[FOM] vagueness in mathematics? -- why so little?
tchow at alum.mit.edu
Sat Feb 25 16:03:18 EST 2017
Dustin Wehr wrote:
> A QUESTION for the list: why do you think proof-focused mathematicians
> do not engage in such work, even if only in their free time?
I gained some insight into this question by reading "Equations From God"
by Daniel Cohen. Cohen explains that in Victorian England, many
mathematicians regarded mathematics as being highly relevant for
reasoning about religious and social issues. I give a brief summary of
the book here:
Part of the answer to your question is that our thinking nowadays is
highly conditioned by the professionalization of knowledge. Even the
statement of your question shows some signs of this; you speak of
"proof-focused *mathematicians*" and not just "people who understand
mathematical proofs." There is a tacit agreement in society about which
kinds of discourse are appropriate in which contexts, and any attempt to
transgress those conventional boundaries is guaranteed to meet with
A second reason is that experience has shown that long chains of
reasoning are highly sensitive to even minute doses of vagueness. If
you try to apply mathematical-type reasoning to something that is not
*strictly* mathematical, then typically, after a few steps, your
argument will no longer be water-tight and someone who doesn't like your
conclusion will be able to find some assumption to disagree with. What
gives mathematics its power is the ability to sustain indefinitely long
chains of reasoning, and this is usually lost instantly as soon as you
step outside the carefully circumscribed box of pure mathematics.
A third reason is that mathematicians are accustomed to having their
arguments accepted universally as long as they are technically correct.
This luxury is generally not available outside of the mathematical
community, indeed some mathematicians may have been partly influenced to
pursue their profession because of this luxury. So they may be
frustrated if they work hard on some argument and it is rejected.
On the bright side, mathematicians are generally eager to work for free
on problems that are recognizably mathematical. If you encounter
difficult mathematical problems in your work and are able to phrase them
in a purely mathematical way, then you have a good chance of being able
to enlist help. The trick is being able to isolate the purely
mathematical aspect of your problem, without dragging in a lot of side
issues that the mathematician doesn't really need to be concerned with.
The skill of being able to abstract the mathematical core of your
problem is not an easy one to acquire, but given your interests, it
seems to me that it is a skill that will pay great dividends and hence
is worth working on.
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