 5 (1958): 679–680; Dana Scott, “On Constructing Models for Arithmetic,” Infinitistic Methods (Oxford: Pergamon Press, 1959) 235–255; Solomon Feferman, Dana Scott, and S. Tennenbaum, “Models of Arithmetic via Function Rings,” Notices of the American Mathematical Society 6 (1959): 173.
See Jaakko Hintikka, “Form and Content in Quantification Theory,” Acta Philosophical Fennica 8 (1955): 5–55.
See Jacques Herbrand, Logical Writings, ed. Warren D. Goldfarb (Cambridge: Harvard University Press, 1971).
Stanislaw Ulam, “John von Neumann, 1903–1957,” Bulletin of the American Mathematical Society 64 (1958, May Supplement): l-49.
This title is of course a variant of the title of Gödel’s famous paper, “Über eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes,” Dialectia 12 (1958): 280–287; English translation (with a bibliography) in Journal of Philosophical Logic 9 (1980): 133–142.
See any good introduction to model theory, e.g., C. C. Chang, and H. J. Keisler, Model Theory (Amsterdam: North-Holland, 1973).
Hilary Putnam, “Models and Reality,” Journal of Symbolic Logic 45 (1981): 464482. Actually, Putnam’s arguments are subtler than what I have indicated, turning on the implications of Skolem—Löwenheim theorems rather than on those of incompleteness. This additional sophistication does not change the overall picture, however.
Leon Henkin, “Completeness in the Theory of Types,” Journal of Symbolic Logic 15 (1950): 81-91 (Cf. Peter Andrews, “General Models and Extensionality”, ibid. 37 (1972): 395-397).
Jaakko Hintikka, “Standard vs. Nonstandard Logic,” ed. Evandro Agazzi, Modern Logic: A Survey (Dordrecht: D. Reidel, 1981) 283–296.
Gottlob Frege, The Foundations of Arithmetic, German text with a translation by J. L. Austin (Oxford: Basil Blackwell, 1959) iv.
H. Poincare, Science and Hypothesis (New York: Dover, 1952) 1–16.
See Penelope Maddy, “Believing the Axioms I-II,” Journal of Symbolic Logic 53 (1988): 481–511 and 736–764; “New Directions in the Philosophy of Mathematics,” PSA 1984, ed. P. Kitcher (East Lansing, MI: Philosophy of Science Association, 1985) 425–447.
The excerpt is from Hartley Rogers’ mimeographed lectures at MIT in 1956. Most of these notes were published later as Hartley Rogers, Jr., Theory of Recursive Functions and Effective Computability (New York: McGraw-Hill, 1967).
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Hintikka, J. (1998). Is there Completeness in Mathematics after Gödel?. In: Language, Truth and Logic in Mathematics. Jaakko Hintikka Selected Papers, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2045-8_4
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