For example, I am confident that very few students majoring in pure mathematics can write a complete proof to the Abel–Ruffini theorem (there is no algebraic solution to general polynomial equations of degree five or higher with arbitrary coefficients) by the time of their graduation. I suspect many students with a Master's degree or Doctorate in pure mathematics could not prove this theorem either. They may know the conclusion, but may not be able to sketch an idea of the proof, let alone give a complete proof.

My question is: should we educate pure mathematics major students in such a way that they should know how to prove most of the classical results in mathematics such as the Abel–Ruffini theorem and the Fundamental Theorem of Algebra before getting their bachelor's degree, or at least their master's Degrees?

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    $\begingroup$ The situation is even worse: many bachelors students cannot prove the quadratic formula. $\endgroup$ – Steven Gubkin Jan 21 at 19:33
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    $\begingroup$ @DanFox It would be interesting for you to pull aside a few random senior math majors and ask them to derive it. Would you be willing to conduct this experiment and report back with your findings? $\endgroup$ – Steven Gubkin Jan 22 at 2:47
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    $\begingroup$ What makes this particular theorem so important? I'm pretty sure that, at the time of graduation, few of those (in my university) who specialized in functional analysis, dynamical systems, statistics, or game theory could even tell you what the theorem is. And all I could have told you about Fubini's theorem (for example) is that it has something to do with multiple integrals. And who can give a proof of Pascal's theorem without making a mess? $\endgroup$ – Servaes Jan 22 at 13:08
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    $\begingroup$ To be clear, I doubt that most university mathematics professors can produce a proof of the Abel-Ruffini theorem on demand. $\endgroup$ – Dan Fox Jan 22 at 14:48
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    $\begingroup$ @JamesS.Cook Results are in. Out of a total of 31, 9 could supply a complete proof (of those 5 completed the square, and 4 worked backwards). 7 had some idea: they said the words "complete the square", or started doing it but could not complete the algebra. 15 were either completely blank or had completely wrong starts (like dividing both sides by x). $\endgroup$ – Steven Gubkin Jan 30 at 13:53

For the project, you would need to define what are "classical results in mathematics". I suspect that different people would disagree on the classicalness of various results.

Furthermore, it is doubtful if everyone should learn the same classical results - mathematics is a big field. You can, of course, cut it down by a sufficiently strong definition of "pure mathematics", in which case, maybe everyone should know the same results.

I think it would be more fruitful to ask what results a particular university or study programme should consider as key results, concepts and tools. This nicely sidesteps the logistical issue of coordinating mathematics study programmes worldwide and the potentially ugly definitional issue of who qualifies as a pure mathematician. Furthermore, since it retains the current situation of different mathematicians knowing different tools, it widens the scope of overall knowledge. Collaboration with and research periods at foreign universities, or at least universities with different priorities, is useful partially due to such differences.

There is the further issue of how much one can realistically ask of bachelor (or even master) students. As Steve Gubkin mentioned in a comment, many bachelor students have problems with even simple proofs. Depending on how wide one wishes to cast the net of "classical results", it might be unfeasible to teach them to everyone, without making "everyone" a much smaller set then nowadays.

  • $\begingroup$ I suspect that different people would disagree on the classicalness of various results. Yep. Is "classical" even a good thing? In 1750, "classical" mathematics might have included topics like woolly philosophizing on the nature of the continuum, or attempts to prove the parallel postulate. Today we would consider those to have been dead ends. I took the required undergrad course in fields and Galois theory, and my impression was that Galois theory was of zero importance except historically. Not true? The text and the professor never tried to convince me otherwise. $\endgroup$ – Ben Crowell Feb 8 at 4:56
  • $\begingroup$ @BenCrowell: Aside from the "classical", I don't think that philosophizing or attempts at the parallel postulate would count as "results" for the purposes of this question. $\endgroup$ – Daniel R. Collins Feb 8 at 6:21

This is an interesting question, but, understandably, confounds at least two different things. E.g., is it really the case that to "know" a true mathematical fact is to be able to produce its proof on command? I think not. Another diagnostic question: must we understand thermodynamics and the Carnot cycle to drive a car usefully? Must we be able to prove the stability of the proton before setting our coffee cup on the table? Yes, of course, I'm exaggerating... but my exaggeration is in the direction I think is relevant.

Namely, awareness is the key point (and assimilation of the facts into one's world-view... to the extent that they might have some impact and affect one's own decisions).

My opinion on this is in the same vein as my objection to people being told to do every exercise before moving forward: not only are many of those exercises either make-work or pranks, but many are also incomprehensible without understanding what happens in the sequel... which one will not see until after? A bit perverse. Sure, some such pranks are "fun" in Math Olympiads and Putnam and such, but...

The problem that I see is that undergrads are too often conditioned to be paranoid that there's some unfathomable flaw in what they've written... that can only be adjudicated by the oracular professor. One of the worst corollaries of this is that kids are very inhibited about broadening their scope, because they're already worried about defending themselves with regard to a tiny, trivial "plot of land", and are taught to give no credence to their own critical faculties.

So, yes, I think this question raises some good issues, but is literally a bit mis-aimed in its assumptions.

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    $\begingroup$ The penultimate paragraph is powerful. Too often I have felt this feeling of paranoia hanging over our heads during my graduation. Thank you for putting something so nebulous into such relatable words. $\endgroup$ – Brahadeesh Jan 24 at 11:21

It just depend on what do you mean by "knowing very well".

To me "knowing very well" means: if you give me 1 hour I can tell you the basic ingredients that go in the proof (and the reason why the result is relevant) and if you give me 1 full day I can sketch a reasonably detailed proof.

This means:

  • I know how to fit the result in an area of math;
  • I know on which books I'd rather look for it (which is very personal, depends on your own background);
  • I am capable of looking back at the relevant prerequisites and recover them quite fastly.

Then yes, that's what we hope from our students. And it is already a quite high level of demand.

(as for the example, I got a PhD in math without ever being exposed to a proof of Abel-Ruffini, so "classical" means different things to different people. I think to grasp the proof of it I'll probably need a week)

  • $\begingroup$ I agree with your description of "knowing very well". I suspect less than 5% of students with bachelor's degree in mathematics can sketch a detailed proof of Abel–Ruffini theorem in 1 day. For university full professors in pure mathematics, perhaps 50% of them could? $\endgroup$ – Zuriel Jan 25 at 1:35
  • $\begingroup$ Minor point: 'fastly' isn't a word. Try 'quickly'. $\endgroup$ – Jessica B Feb 7 at 23:22
  • $\begingroup$ Thanks Jessica, one always learns something... english.stackexchange.com/questions/1413/… $\endgroup$ – Nicola Ciccoli Feb 9 at 9:11

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