Foundations of Mathematics and Set Theory As Viewed From a Historical Viewpoint


Prof. Antonino Drago – TRANSCEND Media Service

Along centuries the foundations of Mathematics (FoM) have been considered as constituted by some mathematical notions, e.g., number, point, infinitesimal, limit, etc., each one uniting in itself mathematical and philosophical meanings. But the crisis of non-Euclidean geometries forced mathematicians to answer which are the very FoM.

According to their dominant philosophy, they re-interpreted past history of Mathematics from the inside only; they purged from any philosophical import the basic notions of previous theories, e.g. point and line in Euclidean geometry, infinitesimals in calculus, limit in Cauchy-Weierstrass’ reform of calculus (still including the actual infinity in implicitly picking up the final, single point inside the approximating intervals), etc.. In addition, no more a common notion was enough to cover the great number and variety of the mathematical theories.

The mathematicians explored the other possibility too, i.e. to consider a single theory as constituting the founrdation for all others. The last attempts of this attitude have been performed by Frege and Cantor that suggested respectively Logic and Set theory, the former one by deliberately dismissing the philosophical imports of his basic notions (Why LEM?), the latter one by even more extending the implicit philosophical import of the basic notion of his theory (i.e. set), so to include the notion of infinity. In fact, both his first notions (“set” and “belong to”) – which never have been accurately defined inside the mathematical realm only – and its methodology for dealing with the degrees of infinity actually chose the philosophical notion of actual infinity. Owing to this incorporation of a new kind of philosophy this theory obtained much more than a technique (as previously the philosophical notion of infinitesimals did when generated calculus).

However, both these new suggestions failed few years after their births, owing to the discoveries by Russell’s and others’ of basic contradictions.

Two other mathematicians, Hilbert and Brouwer, apperceived that the search for the FoM has to be performed without inventing a new theory as the basis of all others. Rather, both wanted to re-construct the entire body of Mathematics according to the following tenets: i) past mathematicians built theories without accurately define the FoM; ii) a successful re-construction of all possible theories will obtain an accurate definition of FoM; iii) this re-construction requires the definition of a bit of philosophy, i.e. a specific methodology of this re-construction program. Hence, both actually explored the re-construction of the entire corpus of Mathematics according to a basic problem, to exhaustively recognize its foundation.

Hilbert deserves the merit to have made clear (although after a long meditation of twenty five years) the entire methodology of his program. Instead Brouwer suggested a (subjective) methodology about the starting point only of his program; he left to subsequent times its complete clarification (for instance, which specific kind of logic had to be adopted).

After the presentation of the two programs a great philosophical debate followed; but even a century after, it resulted to be inconclusive.[i] Moreover Goedel’s theorems stopped the original Hilbert’s program. On the other hand, Brouwer’s program was unsuccessful both in convincing the mathematicians’ community to discuss the formal theories and in consistently develop his own program (since some notions including the actual infinity, e.g. spreads, and formal axiomatizations, e.g. Heyting’s intuitionist logic, have been introduced).

However, after the half of the 20th Century, the works by Markov and Bishop allowed to conceive the FOM as diverging according to a specific dichotomy; either the constructive tools assuring (almost consistently in each author) the existence of a mathematical object through a specific finite algorithm or evenly making use of no more than the notion of potential infinity (PI), or the tools of classical mathematics using freely actual infinity (AI) provided that a contradiction is not met.

