# Papers on history and philosophy of mathematics suitable for master's students

In the fall, I will give a course called "Perspectives in Mathematics". This is a mandatory course at our master's program in mathematics (including applied mathematics and statistics). The course contents are rather vague, but we are supposed to cover historical and philosophical perspectives on mathematics, as well as the role of mathematics in society.

One thing I would like to do is to ask the students to read some short texts on topics such as history, philosophy and perhaps sociology of mathematics and discuss them in class. I would like these to be more than just popular science but still reasonably accessible. Also they should contain some interesting perspectives and perhaps challenge the beliefs of the students, so there is something to discuss. Finally, I want to cover not just the classical history of pure mathematics, but also applied mathematics and its role in society.

I would be really grateful for tips on research papers or other short texts that you think may be suitable.

• I think this is quite a broad question -- it is not too different from asking for tips on suitable research papers or other short texts on "pure mathematics". The history of mathematics is also a very, very broad area, so to say that you want to cover "not just the classical history of pure mathematics" is a bit like saying that you want to cover "not just all of analysis". That said, here's two ideas: you can find many good such texts in the works by Peter Neumann. Also, Chris Hollings has written about the development of how we came to define groups in the way that we define them today. Jun 30, 2021 at 14:41
• As Carl-Fredrik said, this question is very broad; but one short article on math history, written not by a historian but by a mathematician, that I really like is Stanley's paper on Hipparchus-Schröder numbers: math.mit.edu/~rstan/papers/hip.pdf. It led to a reappraisal of combinatorics in the ancient world (for a more academic/historical account see the paper of Acerbi). Jun 30, 2021 at 14:44
• Tymoczko’s anthology on New Directions in the Philosophy of Math is probably a good collection for this purpose. (Note it is no longer “new” — it was last revised 25 years ago.) If anyone can point me to an online table of contents, I could recommend particular parts. Jul 1, 2021 at 11:10
• If you want to show that historical perspective can be useful to a mathematician, then including something by Harold Edwards is probably essential. A combination of an overview like “Read the Masters!” and an example like “A Normal Form for Elliptic Curves” might work well. Jul 1, 2021 at 11:26
• I think that Felix Klein's "Development of mathematics in the 19th century" is very good.
– Z. M
Jul 2, 2021 at 19:42

In hopes that some commenters will post their suggestions as answers, let me post this recommendation here.

An increasingly important aspect of contemporary mathematics is the role of computers and experimental mathematics. One article addressing this is Marjin Heule and Oliver Kullmann's 2017 The Science of Brute Force in Communications of the ACM (and included in the next year's Best Writing on Mathematics).

The motivating example for the article is the authors' resolution of the boolean Pythagorean triples problem: Starting from 1, how far can the integers be colored red or blue with no monochromatic solutions to $$x^2 + y^2 = z^2$$? Their 200 terabyte proof based on SAT solvers shows that $$\{1, \ldots, 7824\}$$ can be 2-colored to avoid monochromatic Pythagorean triples, but $$\{1, \ldots, 7825\}$$ cannot---previously, it was not known whether there was a finite limit.

The larger discussion brings in Appel and Haken's proof of the four color theorem, Lam's proof that there is no projective plane of order 10, the interplay between computation and theoretical results to resolve the Erdős discrepancy problem, and the notion of "alien" mathematical statements. The article is a well written mix of mathematical detail and philosophical considerations with quotations from David Hilbert and Oscar Wilde.

For the philosophy of mathematics, a standard reference is Benacerraf and Putnam's anthology Philosophy of Mathematics: Selected Readings. This will provide a good introduction to some of the "traditional" topics in the subject such as logicism, intuitionism, formalism, and platonism. There are of course many important topics not covered. I'd recommend that you also consider something written by Imre Lakatos, such as Proofs and Refutations, as well as Humanizing Mathematics and its Philosophy: Essays Celebrating the 90th Birthday of Reuben Hersh.

