[FOM] Convincing math-blind people that math is different
Timothy Y. Chow
tchow at alum.mit.edu
Sat Dec 27 16:58:38 EST 2014
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
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