Abstract
We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics.
This paper is loosely based on the Skolem Lecture that I gave at the University of Oslo in June, 2010. The present paper has profited, both with respect to what it now contains and with respect to what it no longer contains, from the discussion following that lecture.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
The finitist types are called the finitist types of the first kind in “Finitism” (Tait 1981). The finitist types of the second kind are the types, corresponding to a constant equation, of the computations proving the equation. We need not discuss these here.
- 2.
Concerning Hilbert’s flirtation with logicism, see Hilbert (1918), his lecture “Prinzipien der Mathematik” in his 1917–1918 lectures on logic Hilbert (2011, Chap. 1) and Mancosu (1999), Sieg (1999a). Concerning his rejection of it, see his lecture “Probleme der Mathematischen Logik” in Hilbert (2011, Chap. 2).
- 3.
The fact that non-primitive recursive functions such as the Ackermann function, defined by twofold nested recursions, are definable using primitive recursion of higher type was already shown by Hilbert in (1926).
- 4.
The difficulty of dealing with equations of higher type was avoided by Spector in (1962) by restricting equations to those between numerical terms and restricting formulas to equations. Equations s = t, standing alone, between terms of type A → B are taken to be abbreviations for sx = tx, where the variable x of type A does not occur in s or t. \(s = t \rightarrow u = v\) is then taken to be an abbreviation for \((1 - sgn\vert s - t\vert ) + sgn\vert u - v\vert = 0\), where | x − y | denotes absolute difference.
- 5.
Gödel seems to have tried over and over again (see Gödel 1972, footnote h), but I think unsuccessfully, to eliminate this logical complexity.
- 6.
See Tait (2006) for further discussion.
- 7.
Jens Erik Fenstad suggested in conversation that it was perhaps Skolem’s close study of E. Netto’s Lehrbuch der Combinatorik (1901), a work of which he later collaborated in producing a second edition, that led him to his conception of foundations of arithmetic. Again, it is true that the work abounds in proofs by mathematical induction, but nowhere is a function or concept defined by induction.
- 8.
It is worth noting that in Dedekind’s case the impredicativity is limited to (1) constructing from a Dedekind infinite system ⟨D, 0, ′⟩ (i.e. where 0 ∈ D and x↦x′ is an injective operation on D whose range does not contain 0) the simply infinite subsystem \(\langle \mathbb{N}, 0, \prime\rangle\), and (2) defining the ordering < of \(\mathbb{N}\). According to his well-known letter to Keferstein (1890), (1) was simply part of his proof that the theory of simply infinite systems is consistent. His development of arithmetic from the axioms of a simply infinite system together with the axioms of order ( < ), in particular his familiar bottom-up derivation of the principle of definition by recursion (as opposed to Frege’s (1893) top-down derivation), is predicative. (2) is worth noting because < is primitive recursive, but the natural definition of its characteristic function is not by iteration; and the principle of definition by induction that Dedekind proved was restricted to definition by iteration. Dedekind defined the ordering n < x essentially as the least set which contains n′ and is closed under x↦x′. Dedekind needed the relation < in his derivation of the principle of definition by iteration.
- 9.
Frege (1879) and Dedekind (1888). It is worth noting that a version of Dedekind’s monograph under the same title can be found in his notebooks dating, as he himself indicates in the Preface to the first edition, from 1872–1878. A reference to this manuscript can also be found in a letter to Dedekind from Heinrich Weber, dated 13 November 1878. (See Dugac (1976).)
- 10.
It is usual to say that, whereas Frege understood the numbers as cardinals, Dedekind took them to be ordinals. Although there is some justice in this, there is also an objection: when Cantor introduced the concept of ordinal number, it was as the isomorphism types of well-ordered sets, just as he introduced the cardinals as isomorphism types of abstract sets. Just as Cantor did not refer to his transfinite numbers in (1883a) as ordinal numbers (see Tait (2000) for further discussion of this), Dedekind did not define the (finite) numbers as order types nor did he refer to them as ordinals. A significant thing about both systems of numbers, Cantor’s transfinite numbers and Dedekind’s finite numbers, and as opposed to Frege’s, is that they are defined intrinsically, without reference to the domain of either sets or well-ordered sets.
- 11.
Such a reduction is carried out in Robinson (1947) for the case in which D is finitary type, without the introduction of product types A ×B. Instead, she introduced the primitive recursive coding of pairs of natural numbers and then showed that all other primitive recursions could be reduced, using this coding, to iteration.
