Low Dimensional Topology

March 18, 2017

A new “Mathematician’s Apology”

Filed under: Misc. — Jesse Johnson @ 8:18 am

In the two and a half years (or so) since I left academia for industry, I’ve worked with a number of math majors and math PhDs outside of academia and talked to a number of current grad students who were considering going into industry. As a result, my perspective on the role of the math research community within the larger world has changed quite a bit from what it was in the early days of may academic career. In the post below, I explore this new perspective.

 In “A Mathematician’s Apology”, published in 1940, G. H. Hardy argued that the study of pure mathematics could be justified entirely by its aesthetic value, independent of any applications. (He used the word “apology” in the sense of Plato’s Apology, i.e. a defense.) Of course, Hardy never had to apply for an NSF grant and his relatives probably never asked him why someone would pay him to solve problems without applications.

In the following decades, mathematics helped win the Second World War and send astronauts to the moon. Many mathematicians began to justify their work in abstract research by pointing to examples such as number theory in cryptography, where ideas from abstract mathematics that were developed based on aesthetics proved to be unexpectedly useful for real world problems. In the 1960s, as baby boomers headed off to college and PhD programs struggled to keep up with the need for new faculty members, some mathematicians began to argue that teachers who were involved in active research would be better equipped to teach students how to think mathematically.

But today, now that graduate programs produce more PhDs than can fill the available research and teaching positions, the reality has set in that most mathematics PhD students will not go on to careers that involve teaching, let alone abstract research. Moreover, the economic slowdown that followed the post-war boom has made it harder for governments to justify investments, whether in the form of grants or tenure lines, for research whose value won’t be apparent for decades or even centuries.

So the mathematics community faces a choice: either accept the new reality by cutting back PhD programs or rethink the way that abstract mathematics should fit into society.

In this post, I will argue that by changing the way we justify mathematics research and the ways we think about the role of the research community in the wider world, we can sustain or even increase graduate programs and research funding without changing our core values or the fundamentals of graduate education. I won’t attempt to distinguish between “pure” and “applied” mathematics. The term abstract research is intended to imply both. I will argue three points:

  • The background one gets from a graduate degree in abstract mathematics is extremely, and increasingly, valuable in a wide range of non-academic careers, beyond the stereotypical security/military and financial sectors.
  • The value of this background comes from time spent working within a large, active, academic community engaged in abstract research and is much greater than the external value of the research itself.
  • Embracing this perspective will not cause a massive exodus of mathematicians from academia, but will instead cause an increase in the number and diversity of students entering graduate programs.

This perspective argues that students leaving academia for industry are the most valuable contribution that math PhD programs make to the rest of society. The changes the community would need to make in order to embrace this new perspective are not simple or easy, but they are mostly peripheral. The value of a background in mathematics comes from the way that students currently learn the ideas, research practices and thought processes. The required changes have to do with the way we recognize this value: The ways we talk to students about potential careers, the ways that we approach professional development and the ways that we talk to each other and to non-mathematicians about how our research fits into the rest of the world.

In particular, when we discuss the external value of mathematics research, we should de-emphasize the theorems we prove, and focus on the diversity of perspective that members of the research community bring to non-academic organizations. A great deal of research in the past few years has demonstrated the value of diversity in teams, and while most of the discussion has focused on ethnic and gender diversity, the same principle applies to intellectual diversity. In fact, a major benefit of ethnic and gender diversity is that it’s a proxy for diversity of perspective. Similarly, the perspective that one forms from engaging in mathematics research can be invaluable to a team of mostly non-mathematicians, not because it’s objectively better than any other perspective, but because it’s different.

While it can be hard to pin down exactly what makes a mathematical perspective different, here’s a partial list. Mathematicians are not the only people who can do these things, but engaging in abstract mathematics research trains students to do them well:

  • Thinking at and between different levels of abstraction: Understanding how axioms fit together to form lemmas, then theorems, is good practice for understanding other complex systems that are too large to see all at once.
  • Boiling systems down to their essentials: Abstracting systems into definitions and axioms requires determining what’s fundamental and what’s peripheral.
  • Discovering parallels between unrelated systems: Solving a problem by transforming it into a previously-solved problem works in the real world too.

When combined with a bit of domain knowledge, these skills can be used to translate vague intuition into precise and usable statements, incorporate ideas from a range of perspectives and terminologies into a cohesive system and create a scaffolding that allows a team to reason about a complex system. Acquiring the domain knowledge that makes this possible is non-trivial – at the very least, it requires a number of years of working outside academia – but the mathematical perspective makes it an order of magnitude more powerful.

One can’t form a mathematical perspective from books and lectures alone. It can only come from working on abstract problems within an active research community. For a graduate student, the research problem is a lens that brings all the tools and problems of mathematics into sharp focus. Every conversation with another mathematician becomes a chance to learn how they would approach the problem. Every new idea must be understood well enough to determine whether it can be applied to help solve the problem.

