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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.

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@darij, it seems to me that 74707 is full of "stereotypes about mathematicians and their counterparts on non-mathematicians," but no one seems interested in pointing that out when the stereotypes favor us mathematicians. –  Gerry Myerson Sep 7 '11 at 14:03
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I created a meta tea.mathoverflow.net/discussion/1132/a-pair-of-questions/… –  quid Sep 7 '11 at 14:14
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I saw a spherical cow once! If they die in a warm place with no predators, their guts swell up and they turn into a near-round ball. It takes a few days. –  Ryan Budney Sep 7 '11 at 17:54
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Gerry, with all due respect, beyond the question being off topic in my opinion, several of these suggestions are terribly off the mark. Item 3, in particular, I find incorrect unless you decide to remove all logicians from the discussion and return to XIX century standards of mathematical thinking. Item 1 also seems inaccurate, unless you approach it uncritically (not the point, but there is serious work on fuzzy logic and paraconsistency, to mention just two examples, that deals with these issues routinely). Item 5 is a caricature. Items 2 and 4 seems oversimplifications, to say the least. –  Andres Caicedo Sep 8 '11 at 0:18
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I am mathematician, I do not fit the criteria number 3 and I don't care because of number 2. –  zroslav Sep 8 '11 at 12:48

4 Answers 4

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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A long quote, from which one can extrapolate trivially a tentative answer.

"[T]echnical treatises in science do not generally receive such a license for explicitly personal expression. I believe that this convention in technical writing has been both harmful and more than a bit deceptive. Science, done perforce by ordinary human beings expressing ordinary motives and foibles of the species, cannot be grasped as an enterprise without some acknowledgment of personal dimensions in preferences and decisions – for, although a final product may display logical coherence, other decisions, leading to other formulations of equally tight structure, could have been followed, and we do need to know why an author proceeded as he did if we wish to achieve our best understanding of his accomplishments, including the general worth of his conclusions.

Logical coherence may remain formally separate from ontogenetic construction, or psychological origin, but a full understanding of form does require some insight into intention and working procedure. Perhaps some presentations of broad theories in the history of science – Newton's Principia comes immediately to mind – remain virtually free of personal statement (sometimes making them, as in this case, virtually unreadable thereby). But most comprehensive works, in all fields of science, from Galileo's Dialogo to Darwin's Origin, gain stylistic strength and logical power by their suffusion with honorable statements about authorial intents, purposes, prejudices and preferences."

SJ Gould The structure of evolutionary theory p. 34.

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I am fairly skeptical about the two versions of the question, namely whether mathematicians have much to learn from habits in other academic disciplines regarding how to do mathematics, and also about the suggestion that mathematician tend to use "mathematical habits of thought" which are unsuitable in other areas.

Regarding the first question: of course, I think it can be very beneficial for mathematicians to learn "habits of thoughts" in other academic disciplines, as well as in other areas of mathematics. And having wide horizons can be, at times, beneficial also in one's own research. Adopting blindly habits of thought from other areas into mathematics (or the other way around) is not a good idea although here and there it can be useful.

Specifically, to the suggestions regarding the first question: Technical difficulty and complexity is indeed often a sign of quality, and there is appreciation also for simplicity. It is interesting if heuristic and nonrigorous methods can be of more use in pure mathematics. But one should be rather cautious about them. To some extents they exist e.g. in making conjectures. Certainly they are central and important in applied mathematics. Heavier use of computers is a nice idea, and it raises interesting question about the nature of mathematics and mathematical understanding. But the progress is rather slow and there is no good evidence for the bold prophecies regarding much more central role to computers in general, and automatic theorem proving in particular. It is not clear at all if debates and discussions regarding mathematics can be useful as they are in other areas.

Regarding the second question. I dont see any evidence that mathematicians use improperly in a noticable way "mathematical habits of thoughts" when they are uncalled for. Of course mathematicians vastly differ in the way they handle nonmathematical issues (and there are different approaches among mathematicians to mathematics as well).

Specifically, I do not understand the suggestion about ambiguity; I dont think mathematicians are less open to opinions compared to others, I dont think nonmathematicians are better (or worse) in dealing with shades of grey. I dont know to what extent "the best people alive cannot add fractions", and the advantages and disadvantages of oversimplified model is a very central issue in mathematics as in other areas.

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The questions as stated are extremely general. I hesitate to name anything which is not a social norm or "common sense" (e.g. do unto others as you would have them do unto you, don't spend more than you have), and so these are already known as how mathematicians and other people should conduct themselves in deed and in thought.

One often has to decide one's course of action based on the environment. Thus many suggestions I might make would not find general application for many classes of people, including the class of readers of this posting. Thus the self-referential maxim : "Learn when a rule does not apply, including this rule".

Even after these cautions, much can be said about how mathematics and its complement might further benefit from a "cultural exchange", as long as one is reasonably relaxed about to do with what is gained from such an exchange. In another post I mentioned multiple perspectives as something that could be used to more effect in mathematics; in this post I mention modified reuse.

I hope many have heard the mantra "Reduce; Reuse; Recycle" as a suggestion for how to treat one's environment so that the next generation can have (as many of) the same choices or more that this generation has. Modified reuse can involve no modification, but the idea is to adapt an item or technique for use in a new situation. This is present in mathematics, but I suggest that it be called out more often, and that the next step after completing a sketch of a proof of some result should be "where or how else can this be used", and postpone briefly the step of "how do I fill in the sketch to make sure I got the details right". Often the sketch is the thing worth reusing, and if studied well the sketch can be refined in its reuse, and help with the postponed step. Further, the search for knowledge is no longer about raw fact, but about the connections between such, and a reorganization that will assist future generations in their search; this idea of modified reuse promotes inventing/discovering such connections.

Gerhard "You Know What Goes Here" Paseman, 2011.09.08

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