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So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acquire during or before graduate school.

To give an example (which others may disagree with), one secular (here, secular means "trend over time") change seems to be that mathematicians today are expected to feel a lot more comfortable with picking up a new abstraction, or a new abstract formulation of an existing idea, even if the process of abstraction lies outside that person's domain of expertise. For example, even somebody who knows little of category theory would not be expected to bolt if confronted with an interpretation of a subject in his/her field in terms of some new categories, replete with objects, morphisms, functors, and natural transformations. Similarly, people would not blink much at a new algebraic structure that behaves like groups or rings but is a little different.

My sense would be that the expectations and abilities in this regard have improved over the last 50-60 years, partly because of the development of "abstract nonsense" subjects including category theory, first-order logic, model theory, universal algebra etc., and partly because of the increasing level of abstraction and the need for connecting frameworks and ideas even in the rest of mathematics. I don't really know much about how mathematics was taught thirty years ago, but I surmised the above by comparing highly accomplished professional mathematicians who probably went to graduate school thirty years ago against today's graduate students.

Some other guesses:

  1. Today, people are expected to have a lot more of a quick idea of a larger number of subjects, and less of an in-depth understanding of "Big Proofs" in areas outside their subdomain of expertise. Basically, the Great Books or Great Proofs approach to learning may be declining. The rapid increase in availability of books, journals, and information via the Internet (along with the existence of tools such as Math Overflow) may be making it more profitable to know a bit of everything rather than master big theorems outside one's area of specialization.
  2. Also, probably a thorough grasp of multiple languages may be becoming less necessary, particularly for people who are using English as their primary research language. Two reasons: first, a lot of materials earlier available only in non-English languages are now available as English translations, and second, translation tools are much more widely available and easy-to-use, reducing the gains from mastery of multiple languages.

These are all just conjectures. Contradictory information and ideas about other possible secular trends would be much appreciated.

NOTE: This might be too soft for Math Overflow! Moderators, please feel free to close it if so.

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"but I surmised the above by comparing highly accomplished professional mathematicians who probably went to graduate school thirty years ago against today's graduate students." -- How did you conclude that they were/are less comfortable with abstraction? I am curious about what you found in your detective work. Perhaps a few concrete examples without naming names? – Justin Curry Mar 25 '10 at 21:44
Justin, Without naming names, I know some mathematicians who are averse to category theory -- they aren't fond of observing that any of their beloved objects of study may be susceptible to interpretations in terms of categories. Some others are aware of category-theoretic interpretations but prefer to formulate things in a non-category-theoretic language. The aversion to using or noticing ideas from model theory and universal algebra is more universal (though, to be fair, these are not perhaps very useful). But I may be wrong; would be glad to hear other views. – Vipul Naik Mar 25 '10 at 22:40

11 Answers 11

One thing I'm sure we'll all agree on: every mathematician should know some flavor of TeX!

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Sadly,yes,Kevin-and I have to stop stalling and do it this summer........ – The Mathemagician Mar 27 '10 at 3:23
Yes and it is such a sad thing that compare with asking today programmers to know COBOL. TeX is a typist language : you don't say what you mean but what it should look! – Jérôme JEAN-CHARLES Nov 21 '10 at 2:34
But I like LaTeX! – timur May 11 '12 at 12:58

As mathematics grows and diversifies beyond belief, surely the collection of topics that every mathematician must know is shrinking fast. One can carry out serious mathematical research in one area while knowing very little of another, even when many mathematicians regard that other area as fundamentally important. Thus, the assumption in the question that there is anything substantial in the list of topics that ALL mathematicians must know seems to me unwarranted. Of course, the interdisciplinary work that connects widely separated research areas is often very important (as well as difficult), but a lot of progress is also made within the various specialities without interacting with other areas. But for someone to to insist that every mathematician must know category theory, say, or homology, seems to exhibit just as narrow a conception of mathematics as to insist that every mathematician must know how to program. There have been profound mathematical advances in subjects requiring none of that knowledge. All other things being equal, of course, a mathematician would be better off knowing some category theory or logic or homology or programming, but in practice, all other things are not equal, since we must all choose how best to spend our time, choosing the topics that seem most relevant to the research we seek to undertake.