For the history of mathematics, there is the encyclopedic book A History of Mathematics by Carl B. Boyer and Uta C. Merzbach. In particular, this book has some good chapters on the early history of mathematics in non-Western civilizations, a topic that was largely neglected in the West until relatively recently. Now, there's another take on the history of mathematics, which is to treat it not just as a subject in its own right but as something that can illuminate and inform the work of a research mathematician. As mentioned by someone in a comment, the work of Harold Edwards is exemplary in this regard. If your students have a strong enough mathematical background for it, I'd recommend the book Galois Theory.

A rather neglected topic is the relationship between mathematics and religion. A good anthology is Mathematicians and their Gods: Interactions between mathematics and religious beliefs, edited by Snezana Lawrence and Mark McCartney. There are also the fascinating books Naming Infinity by Loren Graham and Jean-Michel Kantor and Equations from God by Daniel Cohen.

Many of your students may be interested in the topic of women in mathematics. There are several books on this topic, such as Complexities: Women in Mathematics, edited by Bettye Anne Case and Anne M. Leggett.

Finally, depending on how much you're willing to venture into controversial territory, you could consider devoting some time to the question of how to promote diversity, equity, and inclusion in mathematics. Especially in the realm of mathematics education, DE&I is a very hot topic—possibly hotter than you want to touch. I confess that I am not au courant with the literature, so I don't feel qualified to give recommendations, but one example I heard about recently (and which has generated heated debate) was A Pathway to Equitable Math Instruction. But perhaps others who are more knowledgeable than I am can provide better guidance in this area.

• Do the readings in Benacerraf and Putnam count as “reasonably accessible”? I’d say the collection is reasonably accessible only for people who already have more training in mathematical logic than most master’s students. (As a reference, the table of contents is at assets.cambridge.org/97805212/96489/toc/9780521296489_toc.pdf) Jul 2, 2021 at 16:59
• @MattF. It's been a while since I've looked at that book, but as I recall, it's a mixed bag as far as accessibility is concerned. I thought that some articles were pretty accessible. To some extent, some knowledge of mathematical logic is unavoidable when it comes to these particular topics in philosophy. But if Benacerraf and Putnam is too advanced and the students are unfamiliar with Gödel's Theorem, then one possibility would be Torkel Franzén's book, Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Jul 2, 2021 at 20:03

An interesting (and topical!) topic would be the commentary on the status and philosophy of mathematics by Nikolai V. Ivanov a.k.a. Owl, as expounded in his blog Stop Timothy Gowers. While it was active only for a couple of year or so (the last post seems to be dated February 2017), the posts are extremely interesting and thought provoking. It is clear that Owl thought deeply about the topics, and I learned a lot from his posts. In any event, this will certainly fulfil your requirement that

Also they should contain some interesting perspectives and perhaps challenge the beliefs of the students, so there is something to discuss.

• Is the real identity of the blogger who goes by the Owl common knowledge? I was under the impression that they were deliberately pseudonymous, and have not seen Ivanov's name linked before. Jul 1, 2021 at 7:30
• @asahay: (it is not really a secret) -- owl-sowa.blogspot.com/2015/11/… Jul 1, 2021 at 12:46
• Looking through these posts, they mostly seem to be polemics from a very stereotypical conservative point of view on "pure mathematics." E.g. "computers can never capture the human essence of math," "Martin Hairer is an applied mathematician because he studies stochastic PDEs," and so on. What interesting new perspectives have you found here? Jul 1, 2021 at 18:46
• @auniket I just read through a somewhat random selection of posts from this blog, and it strikes me that much of what he says has been said by Reuben Hersh and others in his "school." As for your question about painting, surely the answer is yes. Just ask any random person in the street whether "modern art" seems fundamentally different in nature from, say, Renaissance painting. Jul 2, 2021 at 20:44
• @auniket I just discovered the article Varieties of mathematical understanding by Jeremy Avigad which I think you will find interesting. Jul 7, 2021 at 0:47