How has "what every mathematician should know" changed?

Question

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.

Answer 1

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

Answer 2

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.

Answer 3

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.

Answer 4

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...

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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

Answer 11

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.