I would suggest that variants of critical realism epistemology (including, but not limited to, philosophical pragmatism) are the most robust and defensible positions from a "reflexive" point of view.  There is a 'tolerance' of diversity, correlated with an authentic scientific ethic (desire to enquire into 'le réel').  There is nonetheless a 'weakness' in the sense that, it is not possible to claim a position of absolute virtue or superiority (neither in epistemological nor other moral terms); rather, one must engage in 'immanent critique' (in the sense of Adorno) or in dialogical interrogation (cf. Serge Latouche, Le Procès de la Science Sociale).  One is vulnerable to (simplistic) critiques of 'relativism' and, one is obliged (in a pragmatic or stoic sense) to accept that one is unlikely to convince dogmatic persons to relax their dogmatism....  . 

The philosophy of mathematics has to assess the main contours of the history of mathematics with a view to diverging theoretical orientations up to the present – and then one should attempt to advance an alternative systematic perspective. The history of mathematics shows that this discipline mainly toggled between an over-estimation either of number or of space. The initial arithmeticism of the Pathagoreans, owing to the discovery of incommensurability, was succeeded by a geometrization of mathematics, which by and large lasted (via Descartes Jacobi and others) until the 19th century (Weierstrass, Dedekind and Cantor) with a renewed attempt to arrive at a complete arithmetization of mathematics.

            The most basic problem in the philosophy of mathematics is therefore: how does one understand the relationship between the “discrete and continuous” (recently also confirmed by John Bell in his SIA – smooth infinitesimal analysis – 2006)? Fraenkel et.al. even speak about a “gap” in this regard which have remained an “eternal spot of resistance and at the same time of overwhelming scientific importance in mathematics, philosophy, and even physics” (Fraenkel et.al., Foundations of Set Theory, 1973:213).

            These authors add a significant remark in this regard. They point out that it is not obvious which one of these two regions – “so heterogeneous in their structures and in the appropriate methods of exploring” – should be taken as starting-point. Whereas the “discrete admits an easier access to logical analysis” (explaining according to them why “the tendency of arithmetization, already underlying Zeno's paradoxes may be perceived in axiomatic of set theory”), the converse direction is also conceivable, “for intuition seems to comprehend the continuum at once,” and “mainly for this reason Greek mathematics and philosophy were inclined to consider continuity to be the simpler concept” (Fraenkel et.al., 1973: 213). Recently a number of French mathematicians returned to the priority of space (the continuum – following the last phase in the intellectual development of Frege) – they are even known as mathematicians of the continuum (see Longo, 2001).

What we read in the second edition of Fraenkel's work on the Foundations of Set Theory confirms this. At the second International Congress of Mathematicians, held in 1900 at Paris, Poincaré said that “there remain in analysis only integers and finite or infinite systems of integers. ... Mathematics ... has been arithmetized ... We may say today that absolute rigor has been obtained” (page 14). Paul Bernays, the co-worker of the famous mathematician David Hilbert, explicitly states that we “have to concede that the classical foundation of the theory of real numbers by Cantor and Dedekind does not constitute a complete arithmetization of mathematics. It is anyway very doubtful whether a complete arithmetization of the idea of the continuum could be fully justified. The idea of the continuum is after all originally a geometric idea” (Bernays, Abhandlungen zur Philosophie der Mathematik, 1976:187-188). Bernays prefers to speak of the intuitions of discreteness and continuity when he designates the basic realities of mathematics.

However, since the discovery of Russell's antinomy in 1900 (and independently by Zermelo in the same year), mathematics struggles with its third foundational crisis. Hermann Weyl talks about the negative effect this had, flowing from the fact that “we are less certain than ever about the ultimate foundations of (logic and) mathematics,” and a similar significant remark by Fraenkel et.al. concerning the “third foundational crisis mathematics is still undergoing” is formulated in oppoisiton to a claim made by Poincaré. At the second International Congress of Mathematicians, held in 1900 at Paris, he said that

“there remain in analysis only integers and finite or infinite systems of integers. ... Mathematics ... has been arithmetized ... We may say today that absolute rigor has been obtained”.

