## Mathematical habits of thought and action which would be of use to non-mathematicians

Once again I come to MO for help with something I'm writing for the public.

Which habits of mathematicians -- aspects of the way we approach problems, the way we argue, the way we function as a community, the way we decide on our goals, whatever -- would you recommend that non-mathematicians adopt, at least in certain contexts?

In other words: if you can imagine a situation in which someone came to you for advice, and you said, "Look, I think you should be a little more like a mathematician about this and...." what would be the end of the sentence?

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Sort authors alphabetically! – Mariano Suárez-Alvarez Sep 7 2011 at 3:48
Recognize the lack of definitions. That's the really big one I can think of. Also, we are a far more supportive bunch than most other academic communities - but I'm not sure that's the type of answer you're looking for. – James D. Taylor Sep 7 2011 at 3:53
"more suppportive bunch" is actually good -- I think mathematicians are very good at disagreeing with each other in a productive way. – JSE Sep 7 2011 at 4:03
Most people would definitely not be receptive to a sentence that starts with "you should be a little more like a mathematician about this...". Just present the recommended habit and why it is worth considering and drop that it's one that mathematicians use. – KConrad Sep 7 2011 at 12:05
Please see the following meta post if you want to debate whether or not this question should stay open. Please upvote this comment so it appears above the fold: meta.mathoverflow.net/discussion/1132/… – David White Sep 7 2011 at 15:00

An applied problem is often formulated as follows:

Go 1 mile East, then 1 mile North, then 1 mile West. You'll get to your destination D, and that's your goal.

And they follow the direction like slaves, they go East-North-West. They can't tell the difference between the goal and the description of the goal. It takes sometimes a mathematician to tell them to go North right away.

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 It takes a mathematician to realize that going 1 mile east, north and west is not the same as going 1 mile north. – Ori Gurel-Gurevich May 3 at 20:15 @Ori, U'r so-so sophisticated. – Wlodzimierz Holsztynski May 4 at 21:33 I talked, metaphorically, about a real issue(!), which I observed over years of workings with engineers for different companies. The problem I have mentioned is also common among average students of mathematics (I am not talking about the gifted ones). Somehow, every time there is a not exactly mathematical question there show on MO some negative emotions, it's so silly. – Wlodzimierz Holsztynski May 4 at 21:41 @Wlodzimierz Holsztynski, I got your intention. I only tried to be so-so funny, and apparently failed even at this modest goal. – Ori Gurel-Gurevich May 6 at 5:28 @Ori, it so happens that your comment relates and illustrates still another shortcoming of common thinking (or lack of it) in the world of applications. I lost interest in writing about them but now I may do it. Your comment perhaps has succeeded in 2 downvotes of my answer :-) And in general, no analogy holds water, while they can be useful (illuminating) anyway. – Wlodzimierz Holsztynski May 6 at 23:21

Mathematicians understand that a lack of evidence is not the same as a proof of non-existence.

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Here are some that came to mind:

Equivalence. Basically, the idea that two things can be functionally equivalent (or close to equivalent) even if they look very different (and conversely, that two things can be superficially similar but functionally quite distinct). For instance, paying off a credit card at 10% is equivalent (as a first approximation, at least) to investing that money with a guaranteed 10% rate; once one sees this, it becomes obvious why one should be prioritising paying off high-interest credit card debt ahead of other, lower-interest, debt reduction or investments (assuming one has no immediate cash flow or credit issues, of course). Not understanding this type of equivalence can lead to real-world consequences: for instance, in the US there is a substantial political distinction between a tax credit for some group of taxpayers and a government subsidy to those same group of taxpayers, even though they are almost completely equivalent from a mathematical perspective. Conversely, the mistaking of superficial similarity for functional equivalence can lead to quite inaccurate statements, e.g. "Social Security is a Ponzi scheme".

Counterfactual thinking. The ability to take a counterfactual hypothesis and deduce consequences from it (or, in some cases, absurdity) is common in mathematics (and in a few other disciplines, such as law or fictional writing) but not always among the general public. For instance, to provide evidence of a claim such as "A always leads to B", it is not enough to produce examples in which A and B both hold; one has to show that the counterfactual situation in which A holds and B fails is necessarily either impossible or implausible. Or for a more mundane example: to get a true sense of how impressive it is that, say, your daily horoscope seems to be eerily accurate, one should analyse the plausibility of a counterfactual situation in which the type of statements one typically receives in a horoscope turns out to be clearly inaccurate.

