Non-Deductive Methods in Mathematics

The PMA and APMP jointly held a symposium in the APA Eastern Division group sessions program on January 14, 2022, in Baltimore.

Believing the Riemann Hypothesis: Inductive Justification Beyond Enumeration

William D’Alessandro (Munich Center for Mathematical Philosophy)

The Riemann Hypothesis (RH) is one of the most important unproved statements in mathematics, with deep implications for number theory, geometry, analysis and beyond. RH has been open for over a hundred and fifty years and a proof still seems out of reach. But the general consensus is that RH is probably true. Why do mathematicians believe this? Although billions of cases of RH have been checked without any counterexamples having turned up yet, this sort of simple enumerative evidence is viewed as quite weak justification for RH (for reasons having to do with the slow growth rates of functions that could plausibly produce counterexamples). Instead, mathematicians’ confidence in RH is based mainly on the known truth of RH-like principles in various surrogate models of the rational numbers. Understanding the epistemology of this case thus highlights the underappreciated role of models, analogies and similar heuristics in mathematical reasoning.

Beliefs, Degrees of Belief and ‘Probabilistic Proofs’ in Mathematics

Silvia De Toffoli, Princeton University

It is uncontroversial that non-deductive arguments can justify arbitrary high credences in a mathematical proposition. In particular, randomized arguments (aka probabilistic proofs) are used to solve computationally heavy mathematical problems like determining the primality of large numbers (Rabin 1976). It has been argued that this type of argument gives rise to justified belief (Fallis 1997) and even knowledge (Paseau 2015). In my talk, I discuss the thesis that mathematical knowledge can be achieved without proof, and suggest that it is either trivial (as in the case of deference) or mistaken. I argue that the fact that we are justified in holding an arbitrary high credence in a mathematical proposition via randomized arguments does not automatically imply that if we believe it, our full-on belief will be justified. To do so, I consider the proposal that credences and full-on beliefs play different epistemic roles and are subject to different epistemic norms (Buchak 2014). I elaborate this proposal and apply it to the case of mathematics.

Enumerative Induction in Mathematics: The Lure of Small Numbers

Alan Baker (Swarthmore College)

In a 2007 paper, I argue for the following two-part thesis: i. Enumerative induction ought not to increase confidence in universal mathematical generalizations (over an infinite domain). ii. Enumerative induction does not (in general) lead mathematicians to be more confident in the truth of the conclusion of such generalizations. This skepticism about the justificatory role of enumerative induction in mathematics has been challenged by Walsh (2014) and by Waxman (2017), and discussed in some detail in Paseau (2021). In this talk I use several arithmetical case studies, including the Goldbach Conjecture, to further explore how and why the role of enumerative induction in mathematics is potentially problematic. My main argument is that particular positive instances of a given universal arithmetical conjecture will always be in an important sense “small” and thus that any such collection of instances is potentially biased. Size can make an important difference to the expected properties of numbers, so mere induction is insufficient to bridge from evidence about small numbers to a claim about all numbers.