Ultimately, we need all kinds of mathematicians: some who are deeply specialized, some who know various areas to build the bridges that can connect diverse subjects, some who know how to communicate ideas from one area to another, and others who know how to communicate the deep ideas of one area to the future specialists in that area, or to the public. Perhaps the intersection of the knowledge of all these people is rather smaller than one might think, and this isn't necessarily a problem.

Contemporary mathematical research is indeed a big tent, as Charlie Frohman said in the comments.

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Don't forget the ones who know how to communicate to the public! – Qiaochu Yuan Mar 26 '10 at 0:28
Yes, of course, that is also important, Qiaochu! – Joel David Hamkins Mar 26 '10 at 0:36

Many, many things have changed in the last 60 years. A mathematician of the fifties (in Europe) was required to know descriptive geometry, rational mechanics, maybe some astronomy, and a lot of physics. He (yes!) was supposed to know how to calculate rather difficult primitives and have many tricks at his fingertips for checking the convergence of a series. Masterful use of logarithms tables and slide-rules went without saying. Nomography, the graphical representation of mathematical relationships (I guess even the word is forgotten), was a popular option, etc...

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At some point in history, the mathematicians were expected to help the physicists with their integrals. I believe that included some schools 60 years ago. Now, typical physicists are much better at integrals than most mathematicians are, and so are computers. – Douglas Zare Mar 25 '10 at 22:55
Perhaps a hundred years from now people will be surprised that mathematicians were expected to know how to do actual epsilon-delta proofs or algebraic manipulations with group and ring elements. That will be for the esoteric specialists in the areas, while most mathematicians will be dealing with "infinity-functors from infinity-categories to infinity-categories" or something like that. It seems hard for me to imagine, though. – Vipul Naik Mar 25 '10 at 23:08
Doron Zeilberger would say that that's a good thing: leave the low-level symbolic manipulation to computers so that humans have a chance to do some "deep" mathematics! – Qiaochu Yuan Mar 25 '10 at 23:55
By the way, the mathematicians I describe (and their elders) invented category theory, sheaf theory,stacks, spectral sequences and the computer.... – Georges Elencwajg Mar 26 '10 at 0:37
The advisor on my master's thesis had to take an entire course on the geometry of special curves. Hypocycloids, Archimedean spirals, astroids, the whole schmeer! This was around 1950 in Scandinavia. – engelbrekt Mar 26 '10 at 16:42

I advise against using MathOverflow as a guide to what most young mathematicians do or ought or learn. The last time I saw such a strong bias towards "abstract nonsense" was when I was a graduate student at Harvard (in the early 80's), where if you wanted to do differential geometry rather than derived categories, you felt like a second class citizen.

I do agree with Steve Huntsman that any math Ph.D. student should devote at least some time towards developing some skills in the practical use of mathematics, including some programming. The fact is that most Ph.D.'s do not end up in a research university, so if you want to have more options than teaching at a lower tier school, these practical skills are extremely useful. You can definitely develop them later, but getting at least some feel for what's involved is very helpful.

Beyond that, there are many, many directions to head in, and each one has its own requirements on what you need to know. Today, a certain facility with abstraction can be quite useful, but it is not essential. Knowing a lot of different things also makes it a lot easier to interact with a broader range of mathematicians. This can be extremely useful to your own research, because you will stumble onto unexpected connections and intersections with work that seems completely unrelated.

Most of us are unable to learn everything we want to, so we have to make choices on what we're going to focus on. This is difficult to do, but developing the proper judgement for this is one of the most important stages of becoming a research mathematician. You can't just follow someone else's advice; you have to learn to figure it out, based on all the different and conflicting views you'll get.

