[FOM] Convincing math-blind people that math is different
tennant.9 at osu.edu
Fri Dec 26 11:25:05 EST 2014
Timothy Chow wrote:
"The math-blind people in my thought experiment are idealized. Although they cannot understand what we mean by syntactic rules or logical inferences, they do understand how human beings interact ... ."
I would suggest that such an idealization is incoherent. For these people to be able to "understand how human beings interact" they would have to understand those human beings' use of language. That in turn entails understanding syntactically compound sentences, involving logical operators. And that in turn entails having some grasp of rules of inference governing the logical operators. So, despite being "math-blind", the people in this thought experiment have some appreciation of the 'hardness of the logical must'.
That gives the logicist a starting point (denied him/her by Chow) from which to inculcate in the allegedly math-blind person a beginning grasp of mathematics. In order to cure the blindness in question, it would be enough to convey to this person our basic knowledge of the natural numbers. The logicist contends that anyone who understands reference, identity, predication, the connectives and the usual quantifiers can be introduced to the natural numbers even if s/he is at present innocent of them.
That story is of course too long to be given here (the interested reader can find it in my book Anti-Realism and Logic); but it involves no further fundamental assumptions than those already set out above, in principled rebuttal of Chow's contention. Chow has not set up his thought experiment in such a way as to rule out the logicist's suggested 'way in' to an eventual appreciation of mathematical knowledge as knowledge of propositions that are both necessary and a priori.
Knowledge of number is something that no innumerate communicators can in principle decline or resist coming to recognize, once it is offered to them as a cultural gift. All they need is some logical smarts to start with, and some minor extension of their vocabulary. They need to take on only one new unary predicate, one new unary function sign, and one new name, namely "... is a natural number", "the successor of ..." and "zero". They can be given some rules governing these new expressions in relation to logical expressions whose rules they already know, and the quick-witted among them will be off to the races.
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