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Obviously, mathematics provides essential tools for physicists, biologists, economists, engineers and many others to use in their research. Equally obviously, physics, biology, economy and engineering provide the inspiration for research topics and directions in mathematics. But this isn't what this question is about. Instead, I'm wondering about tools provided to mathematicians by other disciplines.

Are there any examples of tools (excluding software) from another discipline that you personally have used in your mathematical research?

E.g., I find dimensional analysis (as in assigning dimensionfull units to variables) a very useful tool from physics, which can greatly help in finding errors during algebraic manipulation of equations.

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    $\begingroup$ Probably better clarify that you want to exclude "software" as an answer, or at least lump all computers and computer programs into a single answer, so that you don't get a zillion answers that just list out a whole bunch of programs that are useful for math. $\endgroup$
    – Zach Teitler
    Commented Sep 21, 2020 at 7:25
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    $\begingroup$ @ZachTeitler good point! I've edited the question. $\endgroup$
    – gmvh
    Commented Sep 21, 2020 at 7:27
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    $\begingroup$ Depending on what you mean by tools, I've used dictionaries for translations; the Chicago Manual of Style; and plenty of computer hardware -- all produced by companies based on the work of people in other academic disciplines. $\endgroup$
    – user44143
    Commented Sep 21, 2020 at 11:39
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    $\begingroup$ ... but coffee-making equipment doesn't count? $\endgroup$
    – shane.orourke
    Commented Sep 22, 2020 at 8:04
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    $\begingroup$ "An old joke tells that mathemtics is the second-cheapest department at any college because all you need is a pencil, paper and a wastebasket. Philosophy is the cheapest because you don’t need the wastebasket." $\endgroup$
    – Gerry Myerson
    Commented Sep 22, 2020 at 23:01

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The William G. Pritchard Fluids Lab conducts experiments–actual, physical experiments with messy, wet fluids–in the Penn State Math Department. Perhaps this counts as drawing on laboratory techniques of physics, engineering, etc.

I haven't personally used this in my research, though. So I'm marking this answer as community wiki.

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    $\begingroup$ Good point - there is also a fluids lab at DAMTP in Cambridge, damtp.cam.ac.uk/research#Fluid%20and%20Solid%20Mechanics. $\endgroup$
    – gmvh
    Commented Sep 21, 2020 at 7:39
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    $\begingroup$ I wish the Penn State site had more information -- there are broken links for the lab's namesake, for "a lab in the math department!" and "apply"; and neither of the faculty members has posted a CV or explanation of their intellectual background. $\endgroup$
    – user44143
    Commented Sep 21, 2020 at 11:31
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    $\begingroup$ That there is a wet lab in a US math department is probably a historical accident..but the world of fluid dynamics is full of applied math folks collaborating with physicists and engineers...so in itself it isn't unusual. $\endgroup$
    – Piyush Grover
    Commented Sep 21, 2020 at 16:39
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    $\begingroup$ John Bush, at MIT, runs the Applied Mathematics Laboratory (which is a fluid mechanics lab): math.mit.edu/~bush . He did postdoctoral work at Cambridge. $\endgroup$
    – Michael Lugo
    Commented Sep 22, 2020 at 17:51
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Two books in my library (but, sadly, as yet unread) claim to use physical methods (i.e., methods of physics, specifically, mechanics) to solve - or suggest a way of solving - mathematical problems:

The Mathematical Mechanic: Using Physical Reasoning to Solve Problems (Levi, 2009)

INTRODUCTION

...the book does exact revenge—or maybe just administers a pinprick—agsinst the view that mathematics is a servant of physics. In this book physics is put to work for mathematics, proving to be a very efficient servant (with apologies to physicists). Physical ideas can be real eye-openers and can suggest a strikingly simplified solution to a mathematical problem. ...

Some Applications of Mechanics to Mathematics (Uspenskii, 1961) - a slim volume, reproducing a lecture to (Russian) high school students:

FOREWORD

The applications of mathematics to physics (in particular, to mechanics) are well-known. We need only open a school text-book to find examples. The higher branches of mechanics demand a complex and refined mathematical apparatus.

There are, however, mathematical problems for whose solutions we can succeessfully use the ideas and laws of physics. A number of problems of this kind, soluble by methods drawn from mechanics (namely, using the laws of equilibrium) were given by the author in his lecture ...

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I hope I correctly got what you mean by "tools". Google maps is a great tool for some applications of graph theory, optimization and even topology. I have seen several papers that use the map tools for developing optimal or sub-optimal algorithms on finding the shortest routes or statistical analysis etc. For example, a brief search leads to this paper and a handful of others.

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Since this is now CW, I can just as well add some answers.

As Matt F. mentioned in comments, language skills are often useful in mathematical research (e.g. for accessing literature written in French or Russian, or indeed in English in the case of second-language speakers), and those can certainly be considered tools from another discipline.

