Snippets of Mathematical Candor

A while ago I wrote a post in response to a great Slate article reminding us that math - like writing - isn't something that anyone is good at without (at least a little!) effort. As the article's author put it, "no one is born knowing the axiom of completeness." Since then, I've come across a few other snippets of mathematical candor that I found both helpful and encouraging. And since final/qualifying exam season is right around the corner, I've decided to share them here on the blog for a little morale-boosting.

The first comes from a fantastic post written by University of Illinois at Chicago's recent PhD Jeremy Kun (also blogger at the excellent Math ∩ Programming) in which he answers the question What is it 'really' like for a mathematician to learn mathIn short, his answer is contained in the post's title:  "Mathematicians are chronically lost and confused (and that's how it's supposed to be)."

 Of course, he means this in a good way and elaborates by sharing this colorful metaphor of mathematical research by Andrew Wiles (best known for his proof of Fermat's Last Theorem). 

 
wiles quote.jpg
 

I definitely feel like I'm stumbling in the dark whenever I tackle a new subject. (I believe the technical term for this is learning.) And I'm not even at the research stage yet! In response to this occasional feeling of lostness that we students tend to feel, Kun says

 

Sometimes the answer is to pinpoint one very basic question I don't understand and try to tackle that first. Other times it's to learn what I can and move on.

 

This is great advice for graduate students. Personally, I find that there's just not enough time to learn it all, so often I have to be okay with not being able to understand a topic as well as I'd like. But occasionally I end up "wasting" an entire day just grappling with a single problem/idea. Although those days are frustratingly slow, they actually end up being the most productive and satisfying and not a waste at all! So I was encouraged when Kun mentioned that one of his colleagues has similar sentiments:

 

If I spend an entire day and all I do is understand this one feature of this one object that I didn't understand before, then that's a great day.

 
 
 

While we're chatting about the often laborious process of doing math, have you read the Math Overflow profile of the late William Thurston? Here's a screenshot: 

I love that first sentence of his second paragraph! In my own experience, I've found that perseverance really is key as I'm well acquainted with fog, muddle, and confusion. In fact, mathematical fog is one of the main reasons why I started this blog. But if I'm to be honest, there are times when I'm not sure how much of that fog is simply "the nature of the beast" and how much of it is, well, "encouraged."

 
 
Piper Harron's thesis (source)

Piper Harron's thesis (source)

Quite naturally, this brings me to mathematician Piper Harron's epic thesis, without which no foray into mathematical candor would be complete. No doubt you know what I'm talking about. If you don't, then you must go check it out. Seriously. Right now. Just browse through the table of contents. With numerous laysplanations (explanations for the layperson) and chapter titles like "Getting Mathy With It," you can't tell me you won't want to read more! Harron's thesis is quite the epitome of math-made-accessible, and I was especially drawn in by the transparent honesty in her prologue. There she states that her thesis was written for

 

People who, for instance, try to read a math paper and think, 'Oh my goodness what on earth does any of this mean why can't they just say what they mean????' rather than, "Ah, what lovely results!" (I can't even pretend to know how 'normal' mathematicians feel when they read math, but I know it's not how I feel.)

 

This is SPOT ON! As I mentioned above, I'm often in the stumbling-in-the-dark-while-groping-for-the-light-switch phase. But there are instances when it feels like the switch was installed in the most remote, most inaccessible location possible, like, I don't know, behind the drywall of the ceiling in the attic. I can't tell you how many times I've thought Geez! Why can't mathematicians just say what they mean?!  And since this happens so frequently, I can't help but wonder if there's a secret math-committee out there whose goal is to obfuscate math as best as they can. 

But of course that's silly. (Right?) There is no such committee. (Right?!) But mathematics does at times give off an exclusive vibe which, as I hinted above, I don't much like. So I was very happy to read this excellent article by mathematician and freelance writer Evelyn Lamb in which she contrasts Harron's thesis with Shinichi Mochizuki's proof of the abc conjecture. I won't go into detail since Lamb does this in her article, but the bottom line is that for over three years mathematicians have been working - unsuccessfully, so far - to verify a 500-page-long proof of a conjecture in number theory. But why three years, you ask? In a word: inaccessibility. As Lamb notes, Mochizuki's proof (and, from what I've read, his lack of efforts in communicating his results to other mathematicians) can seem deliberately obscure. So I deeply appreciate her candor as she compares the ways Mochizuki's proof and Harron's thesis were received by the larger mathematical community:

 

I'm not quite sure what conclusions to draw from the way the stories where shared, but it feels important. I think it says something about how mathematicians present themselves allow others to present them. On some level, we want others to put us on a pedestal. We don't really mind it when someone gets the message "oh, it's too complicated--you wouldn't understand."

