On Some Properties of Humanly Known and Humanly Knowable Mathematics

Axiomathes 24 (1):81-88 (2014)
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Abstract

We argue that the set of humanly known mathematical truths (at any given moment in human history) is finite and so recursive. But if so, then given various fundamental results in mathematical logic and the theory of computation (such as Craig’s in J Symb Log 18(1): 30–32(1953) theorem), the set of humanly known mathematical truths is axiomatizable. Furthermore, given Godel’s (Monash Math Phys 38: 173–198, 1931) First Incompleteness Theorem, then (at any given moment in human history) humanly known mathematics must be either inconsistent or incomplete. Moreover, since humanly known mathematics is axiomatizable, it can be the output of a Turing machine. We then argue that any given mathematical claim that we could possibly know could be the output of a Turing machine, at least in principle. So the Lucas-Penrose (Lucas in Philosophy 36:112–127, 1961; Penrose, in The Emperor’s new mind. Oxford University Press, Oxford (1994)) argument cannot be sound

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Author Profiles

Jason Megill
Bentley College
Alex Beal
University of Colorado, Boulder
Tim Melvin
Santa Rosa Junior College

Citations of this work

Computability and human symbolic output.Jason Megill & Tim Melvin - 2014 - Logic and Logical Philosophy 23 (4):391-401.

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