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Title: “What's So Special about Deductive Proof?” 

Abstract: Mathematics is famously a *deductive* science (at least as far back as Euclid 300 BCE).  That is, in order to establish that a mathematical claim is true, mathematicians *prove* it.  And this is not just a matter of stylistic preference.  We tend to think that deductive proof is better at securing mathematical *knowledge* than inductive evidence.  The traditional view is that, although mathematicians sometimes make mistakes even when they restrict themselves to deductive proof, deductive proof is *more reliable* than inductive evidence. 

However, in the last fifty years or so, mathematicians have developed many Monte Carlo methods for checking mathematical claims (see Rajeev and Raghavan 1996).  For example, there are “probabilistic proofs” that can establish with arbitrarily high probability that a number is prime (e.g., Rabin 1980).  As a result of these developments, some mathematicians (e.g., Zeilberger 1993, Hersh 1997, 56-59, Borwein 2008) claim that mathematicians should use probabilistic proofs to establish that mathematical claims are true when such extremely reliable inductive evidence is available.  And a few philosophers (e.g., Fallis 1997, Womach and Farach 2003, Paseau 2015) have wondered whether there really is any epistemic (i.e., knowledge-related) value of deductive proof that inductive evidence always lacks. 

 

In response, several philosophers (e.g., Easwaran 2009, Smith 2016, 48-50, Berry 2019, Hamami 2022, Hamami forthcoming) have subsequently tried to identify the distinctive epistemic value of deductive proof.  These proposals can definitely help us to understand why deductive proof is as reliable as it is.  But as I will argue, since the identified properties of deductive proof are *only* valuable as a means to reliability, these proposals are ultimately unsatisfying.  They do not provide us with any reason not to use inductive evidence when it is just as reliable as deductive proof in establishing that a mathematical claim is true.  In other words, we have yet to identify the distinctive epistemic value of deductive proof. 

 

Speaker Bio: Don Fallis is a Professor of Philosophy and Computer Science at Northeastern University. His research interests include epistemology, philosophy of information, and philosophy of mathematics. His articles on lying and deception have appeared in the Journal of Philosophy, Philosophical Studies, and the Australasian Journal of Philosophy. He has also discussed lying on Philosophy TV and in several volumes of the Philosophy and Popular Culture series.

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  • Patricio Pino
  • Najeema Holas-Huggins
  • Beatrice Knight
  • Meklit Abebe

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