Academic journal article Cosmos and History: The Journal of Natural and Social Philosophy

Mathematical Naturalism and the Powers of Symbolisms

Academic journal article Cosmos and History: The Journal of Natural and Social Philosophy

Mathematical Naturalism and the Powers of Symbolisms

Article excerpt

ABSTRACT: Advances in modern mathematics indicate that progress in this field of knowledge depends mainly on culturally inflected imaginative intuitions, or intuitive imaginings--which mysteriously result in the growth of systems of symbolism that are often efficacious, although fallible and very likely evolutionary. Thus the idea that a trouble-free epistemology can be constructed out of an intuition-free mathematical naturalism would seem to be question begging of a very high order. I illustrate the point by examining Philip Kitcher's attempt to frame an empiricist philosophy of mathematics, which he calls "mathematical naturalism," wherein he proposes to explain novelty in mathematics by means of the notion of 'rational interpractice transitions,' only to end with an appeal to science to supply a meaning for rationality. A more promising naturalistic approach is adumbrated by Noam Chomsky who begins with a straightforward acceptance of mind and language as 'natural' or concrete facts which bespeak the need for a linguistic faculty. This indicates in turn that there may also be a mathematical faculty capable of generating and exploiting the powers of mathematical symbolisms in a manner analogous to the linguistic faculty.

KEYWORDS: Naturalism; Perception; Epistemic Aims; Retroduction; Rationality; Intuitions; Faculty; Imagination; Symbolism

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"Let the dead bury the dead, but do you preserve your human nature, the depth of which was never yet fathomed by a philosophy made up of notions and mere logical entities." (1)

MATHEMATICS AND 'MINDING'

The question of what mathematics really is haunts every philosophy of mathematics. This perennial puzzle is bound up with the question of what mathematics actually contributes to the modern quest for genuine understanding of the world. More specifically, if it is acknowledged that at least some theories of mathematics actually throw important light on natural physical events, the philosopher of mathematics seems bound sooner or later to wonder about the power of symbolisms that appear to bear witness to the possibility of forming intimate relationships between minds and nature. But to get an idea of the range of problems needing to be addressed in this line of thought, one need only consider Descartes' fundamental claim that mathematics "is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency, as being the source of all others" in the same light as the claim that "the human mind has in it something that we may call divine, wherein are scattered the first germs of useful modes of thought." (2)

The irony is that Descartes also indicates the impossibility of giving an adequate account of the foundations of mathematical, or indeed any other type of knowledge, without needing to bring in, perhaps sooner rather than later, spiritual forces or powers that operate beneath the surface of thought. He observes, for instance, that "that power by which we are properly said to know things, is purely spiritual " (HR, 38)." Hence Descartes himself raises doubts about whether the mental powers represented by mathematical symbolisms give the lie to his vision of a philosophy rendered into something like a scientific discipline. Why think that these powers do not present the chief obstacles to understanding cognition tout court, a question which puts the very idea of a distinctive epistemology of mathematics into question. He may even be suspected of turning a bad beginning into an absurdity when he insists on detaching the best of his 'mindings' from the contributions of his deceiving body. He is thereby obliged to distort his own experiencing. He maintains, for instance, that he is "not more" than "a thing which thinks"; that is, a thing "which doubts, understands, [conceives], affirms, denies, wills, refuses, which also imagines and feels" (see "Of the Nature of the Human Mind," HR, 153). …

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