Constructivists restrict the kind of entities they are willing to let into the mathematical domain; thus, e.g., Leopold Kronecker did not accept transcendental numbers as well as other entities (see What was Leopold Kronecker's philosophical view which guided his mathematics?).

It seems that mathematical constructivism narrows down much of the 'standard' mathematical domain, and so I wonder: what does it gain by so doing? Or, what are its strengths in compare to non-constructivist approaches? Does constructivism gain more of the desired security of foundations? If so, how?


It does bring in more than (ephemeral) security of foundations, but what it is more of is different for different people. The early intuitionists like Brouwer and Weyl saw mathematics as free play of a Kantian creative subject, and to them "excesses" of classical mathematics were simply unfaithful to the mathematical intuition of that subject and his other cognitive faculties. This is particularly obvious in Weyl's critique of the "atomistic continuum" of classical mathematics versus intuitive continuum that appears not as "an aggregate of fixed elements but as a medium of free ‘becoming’", and his general longing:"Where is that transcendent world carried by belief, at which its symbols are directed? I do not find it [in classical mathematics], unless I completely fuse mathematics with physics and assume that the mathematical concepts of number, function, etc. (or Hilbert’s symbols), generally partake in the theoretical construction of reality in the same way as the concepts of energy, gravitation, electron, etc.", see Is Aristotle's resolution of Zeno's paradoxes vindicated by motion in the intuitionistic continuum?

Bishop, the founder of constructive analysis, later had equally strong feelings. In a 1974 lecture titled Crisis he wrote "There is a crisis in contemporary mathematics, and anybody who has not noticed it is being willfully blind. The crisis is due to our neglect of philosophical issues..." and in his diagnosis of the crisis's cause as idealistic excess remarked that "it is difficult to believe that debasement of meaning could be carried so far" referring to the non-standard analysis, see Katz's Meaning in Classical Mathematics: Is it at Odds with Intuitionism? It is a bit ironic that Cantor, the most Platonist of all Platonists had strong words against infinitesimals too, see What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal? However, Bishop does not simply dismiss classical theorems:"Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give it a constructive proof... Very possibly classical mathematics will cease to exist as an independent discipline."

This kind of passionate attitude is however quite rare among working mathematicians. But many of them view constructive mathematics favorably because it delivers more in the practical sense: explicit constructions and computations, that can be used not just to prove pure existence but to construct and explore explicit examples, error estimates that are needed if theoretical manipulations are to be made practically feasible and implementable, etc. Bishop himself writes:"It is clear that many of the results in this book could be programmed for a computer... As written, this book is person-oriented rather than computer-oriented. It would be of great interest to have a computer-oriented version". But at the other end of the spectrum we find Tait's Against Intuitionism: Constructive Mathematics Is Part of Classical Mathematics, who as the title suggests, holds that one can accept some self-restrictions for the sake of constructive purposes, while remaining a classical mathematician:"But the notion of computability, though important within mathematics and in applications, is a mathematical notion, understood in terms of the notions of number and (extensional) function. The search for constructive proofs and for constructively provable analogues of classics theorems is well motivated. But it is a search within the common domain of mathematics and is not based on some alien circle of ideas."

More recent philosophical intuitionists, like Dummett, Tennant and Wright, are somewhere in between, their concerns are semantic and epistemological, like Weyl's and Brouwer's, but more pragmatic and less Kantian. I find Dummett's What is Mathematics About? quite insightful.

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    Thank you, @Conifold, for this helpful explanation and for the references and papers you attached - will look into them. Your answer made me wonder: Is Intuitionism 'sub-set' of constructivism? (How do they relate to each other and also to finitism)?.. – L.M. Student May 11 '16 at 6:27
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    @L.M. Many use "intuitionism" and "constructivism" almost interchangeably, this is right in projection to practice, both share the view of mathematics as mental construction, and which tools are acceptable for it, e.g. intuitionistic logic. Motivations are quite different however, intuitionists have semantic and epistemological concerns about lack of justification and meaning in classical theories, while constructivists simply find them less useful. Either motivation can be radicalized to finitism, rejecting even simple inductive sets as finished entities. – Conifold May 12 '16 at 6:29
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    if may ask: was Poincare sort of an intuitionist? Is he considered early intuitionist just as Kronecker is considered at times early constructivist? (How historians relate to such mathematicians as Poincare, Kronecker, and also Peano, in terms of the three early twentieth century schools of logicism formalism and intuitionism?...) – L.M. Student Jun 28 '16 at 10:54
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    @L.M.Student Poincare is hard to pinpoint. He was sympathetic to Kant, and emphasized the role of intuition in mathematics contra Russell, Hilbert, etc., but he was also a conventionalist about what intuition delivers, e.g. Euclidean geometry is not a priori but chosen "for convenience". Modern structural realists consider him their precursor, which is probably the closest. plato.stanford.edu/entries/structural-realism – Conifold Jun 29 '16 at 22:39

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