These ideas, from across mathematics, are abstracted from problems in hundreds of other fields, and bring with them artifacts of the thought processes that spawned them. And while many ideas get written into papers and books, the folklore and meta-ideas that surround them are, arguably, much more important. They make the research community a living entity, and while individual mathematicians may turn coffee into theorems, it is the community that turns students into mathematicians.

Meanwhile, students are increasingly aware of the problems with the academic job market. Many promising undergraduates who love the subject never apply to graduate school because they don’t want to become a professor or don’t think they have what it takes. Moreover, women and members of underrepresented group are much more likely to make such a decision because they’re more likely to perceive that the cards are stacked against them. If they never enter graduate school, we never get the chance to convince them that they can make valuable contributions to the mathematics community.

If attitudes change so that an academic career is seen as one of many acceptable career paths that a math PhD can lead to, the research community might lose a few would-be professors, but more importantly, it will gain graduate students who love the subject more than the career path. These students will bring a much broader diversity of background and interests, which will enrich the community with new ideas. Already, many PhD graduates choose non-academic careers late in the process, after they discover how limited their academic career options are. If they can make these decisions sooner, it will benefit them individually and the math community as a whole.

Changing the way the research community thinks about career paths and its relationship with the outside world will not be simple or easy, and a prescription for such change is far beyond the scope of this post. However, they won’t require changing the fundamentals of how we create and teach mathematics, since these are the things that make a math research background so valuable. We should help students to learn about non-academic careers and how mathematicians can fit into them. We should not push students into applied math and statistics courses or make them into “data scientists”. The value that a mathematician brings to a non-academic career comes from engaging with the mathematics community on an abstract dissertation problem. Today, the math PhDs who follow non-academic career paths are individually demonstrating that value. All we, as a community, need to do is find better ways to recognize it.


  1. thx for the thoughts, was just musing on hardys timeless thoughts about abstraction and beauty in math again recently & run across this in cybersynchronicity. sounds like a tough pill to swallow, some bitter medicine, but we do seem to be in a antiscientific cycle at the moment wrt current admin :(

    think the bigger battle is to try to figure out how to resolve the fundamental conflict between science and capitalism (+ politics) in their current forms and try to fix each to coexist with the other. think capitalism is more of the problem/ significantly out of balance right now. but scientists not recognizing that or acting on it could also be a significant part of the problem. aka “all that is reqd for the spread of evil is for good men to do nothing.”

    (found via reddit)

    Comment by vznvzn — March 18, 2017 @ 12:36 pm | Reply

    • Thanks, vzn. I actually wrote this before the election and for various reasons waited to post it. But it does seem even more relevant with the recent talk of major cuts to funding agencies.

      Comment by Jesse Johnson — March 19, 2017 @ 9:50 pm | Reply

  2. […] Read Full Story […]

    Pingback by A new “Mathematician’s Apology” | ExtendTree — March 19, 2017 @ 3:43 am | Reply

  3. I find this too feeble and unspecific: “Engaging with the mathematics community on an abstract dissertation problem” would equally apply to physics, philosophy, or sociology instead of mathematics. Mathematics offers more: rigor of thought, no way to fudge a missing answer, and no way to replace substance with show. Only computers are less forgiving – mathematicians can become good programmers. But above all: clarity, precision, knowing and questioning your assumptions, and caring about the subject rather than yourself, all innate to mathematics, are rare enough and helpful everywhere.

    Comment by Bernhard von Stengel (@bvonstengel) — March 19, 2017 @ 11:20 am | Reply

    • Thanks, Bernhard. I absolutely agree that “Engaging with the * community on an abstract dissertation problem” could equally apply to any field in the liberal arts. There are lots of different ways to characterize what students learn from engaging in math research, and how it differs from other fields. Rigor is certainly a major one. The important thing is to recognize that these skills are as useful outside academia as inside.

      Comment by Jesse Johnson — March 19, 2017 @ 9:59 pm | Reply

  4. it is very interesting! Thank you!

    Comment by beltranslations — March 20, 2017 @ 3:50 am | Reply

  5. […] Here's an interesting blog perspective on the state of Math PhD in today's workforce: https://ldtopology.wordpress.com/2017/03/18/a-new-mathematicians-apology/   jedishrfu, Mar 21, 2017 at 7:53 […]

    Pingback by A New Mathematicians Apology -- Where the jobs are | Physics Forums - The Fusion of Science and Community — March 21, 2017 @ 8:00 am | Reply

  6. Reblogged this on Towards Data Science.

    Comment by Towards Data Science — March 23, 2017 @ 9:16 am | Reply

  7. Reblogged this on Quantum Tunnel.

    Comment by jrogel — March 24, 2017 @ 7:40 am | Reply

  8. […] via A new “Mathematician’s Apology” — Low Dimensional Topology […]

    Pingback by A new "Mathematician's Apology" - Reblog - Quantum Tunnel Website — March 24, 2017 @ 7:45 am | Reply

  9. […] via A new “Mathematician’s Apology” — Low Dimensional Topology […]

    Pingback by A new "Mathematician's Apology" - Reblog - Quantum Tunnel Website — March 24, 2017 @ 7:46 am | Reply

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