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"(in the early 80's), where if you wanted to do differential geometry rather than derived categories, you felt like a second class citizen." Today I hear "Manifold? I think you mean locally ringed space." At a workshop this summer I ran into someone who insisted that Algebraic Topology and Analysis were both "pretty much solved" subjects. When I asked what was left for a new graduate student to PhD in he responded "Algebraic Geometry will never be solved, but apart from that I guess you could do Quantum Algebra." – Justin Curry Mar 27 '10 at 20:52
Justin, your interlocutor at the workshop seems to have some pretty bizarre opinions. ("Pretty much solved." Pfft.) – Todd Trimble Jul 1 '11 at 10:39

The general question of what a professional mathematician should know was asked by Phil Davis at the end of this article. Barry Mazur posted a brief response about a year ago.

I'm too young to have a picture of this question 30 years ago. Perhaps Bourbaki's Éléments de mathématique comprised an appropriate list. Someone who is old enough to know should correct me.

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After reading Barry Mazur's beautiful five pages, I feel (probably temporarily) as though there is nothing more to say on this topic. – gowers Mar 26 '10 at 9:31
Has anyone noticed that all of a sudden 30 years ago is not that long ago? 1980 -- I remember that. I think that the heyday of Bourbaki's books was probably a good 20 or 25 years before that...but I'm not old enough to say with any confidence. – Pete L. Clark Mar 27 '10 at 2:11
I have to agree with gowers. Mazur, who has to be one of the smartest and most knowledgeable mathematicians on earth, does a remarkably good job of saying something that seems applicable to mere mortal mathematicians like me. My only quibble is that I think he shortchanges probability (and other stochastic stuff) by making it just one item under "computation". – Deane Yang Mar 27 '10 at 14:31

I think one way to answer this question would be to get hold of the qualifying exams from University X from 50-60 years ago and compare them to the exams at the same university today.

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An excellent suggestion, Gerry. – Georges Elencwajg Mar 26 '10 at 0:57
I think it may be increasingly common for such exams to allow for much more choice now than earlier. For example, at my instituion (the City University of New York), we have six subject exams, of which students need only take three before proceeding with their dissertation work. So it probably happens that there are pairs of students having no qualtifying exams in common. – Joel David Hamkins Mar 26 '10 at 1:02
Any suggestions for University X whose qualifying examinations over a long time period are easily accessible? – Vipul Naik Mar 26 '10 at 1:07
I think the UC Berkeley qual study guides were available on-line. Also, David Cruz-Uribe and another person published a version of the guides, which contained versions of the exams stretching back at least a few decades (but not as far as 50 years). – Joel David Hamkins Mar 26 '10 at 1:11
I don't feel like the Berkeley prelim or other US prelims have changed that drastically in 40 years. I think a better sample (but falling outside the questioner's 50-60 yr time range) would be the Cambridge Maths Tripos. This would allow a comparison on the order of centuries. The focus on mathematical physics in the 19th century was particularly impressive. – Justin Curry Mar 27 '10 at 20:58

Practically, mathematicians today should know the rudiments of programming in at least one language (Mathematica and MATLAB count). They should know the basics of probability and linear algebra. They should know these three things because if they get jobs outside of academia they will generally be expected to use at least two of these three, and probably all of them.

Mathematicians should know how to use the internet and how to learn there. They need not recall many formulas, as the convenience of having them at one's fingertips can be "outsourced" to the internet. By the same token, they need not even recall most of what they have learned, but instead should be able to refresh their memory quickly.

Classical and complex analysis have clearly (I think) become less important to command in detail. Combinatorics and algebra have become more so. This is because of computers, and the interplay between mathematics and technology more generally.

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Someone who did PDE would have a very different viewpoint than you do. Maybe its just a bigger tent now, which mean the shared experiences are smaller. – Charlie Frohman Mar 25 '10 at 22:29
Like the arXiv, I didn't intend to include PDE in "classical analysis". – Steve Huntsman Mar 26 '10 at 3:44
"Computer interaction" is definitely a part of a mathemagicians life. – Per Alexandersson Jan 7 '13 at 19:06

In Littlewood's Miscellany there is an essay "A Mathematical Education" where he describes the situation before 1907.