On the other hand, the strong positive reaction to Harron's work within mathematics shows that many in the mathematical community are hungry for something different. Even people for whom the status quo has worked (after all, they're still there) recognize that mathematics loses when we build barriers for some groups of people of encourage people to adopt the 'certain attitude' Harron writes about in her thesis.

 

In light of this, Lamb closes the article by urging mathematicians to "think about how they want to explain mathematics and who they are inviting in or leaving out in the process." 

 
 

Speaking of accessibility, this brings me to my new favorite book: Love and Math by Edward Frenkel. (It's a good read!) In the preface, Frenkel acknowledges that math has a reputation of being the "equivalent of watching paint dry" - an "insufferable torment" and "pure torture." But - and I love this - he also knows that

[Math] is too precious to be given away to the "initiated few." It belongs to all of us.

YES! He then proceeds to give the reader an account of his work on the Langlands Program with the intention of not only informing but also of unveiling the oft hidden (because of encouraged fogginess?) beauties of mathematics. Towards the end of the book, Frenkel states the following which really hit home for me: 

 
frenkel quote.jpg
 

He then points to a footnote, reminding the reader of the Gottschalk v. Benson decision, 409 U.S. 63 (1972) in which the U.S. Supreme Court stated, "a scientific truth, or the mathematical expression of it, is not a patentable invention.... He who discovers a hitherto unknown phenomenon of nature has no claim to a monopoly of it which the law recognizes."

Aside from obvious practical implications, I think the above is a fantastic reminder that math does not belong to anyone or to any one group of people, and so there is in fact no math-obfuscation committee. And just in case you (or I!) need more convincing, I strongly recommend this post by Fields Medalist Terry Tao in which he answers the question, "Does one need to be a genius to do math?" with an emphatic "NO." He writes,

 

In order to make good and useful contributions to mathematics, one does need to work hard, learn one's field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the 'big picture.' And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic "genius gene" that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities (emphases original).

 

Since Tao is one whom most of us would without hesitation call a genius, I found his words to be even more significant.

 
 

This brings me right back to that Slate article I mentioned at the beginning. After the author reminds us that "no one is born knowing the axiom of completeness," he goes on to say, "even the most accomplished mathematicians had to learn how to learn this stuff." How true! And I love how math teacher and self-proclaimed bad-drawer Ben Orlin illustrates this in his 'How to Edit your Math Pessimism' series: 

 
How to edit your pessimism when first encountering negative numbers, via Math with bad drawings
 

I think it's comforting to know that we students can replace "negatives" with just about any mathematical concept we've struggled with today and be in such good company!

 
 

More Snippets

  •  In this article, Marcus du Sautoy points out that everyone - not just those whose brains are 'wired' mathematically - is capable of doing mathematics.
     
  • In a similar vein, Benjamin Braun wrote this piece for the AMS Blog in which he addresses "the secret question (Are we actually good at math?)." At the same blog, Evelyn Lamb wrote about math and the genius myth. Both posts contain several great links.
     
  • Though not math-specific, this tweet needs to be tattooed to the backs of eyelids of students everywhere. (Or at least framed and hung on a wall).
     
  • Over at Baking and Math, Yen Duong notes that students too often think they are too "stupid/slow/dense" to do math, while the problem is often in the communicator rather than the mental abilities of the student.
     
  • At the 2013 Joint Mathematics Meetings, Francis Su gave an excellent talk on why teachers should not value students based on their accomplishments and academic performance. I also love this article he recently wrote for the MAA Focus newsmagazine.
     
  • Speaking of performance, both Cathy O'Neil and Dan Lee have written about the ugly side of math competitions.
     
  • This Quora answer to "What is it like to understand advanced mathematics?" and this article from the Princeton Companion to Mathematics emphasize the amount of hard work one must do in order to produce good results. 
     
  • And finally, here's a pretty funny but all-too-real answer to the question "What do grad students in math do all day?"