[Math] Unmathematical habits of thought and action which would be of use to mathematicians

sociology-of-math

In Question 74707, we ask what mathematical habits of thoughts are useful in other areas. It seems only fair to ask also what we can learn from them. It is also fair to ask what they should not learn from us.

The first question is

What habit of thoughts in other areas
can be of use in mathematics.

Here are a few suggestions to the first question:

  1. In many areas the quality of exposition is very important.

  2. In some areas simplicity is considered as an advantage. In mathematics to some extent difficulty is a criterion for quality.

  3. Other areas give more weight to heuristic and non rigourous arguments compared to pure mathematics.

  4. In other areas there is much heavier use of computers.

  5. In some areas discussions and debates are basic part of the academic discipline. This is not the case in mathematics.

The second question is:

What habit of thoughts in mathematics
should be avoided (even by mathematicians) outside mathematics.

Here are a few suggestions (to the second question) for starters.

  1. Mathematicians (as a rule) avoid ambiguity. Nonmathematicians recognize the value (in appropriate circumstances) of ambiguity.

  2. Mathematicians don't care what anybody thinks. We know when our assertions are facts, because we can prove them, and we know how important they are; we don't need anyone's opinion on that. There is a danger that this attitude carries over to our everyday lives. Nonmathematicians recognize the value (in appropriate circumstances) of the opinions of others.

  3. Mathematicians, with all due respect to Godel, think simple declarative sentences are either true or false. Nonmathematicians are better able to deal with shades of gray.

  4. Those of us who teach are constantly judging the mathematical abilities of others. If we are not careful, we start to judge the worth of others by their mathematical abilities. Nonmathematicians know that some of the best people alive can't add fractions.

  5. Mathematicians (and theoretical physicists) consider a spherical cow. Nonmathematicians understand that conclusions based on unrealistically oversimplified models are untenable.

Best Answer

One habit that I have found useful, and for me came from making (maths based) art rather than mathematical training is to think in terms of aesthetics rather than mathematical correctness when exploring a problem. Take approaches that feel interesting, exciting, beautiful even if you also know that they are wrong. On occasion this can lead to the broader understanding needed to either generalise the problem correctly, or simply suggest an unusual (but correct) approach that might not have been considered directly. For me this often takes the form of trying to make images related to a problem, with the only question being whether an image is visually interesting or not.

I think that it is quite possible to come to this technique from within mathematics, but for me it came from outside, so it seems relevant here.

Edit 27/12/11

Thinking about this a little bit more, an additional habit is to seek a historical perspective. Looking at how problems and questions developed and gaining a sense of what ideas have been used to approach them in the past. I would argue that this is justified simply to grow an appreciation of the culture of the subject; however it can also bring benefit. Getting a sense of different approaches broadens the number of ways that you can attack new related problems where an outdated technique might suddenly find itself useful again!

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