The Princeton Companion to Mathematics, which came out just a few month ago, contains a wonderful short section entitled “Advice to a Young Mathematician” with advice from five eminent mathematicians. I was in the need of inspiration this weekend, and found some in these personal statements. Below the fold you will find a few excerpts applicable to any researcher of any age.

Readers: Please help me and other readers of this blog by posting in the comments section pointers to your favorite sources of research advice.

Sir Michael Atiyah: “My own approach has been to try to avoid the direct onslaught and look for indirect approaches. … it can lead to a beautiful and simple proof, which also “explains” why something is true. In fact, I believe the search for an explanation, for understanding, is what we should really be aiming for.”

Béla Bollobás: “Keep your ability to be surprised.”

Alain Connes: “Once a mathematician truly gets to know, in an original and “personal” manner, some small part of the mathematical world, however esoteric it may look at first, the journey can properly start.”

Dusa McDuff: “often one sees further by starting with the simplest questions and examples, because that makes it easier to understand the basic problem and then perhaps to find a new approach to it.”

Peter Sarnak: “Not to learn the tools is like trying to demolish a building with just a chisel. Even if you are very adept at using the chisel, somebody with a bulldozer will have a huge advantage and will not need to be nearly as skillful as you.”

I also loved this quote from Alain Connes: “Mathematicians usually have a hard time explaining to their partner that the times when they work with most intensity are when they are lying down in the dark on a sofa.” For me instead of “lying down in the dark,” it is “staring absently across the room.”

I’ll end with one final quote from Alain Connes that greatly amused me, but also struck me as largely true, at least with respect to certain areas of physics: “in general mathematicians tend to behave like “fermions,” i.e., they avoid working in areas that are too trendy, whereas physicists behave a lot more like “bosons,” which coalesce in large packs.”

There are other gems to be found in these short accounts; I may blog on some others in future posts. In the meantime, you can find the entire “Advice to a Young Mathematician” online, and please add comments with your sources of advice and inspiration.

Sheldonsays:Nice post. Mentioning Connes, and I can’t help also bringing up P. B. Medawar’s wonderful little book, Advice to a Young Scientist. I discuss a little of his wisdom on how we handle–or don’t handle–the inevitable mistakes in a post I wrote some time ago on “The Intellectual Imperative to Screw Up” ( http://www.guerrillascholar.com/cogito/?p=40).

Another excellent resource is Ronald Gross’ book The Independent Scholar’s Handbook, which I describe in passing in this post on a recent visit to the Exploratorium: http://www.guerrillascholar.com/cogito/?p=91.

Thanks for a great post. I will be reading Connes with great interest in spite of the fact that I am neither young nor a mathematician per se.

Eleanor Rieffelsays:Thanks Sheldon. You make a great point that our educational system “does not seem to formally teach students how to respond to mistakes and, most of all, how not to fear them.”

Gene Golovchinskysays:Interesting difference between mathematics (and perhaps other fields with highly structured theory) on one hand, and Human-Computer Interaction (to pick my discipline) on the other. In math it seems that effective work can be done by hard concentration with little support from external representations or tools. In HCI, although I can think for a while about some problem, typically I need to draw diagrams, sketches, prototypes, or make some other kinds of artefacts to capture the idea not just to communicate it, but to be able to deal with the complexity. In part, it is arguably lack of training (or predisposition) to that kind of effort, but in part it seems that not having expressive formalisms makes it more difficult to merely think about problems.

Or at least so it seems to me without having drawn a diagram.

Eleanor Rieffelsays:Gene,

Hmm. I don’t know about that. Mathematicians certainly draw lots of diagrams and sketch out ideas. I almost included the following quote from Béla Bollobás in my original post: “If after a session your wastepaper basket is full of notes of failed attempts, you may still be doing very well.”

External representations are extensively used in mathematics, and choosing good ones is often key to progress. For this reason, mathematicians inventing new areas of mathematics often spend significant time developing good notation. The most famous example of the importance of notation is the decline of British science and mathematics after Newton, especially in comparison with Europe, because the British insisted on sticking with Newton’s inferior notation for calculus instead of using Leibnitz’s far more suggestive and expressive notation.

Note that Connes says mathematicians are working “with most intensity” when they are lying on a sofa in the dark; he does not say they are doing this most of the time.

jrsays:Nice blog about the advice for young Mathematicians. In my general experience, Math was really fun and challenging when the teacher was really good and knew the stuff. I also think there is great empowerment when young mathematicians see the power of theoretical math being applied to solve real world problems. (I.E. subjects like Cryptography, Number Theory, Numerical Analysis, PDE, ODE, Combinatorics, and Statistics) I guess I am just an applied math guy at heart. :)

Maribeth Backsays:I’ve always loved this quote from Isaac Asimov:

“The most exciting phrase to hear in science, the one that heralds new discoveries, is not Eureka! (I found it!) but rather, “hmmm… that’s funny… “

Michael Heaneysays:My favorite quote is:

“A good scientist is someone who works hard enough to make every possible mistake before coming to the right answer.”

-Richard P. Feynman.

