Perceiving Numbers and Numerical Relations

Abstract

A child who has counted his marbles to find the sum of 5 and 7 has thereby improved his epistemic position vis-'a-vis the arithmetical truth that 5 + 7 = 12. Before the counting he did not know this truth, but now he does. Or, at least. he now seems justified in believing it. Does this mean that we can, atleast sometimes, acquire knowledge of arithmetical truths through perception? In this paper I examine some issues arising from the possibility of perceptual verification of arithmetical propositions-at least, some of the simpler ones. J.S. Mill argued that perception yields inductive evidence for arithmetical truths. But this is false; I advance some points to discredit this sort of inductive empiricism concerning mathematical knowledge. In particular, I try to show that perceptual justification of beliefs in simple arithmetical truths lack some of the central properties of the evidential relation between an inductive hypothesis and its ground. I claim that it is best not to view perception as providing any sort of evidence for the arithmetical propositions which it seems to justify. However, Mill was right in his fundamental assumption that numerical properties and relations are accessible to observation and perception. It has been popular in recent years to argue, on the basis of some vague causal requirement on knowledge, that if numbers are construed as abstract things outside the spacetime framework, knowledge of them and their interrelations becomes impossible. This is often taken to be a fatal objection to Platonism about mathematics. I argue that whatever abstract may come to mean, numbers and numerical relations are no worse off, in point of observability and capacity to enter into causal relations, than such sundry physicalproperties and relations as colors, shapes, and comparisons of length. From an appropriate epistemological point of view, perceiving the twoness of these