CORCORAN-MASOUD ON PLATO’S DEDUCTIVISM

CORCORAN-MASOUD ON PLATO’S DEDUCTIVISMJohn Corcoran and Hassan Masoud.  2015. Plato’s mathematical deductivism. Bulletin of Symbolic Logic.  21 (2015) 199.    According to cognitivism, mathematicians know axioms to be true by intuitions and, by logical deductions from known axioms, know theorems to be consequences of axioms; thereby gaining knowledge of theorems. Cognitivism attributes knowledge in the strongest classical sense to mathematicians. Aristotle subscribed to cognitivism [2, pp. 2f].   According to formalism, “axioms” are arbitrarily chosen character-strings and derivations employ arbitrary string-manipulation rules: mathematicians neither know “axioms” to be true nor know “theorems” to be consequences of "axioms". The “axioms” and “theorems”  are meaningless concatenations of meaningless characters: one should not say ‘symbols’ because these characters do not symbolize. Here ‘form’ refers to the shapes of the character concatenations. Formalism, in this strange and misleading sense, has been mistakenly attributed without evidence to David Hilbert.  According to deductivism, mathematicians don’t know axioms to be true but by means of derivations do know theorems to be consequences of axioms: mathematicians know implications but not premises or conclusions. Deductivism attributes to mathematicians an understanding of the meanings of axioms and theorems. It also attributes to them knowledge of the cogency of the deductions. It thus attributes to mathematicians knowledge that the theorems are implications, logical consequences, of the axioms. But it claims that the mathematician does not, indeed cannot, gain knowledge of the truth of the axioms. Mathematicians might believe the axioms to be true but their subjective beliefs can never be objectively verified.  Some deductivists say mathematicians should try not to believe the axioms because that would make mathematics a religion, or at least a faith-based discipline.   Deductivism is intermediate between cognitivism and formalism—in many ways much closer to cognitivism.  We discuss the question of whether Plato was a deductivist.