[FOM] Convincing math-blind people that math is different
Timothy Y. Chow
tchow at alum.mit.edu
Tue Dec 23 13:47:28 EST 2014
On Tue, 23 Dec 2014, Steven Gubkin wrote:
> It is impossible to communicate without shared experience. You can talk
> about the color green all you like, and it will mean nothing without the
> shared experience of grass, leaves, and frogs.
> Tic-tac-toe is a wonderful game for getting across much of the flavor of
> mathematics in a short conversation.
I should clarify that my primary purpose in raising this question is to
gain some insight into the philosophy/foundations of mathematics, not to
find practical methods of converting almost-math-blind people in the real
world to the "religion" of math. 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 and they live in the same physical world that we do.
Your analogy with color is a good one. In the philosophical literature on
consciousness, we find hypothetical "zombies" with no internal experience
of consciousness ("qualia") but who are externally indistinguishable from
ordinary human beings. There is much debate about whether the distinction
between zombies and ordinary people makes any sense, but I want to
sidestep this debate. Zombies are able, in practice, to distinguish green
objects from red objects, and that is all I care about for the moment. I
don't care if the color green "means anything" to them or not.
Similarly, I'm not concerned with whether my idealized math-blind people
can have their internal state altered to match my own. I'm happy if I can
get them to recognize that there is something qualitatively different
about mathematical knowledge.
If it's still not clear what I'm asking, let me pose the question a
different way. Let's line up representatives of different philosophies of
math side by side, ranging from Kripkenstein through ultrafinitism through
formalism through predicativism through set-theoretic platonism. I'd like
to claim that each person in the lineup judges the people to one side of
them as "math-blind" to some aspects of math. Now it should be clear that
no amount of additional mathematical training is going to get, say, Edward
Nelson to "see" that infinity really exists. He can account for
everything that he observes according to his infinity-blind view of the
world. So now what I'm asking is to take a further step back. Can those
who lack understanding of syntactic rules and logical inference give a
fully satisfactory account of everything they observe about the
mathematical community? Or is there something that really defies
explanation from their perspective?
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