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I believe that one shift is that "what every mathematician should know" is nowadays much less a specific body of mathematical facts and much more a facility with navigating the ocean of mathematical knowledge.

For example, I might not need to have advanced computer programming skills, but I do need to have some sense of what kinds of computations are feasible and when it is appropriate for me to do a computation.

I might not need to hold in my head everything that is known about a certain topic, even if that topic is close to my area of specialization, but I definitely need to have the ability to search the literature, assess what is in a certain paper that my search turns up, and know when I should ask an expert and how to formulate a targeted question to ask.

Similarly, I might not need detailed knowledge of fields (seemingly) distant from my own, but I do need to be able to discern when those distant fields might provide relevant tools for my own work.

So far I have been focusing on what a mathematician needs to know in order to be an effective researcher. However, the phrase "what every mathematician should know" carries overtones of what one should know if one wants to earn a reputation for being an educated, knowledgeable, respectable, and attractive representative of the profession. In my opinion this is quite a different question. For this, you need to be fluent in the language of the hot topics du jour, and au courant with flashy announcements of big breakthroughs in all areas of mathematics. While there's some correlation between this kind of knowledge and the knowledge I discussed above, I find it questionable whether, literally speaking, every mathematician should have it.

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I arrived at this question through my frustration that, despite my master degree, I could not come up with the proof of pi's irrationality just like that. So I studied it and wondered, why was this not on the list of things we learnt at university.

The question is different for an active professional mathematician, a high school math teacher or someone who is otherwise orbiting in our society with a mathematical education in the bag.

I would like to be able and answer questions by non-educated but interested people and picture them a background for the facts. The irrationality of Pi is a likely candidate for Christmas Eve questions, as is the infinite number of primes, or even Gödel's theorem. I studied that one too and it made a lasting impression on me.

In terms of relevance for society, an accomplished mathematician these days should be there to point out flaws in logic and bring an enhanced intuition of statistics to the public domain. Newspapers are full of "significant research results" and their interpretations. People are developing certain common knowledge while mostly remaining ignorant about the statistical aspects of that knowledge, as Daniel Kahneman has pointed out.

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If someone who didn't know calculus asked you why $\pi$ is irrational, what would you say exactly? (All proofs I have seen involve calculus to some degree.) – Todd Trimble Jan 7 '13 at 17:03
Good question. I can perfectly reproduce Hermite's proof but still to fail to fully "see" the grand idea, which I can translate to laymen. I would stick with "One can show that, if pi were rational, a row of whole numbers can be constructed which (globally) descends and at some point will end up between 0 and 1, which is absurd. This row is kept under control by the oscillating sine and cosine functions, of which pi and its half are zeros. The construction is however far from trivial." I tend to think our incapacity to make this proof elementary, points to knowledge still to catch. :) – Dieter Jan 7 '13 at 17:51
Dieter, what's a Christmas Eve question? – Tom Leinster Jan 7 '13 at 19:22
@Dieter: I think that's not a bad start! @Tom: somehow I imagine being in the company of cousins, aunts, and uncles you haven't seen for a year, and Uncle John, thinking up something to say to his mathematician nephew, says he learned as a kid that $\pi$ was irrational, but never learned the reason. – Todd Trimble Jan 7 '13 at 20:53
Related to $\pi$ and its being irrational (as well as the infinitude of the primes): See MO 21367 and the fascinating answer of François G. Dorais. – Benjamin Dickman Nov 29 '15 at 6:45

In earlier times, it seems that great emphasis was put on technical calculation, such as checking for convergence, handling logarithms, and so on (this has been pointed out in an earlier answer).

As for today, the only thing I am convinced every mathematician should know is to formulate correct proofs and be able to come up without hesitation with correct and readable proofs for simple statements such as

If $G,H$ are groups, and $f:G\to H$ is a group homomorphism, then $\text{ker}(f) =\{g\in G:f(g) = 1_H\}$ is a subgroup of $G$.

Mathematicians should all be able to handle quantifiers with care -- for instance see the difference between continuity uniform continuity.

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