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The Two Sides of Modern Axiomatics: Dedekind and Peano, Hilbert and Bourbaki

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Axiomatic Thinking I


This chapter focuses on two different facets of axiomatics: 1. the formal-logical side, linked to careful, rigorous establishing of the inferential structure of a theory, and 2. the conceptual-mathematical side, often linked to the establishment of new interconnections in mathematics, or ‘deeper’ ways of grounding some results. We explore this question, first by offering some classic examples in papers due to Hilbert and Bourbaki (and Hausdorff), and then going on to a simple example: the treatment of arithmetic around 1889 in the hands of Dedekind and Peano. In this thumbnail example, one can already find the above-mentioned duality. It is usual to insist on the equivalence of the works of Peano and Dedekind, but we shall argue that they had different aims—Peano focused on elementary arithmetic and its precise formulation in a new artificial language, while Dedekind aimed to systematize and “deepen the foundations” of number theory (elementary or not). We offer arguments for these claims, including a discussion of “modern” number theory in the 19th century, and we close with some philosophical remarks.

No concept is univocal in the mathematical sense”,

not even “the number concept”. (Kronecker [32])

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  1. 1.

    From the lecture course (Kronecker [32], 231). In context: “Will man aber z.B. den Begriff der Zahl erörtern, so muß man denselben im allerengsten Sinn, nämlich als Anzahl fassen und darf ihm nicht beimischen, was ursprünglich nicht darin liegt. Eindeutig kann man freilich den Begriff nicht fixieren, da es überhaupt keinen eindeutigen Begriff im mathematischen Sinne giebt, aber die Vieldeutigkeit muß so gering wie möglich sein.”.

  2. 2.

    On this topic see Corry [8], Anacona et al. [1], Mathias [32]. Let’s disregard the interesting topic of how they treated AC, following Hilbert, the fact is they included the axiom of Choice.

  3. 3.

    Ferreirós [16], Chap. 10.

  4. 4.

    This is the question that Zermelo, like the Bourbaki later, like Whitehead & Russell, was studying at the time.

  5. 5.

    Bourbaki [6], p. 197. More generally see James [29], on Hausdorff’s “classic text” in pp. 17–18, 213ff.

  6. 6.

    We may consider this to constitute the “semantic” aspect of axiomatics, in parallel with A. the syntactic side. But many terms have several meanings in our language: with ‘semantic’ I would not mean formal semantics, but rather cognitive semantics.

  7. 7.

    Hilbert [27], 1107–1109. This notion of deepening is explained below, in Sect. 6.1.2.

  8. 8.

    The list is provisional (Bourbaki [4], 1274): “nothing is farther from the axiomatic method than a static conception of the science. … The structures are not immutable, neither in number nor in their essential contents. It is quite possible that the future development of mathematics may increase the number of fundamental structures, revealing the fruitfulness of new axioms, or of new combinations of axioms”.

  9. 9.

    This word does not appear a single time, in the paper, nor even the basic notion of isomorphism. One also looks in vain for the general idea of a mapping.

  10. 10.

    See e.g. (accessed 4.03.2019); concerning the history of Galois theory, see Neumann [36], Ehrhardt [13].

  11. 11.

    At least in the usual ones, e.g., the versions of structuralism due to Shapiro, etc. The situation is different with category-theoretic structuralism, since category theory is precisely an explicit way of capturing those interrelations.

  12. 12.

    To be more complete, the latter are: 1. Dependence or independence of the sentences of a theory, in particular of the axioms; 2. the role of the axiom of continuity in a given theory;* 3. freedom-from-contradiction of a theory, in a relative or an absolute sense; and 4. the Entscheidungsproblem. Concerning *, he writes: (Hilbert [27], 1110): “For example, if one follows Planck and derives the second law of thermodynamics from the axiom of the impossibility of a perpetuum mobile of the second sort, then this axiom of continuity must be used in the derivation”.

  13. 13.

    On this general topic, see L. Corry [9].

  14. 14.

    Hilbert had a definitely set-theoretic perspective on math: the ZFC axioms ground both the number systems (in particular the complex numbers) and the general theory of fields. In 1910 he had described set theory as “that mathematical discipline which today occupies an outstanding role in our science, and radiates its powerful influence into all branches of mathematics” (cited in Ferreirós [17]).

  15. 15.

    A referee suggests that he may have had in mind the paper Weber [49].

  16. 16.

    V is conventionally employed to denote “the” intended model of axiomatic set theory, the universe of sets—although some set-theorists actually endorse a multi-verse perspective. See Arrigoni [2].

  17. 17.

    I thank an unknown referee for insisting that I should consider this question more carefully.

  18. 18.

    The other half introduces new questions such as the problems of consistency or decidability, see above.

  19. 19.

    For details, see e.g. Kleiner [30], or Wussing [50].

  20. 20.

    For details, see Scholz [42], or Ferreirós [16], Chap. 2.

  21. 21.

    To make a key point explicit, briefly, let me use Russell’s terminology: a set is a class-as-one, whenever a is a set one can write a ε x; but the classes that we find in the work of logicians prior to 1850 are either conceived intensionally, or they can be understood as a class-as-many.

  22. 22.

