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

[Josef Urban]

Hi Tim,

Your example is quite a bit like the futuristic note on formal proof in
law/society/politics that John McCarthy once wrote:
http://www-formal.stanford.edu/jmc/future/objectivity.html . I really like
it.

A class of examples that may persuade some engineers about the merits of
formal proof are misdesigned control systems (like train control) that fail
only in very rare circumstances.

Josef
On Dec 23, 2014 10:49 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

> I have recently been mulling over a certain unusual angle on the
> philosophy of mathematics that leads to what I think is an interesting
> question.  Roughly speaking, the question is whether it is possible to
> convince a math-blind person that mathematical knowledge is qualitatively
> different from other kinds of knowledge.
>
> By a math-blind person, I mean someone who lacks the internal experience
> of mathematical proof and the feeling of certainty and objectivity that
> mathematical proof (usually) confers.  If we want, we could even take the
> extreme Kripkensteinian route: the math-blind person can't comprehend what
> a syntactic rule is, thinks "grue" is a natural concept and is baffled by
> "green," etc.  But we'll assume that the math-blind person is a normal
> person in non-mathematical areas, and in particular has a deep
> understanding of human social interactions and activities.
>
> Those of us who are not math-blind are (mostly) convinced that there is
> something qualitatively different about mathematical knowledge compared to
> other kinds of knowledge.  My question is, is there a way to convince a
> math-blind person that this qualitative difference exists?
>
> Math-blindness blocks perhaps the most natural route.  Instinctively, most
> mathematicians want to talk about the *certainty* of mathematical
> knowledge.  Now, there are familiar difficulties with the concept of
> mathematical certainty, having to do with the difficulty of being certain
> that there is no error in a proof, or the question of which axioms, if any,
> are certain.  But even leaving those difficulties aside, math-blindness
> presents a much simpler obstacle.  The math-blind person has no idea what
> we're talking about when we talk about the subjective experience of
> certainty.  All the math-blind person can see is that mathematicians
> *claim* to be certain.  But religious fanatics also claim to be certain,
> and most mathematicians want to distance mathematical knowledge from
> religious fanaticism.  The math-blind person can, of course, test this
> alleged certainty by seeing whether the community ever changes its mind
> about the "certain" facts.  But on this score, the religious fanatics will
> probably outdo the mathematicians, who must publish errata on an
> embarrassingly regular basis.
>
> Similarly, *unanimity* doesn't seem to be a good litmus test either.  Is
> there unanimity about whether an arbitrary angle can be trisected using
> compass and straightedge alone?  Sure, say the mathematicians.  But wait,
> asks the math-blind person.  What about those angle-trisectors over there?
> Oh, they don't count.  Well, if it is permitted to exclude some groups of
> "wackos" in order to achieve "unanimity," then it seems that there are
> plenty of other areas of knowledge where there is "unanimity" among the "in
> group."
>
> The best candidate I have been able to come up with so far is that in
> mathematics, it is possible on rare occasions for an "outsider" with no
> credentials among the "in group" to be able to make a very brief statement
> about an alleged error in the accepted body of knowledge, and have it be
> immediately acknowledged to be correct, even when the error has significant
> consequences.  I feel that I'm on the right track here, but I'm not sure
> that I've really "nailed it."  For example, an error in, say, Google Maps
> can survive for a rather long time, and anybody can point out the error and
> have it be universally acknowledged after a rather brief verification
> process.  One response might be, "Whether something is a one-way street
> *is* a mathematical fact."  This point of view seems to put us squarely in
> the fictionalist camp, where mathematical facts are agreed-upon
> conventions.  But even if you are a fictionalist, there seems to be a
> distinction between definitional conventions and chains of logical
> inference.  Google Maps is perhaps mathematical insofar as one regards all
> definitional conventions as part of mathematics, but intuitively there is
> still something different about mathematical knowledge and traffic
> conventions that is not captured by the outsider-can-find-an-error
> criterion.
>
> I am wondering if someone can come up with a better criterion, or
> alternatively, argue convincingly that there is no way to convince the
> math-blind person that mathematical knowledge is qualitatively different
> from other kinds of knowledge.
>
> Tim
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