- 12.
By the Strong Normalization Theorem, definitional equality means having the same normal form. This theorem is preserved when the conversions of (pL, pR) and λxtx to p and t, respectively, are admitted.
- 13.
“Here I stop this monotonous series of reasonings.” Sect. IV.
- 14.
Many of the writers that I mention, including Poincaré and Dedekind, in fact take the least natural number to be 1. For the sake of simplicity, since nothing of relevance for us is really at stake in the choice, I will pretend that everyone starts with 0.
- 15.
For a discussion of this see Goldfarb (1988).
- 16.
It has sometimes been suggested that the difference between these two meanings of “intuition”, Kant’s and the particular sense of ‘intuition that’ that we are discussing, deriving from “intuitus”, was created by translating Kant’s “Anschauung” into English (and French) as “intuition”. But what Kant referred to as “Anschauung” in the Critique of Pure Reason, he sometimes parenthetically called “intuitus” and also referred to exclusively as intuitus in his earlier Inaugural Dissertation, written in Latin. (See for example Sect. 10.) Thus, in using the term “Anschauung”, he was merely translating the Latin into German: no new meaning was created by our translation of “Anschauung”; it was already there in his own use of the term “intuitus.” An interesting question, which I won’t attempt to answer here, is why Kant adapted the term intuitus in the way he did.
- 17.
Thus, the intuitive truths in this sense are the a priori truths (i.e. the ‘first principles’) in the original sense of that term.
- 18.
If A, B and C, D are pairs of like magnitudes, then A : B ≤ C : D if and only if for all positive numbers m and n, mB ≤ nA implies mC ≤ nD.
- 19.
One would have, on Kant’s behalf, to admit, given the construction of a number f(X) from the arbitrary number X, the iteration f Y(X) of this construction along the arbitrary number Y.
- 20.
In his discussion of mathematical reasoning in contrast with philosophical reasoning in the Discipline of Pure Reason, he speaks of geometric reasoning and algebraic reasoning but indicates no awareness of the special character of reasoning about the natural numbers.
References
Bernays, P. 1930–1931. Die Philosophie der Mathematik und die Hilbertsche Beweistheorie. Blätter für deutsche Philosophie 4: 326–367. Reprinted in Bernays (1976). A translation by P. Mancosu appears in Mancosu (1998), 234–265.
Bernays, P. 1976. Abhandlungen zur philosophie der mathematik. Darmstadt: Wissenschaftliche Buchgesellschaft.
Cantor, G. 1883a. Grundlagen einer allgemeinen Mannigfaltigheitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen. Leipzig: Teubner. A separate printing of Cantor (1883b), with a subtitloe, preface and some footnotes added. A translation Foundations of a general theory of manifolds: a mathmatico-philosophical investigation into the theory of the infinite by W. Ewald is in (Ewald, 1996, 639–920).
Cantor, G. 1883b. Über unendliche, lineare Punktmannigfaltigkeiten, 5. Mathematische Annalen 21: 545–586. In Cantor (1932).
Cantor, G. 1932. In Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. E. Zermelo. Berlin: Springer.
Curry, H. 1940. A formalization of recursive arithmetic. Bulletin of the American Mathematical Society 263–282.
Dedekind, R. 1872. Stetigkeit und irrationale Zahlen. Braunschweig: Vieweg. In Dedekind (1932). Republished in 1969 by Vieweg and translated in Dedekind (1963).
Dedekind, R. 1888. Was sind und was sollen die Zahlen? Braunschweig: Vieweg. In Dedekind (1932). Republished in 1969 by Vieweg and translated in Dedekind (1963).
Dedekind, R. 1890. Letter to Keferstein. Translated in van Heijenoort (1967), 99–103. Cambridge: Harvard University Press.
Dedekind, R. 1932. In Gesammelte Werke, vol. 3, ed. R. Fricke, E. Noether, and O. Ore. Braunschweig: Vieweg.
Dedekind, R. 1963. Essays on the theory of numbers. New York: Dover. English translation by W.W. Berman of Dedekind (1872) and Dedekind (1888).
Dugac, P. 1976. Richard Dedekind et les fondements des mathematiques (avec de nombeux textes inédits). Paris: Librairie Philosophique J. Vrin.
