Knowledge of Mathematics without Proof

Alexander Paseau
Oxford University
Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that some of what mathematicians take to be deductive knowledge is in fact non-deductive. 1 Introduction2 Why It Might Matter3 Two Further Examples and Preliminaries4 An Exclusive Epistemic Virtue of Proof?5 Analyses of Knowledge6 The Inductive Basis of Deduction7 Physical to Mathematical Linkages8 Conclusion.
Keywords mathematical proof  mathematical knowledge  mathematical justification  deductive knowledge  non-deductive knowledge  nondeductive knowledge  non deductive knowledge  inductive knowledge  empiricism in mathematics
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DOI 10.1093/bjps/axu012
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References found in this work BETA

Is Justified True Belief Knowledge?Edmund Gettier - 1963 - Analysis 23 (6):121-123.
What Else Justification Could Be.Martin Smith - 2010 - Noûs 44 (1):10 - 31.
Content Preservation.Tyler Burge - 1993 - Philosophical Review 102 (4):457.
A Causal Theory of Knowing.Alvin I. Goldman - 1967 - Journal of Philosophy 64 (12):357-372.
Justified Judging.Alexander Bird - 2007 - Philosophy and Phenomenological Research 74 (1):81–110.

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Does Homotopy Type Theory Provide a Foundation for Mathematics?James Ladyman & Stuart Presnell - 2016 - British Journal for the Philosophy of Science:axw006.

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