[FOM] Convincing math-blind people that math is different

[Timothy Y. Chow]

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?

Tim