Blackwell Reference Online

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The following paradox is presented by mathematical knowledge. Mathematics is, historically, perhaps the earliest science. For many thinkers mathematical knowledge, by virtue of its seeming absolute certainty, has served as an ideal or paradigm for all the sciences. For example, the mathematical method was extended by rationalistic scientists like Galileo and Descartes to the realm of what we today call physics. Even if we do not go so far as to regard physics as the “mathematics of motion,” mathematical knowledge seems to be indispensable for modern scientific knowledge – the mathematically illiterate cannot read the papers of Dirac, Einstein, or Feynman. We can say, therefore, that mathematics is at least continuous with scientific knowledge. Yet mathematics seems continuous also with metaphysics. Indeed, mathematics seems not to deal with nature – its subject matter could be variously described as “ideal,” or “abstract”. Figures like triangles and spheres, etc. are ideal – they are perfectly shaped, and have no breadth. They seem to be the limit of some infinite process unattainable in the actual world. Numbers, on the other hand, are abstract: they are not, apparently, the idealization of any actual objects. Furthermore, the very certainty of mathematical knowledge seems to set it apart from empirical knowledge. Kant put the matter polemically in his “good company” argument: ... log in or subscribe to read full text