Abstract
To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide variety of philosophical positions. Once the axioms are generally accepted, mathematicians can expend their energies on proving theorems instead of arguing philosophy. Given this account of the role of axioms, I give four criteria that axioms must meet in order to be accepted. Penelope Maddy has proposed a similar view in Naturalism in Mathematics, but she suggests that the philosophical questions bracketed by adopting the axioms can in fact be ignored forever. I contend that these philosophical arguments are in fact important, and should ideally be resolved at some point, but I concede that their resolution is unlikely to affect the ordinary practice of mathematics. However, they may have effects in the margins of mathematics, including with regards to the controversial “large cardinal axioms” Maddy would like to support.
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Notes
This way of putting the distinction is slightly tendentious, but something like it should make sense to most mathematicians. After all, we recognize a distinction between the debate about whether all rings have an identity element, and the ultrafinitist debate about whether every natural number has a successor. On some structuralist views of mathematics, this distinction turns out to be illusory. All axioms are assimilated to the structural sort. “Mathematics is the study of those structures which arise in different uses but with the same formal properties—and mathematicians aim to carry out that study by using proofs. This view, unlike platonism, also accounts for the ways in which mathematics is used in other sciences.” (Mac Lane 1997, p. 151) This view does exist, but it is not popular, even among structuralists. If this view really is tenable, then it may undermine the entire debate about axioms. But I suspect that it isn’t, because it classes the ultrafinitist debates with merely terminological ones.
I will generally ignore Euclid’s axioms for geometry. It’s unclear whether they were taken to be structural or foundational. And as Hilbert eventually showed in his 1899 Foundations of Geometry, Euclid’s axioms weren’t actually sufficient to prove all the results he claimed. Thus, the Euclidean use of axioms differed substantially from the modern practice. “Hilbert’s axiomatization of geometry furthered this process [making all assumptions explicit] by using a formal axiomatic method as distinct from that of Euclid.” (Moore 1982, p. 309)
Field tried in addition to show that mathematics is in fact false, but his program is widely considered to be unsuccessful so far. See (MacBride 1999) for a discussion of the development of this program.
The case of probability seems to be interestingly complicated. Some philosophers argue that subjectivist, frequentist, and propensity interpretations of probability are all useful, making Kolmogorov’s axioms structural. But others argue that only one of these interpretations is correct, so that the Kolmogorov axioms just represent a foundational agreement among probabilists to work together to prove theorems, while bracketing philosophical questions for later. See (Hájek 2003).
Jules Richard, responding to Hadamard and Poincaré, defended AC by appealing to an alphabetization of the definitions of elements of the sets, as in his paradox of definability. (Moore 1982, pp. 104–105) “From the constructivist viewpoint espoused by Richard, the Axiom [of Choice] was true precisely because the notion of set was restricted to that of containing a definable element.” (Moore 1982, p. 309)
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Easwaran, K. The Role of Axioms in Mathematics. Erkenn 68, 381–391 (2008). https://doi.org/10.1007/s10670-008-9106-1
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DOI: https://doi.org/10.1007/s10670-008-9106-1