However, Fraenkel et al. had to concede:

            Ironically enough, at the same time that Poincaré made his proud claim, it had already turned out that the theory of the ‘infinite systems of integers’ – nothing else but a part of set theory – was very far from having obtained absolute security of foundations. More than the mere appearance of antinomies in the basis of set theory, and thereby of analysis, it is the fact that the various attempts to overcome these antinomies, to be dealt with in the subsequent chapters, revealed a far-going and surprising divergence of opinions and conceptions on the most fundamental mathematical notions, such as set and number themselves, which induces us to speak of the third foundational crisis mathematics is still undergoing (Fraenkel et.al., 1973:14)

The significance of a non-reductionist ontology for mathematics is that instead of attempting to reduce space to number or number to space, it opts for recognizing the uniqueness of number and space as well as their mutual interconnectedness. (This alternative have been worded out in various publications of D.F.M. Strauss.)

Those who may be under the impression that mathematics is “exact” may once more consider the history of this discipline and in addition reflect on the assessment of Brouwer. But first consider what Stegmüller and Beth said: “The special character of intuitionistic mathematics is expressed in a series of theorems that contradict the classical results. For instance, while in classical mathematics only a small part of the real functions are uniformly continuous, in intuitionistic mathematics the principle holds that any function that is definable at all is uniformly continuous” (Stegmüller, 1970:331). Beth also highlights this point: “It is clear that intuitionistic mathematics is not merely that part of classical mathematics which would remain if one removed certain methods not acceptable to the intuitionists. On the contrary, intuitionistic mathematics replaces those methods by other ones that lead to results which find no counterpart in classical mathematics” (Beth, 1965:89).

            However, the most penetrating explanation is given by Brouwer himself. He states “that classical analysis … has less mathematical truth than intuitionistic analysis” (Brouwer, 1964:78) and then continues:

As a matter of course also the languages of the two mathematical schools diverge. And even in those mathematical theories which are covered by a neutral language, i.e. by a language understandable on both sides, either school operates with mathematical entities not recognized by the other one: there are intuitionist structures which cannot be fitted into any classical logical frame, and there are classical arguments not applying to any introspective image. Likewise, in the theories mentioned, mathematical entities recognized by both parties on each side are found satisfying theorems which for the other school are either false, or senseless, or even in a way contradictory. In particular, theorems holding in intuitionism, but not in classical mathematics, often originate from the circumstance that for mathematical entities belonging to a certain species, the possession of a certain property imposes a special character on their way of development from the basic intuition, and that from this special character of their way of development from the basic intuition, properties ensue which for classical mathematics are false. A striking example is the intuitionist theorem that a full function of the unity continuum, i.e. a function assigning a real number to every non-negative real number not exceeding unity, is necessarily uniformly continuous (Brouwer, 1964:79).

Brouwer, L.E.J. 1964. Consciousness, Philosophy, and Mathematics. In: Benacerraf et.al., 1964, pp.78-84.

Benacerraf, P. and Putnam, H. 1964 (eds.). Philosophy of Mathematics, Selected Readings. Oxford: Basil Blackwell.

Beth, E. 1865. Mathematical Thought. Dordrecht: D. Reidel.

Stegmüller, W. 1970. Main Currents in Contemporary German, British and American Philosophy. Dordrecht: D. Reidel Publishing Company, Holland.

Available on ResearchGate:

Strauss, D.F.M.. 2002. Philosophical reflections on continuity, in: Acta Academica, 34(3):1-32.

Strauss, D.F.M. 2011. Defining Mathematics. In: Acta Academica, 2011 43(4):1-28.

Strauss, D.F.M. 2011a. Bernays, Dooyeweerd and Gödel – the remarkable convergence in their reflections on the foundations of mathematics. South African Journal of Philosophy 30(1):70-94.

Thank you for your answers. I read Sayer's work, one of the more accessible CR texts I have found thus far.