Quantification. Cost-benefit analysis is basically impossible to do right unless one has at least a rough order of magnitude for each of the costs and benefits. With only a qualitative understanding of the costs and benefits, one may end up expending far too much time and money to avoid a tiny amount of risk or cost, or conversely skimping on a negligible expense which would protect against a high-probability catastrophic event in the future. Also, because one cannot easily adjudicate between costs and benefits when one has a qualitative mindset instead of a quantitative one, there is a psychological incentive to "simplify" the problem by downplaying or ignoring the costs of actions that one wishes to take, while downplaying or ignoring the benefits of actions that one wishes to avoid.

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 Nice answer, but I think your horoscope example belongs in your qualitative vs. quantitative section. – horse with no name Feb 27 at 12:48

Keep in mind that it is easy to make mistakes.

The most striking thing I learned from doing mathematics is that even in an environment entirely devoid of ambiguities and characterized by precise axiomatic constraints to the point that it became synonymous with it, even when I am doing my absolute best to be completely careful and precise, even when I double check each of my words, then show it to two careful colleagues, then let it simmer for a while, then go through it again with a critical eye, then show it to an authority in the field, then re-read it again; even after this excruciating process of constant self-examination, even after the strength of my arguments has confounded (perhaps in the two meanings of the word) my utmost critical self as well as the objections of several knowledgeable observers, I know that dozens of mistakes, inaccuracies and outright errors still remain.

Doing math is certainly not the only way to come to this bitter conclusion-simply interacting with people is usually enough, as Philipp Roth once famously remarked-yet I can't help to shudder when I sometimes contemplate how many things I must be getting completely and obviously wrong whenever I am outside my tiny bubble of professional rigour, where a prompt and witty remark is more than often enough to obtain general assent.

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Some habits of thoughts in mathematics which are useful outside mathematics are:

Quantitative thinking - Often it is very important to understand quantitative aspects of an issue.

Logical thinking - Mathematics is a very clear platform for precise logical thinking.

Computational thinking - Often we need to make some computations, to write some formulas and manipulate them, in order to address an issue

probabilistic thinking - The ability to think about luck, chance, risks and decisions in probabilistic terms and to draw insights from probability theory is important in many areas

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 +1 for Probabilistic Thinking. Certainly understanding risk and expected value help when making decisions. I've also found subtler uses of probability coming up in my thinking, e.g. my ability to accept uncertainty in general and react accordingly, how much time to devote to unlikely possibilities, etc. I feel like my philosophical thinking changed after I learned basic probability theory, but it was so long ago that I can't quite explain what I mean. I wonder if others had a similar reaction. I'll post another comment if I think of a good way to put this in words. – David White Sep 8 2011 at 13:36

This is a sort of anti-answer. When the Unabomber Manifesto was published by the NYT, someone in sci.math (or sci.math.research) recognised the reasoning as being like that of a mathematician. I don't know if anyone mentioned that to the FBI. It eventually turned out that the unabomber was indeed a mathematician. Btw, I was just in the next room when one of his bombs exploded.

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I think it should be checked whether someone in sci.physics "recognised" the reasoning as being like that of a physicist, someone in sci.computer.science "recognised" it as being like that of a computer scientist, &c. – Jonathan Chiche Sep 9 2011 at 19:57

One of the things that mathematicians spend a lot of their time doing is trying to understand and learn new things. And while there are a lot of different methods for doing this, anecdotally two things that mathematicians do more than other people.

Gerhard Paseman already mentioned multiple viewpoints. Reading from multiple sources and try to approach things from multiple directions. (Often I feel, especially with students, that when they hit a wall with a text they're trying to learn from, they're not aware that certain aspects may be covered better elsewhere or that a different point of view really might make a difference.)

Even more important, though, is taking a step back and let things simmer, without actively thinking about them for more than a little at a time. There's a cycle of work and passive thinking that needs to take place. (Of course, this goes along with the famous story about Poincare's moment of inspiration while stepping on a bus.)

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(or Poincare's work habits in general!) – Kevin Lin Sep 8 2011 at 7:52
I heard (from Serre, I can't help name dropping) that Frobenius wrote to Dedekind that his success with the original group-determinant problem came from (paraphrasing) “following my usual habit with such problems, to pretend I didn't care, and was interested in something else entirely, whereupon the solution came to me.” – L Spice Sep 8 2011 at 14:52
From Lam's article (p. 9) at www.ams.org/notices/199803/lam.pdf, Frobenius wrote "I hope you will not give away the trade secret to anyone. My great work On the Methods of Mathematical Research (with an appendix on caterpillar catching), which makes use of it, will appear after my death." – KConrad Sep 13 2011 at 9:52

Think in a precise context, get an idea geometrically/globally and check your idea logically with a rigor without any concession.