Ewald, W. (ed.). 1996. From Kant to Hilbert: a source book in the foundations of mathematics. Oxford: Oxford University Press. Two volumes.
Frege, G. 1879. Begriffsschrift, eine der arithmetischen nachgebildete Formalsprache des reinen Denkens. Halle: L. Nebert.
Frege, G. 1893. Grundgesetze der Arithmetik: Begriffsschriftlich abgeleitet, Band I, Jena: H. Pohle. Reprinted in 1962 along with Frege (1903) by Hildesheim: Georg Olms.
Frege, G. 1903. Grundgesetze der Arithmetik: Begriffsschriftlich abgeleitet, Band II. Jena: H. Pohle.
Gödel, K. 1958. Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica 12: 280–287. Reprinted with an Englsh translation in (Gödel, 1990, 240–252). Gödel (1972) is a revised version.
Gödel, K. 1972. On an extension of finitary mathematics which has not yet been used. In Collected works, vol. II Gödel (1990), 271–280. Revised version of Gödel (1958).
Gödel, K. 1990. Collected works, vol. II. Oxford: Oxford University Press.
Goldfarb, W. 1988. Poincaré against the logicists. In History and philosophy of modern mathematics: minnesota studies in the philosophy of science, vol. XI, ed. W. Aspray and P. Kitcher, 61–81. Minneapolis: The University of Minnesota Press.
Goodstein, R. 1945. Function theory in an axiom-free equation calculus. Proceedings of the London Mathematical Society 48: 401–34.
Goodstein, R. 1957. Recursive number theory. Amsterdam: North-Holland.
Grassmann, H. 1904. Gesammelte mathematische und physikalische Werke, vol. 2. Leipzig: Druck und Verlag von B.G. Teubner.
Hilbert, D. 1905. Über die Grundlagen der Logik und der Arithmetik. In Verhandlungen des Dritten Internationalen Mathematiker-Kongress. Leipzig: Teubner.
Hilbert, D. 1918. Axiomatisches denken. Mathematishe Annalen 78: 405–15. Reprinted in (Hilbert, 1932-9325, vol. 3, 1105–1115). Translated by W. Ewald in (Ewald, 1996, vol. 2).
Hilbert, D. 1922. Neubegründung der Mathematik: Erste Mitteilung. Abhandlungen aus dem Seminar der Hamburgischen Universität 1: 157–177. English translation in (Mancosu, 1998, 198–214) and (Ewald, 1996, 1115–1134).
Hilbert, D. 1923. Die logischen Grundlagen der Mathematik. Mathematische Annalen 88: 151–165. English translation in (Ewald, 1996, 1134–1148).
Hilbert, D. 1926. Über das Unendliche. Mathematische Annalen 95: 161–90. Translated by Stefan Bauer-Mengelberg in From Frege to Gödel: a source book in mathematical logic, 367–92.
Hilbert, D. 1932–9325. Gesammelte Abhandlungen. Bedrlin: Springer. 3 volumes.
Hilbert, D. 2011. In David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–1933, ed. M. Hallett, W. Ewald, W. Sieg and U. Majer. Berlin: Springer.
Kronecker, L. 1881. Grundzüge einer arithmetischen Theorie der algebraischen Grössen. In Leopold Kronecker’s Werke, vol. 2, ed. K. Hensel, 236–387 New York: Chelsea.
Kronecker, L. 1886. Über einige Anwendungen der Modulsysteme auf elementare algebraische Fragen. In Leopold Kronecker’s Werke, vol. 3, ed. K. Hensel, 147–208. New York: Chelsea.
Kronecker, L. 1887. ber den zahlbegriff. In Leopold Kronecker’s Werke, ed. K. Hensel, 251–274, New York: Chelsea.
Mancosu, P. (ed.). 1998. From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920’s. Oxford: Oxford University Press.
Mancosu, P. 1999. Between russell and hilbert: Behmann on the foundations of mathematics. Bulletin of Symbolic Logic 5: 303–330.
Martin-Löf, P. 1973. An intuitionistic theory of types: predicative part. In Logic colloqium ’73, ed. H. E. Rose and J. C. Shepherdson. Amsterdam: North-Holland.
Martin-Löf, P. 1998. An intuitionistic theory of types. In Twenty-five years of constructive type theory, ed. G. Sambin and J. Smith, 221–244. Oxford: Oxford University Press.
