Math is Like Science, Only Proof-y

The slightly unclear photo and shaky handwriting in this diagram brought to you by moving train.

Math is not science.  Sciences seek to understand some aspect of phenomena, and is based on empirical observations, while math seeks to use logic to understand and often prove relationships between quantities and objects which may relate to no real phenomena.  Scientific theories may be supported by evidence, but not proven, while we can actually prove things in math. On the other hand, math is like science, and emphasizing the difference may really work against math.  I find my students often have no sense of how anyone would actually “do math,” for example, how we think of things to try and prove.  Students do seem to have a sense of how people “do science,” and they find it correspondingly less intimidating.  With this in mind I have been working to convince my students that math often works pretty much like science.  Joseph Silverman lays this out in his book A Friendly Introduction to Number Theory (which I am teaching from this semester) really nicely–in number theory, we gather data by computing a lot of examples.  Then we search for a pattern, make a hypothesis, and test it against additional data (more examples).  If the hypothesis doesn’t match the new data, we revise it.  After some iterations, when new data matches our hypothesis, we finally try to prove the hypothesis.

While this step of full logical proof differs from the scientific method, the next part of the process is the same in math as in science: peer review! Science would be a total mess without peer review, likewise math. Simply because it is possible to logically prove things in math, but human fallibility means that our own mistakes may be invisible to us.  Even logic takes a village.

We all know peer review matters—every time we use a published result without having to painstakingly check the proof ourselves, every time we submit a paper which eventually comes back with miraculous reports, from mysterious people who find a host of mistakes, mostly small but occasionally something major (see Adriana Salerno’s post on referees here).  This has always been a bit amazing to me: at least two anonymous people have taken the time to read and carefully check everything I have ever submitted, without being paid or even identified so I could acknowledge them publicly.  Knowing how long it takes me to read and fully make sense of a paper, this represents a significant outlay of time and energy.  What a public service!  Yay for good referees!

Of course, I’m saying this now because I am myself reviewing a paper for the first time, and it is taking quite a bit of work.  And producing a fair amount of anxiety, I must say. I have been asked to review things before but have not been able to say yes, for various reasons.  This spring I was asked to look at two articles that both looked very interesting and were in areas close to work that I had done.  Shouldn’t be too hard, right?  A few hours each, right?

I am now near the twenty-hour mark on the first article.  Ten of those hours happened on a plane during an epic cross-country flight that was diverted and delayed. At around 1 AM, my red pen started oozing ink after a pressure change and dripped all over the printout I was working with, making it look like my blood, if not necessarily sweat and tears, was literally going into this project. Don’t get me wrong—twenty hours is actually okay, and reviewing a paper is clearly way, way easier than writing one.  This manuscript is well-written, the result is interesting, and I haven’t found any deep issues.  I’ve learned some things by working through it so closely.  Overall this has been really positive; however, I have been at points been racked by terrible doubts.  The early part of the manuscript had some typos and errors in definitions that made many computations impossible to follow.  I was sure that something was wrong, but also sure that I must be missing some obvious reason or solution.  I tried changing the definitions so that the later calculations would hold, but the changes I made seemed to take me in circles.  The authors are mathematicians I really respect and the work is interesting, but my anxiety mounted as I found I couldn’t move forward because I wasn’t sure what the definitions should be.  Eek!  What to do?  Should I send the editor a query to pass on to the author?  Was it okay to ask someone else to look over my work?  I really wanted to do a good job, and I also didn’t want to embarrass myself by making a big deal over something obvious.  Even though I am always encouraging my students to speak up in class, saying there is nothing wrong with asking questions, I really didn’t want to ask what I felt might be a stupid question.

It turned out that after some encouragement from a math friend and a little time away from the article, I found the right small changes to make the paper consistent and was able to move ahead.  All is well.  However, I swore to become a proof-reading fiend in my future writing.  My own errors may be hard for me to see, but they may be even harder for others to fix.  I have a whole new appreciation for referees, and I hope that I can make their lives easier in the future.

This experience also started me wondering about how reviewing/refereeing papers is appreciated or rewarded by the larger mathematical community.  I am happy to do this, and I think it’s important to do this.  I benefit because this made me read an an article I would have wanted to read anyway.  I now know more about my area and that I am just a slightly better mathematician for having worked through this paper carefully.  I have to wonder, though: how much do departments and tenure committees appreciate refereeing as either scholarly or service work?  Do editors take reviewers who write thoughtful reports more seriously?  For a pre-tenure professor, how much referee/review work will benefit a career and how much is too much?  I honestly don’t know any of these things, and I would love some reader feedback on this.

Returning to my starting thoughts: math is not science, but their fates are inextricably linked. We support the communities from within by peer-reviewing, but we have to work together as a larger community to secure support from the outside.  The National Science Foundation’s support for mathematical research has become more and more essential as other funding sources (like the National Security Agency’s Mathematical Science Program) disappear.  Also, as a lover of logic and believer in the importance of using the power of mathematical thinking to do good in the world, I want to advocate for science-based policy for the common good.  Karen Saxe, director of the Washington office of the AMS, has written some excellent posts about NSF funding for mathematical sciences and even provided a template email to send to your representatives and senators. Though one appropriations request deadline has passed, it is still important and never too late to let them know that you fervently support funding for science and mathematics.  That’s why I’m going to the March for Science this Saturday in Washington DC.  In a lucky or well-planned concomitance, the National Math Festival is also happening on Saturday, about a mile from the site of the march.  I’m headed to both—anyone else?  Maybe we can even lead the march back for some math fun afterward! Sounds like a great weekend for celebrating and supporting math and science. So I’d better get busy and finish my referee report.

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And now for something slightly different, part I

Christelle Vincent and her pet icosahedron.

