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I know, "practice, practice, practice." But is that really what professional mathematicians do? Do you actually solve a differential equation every day, do a contour integral every day, do a SVD of a matrix, compute a sum of random variables, work a detailed vector calculus problem, a group theory problem, a differential geometry problem? That's just listing the subjects I once knew, and it seems as if that would take hours every single day. Plenty of people here seem to know a great many more mathematical subjects as well. How do you hold it all in your head at once?

I have written out many problems and saved them, and outlined books in great detail (Spivak's Calculus on Manifolds, Korevaar's Mathematical methods book, Linear Geometry by Rafael Artzy, a quantum mechanics text by Bransden and Joachaim, Wackerly's Statistics text, etc.)and try to reread the notes, but it still seems that the knowledge decays just as fast as I can add it or restore it. And a subject like differential equations! When working a single problem takes 15-30 minutes, I don't understand how anyone gets real mastery of that in the first place!

Is it just a matter of my lacking sufficient talent? Is hours of review every day simply a normal part of being a mathematician that nobody ever mentions? Or is it something else?

In short, am I doomed, lazy, or just ignorant of something very important?

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    $\begingroup$ I think teaching is the one of the main things that helps professional mathematicians to keep their mathematical knowledge fresh. As someone who does not teach mathematics, I find that engaging with sites like MSE helps to keep me on my toes. $\endgroup$ – Rob Arthan Apr 17 at 22:08
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    $\begingroup$ Go to college and major in mathematics. The courses build on each other so you end up with a lot of overlap on the important ideas. If you go to graduate school you will most likely teach some of the courses from your undergraduate studies and that experience is what generally leads to mastery of those basic topics. Self learning is a fool's errand for all but the most talented and dedicated. $\endgroup$ – John Douma Apr 17 at 22:09
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    $\begingroup$ Ironically, as a self-learner for several decades, I completely agree with John Douma's comment, and disagree with Adam Rubinson's comment. First of all, you will have a great deal of trouble finding just the right book for yourself, on a particular subject. A book right for one person won't necessarily be right for another. Secondly, you definitely will need a support system, which can be occasional meetings with a math prof (perhaps without even being enrolled anywhere) or frequent mathSE postings. Books have errors. Also, the conceptual intent of the writer may be unclear. $\endgroup$ – user2661923 Apr 17 at 22:32
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    $\begingroup$ I spend a lot of time revisiting material to try to understand things more clearly or deeply, to the point that I can see why the idea is "secretly obvious" even though it might have seemed quite complicated or unmotivated the first time I saw it. For example, I was recently pondering what is the best way to think about the cross product. Given $b, c \in \mathbb R^3$, the function $L(x) = \det \begin{bmatrix} x & b & c \end{bmatrix}$ is linear, so there exists a vector $a$ such that $L(x) = a^T x$. This vector $a$ is by definition $b \times c$. Some people think this is the best viewpoint. $\endgroup$ – littleO Apr 17 at 23:46
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    $\begingroup$ @user2661923 Oh, I know about textbook errors! I was foolish enough to tackle Von Neumann's Mathemtical Foundations of Quantum Mechanics and I think I found over 130 errata in it! $\endgroup$ – RobertTheTutor Apr 18 at 2:00
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Even among the best mathematicians, I'd wager that it is really rare to have someone who is proficient in more than a couple of areas. Sure, an analyst would know the very basics of group theory and an algebraist can work out an epsilon-delta proof without much issue.

But it sounds like you are missing the point of obtaining breadth in mathematical education somewhat there. I am much more analysis/applied math inclined myself and I cannot recall the proofs of the Sylow's theorems at this point even though I have taken graduate algebra sequence. However, if that becomes necessary for me, then I can grab Dummit and Foote and refresh myself really quickly on the Sylow's theorems. Similarly, a topologist or a logician would also very quickly be able to relearn/remember how to compute SVD of a matrix and do some basic linear algebra proofs etc. It is that mathematical maturity you obtain through the math courses that enables you to pick up anything relatively quickly if need be. Every now and then, I teach multivarible calculus/vector calculus for engineering students and I review Stoke's/Gauss' Theorems, div/grad/curls and their applications to physics etc everytime.

Last but not least, if you spend no more than $30$ minutes on any differential equation problem, then maybe you are only focusing on purely computational ones and do not bother attempting more advanced ones which are kind of necessary to build more intuition. Some textbook problems can and will definitely take more than $30$ minutes to understand and digest.

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    $\begingroup$ Well, the diff eq's used in quantum mechanics have probably seen many hundreds of my hours poured into them! But yes, I was just thinking of the straight computational ones--which is the focus in an intro class and I've never gotten to take a second course in diff eq's. $\endgroup$ – RobertTheTutor Apr 18 at 2:03
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It’s a nice question, but I think that this is not just related to mathematics, it is related to our memory process and it goes under the name of “decay theory”. Let me cite Wikipedia: decay theory is a theory that proposes that memory fades due to the mere passage of time. Information is therefore less available for later retrieval as time passes. When an individual learns something new, a neurochemical "memory trace" is created. However, over time this trace slowly disintegrates.

It is physiological. Going deeper in the study of psychology and cognitive sciences of memory one can find “tricks” to restore periodically (there are indeed more helpful intervals of time in which one can fruitfully refresh some old knowledge) specific concepts that, day after day, month after month, year after year, become solid and easier to remember.

Overall, the same difficulty a mathematician may find in remembering how to solve a second order linear differential equation or how to apply Stoke’s theorem is the same an art historian may have in remembering the date in which Titian fled from Venice due to the plague or the same a philosopher may find in recalling the exact subdivision of Hegel’s Phenomenology of the spirit. Moreover, in conclusion, I see that for mathematicians it is, sometimes, even easier than other subjects since when you dig deeper in one field, you usually need and use extensively previous concepts: hence, you implicitly revise them and - let me add this - you understand them more and more every time.

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    $\begingroup$ "All knowledge has a half life" is something I tell my students often! $\endgroup$ – RobertTheTutor Apr 17 at 23:00

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