Netto, E. 1901. Lehrbuch der Combinatorik. Leipzig: Verlag von B.G. Teubner.
Poincaré, H. 1894. Sur la Nature du Raisonnement mathématique. Revue de métaphysique et de morale 2: 371–84. Translation by George Bruce Halsted in Ewald (1996), vol. 2, 972–982.
Poincaré, H. 1900. Du rôle de l’intuition et de la logique en mathématiques. In Compte rendu du Deuxiéme congrès international des mathématiciens tenu à Paris du 6 au 12 août 1900, 210–22. Pais: Gauthier-Villars. Translation by George Bruce Halsted, reprinted Ewald (1996), vol. 2, 1021–1038.
Poincaré, H. 1905. Les mathématiques et la logique. Revue de métaphysique et de morale 13: 815–35. Translation by George Bruce Halsted in Ewald (1996), vol. 2, 1021–1038.
Poincaré, H. 1906a. Les mathématiques et la logique. Revue de métaphysique et de morale 14: 17–34. Translation by George Bruce Halsted in Ewald (1996), vol. 2, 1038–1052.
Poincaré, H. 1906b. Les mathématiques et la logique. Revue de métaphysique et de morale 14: 294–317. Translation by George Bruce Halsted in Ewald (1996), vol. 2, 1052–1071.
Poincaré, H. 1906c. A propos de la logistique. Revue de métaphysique et de morale 14: 866–868.
Poincaré, H. 1909. Le llogique de l’infin i. Revue de métaphysique et de morale 17: 461–82. Translation by George Bruce Halsted in Ewald (1996), vol. 2, 1038–1052.
Robinson, J. 1947. Primitive recursive functions. Bulletin of the American Mathematical Society 53: 925–942.
Sieg, W. 1999a. Hilbert’s programs: 1917–1922. Bulletin of Symbolic Logic 5: 1–44.
Skolem, T. 1923. Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre. Matematikerkongressen in Helsingfors 4–7 Juli 1922, Den femte skandinaviske matematikerkongressen, Redogörelse, 217–232. Helsingfors: Akademiska Bokhandeln.
Skolem, T. 1947. The development of recursive arithmetic. In Copenhagen: proceedings of the tenth congress of scandinavian mathematicians, 1–16. Reprinted in Skolem (1970), 499–514.
Skolem, T. 1956. A version of the proof of equivalence between complete induction and the uniqueness of primitive recursions. Kongelige Norske Videnskabsselskabs Forhandlinger XXIX: 10–15.
Skolem, T. 1970. In Selected works in logic, ed. J.E. Fenstad. Oslo: Universitetsforlaget.
Spector, C. 1962. Provably recursive functionals of analysis: a consistency proof of analysis by an extension of the principles formulated in current intuitionistc mathematics. In Recursive function theory, proceedings of symposia in pure mathematics, vol. 5, ed. J. Dekker, 1–27. Providence: American Mathematical Society.
Tait, W. 1981. Finitism. Journal of Philosophy 78: 524–556.
Tait, W. 2000. Cantor’s Grundlagen and the paradoxes of set theory, 269–90. Reprinted in Tait (2005b), 252–275.
Tait, W. 2005a. Proof-theoretic semantics for classical mathematics. In Proof-theoretic semantics for classical mathematics, ed. R. Kahle and P. Schroeder-Heister. Special edition of Synthese.
Tait, W. 2005b. The provenance of pure reason: essays in the philosophy of mathematics and its history. Oxford: Oxford University Press.
Tait, W. 2006. Gödel’s interpretation of intuitionism. Philosophia Mathematica 14: 208–228.
van Heijenoort, J. (ed.). 1967. From Frege to Gödel: A Source Book in Mathematical Logic. Cambridge: Harvard University Press.
Weyl, H. 1921. Über die neue Grundlagenkrise der Mathematik. Mathematische Zeitschrift 10: 39–79. Translated by P. Mancosu in Mancosu (1998).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media Dordrecht.
About this chapter
Cite this chapter
Tait, W.W. (2012). Primitive Recursive Arithmetic and Its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections. In: Dybjer, P., Lindström, S., Palmgren, E., Sundholm, G. (eds) Epistemology versus Ontology. Logic, Epistemology, and the Unity of Science, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4435-6_8
Download citation
DOI: https://doi.org/10.1007/978-94-007-4435-6_8
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-4434-9
Online ISBN: 978-94-007-4435-6
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)