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This question comes from the perspective of an undergraduate math major who feels that much (although not all) of the mathematical discipline is a liberal art, rather than a science, and should be presented as such. I regularly interact with bright, creative peers, whom I think could make terrific mathematicians with training, but who immediately dismiss the study of math. This is often due to poor experiences with math courses in primary and secondary school, or "I'm just more of a humanities person." My interest is more in the latter response: I too am "more of a humanities person"! I would much rather spend a semester doing nothing but literature study than nothing but biology; mathematics is for me and many in my department (and, I'm sure, many people on this forum and math.SE and mathoverflow) artistic, creative, and self-cultivating in the way humanities studies are.

How, then, can we present the mathematical discipline, in its "true form", to students who may otherwise cast it aside as "hard," "cold," and "calculating"? Certainly, there are students who just won't like math, and that's perfectly fine. However, I worry often that there are many students who would flourish while studying math but are not shown the breadth of study that the field encompasses. Perhaps I'm just rambling now, but this does interest me. Thoughts?

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    $\begingroup$ I am sure the biologists would say the same thing about their field. I am sure it is artistic, creative, and self cultivating to do biology. I just wish that we all recognized the depth of all fields of human study, and could appreciate them without putting others down. It is also not necessary for everyone to study every field to depth. For me, math, education, BJJ, and permaculture are enough (and all about equally deep). $\endgroup$ – Steven Gubkin Jan 20 '17 at 3:09
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    $\begingroup$ I would start with Plato's Meno and discuss the dialogue with the slave. $\endgroup$ – mweiss Jan 20 '17 at 3:45
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I am going to answer this question from the perspective, ironically, of a biology major who was convinced to look into mathematics after some poor experiences in both early undergrad and secondary school. But I was a humanities-heavy biology student:

  1. Set a realistic objective - odds are you're not going to get very many people to switch to math whole-hog. Rather, go for something closer to "appreciation" and a few math classes, with the hope of something more.
  2. Don't push "Math is Beautiful" too heavily. This was an approach a lot of people took with me, and to be frank, it really didn't resonate. It took a long time until I started appreciating elegance in mathematical concepts, and in the timeframe you're talking about, you're facing the problem that they probably consider their field to be beautiful right now.
  3. Instead, I'd suggest showing how it can be relevant to their studies. How even a little bit of mathematical background can help push things forward. I'll give two examples of this, but for me, "This can make Biology cooler..." was the entry, rather than "Math is amazing in its own right..."

Example 1: I met a medieval historian whose background was in civil engineering. Her project was to take the semi-hysterical accounts of the number of Viking ships involved in the invasion of Ireland, typically in the form "ALL THE SHIPS! THOUSANDS!", and try to actually put realistic bounds on those numbers, based on things like the population of Viking settlements, how many ships could have been supported, etc. That's not terribly difficult math, but it was an entirely different way of looking at a problem.

Example 2: The Three-Body Problem by Cixin Liu and Ken Liu. This is an amazing novel spanning the Cultural Revolution to the modern age, all built around the problem in the title. It's a good book without knowing what the Three Body Problem is and just using "It's hard math" as a standin. But it's actively more enjoyable if you get the math behind it - and again, the math isn't particularly advanced.

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    $\begingroup$ + for "Don't push 'Math is Beautiful' too heavily." $\endgroup$ – Daniel R. Collins Jan 21 '17 at 22:38
  • $\begingroup$ Your "semi-hysterical" typo was hysterical. $\endgroup$ – KCd Feb 1 '17 at 11:07
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Many humanities students find it unexpected that mathematicians think about fairness questions - cost sharing, fair elections, fair games, fair division questions of different kinds, etc. Many of these areas can be discussed without much background other than arithmetic, and students find both the questions and the mathematics rather appealing. If you are not familiar with this "domain" you might look at the book:

H.P. Young, Fairness in Theory and Practice, Princeton U. Press.

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A natural point of contact with the humanities is found in philosophy. Many philosophical questions are rooted in problems of the philosophy of mathematics. For example, philosophical pragmatism, which in present times is often applied as general cultural critique, has roots in the work of Charles Sanders Peirce...son of Benjamin Peirce, a leading Harvard mathematician of his day. This is more of a comment, but I think that talking about the associated philosophical problems can lead to an interest in the mathematics...

Questions about the nature of human thought and the mind can be connected to the nature of proof, a la Hofstadter's "Godel, Escher, Bach"...

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