Intuition does play a role in making arithmetic evident to the degree that it is, in that there is a ground level of arithmetic, not extending very far, that is intuitively evident. Furthermore, the objects that play the role of numbers in this low-level arithmetic can continue to do so in a more full-blooded arithmetic theory.After noting that logical notions allow this further extension, he insists that
the role of intuition does not disappear, because it is central to our conception of a domain of objects satisfying the principles of arithmetic ... an intuitive domain witnesses the possibility of the structure of the numbers (336).Here, then, we have a definite epistemic role for intuition of objects. It helps us to explain what is different about arithmetic, or at least the fragment of arithmetic that is closely related to these intuitions. (In chapter 7, this fragment is said to not even include exponentiation, so it fars fall short of PRA.)
While this objection to holism is quite persuasive, Parsons is at pains to emphasize how modest it really is. He offers some additional discussion of the implications for set theory, but the book seems primarily focused on what distinguishes arithmetic from other mathematical theories. It is an impressive achievement that I am sure will frame much of philosophy of mathematics for a long time.
2 comments:
I'm not sure I follow what the objection here is to holism. Could you sketch it a little more? The idea seems to be that intuition of strokes contributes to how we learn basic arithmetic, but I'm not sure how to carry this much further. Thanks!
No problem. Quinean confirmational holism about mathematics is the view that our mathematical beliefs are only justified as part of our best scientific theory of the world. The whole theory, including the mathematics, faces the 'tribunal of experience' together. Parsons has long objected to this that it clashes with our actual experience of doing mathematics. Here Parsons makes this point in more detail by clarifying a domain of "intuitively evident" mathematics. Our perception of concrete tokens of strokes yields intuition of abstract (or quasi-concrete in Parsons' sense) types of these strokes. This, in turn, justifies our belief that there are natural number structures.
So, it is not supposed to be a point just about learning arithmetic, but about its justification. If Parsons is right, then at least some parts of mathematics are justified independently of their role in our best science.
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