THERE IS GREAT MOTIVATION WITHIN FREGE'S THEORY TO\nCONSTRUE THE CARDINAL NUMBERS AS QUANTIFIERS, WHICH ARE\nHIGHER LEVEL CONCEPTS. BUT FREGE ARGUED THAT THE CARDINAL\nNUMBERS ARE OBJECTS, NOT CONCEPTS, AND DEFINED THEM\nACCORDINGLY. MOREOVER, FREGE'S HIERARCHY OF CONCEPTS\nPREVENTED HIM FROM CONSTRUING THE NUMBERS AS CONCEPTS. MY\nPURPOSE IS TO BRING OUT THE QUANTIFICATIONAL NATURE OF THE\nNUMBERS IN THE FACE OF THESE OBSTACLES. THE PAPER PRESSES\nTHE QUANTIFICATIONAL VIEW ONTO FREGE'S CONCEPT OF NUMBER AS\nIT TRACES ITS DEVELOPMENT FROM THE "BEGRIFFSSCHRIFT",\nTHROUGH THE 1880S, INTO ITS FORMALIZATION IN (...) THE\n"GRUNDGESETZE". THE THEORY YIELDS SURPRISINGLY EASILY UNDER\nPRESSURE FROM THE QUANTIFICATIONAL VIEW, AND ONE\nSYMPATHETIC WITH THAT VIEW MAY TAKE ENCOURAGEMENT FROM THE\nFACT THAT IT WAS FREGE'S HIERARCHY OF CONCEPTS, NOT HIS\nACTUAL ARGUMENTS, THAT PREVENTED HIS CONSTRUING NUMBERS AS\nQUANTIFIERS. (shrink)

Philosophers have recently expressed interest in accounting for the usefulness of mathematics to science. However, it is certainly not a new concern. Putnam and Quine have each worked out an argument for the existence of mathematical objects from the indispensability of mathematics to science. Were Quine or Putnam to disregard the applicability of mathematics to science, he would not have had as strong a case for platonism. But I think there must be ways of parsing mathematical sentences which account for (...) applicability of mathematics and also do not require us to believe in entities we have no evidence for, other than through reading these sentences literally. We will explore a particular way to interpret sentences of arithmetic which promises to account for their applicability without bringing in metaphysics not also brought in by science. The investigation will be limited to the arithmetic of cardinal numbers. The general strategy is to argue for the analogy between arithmetic and science, rather than to argue for one case having a particular characteristic independently of the other. (shrink)

Theorems of arithmetic are used, perhaps essentially, to reach conclusions about the natural world. This applicability can be explained in a natural way by analogy with the applicability of statements of law to the world. ;In order to carry out an ontological argument for my thesis, I assume the existence of universals as a working hypothesis. I motivate a theory of laws according to which statements of law are singular statements about scientific properties. Such statements entail generalizations about instances of (...) those properties. My task, then, is to show that theorems of arithmetic, in application, are similar to statements of law in this respect. On the basis of my working hypothesis, I argue that there are inclining reasons to think that cardinal numbers are scientific properties whose instances occur in the natural world. The works of Mill, Frege and Benacerraf play central roles in these arguments. ;In general, I urge that there are no conclusive epistemological or ontological factors which preclude arithmetical truths from being laws of nature. Consequently, I think we can extend any semantic theory which is adequate for statements of law to theorems of arithmetic. This would apply even to a theory of predication which does not involve commitment to universals. So, to the degree that my argument succeeds with the working hypothesis, it also supports the unconditioned thesis that arithmetical truths are laws of nature. (shrink)