Readers, I realize that you may be tired of hearing about my life.  I mean, PhD+Epsilon is about early-career mathematical life, but when I write, it’s usually about my life/career, which is only one of many options. Thus, this week we have a blog experiment—I ask someone with a slightly different job a few questions about their life.  In this first installment (hopefully there’s more!), I check in with Christelle Vincent. She was in town last week to talk math (torsion points on the Jacobians of Picard curves) but we also talked math life. Christelle and I met in 2013 at a Sage Days workshop and have been part of many workshops together since.  After graduating from the University of Wisconsin in 2012, Christelle spent three years as a post-doc at Stanford.  She then spent a semester at ICERM before starting a tenure-track position at the University of Vermont in January 2016. She’s at a state school, is currently supervising a Masters student, and will be taking PhD students sometime in the next couple of years.  I asked Christelle a few questions, and here’s what she had to say:

Have you realized anything surprising in the last few years of your math career?

I remember vividly being in the audience of a career panel the year after I graduated and someone asked how one thinks of new problems to work on. That was an anxiety that I very much shared. I felt that the biggest threat to my budding career was to run out of problems and that one day I would just not be able to publish anymore because I ran out of problems to solve. Five years later, that is not an anxiety I have anymore. I do worry about working on problems that are “hard enough” to get grants and recognition from the community, but even that doesn’t feel so bad. I think it’s because having the experience of progressing from working on exactly one problem (my thesis) to working on enough problems that I feel that I could keep going on for a few years at this point, I feel like it’s likely that naturally, if I keep working and doing what I’m supposed to, I will find myself working on harder problems and developing more of a program.

What’s the best part of your job so far?

I feel very valued at my job. As a professor, I am more involved in the department life and I feel that I am really contributing to making our department a better place for my colleagues and our students. My colleagues value my research, ask me about it, and support me when I need to travel. My students enjoy my classes and what I do for them. I feel like I’m coming into my own a little bit more, I’ve shed a lot of the insecurities I had about being good enough to “make it” in academia. For me, being a math professor has been the endgame for a very long time, and finally getting there is really enjoyable.

What are some big issues in your math life/career?

As much as I am enjoying taking some time to enjoy where I am after getting a tenure-track position, I know that very soon I will start to worry about getting tenure. Right now I feel like I need to find time to do even more research, to carve out and protect that time. It’s something that I’m struggling with a little bit. It’s easy for me to get caught up in a bunch of little things that leave me exhausted and research has to happen before all of those little things. I’m still very new at my job so I think the rhythm will get easier with time.

Thanks to Christelle for taking the time to talk with me (math and blogwise)!

So, now my questions are for readers—what kinds of early-career math lives would you like to read about, and what would you want to ask? Let me know in the comments!

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Growing Roots


I don’t have much to say about math at the moment. Classes are proceeding. Papers are too. Committee work has continued to intensify. Hiring for two positions has taken an enormous amount of time and energy from the whole department. But I have to admit these things haven’t really been at the front of my mind.

In speaking to many of my other early-career colleagues, one of the strangest things about the tenure track is its semi-permanence. Sure, people change schools sometimes. There’s a open letter or blog post every other week about somebody leaving academia entirely for greener pastures. And there’s always the looming worry that “tenure track” won’t turn into “tenured.” But I think most people don’t start their tenure track job with one foot out the door. We mean to take off our coats and stay awhile.

Which is the polar opposite of the lifestyle we’ve grown accustomed to. Our entire adult lives are spent as transients: off somewhere for undergrad, somewhere else for grad school, and increasingly a series of postdocs or visiting positions or adjunct gigs. No time to put down roots or develop a real sense of community, at least outside the walls of the department. Little time or energy for volunteer work. No money for a house to fix up, or a garden to plant. Your friends are a revolving door of graduating PhDs, fresh-faced new grad students, and faculty members at schools where you’ll spend a year or two at best.

A colleague in another department realized last spring that the reason she was getting anxious as summer approached was because it felt like it was time to move again, since that’s what she’d done over so many summers before. And I’ve felt that way once or twice too these past couple of years, like it’s automatically getting to be time to move on.

I loved the nomad life while I lived it, and given the academic job market I don’t think I ever really thought it would stop. But now I also love my job and my colleagues and my students and my city. Even though I’m getting itchy feet out of pure habit, I can’t imagine leaving.

Monday afternoon, we close on a house. Our first. I honestly don’t think I ever thought we’d own a home. I figured we’d end up in a large-ish city with a too-high cost of living, or we’d be so transient that it would never be worth the investment. But then we moved to Frederick, where, at least compared to neighboring DC, a house is pretty reasonable. And we have no plans to leave anytime soon. So next week we’ll have a home, with a garden and a tire swing (really!), and maybe we’ll finally get around to finding furniture that didn’t come from the Ikea scratch-and-dent section or off of somebody else’s curb.

I’ve also thrown myself into the community. I used to do a ton of volunteer work when I was younger, but then graduate school took up all my energy. Even during my postdoc I didn’t do much, though I had the time. I just wasn’t compelled to invest a ton of work in a community I’d be leaving so soon. But now I’m planting trees and pulling invasive plants in the park near my house. I’m getting involved in local politics, even organizing educational events in town. I think I even promised to go to a city budget meeting to advocate for a cause in a moment of weakness. I’ve met almost every one of my political representatives, from the local to the federal level. I feel a part of this town in a way I haven’t since I was a kid. I can’t even go to the grocery store without bumping into somebody I know, and somehow that doesn’t sound as bad as I would have thought it was a year or two ago.

So we’re fighting the weird career wanderlust for now. None of this fixes the worries about disaster striking come dossier time. But I don’t see a reason to spend the next five years with my bags half-packed. And I think I’ll be even more productive now that I can take breaks in my sweet tire swing.

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