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What does it mean to get an idea geometrically in a non-mathematical setting? – L Spice Sep 8 2011 at 14:53

The Australian writer Clive James, after several decades of experience, came to the conclusion that "Writing is essentially a matter of saying things in the right order" (see his Unreliable Memoirs, p. 162). Mathematicians have a head start on writers in this respect, because we have to say things in the right order from day one!

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Yes. Even in the 21st century, all known languages are not commutative. – Dror Speiser Sep 7 2011 at 16:01
I hear there's even a tribe somewhere whose language is a Moufang loop... – Mariano Suárez-Alvarez Sep 7 2011 at 16:04
Latin is kinda commutative, though. Incidentally, it has always stricken me as peculiar that languages seem to evolve in the direction complicated => simple, rather than the other way round like everything else. Just after we invented fire and the wheel, there must've been a genius somewhere who invented a zillion of cases, declensions and verb tenses. Must've been either the aliens or Atlantis. – Federico Poloni Sep 7 2011 at 19:55

This is really just an extension of James D. Taylor's comment on the question, but recognizing the value of definitions, and the inherent ambiguity without them, is ridiculously helpful.

I recently saw a talk on how to teach students to write mathematics well. The advice "think like a lawyer" was given, which I totally agree with when writing mathematics. Anyone here who has read a quality legal document will know the similarities which this analogy is getting at. Both mathematicians and lawyers define their terms clearly at the beginning (of a debate/proof/court case) in order to eliminate as much ambiguity as possible from the words being used.

This is a great skill to have in order to cut through the BS in lots of other situations. E.g. Time and time again you see opinion pieces in which writer has no real point, but just trivially exploits the lack of a definition for a certain word. This really annoys me. Or you can be much better at seeing when an argument is a genuine difference of opinion or just a confusion arising from two people having different definitions.

If you take this too far, you end up as a bit of nihilist in that respect though: There's nothing to argue about because either people have different opinions (and there will always be people with different opinions) or people have different definitions... and arguments end trivially and inconclusively (I know I've done this many times to end boring arguments:) "If we accept your definition of [e.g.] feminism, then you're right and if we accept mine, then I'm right". (I suppose seeing to the core of an argument like this is similar to David White's first point).

On the other hand, you can debate quite freely things you have no clue about, just by deciding on a few axioms/vaguely reasonable assumptions and working from the definitions! (this is sort of the skill of debating competitions).

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Reduce the clutter in a discussion: I'm not saying everyone should abstract everything away, but I've found mathematicians have a knack for seeing what is pertinent to the discussion and putting the other things aside for the time being. I see this particularly on university committees, where mathematicians seem to be the best at both getting things done and getting near optimal solutions. All over the world we see parties in a discussion holding grudges and settling into opposing camps based on historical disagreements. I see the same behavior on university committees but mathematicians seem to be better about ignoring the disagreements of last week and focusing solely on the problem at hand.

In a similar vein, mathematicians seem to be able to take positive action quickly in these discussions rather than getting disheartened by the difficulty of the problem at hand. I suspect this is because we train ourselves to build theorems out of lemmas and to see the whole structure of a proof before starting. So I suppose this point is more about making an outline or plan of attack (and many fields train you to do this), but we seem particularly good at it because we do such problem-solving for a living and we know how to break a problem down into easier pieces.

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+1 for "we know how to break a problem down into easier pieces." – Joel Reyes Noche Sep 8 2011 at 0:27

Since people seem to be pussyfooting around the obvious:

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I never thought I would see the day LOLCats will appear on MathOverflow! – Asaf Karagila Sep 7 2011 at 12:32
I agree with the spirit of this response, but I can't help but think that its very unclear what, exactly, would constitute a "proof" in a nonmathematical context. – alex Sep 7 2011 at 16:40
Why so surprised? In the long run everything on the internet will be pictures of cats. – Noah Snyder Sep 7 2011 at 19:00
I can uses teh axiom of choice? – J.J. Green Sep 7 2011 at 19:27
I admit this is really getting off-topic, but there's a great mathematical LOLtheorist: loltheorists.livejournal.com/124774.html – Henry Cohn Sep 7 2011 at 21:43
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I recently talked to a friend of mine about this. She has a PhD in Mathematics and works in the software industry. In Math it is a common practice that you present a proof to your colleagues and discuss it with them. In her company she suggested a similar approach for pieces of software that people have written. Her colleagues, who were mainly trained in Computer Science and in particular not at a university but rather at a technical school, found this completely unacceptable, since they were not ready to face the possibility of being somewhat publicly criticised.

