[FOM] vagueness in mathematics? -- why so little?

[tchow]

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:

http://alum.mit.edu/www/tchow/reviews/dcohen.html

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 
resistance.

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.

Tim Chow