Two excellent books are: “A Ph.D. is Not Enough,” by Peter Feibelman; and “Advice to a Young Investigator,” by Santago Ramon y Cajal.

Andrew Newellsays:Eleanor,

Thank you for bringing our attention to this section with advice to young mathematicians. A comment in the introduction caught my eye: “… there was remarkably little overlap between the contributions.” That is why it is valuable to have so many perspectives – not everything works for any given person.

RIchard Feynman, the Nobel Prize-winning physicist, was aware of this. In the collection of his letters, “Perfectly Reasonable Deviations From the Beaten Track”, he sometimes answers letters from high school students or undergraduates asking for advice. He never prescribes specific subjects or textbooks. A typical answer is the following, to a student who built a cloud chamber for a science project but was afraid that he didn’t have the mathematical aptitude for physics: “If you have any talent, or any occupation that delights you, do it, and do it to the hilt. Don’t ask why, or what difficulties you may get into.”

I can’t resist adding this comment, to a teacher who relayed a student’s “foolish” question – with a far from obvious answer: “Simple questions with complicated answers are always asked by dull students. Only intelligent students have been trained to ask complicated questions with simple answers – as any teacher knows (and only teachers think there are any simple questions with simple answers).”

Eleanor Rieffelsays:Maribeth – that is a great quote from Asimov! I hadn’t heard it before, but totally agree with it. Particularly delightful are the cases where it is funny (Ha, ha!), not just funny (odd). A lot of the pleasure of research is being surprised!

Michael, Andrew – Sounds like I should read more Feynman! A number of years ago, I had read (and enjoyed) some of his lighter books as well as some of his physics lectures, but I haven’t read “Perfectly Reasonable Deviations From the Beaten Track.” Michael – where is your quote from?

jr – It is interesting that Applied Math and applied math do not mean the same thing. Number theory is Pure Mathematics, not Applied Mathematics, but as you say it has wonderful applications in cryptography. I love applications of mathematics, but many of my favorite areas are considered Pure Mathematics. I’d be curious to know the history as to why people started calling certain areas Pure and others Applied. It seems remarkably short-sighted not to realize that future uses of mathematics would rapidly and radically change which subject would be viewed as having important applications.

Sheldonsays:One other note regarding Gross’ book The Independent Scholar’s Handbook:

You can now download the book for free as a .pdf file here:

http://www.sfu.ca/independentscholars/isbook.htm

The book is kind of dated in some respects (I think the latest print issue was in 1993), but it is still full of useful advice and has some wonderful examples of doing good intellectual work in unusual ways.

Forrestsays:Andrew – I like the sentiment expressed in the Feynman quote, a variation on follow your bliss, as much as anyone. I will add that it has worked well for me. However, I am skeptical that it is good advice on average. We always hear such quotes from highly successful people, but no one ever goes out and collects the false positives. I submit that there is an army of people following their bliss into entertainment and sports who will end up disappointed, and another army who didnt follow this path who are better off for it. A few of us are lucky enough to have talents that delight us *and* pay the bills. But I’ve seen no happy advice for those whose interests have extremely low average returns.

Sheldonsays:Before we leave this topic entirely, here is a huge collection of quotes about math and mathematicians:

http://www.quotemaniac.com/Mathematics.html

Enjoy!

Constantinesays:If you liked these five essays, you might also like:

a video of Atiyah talking about math – http://video.google.com/videoplay?docid=-5911099858813393554&ei=mrzZSdaGA4b-

an interview of Atiyah and Singer upon receiving the Abel prize – http://www.abelprisen.no/en/prisvinnere/2004/interview_2004_1.html

biographic material pertaining to Karen Uhlenbeck – http://www.ma.utexas.edu/users/uhlen/vita/bio.html

Another source of inspiration are historical accounts, like “Never at Rest” (a biography of Newton) or “Men of Mathematics” (a biography of many mathematicians). I have not read Constance Reid’s books, but they always get good reviews. I found “A Beautiful Mind” (about Nash) worth reading. Reading about the travails of mathematicians of the past helps me put the challenges of my own life in perspective.

I enjoyed all five essays. While the styles and points of emphasis varied significantly, the authors are all established academics, and I think that there is some value in listening to others’ viewpoints. For example, http://mathgradblog.williams.edu/ is a blog by and for math graduate students.

In another direction, there is non-academic mathematics. My experience is that the rules and goals in other settings (e.g., industry, government) are quite different. For that matter, if one adds in amateur mathematicians to the mix, then things are different there too. For example, the consequence of making a mistake in a calculus class is an entirely different matter than making one working as a quant at a hedge fund. Another example: academics like to pursue interesting ideas, while workers in the private sector like to pursue ones that are (or are thought to be) valuable (usually in the short term).

Personally, when I want inspiration on mathematics, I try to remind myself of why I got interested in the subject in the first place, either by looking over theorems and papers that I find compelling (even after many years of thought!) or reading accounts of people that I respect highly.

