Geoffrey Hellman

As argued in Hellman (1993), the theorem of Pour-El and Richards (1983) can be seen by the classicist as limiting constructivist efforts to recover the mathematics for quantum mechanics. Although Bridges (1995) may be right that the constructivist would work with a different definition of 'closed operator', this does not affect my point that neither the classical unbounded operators standardly recognized in quantum mechanics nor their restrictions to constructive arguments are recognizable as objects by the constructivist. Constructive substitutes that may still be possible necessarily involve additional 'incompleteness' in the mathematical representation of quantum phenomena. Concerning a second line of reasoning in Hellman (1993), its import is that constructivist practice is consistent with a 'liberal' stance but not with a 'radical', verificationist philosophical position. Whether such a position is actually espoused by certain leading constructivists, they are invited to clarify

Three questions are highlighted concerning the scope and force of indispensability arguments supporting classical, infinitistic mathematics. The first concerns the need for non-constructive reasoning for scientifically applicable mathematics; the second concerns the need for impredicative set existence principles for finitistic and scientifically applicable mathematics, respectively; and the third concerns the general status of such arguments in light of recent work in mathematical logic, especially that of Friedman et al. and Feferman et al. Some recent results are then presented bearing on the first question on the need for non-constructive analysis, especially for quantum physics. Despite the impressive work of Bishop et al. in constructive analysis, Hilbert's objection to intuitionism still carries significant force, and may be decisive depending in part on one's conception of "physics"

Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects and relations. MS, in contrast, overcomes or avoids both sets of problems. Finally, it is argued that the modality of MS or a close cousin appears at crucial junctures in both STS and SGS, so that the above outcome is not obviously tendentious.