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Since you mentioned dimensional analysis, you might enjoy the book Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving by Sanjoy Mahajan. From the book description: "Sanjoy Mahajan builds, sharpens, and demonstrates tools for educated guessing and down-and-dirty, opportunistic problem solving across diverse fields of knowledge—from mathematics to management. Mahajan describes six tools: dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy."

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I have built physical models (wood, paper, brass, plastic) of geometric objects (polyhedra, dissections, tensegrities) that have helped me see and prove theorems.

Steinhaus's Mathematical Snapshots is an inspiration. Dover offers the ebook. Better: find a used hard copy.

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    $\begingroup$ I love models, but I don't see how they're a tool from other disciplines. $\endgroup$
    – user44143
    Commented Sep 22, 2020 at 16:19
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    $\begingroup$ @MattF. Fair comment. The "other discipline" might be art, or carpentry. $\endgroup$
    – Ethan Bolker
    Commented Sep 22, 2020 at 16:25
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    $\begingroup$ or, these days, 3D printing. $\endgroup$
    – Michael Lugo
    Commented Sep 22, 2020 at 17:53
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    $\begingroup$ @MichaelLugo I do that too, but left if out since computer applications were discouraged). You can find stuff here cs.umb.edu/~eb/slicecubes and here gallery.bridgesmathart.org/exhibitions/… . $\endgroup$
    – Ethan Bolker
    Commented Sep 22, 2020 at 20:17
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    $\begingroup$ BTW, this is the easiest possible away to understand Desargues theorem (in 3 dimensions) ... - Also - the most astonishing and unexpected thing I have ever seen: I moved to a new neighborhood about 15 years ago and while driving around saw in someone's front yard a bunch of large stick objects made of bamboo sticks (he had lots of bamboo plants). Each was about 4' tall. All of the platonic solids, of course. Quite a few Archimedean solids. And a proof of Desargues' theorem!!! 6' high made of bamboo in his front yard! $\endgroup$
    – davidbak
    Commented Sep 23, 2020 at 0:13
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I find that thinking about types (as in object-oriented programming) is really helpful. This can help clarify things (and in particular, help students). For example, one cannot usually add a matrix-type and a partition-type. I suppose this is closely connected to category theory and/or type theory, but one does not need any background in those fields, in order to think about types; perhaps this is similar to the "units"-connection mentioned in a different answer.

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    $\begingroup$ Even though OOP is the first place where one encounters the concept of types conciously (say, as a beginning CS student), and so this is totally fine for the scope of the question, I still want to point out that having types does not require being object-oriented. In fact, I would argue that the programming languages which most cleanly embrace the concept of types are typically not object-oriented (Haskell is probably the best example). In a way, the object-oriented paradigm forces you to lump together the types (i.e. what objects are) with all the things you can do with them. $\endgroup$
    – Achim Krause
    Commented Sep 30, 2020 at 13:47
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Sorry to post yet another self-answer, but this is completely separate from my other answer.

Polymath-style projects or sites like MathOverflow are tools for mathematical research that could be considered applications of social science to mathematics research. This is especially true if incentive systems like reputation scores or badges are being used, which are ultimately based on insights from psychology and/or economics (which can to some extent be understood to be the study of incentives).

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    $\begingroup$ Is this accurate — was StackOverflow based on applying social science? This article on the software from 2008 already shows reputation and badges, and I don’t see any evidence that the programmers who built it used any social science in its construction. readwrite.com/2008/09/10/stackoverlow $\endgroup$
    – user44143
    Commented Sep 25, 2020 at 22:36
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    $\begingroup$ Jeff Atwood, one of the founders of StackOverflow, shows his intellectual influences on his page of recommended books for developers, with plenty of books by computer scientists, and none by economists or psychologists: blog.codinghorror.com/recommended-reading-for-developers $\endgroup$
    – user44143
    Commented Sep 25, 2020 at 23:08
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    $\begingroup$ He lists lots of works by Edward Tufte, who (while his background is in statistics) worked a lot on political economy. The book "Peopleware", which he lists, while written by programmers, starts from the statement "The major problems of our work are not so much technological as sociological in nature". And of course UI design is ultimately based on perceptual psychology, even if most people writing books about it are computer scientists. It wouldn't work if it weren't. $\endgroup$
    – gmvh
    Commented Sep 26, 2020 at 7:33
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It's not clear how useful this would actually be, but the idea of using "citizen science" (a somewhat sociology-based notion which has been developed into a useful tool for — primarily observation-based ­— scientific research by astronomers, botanists and zoologists, among others) for mathematics research has been floated here on MO: Can pure mathematics harness citizen science?. It would be nice to know whether any of the projects proposed in answers to that question ever came to fruition.

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