The Limits of Human Mathematics
Salmon (Nathan)
Source: Salmon (Nathan) - Metaphysics, Mathematics, and Meaning: Philosophy Papers 1
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    Godel derived a philosophical conclusion from his second incompleteness theorem: Either the human mind infinitely surpasses any finite machine, or there are mathematical problems that are humanly unsolvable in principle; and therefore, either the human mind surpasses the human brain or it is not responsible for the creation of mathematics. Godel's derivation is here examined, reformulated, and defended. Remarks of Alonzo Church to the effect that nothing is a mathematical proof unless it is effectively decidably so are critically examined and found to be unconvincing. The investigation makes room for Godel's optimism that in principle any mathematical problem is solvable. (OSO): Gödel’s claim is defended that his famous incompleteness theorems yield the result, as a mathematically established fact, that the mathematical problem-solving capacity of the human mind either exceeds that of any finite machine or is incapable of solving all of mathematics’ mysteries. The issue turns on the nature of mathematical proof by the human mind. Of particular relevance is the question of whether the deductive basis of human mathematics is decidable.
    (???): Discusses insights into the mathematical ability of the human brain derived from Godel's incompleteness theorems. Superiority of the structure and power of the human mind over any non-living machines; Non-algorithmic role of consciousness in the formation of mathematical judgments; Capacity for a priori certainty.


Philosophical Perspectives, Vol. 15; Metaphysics; Oct2001, p93, 25p

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