Arithmetic, set theory, reduction and explanation

Abstract

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important and well-known case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer some evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate and successful foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences.[I]n the philosophy of science the notions of explanation and reduction have been extensively discussed, even in formal frameworks, but there exist few successful and exact applications of the notions to actual theories, and, furthermore, any two philosophers of science seem to think differently about the question of how the notions should be reconstructed. On the other hand, philosophers of mathematics and mathematicians have been successful in defining and applying various exact notions of reduction (or interpretation), but they have not seriously studied the questions of explanation and understanding. I hope the rest of the paper will show that this neglect is unwarranted. As philosophers of science have long known in the empirical context, understanding the link between reduction and explanation is a vital part of understanding theory succession, mathematical progress, and the nature and purpose of foundations of mathematics. I aim to illustrate these points by examining a particularly important single case: the reduction of arithmetic to set theory. Although a number of authors have expressed views about the explanatory significance of this reduction, the issues involved haven't yet been weighed as carefully as they deserve to be. This paper sets out to do so, and to draw some considered conclusions.My thesis is that the reduction of arithmetic to set theory is unexplanatory. After clarifying this claim and giving some positive reasons to believe it, I devote the larger part of the paper to showing that the most serious arguments to the contrary-due to Steinhart, Maddy, Kitcher, and Quine-are unsuccessful. Finally, I discuss some consequences of the view for philosophy of mathematics, philosophy of science and the theory of reduction. Intertheoretic reduction and explanation in math...

Explanation in mathematics: Proofs and practice

“…Another reason to study ME, then, is because doing so promises to enrich (and perhaps correct) our ideas about mathematical practice. See for instance Detlefsen (1988), Mancosu (1999), Tatzel (2002), Harari (2008), Mancosu and Hafner (2008), and D'Alessandro (2018) for more on the role of explanatory concerns in historical and contemporary mathematics.…”