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In this paper we examine the pragmatic meaning of mathematical necessity. While Peirce writes extensively on the pragmatic meaning of a "would be", he does not provide a developed account of the pragmatic meaning of a mathematical "must be". After examining several possibilities such as information relative accounts, conceivability, the ideal world, and possible worlds accounts, we consider mathematical necessity in terms of his overall pragmatism whereby metaphysical notions are understood in terms of their pragmatic effects, and, in particular, their role in inquiry. We argue that mathematical necessity should be considered a regulative hypothesis. Peirce often addresses questions of modality, such as whether to posit real possibility or physical necessity, by appealing to what is good for inquiry. Just as the hypothesis of the real world constrains and directs inquiry in science, positing the reality of mathematical possibility (or what Peirce sometimes calls " substantive possibility") can also be seen to constrain and direct mathematical inquiry as well as natural science. The necessity of mathematics may, in the end, be stipulated in the form of a hypothesis, just as the reality of the natural world functions as a hypothesis in Peirce's system. The effect of positing this ideal world of substantive possibility lies both in its effects within mathematical inquiry, and ultimately within the natural sciences as mathematical conclusions constrain the hypotheses we see fit to entertain.

The Pragmatic Maxim and the Idea of Necessity in Mathematical Inquiry