# Probabilistic Proof

Edited by Jordan Bohall (University of Illinois, Urbana-Champaign)
 Summary Probabilistic proofs involve non-deductively valid proof methodologies, which focus on establishing that conclusions have a high probability of being true. These methods often involve experimental practices (in a literal sense) and precise calculations to determine that some result is probably true.
 Key works Adleman (1994), Fallis (1997), and Fallis (2002)
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1. Transferable and Fixable Proofs.William D'Alessandro - forthcoming - Episteme:1-12.
A proof P of a theorem T is transferable when a typical expert can become convinced of T solely on the basis of their prior knowledge and the information contained in P. Easwaran has argued that transferability is a constraint on acceptable proof. Meanwhile, a proof P is fixable when it’s possible for other experts to correct any mistakes P contains without having to develop significant new mathematics. Habgood-Coote and Tanswell have observed that some acceptable proofs are both fixable and (...)

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2. Probabilistic Proofs, Lottery Propositions, and Mathematical Knowledge.Yacin Hamami - 2021 - Philosophical Quarterly 72 (1):77-89.
In mathematics, any form of probabilistic proof obtained through the application of a probabilistic method is not considered as a legitimate way of gaining mathematical knowledge. In a series of papers, Don Fallis has defended the thesis that there are no epistemic reasons justifying mathematicians’ rejection of probabilistic proofs. This paper identifies such an epistemic reason. More specifically, it is argued here that if one adopts a conception of mathematical knowledge in which an epistemic subject can know a mathematical proposition (...)

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3. Bayesian perspectives on mathematical practice.James Franklin - 2020 - Handbook of the History and Philosophy of Mathematical Practice.
Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and (...)

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4. Von Neumann in 1932 was the first to outline the possible non-existence of dispersion free ensembles in quantum mechanics, and he used this basic evidence to give a preliminary proof on incompatibility between quantum mechanics and local hidden variables theory. In the present paper, we give a detailed theoretical elaboration on the manner in which such a fundamental subject could be explored at perceptive and cognitive levels in humans. We also discuss a general design of the experiment that we have (...)

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5. Probabilistic Proofs and the Collective Epistemic Goals of Mathematicians.Don Fallis - 2011 - In Hans Bernhard Schmid, Daniel Sirtes & Marcel Weber (eds.), Collective Epistemology. Heusenstamm, Germany: Ontos. pp. 157-175.
Mathematicians only use deductive proofs to establish that mathematical claims are true. They never use inductive evidence, such as probabilistic proofs, for this task. Don Fallis (1997 and 2002) has argued that mathematicians do not have good epistemic grounds for this complete rejection of probabilistic proofs. But Kenny Easwaran (2009) points out that there is a gap in this argument. Fallis only considered how mathematical proofs serve the epistemic goals of individual mathematicians. Easwaran suggests that deductive proofs might be epistemically (...)

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6. Probabilistic proofs and transferability.Kenny Easwaran - 2009 - Philosophia Mathematica 17 (3):341-362.
In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs have. I also consider what this (...)

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7. Randomized arguments are transferable.Jeffrey C. Jackson - 2009 - Philosophia Mathematica 17 (3):363-368.
Easwaran has given a definition of transferability and argued that, under this definition, randomized arguments are not transferable. I show that certain aspects of his definition are not suitable for addressing the underlying question of whether or not there is an epistemic distinction between randomized and deductive arguments. Furthermore, I demonstrate that for any suitable definition, randomized arguments are in fact transferable.

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8. The Reliability of Randomized Algorithms.D. Fallis - 2000 - British Journal for the Philosophy of Science 51 (2):255-271.
Recently, certain philosophers of mathematics (Fallis [1997]; Womack and Farach [(1997]) have argued that there are no epistemic considerations that should stop mathematicians from using probabilistic methods to establish that mathematical propositions are true. However, mathematicians clearly should not use methods that are unreliable. Unfortunately, due to the fact that randomized algorithms are not really random in practice, there is reason to doubt their reliability. In this paper, I analyze the prospects for establishing that randomized algorithms are reliable. I end (...)

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9. The Epistemic Status of Probabilistic Proof.Don Fallis - 1997 - Journal of Philosophy 94 (4):165-186.

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10. Mathematical Proof and the Reliability of DNA Evidence.Don Fallis - 1996 - The American Mathematical Monthly 103 (6):491-497.
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11. Non-deductive logic in mathematics.James Franklin - 1987 - British Journal for the Philosophy of Science 38 (1):1-18.
Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as subjective', heuristic' or (...)

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