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

[Timothy Y. Chow]

Thanks to all who have responded.  I'll summarize what I have concluded 
from this discussion.

It has been rightly pointed out that it's important for my thought 
experiment exactly what "math-blindness" is, and that I haven't been 
sufficiently clear on this point.  I believe that A. Mani's remarks come 
closest to what I would like to say.  I imagine the math-blind person not 
to be entirely incapable of logical reasoning, but to suffer from some 
serious limitations.  We could model those limitations in various ways, 
e.g., the math-blind person might have severe computational limits, or 
have a very high error probability, or might be able to grasp only vague 
concepts such as "a heap of sand" for which long chains of logical 
reasoning break down.  I would submit that such a person could not only 
function normally in human society, but could understand (or at least 
appear to understand) most human discourse and even appear intelligent 
according to many measures of intelligence.

The goal of my thought experiment is not to "cure" the math-blind person 
of math-blindness, but merely to convince such a person that there is 
something qualitatively distinct about mathematical knowledge compared to 
other kinds of knowledge.  By "qualitatively distinct" I mean not just 
that the subject matter is distinct; I want something more than the 
trivial observation that mathematicians study mathematics while art 
historians study art history and religious scholars study religion. 
Rather, it is qualitatively distinct in the sense that we can give a 
demonstration of that knowledge that has no analogue in other fields of 
knowledge.

I believe that the results of complicated but concrete mathematical 
computations, such as I sketched previously and such as Harvey Friedman 
also suggested, are probably the best one can do in this direction.  The 
key point is that math-blind people, with their severely limited reasoning 
abilities, are still capable of *verifying* the results of a complicated 
computation that they cannot directly mimic.  Note again that the 
demonstration is not intended to cure the math-blind people of their 
math-blindness.  It is intended only to demonstrate to them the existence 
of a certain capability.

Does this kind of demonstration uniquely distinguish mathematics from 
other fields of knowledge?  I already said that I don't think it clearly 
distinguishes mathematics from, say, physics.  It certainly doesn't 
distinguish mathematics from logic (for those who think there is a 
distinction).  To the extent that long chains of logical reasoning can be 
carried out in other fields of study (law, for example), those other 
fields of study are also not distinguished from mathematics (though maybe 
this is actually fine, because if long chains of reasoning are possible 
then it is presumably because there are highly precise concepts involved, 
and so the reasoning can perhaps be regarded as "mathematical" in nature 
even though the subject nature is not what we normally think of as 
mathematics).  Still, I think that being able to demonstrate that there is 
something distinctive about mathematical and/or scientific knowledge to 
math-blind people is a qualified success.

Does the demonstration show that there is really such a thing as ironclad 
logic or infallible mathematical deduction?  I don't think so.  If you 
believe, as I do, that there is something to Kripkensteinian/grue 
skepticism, then no amount of finite information can demonstrate something 
that is supposed to have an infinitary quality.  So in a sense, I've 
already pre-empted any possibility of a "strong" demonstration of 
mathematics just by the way I've set up the rules of the thought 
experiment.

I suspect that this is about as much as my thought experiment can deliver. 
Now let me say a few words about how I think the results of this thought 
experiment can be "applied" to familiar questions in the philosophy or 
foundations of mathematics.

There is a longstanding debate about whether a careful examination of the 
actual practice of doing mathematics undermines what we might call a 
"formalist" view of proof.  Here, by "formalist," I do not mean to refer 
to the debate over platonism or the reality of infinity.  I mean the more 
mundane view that any valid mathematical proof can be formalized as a long 
series of mechanically-verifiable deductions from a specified list of 
axioms.  Although widely held among mathematicians with training in the 
foundations of mathematics, this view has been challenged.  Critics point 
out that notions of mathematical rigor have changed over time, that the 
mathematical community relies on human authorities, that the prevalence of 
errors undermines any claim of certainty, that the idealized proofs are 
almost never actually written down and may never be written down, that 
even axioms and definitions are subject to revision and redefinition, and 
so forth.  All these things are true, but I do not conclude that 
mathematics is really just a social activity like any other social 
activity and that formal proof is a polite fiction.  It seems to me---and 
now I finally come to the point of my thought experiment---that the most 
cogent way to rebut this line of reasoning is *not* to appeal to platonism 
or to absolute certainty, but rather to appeal to the empirical 
demonstration of long verifiable computations.  If mathematics is "just" a 
social activity, and long chains of formal proof are a fiction, then how 
does one account for the results of such demonstrations?  Wouldn't one 
have expected such a long chain of reasoning to have caused the heap of 
sand to fragment into competing schools of thought with different 
authorities arguing different points of view?

I don't expect that this analysis will end all debate on the matter, but 
that was not my goal.  My hope was just to help focus the debate about the 
nature of proof on what I believe is the crux of the matter, instead of 
getting sidetracked on intractable arguments about the nature of infinity 
or about the alleged absolute certainty of mathematical knowledge.

Tim