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

[Timothy Y. Chow]

On Wed, 24 Dec 2014, Auke Booij wrote:
> So here's a radical idea: mathematical knowledge is not qualitatively
> different from most other kinds of (academic) knowledge.

I am glad that you have taken the time to articulate this point of view. 
I think you have pinpointed the crux of the matter, that anyone who wishes 
to argue for the distinctiveness of mathematical knowledge must address.

However, first I would like to back up one step.  It seems to me that you 
might agree that there is at least a tangible distinction between what 
I'll call "mathematical/scientific knowledge" and other kinds of 
knowledge, even if you deny any kind of sharp boundary within that 
category (namely, between mathematical knowledge and scientific 
knowledge).  In any case, whether or not you believe this, I think I can 
sketch a way that such a distinction could be demonstrated to a math-blind 
person pretty convincingly.  I will do this now.  The description may also 
help to clarify, by way of example, what kinds of capabilities I envision 
the idealized math-blind person to have.

The demonstration is simple to describe.  I build a computer and implement 
an algorithm that prints out, on paper, a million digits of some constant 
that hasn't been explicitly computed before---say, sqrt(12523599347). 
Then I build a completely different kind of computer and implement a 
completely different algorithm.  I announce that my new system will print 
out exactly the same million digits.  Then I hit the "go" button and the 
machine duly churns out the predicted million digits.  The math-blind 
person can verify that the million digits are indeed the same.

This sort of demonstration would seem to have no analogue in other fields 
of knowledge.  For example, we might be able to find two people who are 
able to recite the entire Koran word-for-word, but this is because the 
Koran has already been written out explicitly for all to examine.  The 
first million digits of sqrt(12523599347) have not been written out 
before, as far as the math-blind person can see.  All non-scientific 
examples I can think of that involve agreed-upon conventions (e.g., laws, 
works of art) require that a community spend considerable time drawing up 
the conventions explicitly, and explicitly disseminating that knowledge. 
The way in which an algorithm encodes an enormous number of digits seems 
to be a uniquely mathematical/scientific phenomenon.

Although I find this to be a convincing demonstration of the qualitative 
difference between mathematical/scientific knowledge and other kinds of 
knowledge, I am not sure that it serves as a demonstration of the 
distinction between mathematical knowledge and scientific knowledge.  We 
are, after all, still in the realm of finite predictions of finite 
experiments with finite results.  If, as I would like to propose, a 
math-blind person lacks the ability to *extrapolate* or *abstract*, then 
it seems to me that we are stymied at this point.

Tim