# Math is Like Science, Only with Proof

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, and he says this in chapter 1, accessible here) 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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### 7 Responses to Math is Like Science, Only with Proof

1. Greg Friedman says:

You’re right that refereeing can seem (and perhaps be) thankless, and it’s never quite clear how much credit one gets for it when it comes to tenure/promotion/raise time (probably some, but probably not much). But I nonetheless tend to think of refereeing as something of a moral obligation. For one thing, someone is refereeing my papers, so it’s something I owe back to the community. But more than that, I think we all, as referees, have an obligation to try to make the papers we recommend accepting into things that are useful for our communities. If I think a paper has errors or is difficult to read in places but don’t try to require making improvements, then some other poor reader down the line is going to be using incorrect results or struggling to understand correct ones. I don’t want to be that reader down the line, and so here’s my opportunity to try to prevent that for others (and for myself).

That said, there are other reasonable considerations. The ultimate responsibility for a paper lies with the author, and if the author has done a shoddy job in some way, I believe it’s completely acceptable to provide a report that says a more detailed version of “I’ve tried to read this and I’ll mention some corrections, but overall the paper is not up to snuff and the author needs to fix it up first before I’ll put more time into it.” You also have to know your own limits and not be afraid to turn down a referee job that you’re just not qualified to do because the field isn’t right. Remember that editors can’t be experts on everything, and so sometimes they’re just making an educated guess about who would be a suitable reviewer. I think it’s reasonable to say “this just isn’t something I’ll be useful for, but please send me something else closer to my field when you have it and then I’d be happy to help.” Of course it’s really great if you can suggest some other referees who might be better for the job.

• Beth Malmskog says:

Thanks for your thoughts on this! Any guess at how many papers an “average” mathematician referees per year?

2. Allen Knutson says:

I hope you realize that your initial referee report on a paper is unlikely to be your last one. Its primary purpose is to let the editor judge whether the paper is worth pursuing at all, and it can certainly include “I was confused by the following issues and would like the authors to address them before I continue review”.

• Beth Malmskog says:

This is good to know–on the first go round, it’s unclear what the role of my report will be or even the depth that is expected. I wrote a report at a similar depth to what I have received on my submitted papers, which I expected to be probably my only communication with the authors. It didn’t occur to me that this was likely be more of an iterative process. I guess I did have one paper that required some more major changes and involved a second referee’s report, but I didn’t realize that was usual.

• Greg Friedman says:

Maybe this varies from field to field, but I’d say that multiple serious referee’s reports is more “not unusual” than it is “usual.”

3. BP says:

When something in a paper I’m refereeing makes it difficult for me to proceed, I usually communicate with the authors right away, via the editor, and tell the editor that I would like to see a revised version before making a final recommendation. I think it is better for everyone this way! The editor will be happy to be able to respond to the authors quickly, even if the report is only partial. The authors will be happy that they got a timely response, and will be happy to learn how they can make their paper more readable. And I am happy because I saved myself from spending hours being stuck on some point that probably the authors could have made clearer. Sometimes it is a point I should have been able to figure out, but even then there is not much to be embarrassed about since this kind of thing happens all the time, and usually only the editor knows who was being stupid, and even the editor will soon forget this and remember only my timeliness.

• Beth Malmskog says:

That’s a really nice way to think about it!