{"id":1966,"date":"2017-04-17T19:55:56","date_gmt":"2017-04-17T23:55:56","guid":{"rendered":"http:\/\/blogs.ams.org\/phdplus\/?p=1966"},"modified":"2017-05-20T12:21:52","modified_gmt":"2017-05-20T16:21:52","slug":"math-is-like-science-only-proof-y","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/phdplus\/2017\/04\/17\/math-is-like-science-only-proof-y\/","title":{"rendered":"Math is Like Science, Only with Proof"},"content":{"rendered":"
\"\"<\/a>

The slightly unclear photo and shaky handwriting in this diagram brought to you by moving train.<\/p><\/div>\n

Math is not science.\u00a0 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.\u00a0 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\u00a0difference may really work against math.\u00a0 I find my students often have no sense of how anyone would actually \u201cdo math,\u201d for example, how we\u00a0think of things to try and prove.\u00a0 Students do seem to have a sense of how people \u201cdo science,\u201d and they find it correspondingly less intimidating. \u00a0With this in mind I have been working to convince my students that math often works pretty much like science.\u00a0 Joseph Silverman lays this out in his book A Friendly Introduction to Number Theory<\/a><\/em> (which I am teaching from this semester, and he says this in chapter 1,\u00a0accessible here<\/a>) really nicely–in number theory, we gather data by computing a lot of examples.\u00a0 Then we search for a pattern, make a hypothesis, and test it against additional data (more examples).\u00a0 If the hypothesis doesn\u2019t match the new data, we revise it.\u00a0 After some iterations, when new data matches our hypothesis, we finally try to prove the hypothesis.<\/p>\n

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.\u00a0 Even logic takes a village.<\/p>\n

We all know peer review matters\u2014every 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<\/a>).\u00a0 This has always been a bit amazing to me: at least two anonymous people have taken\u00a0the time to read and carefully check everything I have ever submitted, without being paid or even identified so I could acknowledge them publicly.\u00a0 Knowing how long it takes me to read and fully make sense of a paper, this represents a significant outlay of time and energy.\u00a0 What a public service!\u00a0 Yay for good referees!<\/p>\n

Of course, I\u2019m saying this now because I am myself reviewing a paper for the first time, and it is taking quite a bit of work.\u00a0 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.\u00a0 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.\u00a0 Shouldn\u2019t be too hard, right?\u00a0 A few hours each, right?<\/p>\n

I am now near the twenty-hour mark on the first article.\u00a0 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\u2019t get me wrong\u2014twenty hours is actually okay, and reviewing a paper is clearly way, way easier than writing one. \u00a0This manuscript is well-written, the result is interesting, and I haven\u2019t found any deep issues.\u00a0 I\u2019ve learned some things by working through it so closely.\u00a0 Overall this has been really positive; however, I have been at points been\u00a0racked by terrible doubts.\u00a0 The early part of the manuscript had some typos and errors in definitions that made many computations impossible to follow.\u00a0 I was sure that something was wrong, but also sure that I must be missing some obvious reason or solution.\u00a0 I tried changing the definitions so that the later calculations would hold, but the changes I made seemed to take me in circles.\u00a0 The authors are mathematicians I really respect and the work is interesting, but my anxiety mounted as I found I couldn\u2019t move forward because I wasn\u2019t sure what the definitions should be.\u00a0 Eek!\u00a0 What to do?\u00a0 Should I send the editor a query to pass on to the author?\u00a0 Was it okay to ask someone else to look over my work?\u00a0 I really wanted to do a good job, and I also didn\u2019t want to embarrass myself by making a big deal over something obvious.\u00a0 Even though I am always encouraging my students to speak up in class, saying there is nothing wrong with asking questions, I really didn\u2019t want to ask what I felt might be a stupid question.<\/p>\n

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.\u00a0 All is well.\u00a0 However, I swore to become a proof-reading fiend in my future writing.\u00a0 My own errors may be hard for me to see, but they may be even harder for others to fix. \u00a0I have a whole new appreciation for referees, and I hope that I can make their lives easier in the future.<\/p>\n

This experience also started me wondering about how reviewing\/refereeing papers is appreciated or rewarded by the larger mathematical community. \u00a0I am happy to do this, and I think it’s important to do this. \u00a0I benefit because this made me read an an article I would have wanted to read anyway.\u00a0 I now know more about my area and that I am just a slightly better mathematician for having worked through this paper carefully. \u00a0I have to wonder, though: how much do departments and tenure committees appreciate refereeing as either scholarly or service work?\u00a0 Do editors take reviewers who write thoughtful reports more seriously?\u00a0 For a pre-tenure professor, how much referee\/review work will benefit a career and how much is too much?\u00a0 I honestly don\u2019t know any of these things, and I would love some reader feedback on this.<\/p>\n

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.\u00a0 The National Science Foundation\u2019s support for mathematical research has become more and more essential as other funding sources (like the National Security Agency\u2019s Mathematical Science Program<\/a>) disappear. \u00a0Also, 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. \u00a0Karen Saxe, director of the Washington office of the AMS, has written some excellent posts about NSF funding<\/a> 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.\u00a0 That\u2019s why I\u2019m going to the March for Science<\/a> this Saturday in Washington DC.\u00a0 In a lucky or well-planned concomitance, the National Math Festival<\/a> is also happening on Saturday, about a mile from the site of the march.\u00a0 I\u2019m headed to both\u2014anyone else?\u00a0 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\u2019d better get busy and finish my referee report.<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

Math is not science.\u00a0 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 … Continue reading →<\/span><\/a><\/p>\n

<\/div>\n","protected":false},"author":90,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[49],"tags":[219,220,218,216,217],"class_list":["post-1966","post","type-post","status-publish","format-standard","hentry","category-refereeing","tag-march-for-science","tag-national-math-festival","tag-nsf","tag-peer-review","tag-refereeing"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3c1jI-vI","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/phdplus\/wp-json\/wp\/v2\/posts\/1966","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/phdplus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/phdplus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/phdplus\/wp-json\/wp\/v2\/users\/90"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/phdplus\/wp-json\/wp\/v2\/comments?post=1966"}],"version-history":[{"count":5,"href":"https:\/\/blogs.ams.org\/phdplus\/wp-json\/wp\/v2\/posts\/1966\/revisions"}],"predecessor-version":[{"id":2002,"href":"https:\/\/blogs.ams.org\/phdplus\/wp-json\/wp\/v2\/posts\/1966\/revisions\/2002"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/phdplus\/wp-json\/wp\/v2\/media?parent=1966"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/phdplus\/wp-json\/wp\/v2\/categories?post=1966"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/phdplus\/wp-json\/wp\/v2\/tags?post=1966"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}