    I mentioned the differences between Dedekind and Peano already in Ferreirós ([15], 622), but my indications there—important in my view—were probably too quickly explained. For further details see also Gillies [23], Segre [42], my paper, or Skof [43].

  23. 23.

    Even if Grassmann’s intention was not to provide an axiomatic foundation of arithmetic (probably because he had in mind a deeper grounding), he should have clearly stated the general principle of induction. See Grassmann [25], 6–7 where the first “inductorische” proof can be found. His textbook aimed to present arithmetic “in its most rigorous form” (Preface, vi).

  24. 24.

    I replace some dots by parentheses, employ the symbol ‘∧’ for conjunction (‘and’), and replace an occurrence of ‘=’ in 7., which denotes ‘if and only if’, by ‘↔’. This first version also intermingled the principles of the logic of identity (restricted to number objects—axioms 2–5) with the famous five axioms (Peano [37], 94).

  25. 25.

    In the usual modern way, ‘a+’ would be written ‘+(a)’, and ‘s+’ written ‘+(S)’.

  26. 26.

    See McLarty [35]. As Ø. Ore wrote in 1935, Noether’s tendency in the discussion of the structure of algebraic domains was this: “one is not primarily interested in the elements of these domains but in the relations of certain distinguished subdomains” (cited in McLarty [35], 194). Also W. Krull underscore another Noether’s principle: “base all of algebra so far as possible on consideration of isomorphisms” (op. cit. 194). Both ingredients can be found in Dedekind already—thus I’d qualify what McLarty says in [35], 193 (see also Ferreirós [20]).

  27. 27.

    Except for the well-known problem of the Induction principle, which must be turned into an axiom-schema, as T. Skolem suggested for the first time.

  28. 28.

    A set S is Dedekind-infinite if and only if there is a proper subset T of S and a one-to-one map g: T → S (i.e., the subset T is bijectable with all of S).

  29. 29.

    Here I modernize and improve his notation, in fact Dedekind wrote ‘so(1)’ or ‘1o’, exploiting a (dangerous) ambiguity between 1 and {1}. He explained the danger himself in some manuscripts.

  30. 30.

    An example is the law of asymptotic distribution of prime numbers among the naturals, the Prime Number theorem proved in 1896 by Hadamard and de la Vallée Poussin (see below).

  31. 31.

    This means that the topic is no longer the natural numbers, but the integers in any number-field \( K \subset \mathbb{C}\), a simple example being the Gaussian integers a + ib (where i is the complex unit).

  32. 32.

    He actually says: “the other extensions may easily be carried out in the same way,” and he reserves himself the right [ich behalte mir vor] “sometime to present this whole subject in systematic form” (Dedekind [10], 792).

  33. 33.

    The key idea is implicit in Dedekind’s well known thesis that “arithmetic (algebra, analysis) is only a part of logic”—in a word, all of pure mathematics stands on the same foundation, whose basis is laid in Dedekind [10]. This is not the place to discuss Dedekind’s notion of logic, nor his form of logicism and its place in history (see Benis Sinaceur [3], Ferreirós [18], Reck [31], Klev [40]).

  34. 34.

    “Der von mir gefundene Beweis … ist nicht rein arithmetisch, sondern beruht zum Theil auf der Betrachtung stetig veränderlicher Gröſsen”.

  35. 35.

    Hardy’s 1921 Lecture to the Mathematical Society of Copenhagen, quoted in Chudnovsky et al. [7], p. 181. Hardy was later proved wrong by the work of A. Selberg.

  36. 36.

    On the varieties of arithmetization, see Petri and Schappacher [39].

  37. 37.

    “Es ist mir gelungen, die Theorie derjenigen complexen Zahlen [...] zu vervollständigen und zu vereinfachen; und zwar durch Einführung einer eigenthümlichen Art imaginarer Divisoren, welche ich ideale complexe Zahlen nenne;”

  38. 38.

    Mémoire sur la théorie des nombres complexes composés de racines de l'unité et de nombres entiers, 1851, in Kummer [33]. Kummer was sympathetic to Hegel’s ideas, and the choice of name “ideale Zahlen” may be an implicit reference to German idealism.

  39. 39.

    All of this had an impact on Cantor, who started his career in Berlin as a number theorist. See Ferreirós [16].

  40. 40.

    See Goldstein et al. [24].

  41. 41.

    Many years later, it turned out that Dirichlet’s theorem on primes in arithmetic progressions could be proved elementarily; this was done by A. Selberg. But the problem whether a given theorem in advanced number theory allows for an elementary derivation is often a very difficult open question, even today. Best known is the case of Fermat’s last theorem.

  42. 42.

    In fact, Peano’s Arithmetices principia went beyond the theory of natural numbers, including sections on the integers, the rationals, the real numbers. But the above statement remains true.

  43. 43.

    Peano was well aware of current work on number theory, and this applies both to analytical number theory and to Dedekind’s ideal theory, which he had followed closely.


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Ferreirós, J. (2022). The Two Sides of Modern Axiomatics: Dedekind and Peano, Hilbert and Bourbaki. In: Ferreira, F., Kahle, R., Sommaruga, G. (eds) Axiomatic Thinking I. Springer, Cham.

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