I am reading Stewart Shapiro's book Thinking About Mathematics (2000 Oxford University Press) and in the second chapter he discusses the ontological theories of mathematical objects. That is, are mathematical objects such as numbers, points, lines, and sets real in the sense that they exist independently of any mathematicians? This is certainly my worldview, the alternative being that mathematical objects are not really real but are made up by mathematicians-- that if there were no mathematicians, there would be no mathematical objects. The first view is called realism, while the second is called idealism. This is not the space to discourse about the merits of either view, but I think it really depends on your worldview and your belief structure. The realist worldview is what comes out of Plato's teachings, and that is what excited me the most.
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| Plato the realist, point up to the World of mathematical Forms. Aristotle the idealist, pointing outward to the phyiscal world of Matter. |
Here is what Shapiro says on p. 27:
The scientific literature contains no reference to the location of numbers or to their causal efficacy in natural phenomenon or to how one could go about creating or destroying a number. There is no mention of experiments to detect the presence of numbers or determine their mathematical properties. Such talk would be patently absurd. Realism in ontology is sometimes called 'Platonism', because Plato's Forms are acausal, eternal, indestructible, and not part of space-time.This of course opens the door to many questions, such as how do science and mathematics differ, and what can be said about how theoretical physics differs from experimental physics? Is theoretical physics just a branch of mathematics? If so, how does its application lead to experimental results? What is the relation between the Platonic world of Forms and the physical world of matter?
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