Independently, Beth advised that the current development of Mathematics is biased by the common use of only one model for the systematic organization of a mathematical theory, i.e. the deductive (axiomatic) one (AO)[ii]. Weyl, the same Beth, and then van Heijenoort, Kreisel and Hintikka interpreted Goedel’s theorems as a suggestion for looking for an alternative model of a theory organization[iii]. Actually, some founders of important scientific theories (e.g. L. Carnot, Lavoisier, S. Carnot, Lobachevsky, Galois, Boole, Klein, Brouwer, Kolmogorov, Markov) did not presented them according to the deductive model. A comparative analysis of these theories suggests the following characteristic features of their kind of organization.[iv] I call it a problem-based organization (PO) since the theory starts from a universal problem, then it looks for a new scientific method capable of solving this problem. It then argues through doubly negated propositions, each being not equivalent to its affirmative version; thus, such a kind of proposition belongs to a non-classical logic (e.g., intuitionist logic). In the original texts these doubly negated propositions are grouped in some cycles of argument, each posing a sub-problem and then solving it by means of an ad absurdum proof concluding no more than a doubly negated proposition.[v] A final ad absurdum proof concerning all cases of the main problem concludes a doubly negated predicate ØØT. At this point, the author, in the belief to have collected enough argumentative evidence, converts the above conclusion to the corresponding affirmative predicate T; from which he then draws in classical logic all consequences. This change of both predicate and the whole logic amounts to jump from a problem-based organization theory to a subsequent deductive theory.

All in the above leads to consider the FOM as constituted by two dichotomies of both philosophical and mathematical imports; 1) either the PI or the AI, on which are formally developed the two respective kinds of mathematical tools, i.e. either the constructive ones or the classical ones; 2) either a AO theory or a PO theory, which are formally developed according to the two respective kinds of logic, either the classical one or the non-classical one.

Cantorr’s Set theory is recognised as relying on the (declared) two choices for AI and AO, i.e. the most powerful and promising ones for a mathematician. No surprise if it took side with Hilbert’s program, which claimed the same two choices and to want to remain for ever inside the “Cantor’s Paradise”. Instead Brouwer’s program declared the basic choice for PI and moreover distrusted classical logic; hence, though implicitly, he chose PO.

Being the two alternatives in each dichotomy exclusive in nature, the comparison of the two above programs, differing in both basic choices, generated an incommensurability phenomenon; in other terms, no common language was possible since their common notions (e.g., number one, infinity, LEM, etc.) presented radical variations in meanings or even have been claimed by the opponent scholar as inexistent (e.g. intuition, logical alternative, formal, etc.).

In retrospect, This phenomenon of incommensurability gives reason why along a century these two programs gave rise to irreducible conflicts. Moreover, it gives reason why past mathematicians met great difficulties to induce from the basic mathematical notions – of a subjective or an objective nature – the corresponding structural features, i.e. the two dichotomies constituting the FOM.

However, both programs may be recognised as specific contributions in order to achieve the recognition of FOM. In fact, both either manifested or closely approached, although in an obscure way, the recognition of all the four basic choices of the FoM. As a consequence, at present it is no more possible to go back to any 19th Century conception of the FoM.

In philosophical terms, the very novelty of this long research for recognizing FoM is its final result; which is not the recognition of a winner program on all others, but a pluralist conception of FOM; they include both classical logic and intuitionist one, both classical mathematics and constructivist one.[vi] However, once the specific philosophical import of each couple of choices on the two dichotomies is recognised, then to develop a theory according to the corresponding formalizations is a purely mathematical task.

After Russell paradoxes, Cantor’s theory was re-formulated through an ambitious operation in the ZFC set theory. However, Zermelo’s and Fraenkel’s innovations, although presented as more improved axioms pertaining to a deductive theory, are easily recognised to be new methodological principles (to allow an idealised choice, to exclude some specific sets) for conceiving set theory as a new, albeit implicit, program for re-constructing the whole mathematics.