Surely a large part of success in mathematics (or for that matter, any endeavor) comes from being able to match opportunities in the real world with one’s personality. (Let me add that I think that women are particularly good at identifying such matches.) Some of the differences were mentioned in the articles, e.g., Atiyah talking about problem-solvers and theorists or Grothendieck contrasting analysis with algebraic geometry. Dyson likes to classify mathematicians as “frogs” or “birds” – see http://www.ams.org/notices/200902/rtx090200212p.pdf The questions to ask oneself seem to be of the following sort: Do I like to collaborate, or do I want to work alone? How competitive am I; how often do I want to prove myself? Do I want to work in fashionable or established fields? Do I like hard calculations, sweeping statements, or counterexamples? Do I want to stay within one clearly-defined field, or do I like interdisciplinary projects? What tradeoff between accuracy, quality, and expense suits me best, e.g., do I prefer to solve lots of fairly simple problems quickly (say, within a few days or a week) or work on hard problems for years? Would I rather be in an environment where people brainstorm without commitment or one where anything publicly stated is expected to be accurate? Do I want immediate feedback, or do I want to receive it only occasionally? Do I value appreciation or money more? Do I prefer to ask questions or give answers? Do I like to risk my time and energy on longshots or sure things? Do I want to be responsible to a small group of experts or a large group of users? Do I want my work to be public or secret? Do I like communicating in words, equations, or pictures? What kinds of mistakes do I find funny?

Also, I could not help noticing that all five essays had parts that were inspiring and also had parts that were comforting or even cautionary. I think that math is a tough business; there is no shortage of sad stories among historical figures (Abel, Archimedes, Chebychev, Galois, Hypatia, etc.), and I get the sense that even people like Newton and Gauss were quite unhappy at times. While everyone agrees that math can be difficult because of technical issues, I think that much of the challenge stems from the difficulties communicating mathematics. As Paul Halmos says, “It saddens me that educated people don’t even know that my subject exists.” And even fellow mathematicians may find understanding ideas outside of their field an uphill battle. So it becomes very hard to evaluate work that might well have been difficult to produce, and the delay in recognition for one’s hard efforts can be frustrating. Regardless of the field one is in, when someone tries to evaluate someone else’s work, the following three questions must be answered: “Is it right?” “Is it important?” “Is it new?” We would all like to live in a world in which answering these questions is an objective process, but alas, it is not so.

I was also struck that Atiyah and Connes both advised choosing your own thesis problem. I happen to have chosen my own thesis problem, and while that decision was the right one for me, I cannot say that I would recommend it for most people. At the very least, the advisor has to be receptive to this choice, and one should be very clear on what the criteria are for success. Also, one must be willing to shoulder much more responsibility.

Research always entails lots of dead ends, and finding a way to manage them is a large part of being able to succeed in the long run. Some people like to think about lots of projects, so that when one is not going well, others can be worked on; alternatively, if one is stuck, it might be a good time to do something completely unrelated (like watch a movie, etc.). Another strategy is to talk about it with another mathematician, perhaps even a collaborator.

But I think it is important to keep in mind what success can really mean; otherwise, what is the point? What keeps me going most reliably is thinking about my favorite formula, my favorite paper, my favorite mathematician, etc.

Eleanor Rieffelsays:Forrest – Like you, I both enjoyed and was troubled by the Feynman quote. For most people, it simply isn’t possible to make a living pursuing their primary passion. I do know, however, a number of people who are pursuing non-lucrative passions more or less successfully while holding down another job. For many of them, that is incredibly rewarding, but it isn’t easy. I don’t know of good sources for advice for successfully balancing the two. Does anyone else have suggestions?

The Independent Scholar’s Handbook that Sheldon recommends has lots of advice on how to be an effective scholar outside of academia, but it only has a short and, in my opinion, insufficient chapter on how to support yourself while doing so. (In other respects, it looks like a great read.) One of my sisters loves the book The Four Hour Work Week, which certainly has advice directed at exactly this subject. I found it helpful in other ways, but on the subject of making money to support other interests, I didn’t feel it was realistic for most people including me. I have always felt incredibly fortunate that my scientific passions have enabled me to work jobs that pay well and that I enjoy (and even leave me some time to pursue my passion for music).

Eleanor Rieffelsays:Constantine – Thank you for your thoughtful comments.

One of my all time favorite books is Constance Reid’s Hilbert. A number of my non-mathematical friends have enjoyed it very much also, on my recommendation. Hilbert is, of course, a strong central character, and the times he lived in and the community he was part of are both fascinating. Reid does a great job communicating what underlies his passion for mathematics in a way that is accessible to all. I particularly loved the descriptions of his collaborations. I still cry when I read about his collaboration and friendship with Minkowski. It is sad to read about the destruction of the vibrant Gottingen mathematics community by the Nazis. You speak of going back for inspiration to your favorite papers or your favorite mathematicians. “Hilbert” is a big source of inspiration to me.

Like you, I also was struck at how the essays discussed the frustrations of research and the common feeling of discouragement, while sometimes the same essay would contain an admonition to enjoy the work. If I can put into words how I think we reconcile the two – which we do! – I will post on the subject. Thoughts, anyone?

Michael Heaneysays:The earlier entries roused my curiosity about independent scholars. I just ordered “The Independent Scholar’s Handbook.” I’ll review it here after reading.