III. Can the Struggle within Mathematics be Resolved?
Much as economics is riven with vigorous debates between different schools of thought, so it is the case within mathematics, even if there is a dominant school that is often labeled “classical.” Much as heterodox economics schools struggle with and against the dominance of neoclassical orthodoxy within economics, so do the constructivists struggle against this dominant classical approach within mathematics, even as this oversimplifies the lineup and classification of schools within mathematics. Indeed, within logic, the classical approach is usually thought to contain two schools, the “logicistic” of Russell and Whitehead and the ‘”formalistic” of Hilbert and von Neumann, with the intuitionists being the third school that is more on the outside within the broader constructivist camp (, pp. 46-53). However, within mathematics more broadly there are other schools as well, just as the division between “heterodox” and “orthodox” within economics is a drastic oversimplification .
Within this major paper , Velupillai makes the case for constructivism by appealing to certain mathematical ideas invented by physicists that were looked down upon by rigorous, classical mathematicians,12 but which in the end have come to be accepted, and were used by physicists and engineers before that happened because of their practicality. One is the idea of path integrals developed by Feynman  and the other is the Dirac delta function . Of these two, Feynman was more concerned with the practicalities involved rather than the philosophical or strictly mathematical implications compared to Dirac, who developed quite strong views about these latter matters. However, for Velupillai the great appeal of both of them is their willingness to accept approximations that work rather than being obsessed with an unattainable idea of idealistic perfectionism.
Feynman’s sin was to solve for quantum mechanics an equation by simultaneously carrying out integrals over all space variables at each point in time, giving the “integral over all paths,” a concept that remains to this day non-axiomatized, with some labeling it as “mathematically meaningless” if still “impressive for the geniality of the physical intuition underlying it” and “one of the greatest achievements of 20th – century’s theoretical physics” (, p. 3). Velupillai (, p. 9) argues that it is best interpreted as being a “rule of thumb algorithm.” This certainly places it into the constructivist tradition, even if Feynman himself cared little about such debates.
Dirac’s delta function is perhaps even more troublesome from the standpoint of much of classical mathematics, derived as it is from the earlier (and also intuitive) Heaviside function (which is the integral of the Dirac function). So, the Heaviside function is a flat function except for a discrete step at one point, while the Dirac delta function is zero everywhere except for at the point of the step in the Heaviside function, where it is infinite. It does not look like a meaningful function at all, and it would be criticized by the likes of von Neumann  until it was rescued from non-respectability by Laurent Schwartz  inventing the concept of generalized functions and their distributions within which the delta function could fit. Again, like Feynman after him and Oliver Heaviside before him, Dirac was driven by intuitions about physics rather than mathematical rigor. Dirac was also motivated by aesthetics, preferring ideas that seemed possessed of beauty, thus perhaps agreeing with Hardy in being a follower of John Keats and his Ode on a Grecian Urn.
Curiously, besides interpreting the Dirac delta function as being generalized, yet another school of mathematical thought allows for the possibility of its being an exact and well-defined function, namely nonstandard analysis, with such a possibility being established by Todorov . Developed by Abraham Robinson , nonstandard analysis looks on the surface to be the exact opposite of constructivist mathematics, and many consider this to be the case, with Errett Bishop  leading a charge specifically against using it in the teaching of elementary calculus as proposed by Keisler (, ). Bishop argued forcefully that it involved a debasement of numerical meaning, as well as an excessive reliance on the use of the Axiom of Choice in its development by Robinson.
Another problem is that whereas constructivism in its intuitionist formulation eschews the infinite, nonstandard analysis positively glories in it. Not only are the transfinite cardinals accepted, but an entirely different set of infinite aggregates are allowed, hyperreal numbers that are infinite. The real payoff for allowing these is that their reciprocals are infinitesimals, numbers arbitrarily close to zero, but not equal to it. Indeed, Robinson did not invent this idea, rather he resurrected it, arguing reasonably that when Leibniz originally developed his version of the calculus, he conceived of derivatives as ratios of infinitesimals, rather than as limits of sequences of finite numbers. His great rival, Newton, apparently used both ideas in his development of “fluxions,” with Robinson (, p. 280) suggesting that Newton was reluctant to admit to infinitesimals in the face of attack from Berkeley who denied the existent of actual infinities.13 However, Robinson (, p. 281-282) notes that it is possible to use infinitesimals, which certainly appeal to the intuition of many introductory calculus students, without necessarily believing that they really exist, saying that “the intuitionists and other constructivists” might agree with Leibniz who declared regarding infinitely small and large numbers, “que ce n’étaient que des fictions, mais des fictions utiles.” He then argues that while the “majority of mathematicians” agree with the “platonistic” classicism of Cantor that infinities are ontologically real, for a logical positivist the question is meaningless, even while one “would concede the historical importance of expressions involving the term ‘infinity’ and of the (possibly, subjective} ideas associated with such terms.”
Now it would appear in this paper by Velupillai  that he recognizes nonstandard analysis as a possible alternative to constructivism and computable approaches, but he does not provide any indication of any enthusiasm for it. This may reflect how it has been used in the past in economics, largely for proving minor extensions of standard proofs of existence of general equilibrium, allowing for conceptually larger numbers of agents, but still based on such non-constructive concepts as the Axiom of Choice (, , ). Nevertheless, in his paper Velupillai cites some items that include some efforts to reconcile constructivism and nonstandard analysis, notably Schechter .
Indeed, that effort has been a gradually building enterprise, perhaps initiated by Wattenberg , who noted that the effort at reconciliation was coming more from advocates of nonstandard analysis than from the harder line constructivists such as Bishop. While Robinson and Keisler developed it to be essentially an extension of classical formalism, not contradicting the conventional view in any way, the newer efforts make this effort to be more consistent with constructivism, especially in its intuitionist formulation. Thus, Geoffrey Hellman  points out that the essential argument about infinitesimals involves the very intuitionistic idea that they are both equal to zero while also not being equal to zero. In the end, Hellman argues for a “pluralism” of mathematical systems, in this regard reflecting the arguments of some supporters of more heterodox approaches to economic theory.