[FOM] Convincing math-blind people that math is different
Auke Booij
auke at tulcod.com
Wed Dec 24 12:33:47 EST 2014
Disclaimer: I am a constructivist (to be more specific, I believe that
constructive logics are sufficient for expressing mathematical truths
that help technological (in the broad sense) progress), and would like
to contest your thesis that mathematical knowledge is special in the
way you describe.
I like to believe that if something so fundamental is true, then it
should be easy to explain. And this particular idea seems so deeply
rooted in mathematics, and so many people have made an attempt at
explaining it clearly, that something must be wrong.
So here's a radical idea: mathematical knowledge is not qualitatively
different from most other kinds of (academic) knowledge.
So let's start by specifying some kinds of knowledge that I claim
mathematical knowledge is similar to (for the moment conflating "not
different" with "similar"): physics, engineering, arts.
Since this would normally be the point where opinions diverge, let me
try to convince you. To do that, I am going to make a guess as to what
you might believe constitutes mathematical knowledge.
Especially if you are a platonist (or if you otherwise believe in "the
existence" of mathematical objects), this is easy: mathematical
knowledge is some collection (e.g. a book) of properties of otherwise
inaccessible objects in some higher world.
And if this is what you believe, then I have to admit that
mathematical knowledge is in some sense special, and you can explain
it by referring to this external universe of mathematical objects,
reducing the explanation problem to a different one (though not one
that I could solve, but this was not your original question).
However, I know that there are many people who have doubts about such
forms of platonism. In particular, on the other end of the scale,
there is constructivism, which we can see as being all about building
machines (although obviously, philosophy of constructivism is a field
of its own - this is merely my interpretation): indeed, in the
computational semantics, proofs (a.k.a. truths) in constructive
mathematics correspond exactly to computer programs, and in a more
general viewpoint, constructive truths correspond exactly to the
things we can see, measure, build, and construct. We can measure the
difference between the numbers e and pi because at some finite level
of precision, some physical measurements have different outcomes. Only
in specific cases can we construct roots of a physical input-output
relation (read: a machine executing a mathematical function), and not
in general of every continuous function: but luckily, in the case of
physics, all functions are smooth.
And it *looks* like all mathematics can be expressed constructively.
In particular, looking at it extremely syntactically, it is decidable
(read: constructive) whether a list of formulae is a valid deduction
of the last formula in the list. So in some sense, even
nonconstructive mathematics can be expressed constructively, albeit
only syntactically.
And so in some sense, the engineering of machines (electrical
circuits, say) corresponds to the implementation of mathematical
truths. Conceiving of an experimental setup that will teach you
something about physics corresponds to conjuring up a mathematical
measurement. And indeed, the arts are in some respect just a body of
knowledge that teach you how to respond to ideas in society (and I do
believe there are artists whose behavior could very well be modeled by
a flow chart). There is no need to talk about some external "truth" of
theorems.
At this point one might argue (or not) that these arguments are
invalid, since in the case of non-mathematical sciences and arts, we
can "see" if a certain machine/concept/construction works. We can
measure it, whereas mathematical truths are considered true without
experiments: you can assert their truth without doing any experiments.
But then I bring up the fact that (under a materialistic viewpoint)
mathematicians are just walking and talking computers, and these
machines *are* what do the measurement. Whenever a mathematician sits
down to verify a theorem, we can interpret this as a computer program
going through some steps to (hopefully) reach some final state.
*Doing* mathematics is as much a physical experiment as is any other
physical experiment.
So we reach the conclusion that, under this point of view,
mathematical knowledge is not so dissimilar from other sciences and
arts. And I have now definitely reached the subject of philosophy of
mathematics, which I am not sure belongs on this mailing list.
Auke.
On 23 December 2014 at 04:52, 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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