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Since you are a mathematician, you won't mind if I publicly criticize (or even criticise) your spelling. – Gerry Myerson Sep 7 2011 at 13:11
I believe peer-review of code has been recognized as best-practice. I don't know how widely implemented it is, though. – Thierry Zell Sep 7 2011 at 14:37
@Gerry Myerson: Indeed, I don't mind so much and I am only slightly embarrassed. I edited it. – Stefan Geschke Sep 7 2011 at 18:29
[@Gerry: I spell criticise with an s. Perhaps it's a UK/US thing? ] – Kevin Buzzard Sep 7 2011 at 21:48
@Kevin Buzzard: The typo was more embarrassing than that. – Stefan Geschke Sep 7 2011 at 22:06
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Questioning assumptions is one aspect of mathematical endeavour that is useful in other areas of society to improve or understand why things are the way they are, or change to how they might be. Having faith in logic and one's conclusions is also useful, and is often used in diagnosing problems and finding what are root causes of certain situations, or solutions to certain problems.

Another aspect which may be omnipresent outside of mathematics and which mathematicians and others alike would do well to use is multiple perspective. Being able to process several views of a situation often leads to an increase in understanding, an abillity to modify, and possibly achieve an outcome which is desired in many of the viewpoints.

And, of course, trying to communicate well the particular sequence of ideas requires a well-timed orchestration of abstraction, generalization, specialization, and shifting of viewpoints. However, it might be even better to relate such information by telling a story ("You mean my English literature class applies to real life?").

Gerhard "Yes, It Really Does Apply" Paseman, 2011.09.06

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I agree completely with "questioning assumptions...to improve or understand." It's amazing how many times I get into discussions, even with very smart people, where hidden assumptions (and lack of good definitions) are all over the place. Sadly, questioning such things--even for the right reasons--is usually taken as criticism and ironically this prevents us from reaching an improvement or shared understanding. I wonder if this negative reinforcement ever stops mathematicians from questioning assumptions in this way, at least in everyday conversation. – David White Sep 7 2011 at 13:25

Edit: I have taken away my comments comparing my experience in mathematics and math ed departments. I still think there is something true about the kind of character that emerges from working on hard problems which demand a certain kind of rigor, and I also think there is something special (and not known to the general public) about the strong social bonds that exist in the mathematical community, but I don't think that one needs to contrast math and math ed groups to make that point.

My short answer (to which Gil alludes below) was: Persistence and humility.

Though I'm not sure this applies only to mathematicians, but in general to people who work on hard, and perhaps in some ways technical, problems.

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I wonder if (2) and (3) might have to do with funding. I've never been in a department in any other science but I imagine physicists, chemists, etc need to compete for funding for their labs. – David White Sep 7 2011 at 13:27
Manya: You are just listing some of your personal experiences. It's anecdotal. No all math departments are as the one you describe. Not all math ed departments are as the one you describe... – André Henriques Sep 7 2011 at 14:34
I think Manya makes that extremely clear with phrases like "my math colleagues" and "groups I have been with". – Allen Knutson Sep 7 2011 at 15:48
Thank you, Allen. But I think André has a point. Actually I had come back to this site in order to remove this comment because I wasn't sure these observations generalized, not even completely from my experience. – Manya Sep 7 2011 at 20:09

In my experience, mathematicians will frequently argue (in general, not just in mathematics) by passing to an extreme case at the beginning. Non-mathematicians (again in my experience) sometimes object to such a mode of argument as invalid or irrelevant because such extreme hypotheticals are clearly unrealistic.

I think that the mathematical idea of first setting all the parameters to their maximal, or minimal, values, and understanding that case, before trying to tune them to a more realistic choice of values (and seeing how the solution/context changes with the parameters) can sometimes be valuable (even though it involves as a first step considering a situation that may be very unrealistic).

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I think this is an extension of being able to recognize definitions, or lack thereof. Going to extremes sets the parameters of the definitions in discussion. – James D. Taylor Sep 7 2011 at 4:41
I've definitely done this in my non-academic job, and it surprises me that no one else thought to consider such situations (and it wasn't necessarily an extreme hypothetical). – David Roberts Sep 7 2011 at 5:27
+1. Yes!; well-articulated. I do this all the time when talking with my girlfriend and she nearly always argues that the extremity of my example makes it invalid and irrelevant to the discussion, whereas I see the extremity as potentially highlighting the salient points of the general case. – Spencer Sep 7 2011 at 14:33
Actually,In areas outside mathematics, passing to extreme cases first and considering unrealistic situations is a habit of thought and action which is more often damaging than useful. – Gil Kalai Sep 8 2011 at 6:21
The Monty Haul problem is a good litmus test in this regard. Resolving that problem by using an extreme case (1,000,000 doors!) to illustrate the main issue tends to be convincing if and only if one has had exposure to mathematical thinking. – Terry Tao Sep 10 2011 at 5:49