I have read from Experimental Mathematics by V. I. Arnold that if someone wants to study with Vladimir Arnold, they must solve Arnold's Trivium first in order to "understand mathematics". Arnold himself had said that solving the Trivium was a way to truly understand mathematics.

Why? What does Arnold mean with "truly understand mathematics"?

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    $\begingroup$ It might help if you were to tell us what the Trivium is. $\endgroup$ – Gerry Myerson Jun 21 '17 at 7:19
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    $\begingroup$ After a short look into the paper, it mainly seems to consist of a set of tasks. I guess the point of those tasks is not to gain mathematical understanding, but to to test mathematical understanding. That is, it's not that you need to have completed those tasks to obtain mathematical understanding, but that by being able to solve those tasks, you demonstrate understanding of mathematics. $\endgroup$ – celtschk Jun 21 '17 at 7:42
  • $\begingroup$ Do you have any source for these quotes, or paraphrases? Particularly the claim you're attributing to Arnold? I think you're misrepresenting Arnold's purpose in constructing this collection of problems. $\endgroup$ – user2357112 supports Monica Jun 21 '17 at 17:50
  • $\begingroup$ books.google.com.mx/… $\endgroup$ – HeMan Jun 21 '17 at 18:22
  • $\begingroup$ In p.9, he says that Adriana Ortiz "started making sense of mathematics" $\endgroup$ – HeMan Jun 21 '17 at 18:23

(Too long for a comment.)  Quoting author's motivation from the source (page 2 of the linked pdf):

In Feynman's words, these students understand nothing, but never ask questions, so that they appear to understand everything. [...] The students reach a state of "self-propagating pseudo-education" and can teach future generations in the same way. But all this activity is completely senseless, and in fact our output of specialists is to a significant extent a fraud, an illusion and a sham: these so-called specialists are not in a position to solve the simplest problems, and do not possess the rudiments of their trade.

Thus, to put an end to this spurious enhancement of the results, we must specify not a list of theorems, but a collection of problems which students should be able to solve. [...] The compilation of model problems is a laborious job, but I think it must be done. As an attempt I give below a list of one hundred problems forming a mathematical minimum for a physics student.


There's nothing inherently special about the problems (and they're not even, say, math-contest-style problems); they're just intended to be reasonably comprehensive (with respect to an undergrad math education) and demonstrate the virtues of a written rather than oral examination. There's nothing there that anyone with a undergrad math degree should have any problem with, possibly modulo looking up a couple of specific formulas. He published a note after the problem set comparing it to other such examinations, if that's helpful to you.

That having been said, the subjects chosen aren't what I would expect. There's a lot of involved calculation, including some questions on numeric approximations. There are pages and pages and pages of questions about differential equations; in fact, the questions are heavily skewed to applied math in general. There are a couple of desultory questions about group theory and probability at the end, but that's about it; the rest is real and complex analysis (including a bunch of computations of specific integrals) and dozens of questions about differential equations. There's nothing about set theory, ring theory, commutative algebra, Galois theory, topology (beyond one question about Betti numbers and a couple about Riemann surfaces), Lie algebras (Problems #89 and #90 don't count), representation theory, any group theory that one wouldn't get in a physics class, etc. That would make more sense if Arnol'd was using it as a way to weed out prospective grad students working specifically under him, but it focuses on an extraordinarily narrow and applied curriculum. It doesn't strike me as a great way of demonstrating mathematical understanding.

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    $\begingroup$ "There's nothing inherently special about the problems (and they're not even, say, math-contest-style problems)" This might be your opinion but then you should be aware that Arnol'd and quite a few other people strongly disagree with it (but you are right that the problems are not math-contest-style... in fact they are much more interesting than that). $\endgroup$ – Did Jun 21 '17 at 7:54
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    $\begingroup$ I think a key point which @dxiv highlights is that Arnold saw it as a "list of one hundred problems forming a mathematical minimum for a physics student'. As such, it is no surprise that it focuses on a narrow and applied curriculum. $\endgroup$ – James Jun 21 '17 at 8:02
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    $\begingroup$ @James: Sure. I just wouldn't hold it up as an exemplar of what a math undergrad should know. $\endgroup$ – anomaly Jun 21 '17 at 8:05
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    $\begingroup$ @Did: If you strongly disagree, then show what's interesting about them. To take a couple at random: #13 is unremakrable; #89 and #90 really want to be questions about Lie algebras but aren't (and one can just compute them directly); #46 is the tedious sort of problem that's best solved by looking up a table of such things (though this particular one is easy enough, and it's not a great question to discuss the Riemann mapping theorem) ; #50 is the sort of real-integral-by-Cauchy's-theorem question that's on every final exam in intro complex analysis; and so on. $\endgroup$ – anomaly Jun 21 '17 at 8:15
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    $\begingroup$ @MassimoOrtolano: The anecdote about Adriana Ortiz above suggests that it was designed for third- or fourth-year undergrads wanting to work with Arnol'd. None of the problems there are particularly difficult or require a huge amount of mathematical background, but I wouldn't expect freshman physics students to know anything about, say, Betti numbers. $\endgroup$ – anomaly Jun 22 '17 at 15:40

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