As a fact, it obtained a new language aimed to both cover and improve the (almost) whole body of the mathematical theories. Its cost is to approach very near the contradictions, as the reverse mathematics showed.[vii] Moreover, this language is surely an artefact when dealing with the constructive mathematics (PI); thus it cannot say nothing of interesting about Goedel’s theorems. Even less it can say something about non-classical logic (PO). Hence, Set theory may be considered as an extraordinary attempt to grasp through a single mathematical theory two philosophical notions, not only infinity but also wholeness; which however resulted in a more modest mathematical aspect, i.e. a mathematical language suitable for most theories.[viii]

I conclude by reiterating in specific terms for mathematicians what Burtt wrote in 1924 for the scientists in general: “Metaphysics [the mathematicians] tended more and more to avoid [from Mathematics], so far as they could avoid it; so far as not, it became an instrument for their further conquest of the mathematical world.”[ix]

Only by exiting out this metaphysics, mathematics will no more obstruct the conflict resolution; otherwise Leibniz’ dream of “Calculemus!” or other appeal to scientific techniques will exclude the personal engagement for achieving a supportive peace. Only by assuming a pluralist attitude – as all the above presents – mathematics will contribute to peace which can only be a pluralist enterprise and a pluralist living together nevertheless the lasting differences.


[i]               P. Martin-Loef: “The Hilbert-Brouwer controversy resolved?”, in M. van Atten et al. (eds.): One Hundred Years of Intuitionism (1907-2007), Birkhaueser, Berlin, 2007, 245-256.

[ii]               E.W. Beth: Foundations of Mathematics, North-Holland, Amsterdam, 1959, ch. 1. 2.

[iii]              H. Weyl: “Mathematics and Logic”, Am. Math. Monthly, 53 (1946) 2-10; E.W. Beth: “Fundamental Features of Contemporary Theory of Science”, Brit. J. Phil. Science, 1 (1950) 89-102, , p. 102; J. van Heijenoort: “Goedel’s theorem”, in The Encyclopedia of Philosophy, MacMillan, London, 1967, p. 356; G. Kreisel: “Logic aspects of axiomatic method”, in H.-D. Ebbinghaus et al. (eds.): Logic Colloquium ’87, Elsevier, 1989, 183-217. J. Hintikka: “Is there completeness in Mathematics after Goedel?”, Phil. Topics, 17 (1989) no. 2, 69-90. Yet, two centuries before this alternative has been already described by Leibniz, D’Alembert and Lazare Carnot; to the “rational theory” they all contrasted an “empirical theory”.See the more detailed description in L. Carnot: Essai sur les Machines en général, Defay, Dijon, 1783, 101-103. In last century a similar dichotomy, between “constructive theories” and “principle theories”, was suggested in theoretical physics by Poincaré, and Einstein too.H. Poincaré: La Science et l’Hypothèse, Flammarion, Paris, 1902, ch. “Optique et Electricité”. A.I. Miller: Albert Einstein’s Special Theory of Relativity, Addison-Wesley, Reading, 1981, 123-142.

[iv]             A. Drago: “Pluralism in Logic: The Square of Opposition, Leibniz’ Principle of Sufficient Reason and Markov’s principle”, in J.-Y. Béziau and D. Jacquette (eds): Around and Beyond the Square of Opposition, Birkhaueser, Basel 2012, 175-189.

[v]               Incorrectly many scholars considered this kind of proof as reducible to a direct proof since they implicitly applied to its conclusion the classical law of the double negation. See e.g. J.-L. Gardiès: Le raisonnement par l’absurde, PUF, Paris,1991.

[vi]              Actually, this pluralist framework has been already initiated by several mathematical theories of French revolution, in particular by L. Carnot’s and Lobachevsky’s theories. N.I. Lobachevsky: “Introduction” to New Principles of Geometry, 1835-38.

[vii]             S.G. Simpson, Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, Cambridge, 2009.

[viii]             For a short, but detailed summary of the criticisms to contemporary Set theory, see S. Feferman: In the Light of Logic, Oxford U.P., Oxford, 1998, p. 288.

[ix]              E. Burtt: Metaphysical Foundations of Modern Science, Routledge and Kegan, London, 1924, p. 303.


Antonino Drago – Member of the TRANSCEND Network. Formerly at University of Naples –


This article originally appeared on Transcend Media Service (TMS) on 21 Apr 2014.

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