# "Out of fashion" topics in degree level math

I just had a look at the curriculum of a university's math faculty 100 years ago. Most of the topics there are the same as the topics taught today, including complex analysis, differential equations, etc, but there are some major differences:

1. Functional analysis has not developed fully at that time, so this is not in the curriculum. Real analysis is also not in it.
2. There seemed to be a course about special functions such as $$\Gamma,$$ beta function in the past. But now this seems to be out of fashion - students don't learn it until they have to use it.

I am just wondering which course is more old fashion than which. For example, functional analysis and quantum mechanics are definitely more modern than gamma functions and beta functions.

Question: What other topics are added/excluded in the past 100 years? For what reason they are added/excluded?

(I would be interested if anyone can predict what topics will be added in the future as well, although this might be too opinion-based.)

• What makes you think there is a fixed list of topics 'in fashion' now? Upper level modules are likely to vary according to the research interests of the department. Jul 9 '19 at 15:58
• I presume it depends on whether you need to figure out the trajectory of a cannonball, develop lethal gas, model nuclear fission, calculate the path of a ballistic missile or design the SkyNet. Jul 9 '19 at 16:47
• How do you define "out of fashion" topics in degree level maths, @MaJoad ? Jul 13 '19 at 15:01

I'm not a historian nor would I trust one unless they were also a mathematician, but it seems to me Abstract or Modern Algebra was probably not part of the undergraduate curriculum at non-elite schools. This is in part due to the fact that parts of what we teach now were not yet discovered. For example, Noether's treatment of ring theory brought out into the open some things which were probably only known to experts before her time.

I think the Introduction to Proofs course we find a lot of places is also a relatively modern invention.

Differential Equations as an introductory course has changed a lot in terms of rigor. If you look at older texts you find much more attention to analysis and/or computational generality. Now we have books like Blanchart and Devaney and Hall which go so far as to relegate power series solutions to an Appendix. Forget about any real attention to orthogonal polynomials or the Frobenius method. Of course this is done in the name of giving attention to modelling, but I can't shake the feeling it is just a ploy to not face the music of student's inability to hack real calculus calculations. I suspect, the older DEqns course was more of a junior or senior level course where much more mathematical maturity was assumed. Or, it was a course that was taken by engineering students who were more elite having passed through a serious matriculation process. Anyway, my point is just that this course may have the same name now and in 1919, but it is a radically different course.

In fact, now that I think about your question a little while, it occurs to me the real answer is that none of the courses are the same. We have introduced new notation, philosophy and technology to all subjects which change the very nature of each course. Furthermore, even in the most pure of topics, surely we learned something in the last century to improve how we teach the material.

1. Spherical trig: huge one that was very much the norm, through WW2 (heavy navigation applications), now not a part of the standard course.

2. Solid geometry: still a part of the standard course, but emphasized more back in the day.

3. Projective geometry: (See bullet 2, but even more so.)

4. As for "real analysis", I think advanced analysis concepts (epsilon delta, continuity, etc.) were out there, if not emphasized the same way (and not a topological interpretation).

5. It's probably also useful to note that the Tripos emphasized a very computational, puzzle-y, problem-solving type of approach to math.

This is not all good or bad. It's probably still a mildly better course for people going into applied areas (physics/engineering) where "the 19th century math is alive and well" as opposed to the extreme rigor/proof emphasis that graduates students who can't solve AP Chemistry stoichiometry problems with two equations in two unknowns and a few conversion factors. Also, note that Ramanujan got massive inspiration from an encyclopedic mechanistic prep book for Tripos students (of the most abhorrent kind to those who obsess over abstraction at the expense of all else).

All in all, I think the way to look at "old school" stuff is as a source of hidden/forgotten gems. But not to rely on. Perhaps not to be returned to completely (for one thing the writing can be too flowery), and also of course, there are huge overlaps with current practice or things you miss, or don't want. But still dipping into the old stuff can be a great idea starter. Edwards Integral Calculus Treatise (a supposedly starter calculus course!) has freaking polylogs in it, with references to 1700s papers and such. And many problems can be found in old books that are good homework (for advanced students).

In general, older books (especially in English or from England) will click more with mathematical physicists or applied mathematicians and less with "don't want to do algebra because it's too trivial for me to do the work on" abstraction lovers and definition/caveat eagle-eye types. But even for them, the writing can be a hurdle if too flowery (a fraction are not flowery...but many are.)

Also, note that there are (usually) modern treatments of the less fashionable topics. So you are not forced to use books that are sometimes hard to read. All that said, I think occasionally dipping into old stuff is a good way to find ideas.

P.s. I think it's very fair to find many areas where the course has not changed much. If you had a great knowledge of algebra 1, 2, trig, geometry, and first year calculus, and ODEs/PDEs, you would be just as good as someone from nowadays at least in terms of engineering math. Maybe within that course, the one thing lacking would be strength in vectors (not even vector analysis) as it is emphasized more now and earlier.

• Engineering math now covers Bode plots for frequency response, and the Macauley notation for integrals and derivatives of the ramp function. Steps, singlets, and doublets are among the derivatives. I suspect that engineering math now places less emphasis on Lagrangians. Jul 9 '19 at 20:44
• What do you mean by tripos? Cambridge tripos? Jul 9 '19 at 22:16
• I've deleted a comment thread here that strayed far from providing constructive criticism or requesting clarification of this answer. Jul 12 '19 at 16:08
• @ChrisCunningham, as in my now-vanished comment: overlying some otherwise accurate assertions, there is a false dichotomy implicit in this answer, that computation and heuristics are the opposite of rigor and proof... or something of this sort, not to mention some name-calling thrown in. This answer would be better, and would lose nothing, if this were remedied... but I am not going to make those edits myself. Jul 12 '19 at 17:45
• @paulgarrett Thanks for coming back despite the vanishing comment. I can't really remove the false dichotomy (and I think that's a good topic for discussion) but I agree the name-calling was unnecessary; I've spent some effort making an edit to remove the name-calling while keeping the spirit of the answer intact. Cheers! Jul 13 '19 at 15:07
• Actually, I thought inequalities (without getting into very technical types) would still be very important, as those are what you work with quite often in most areas of analysis. For instance, to show something approaches zero you often look for something larger that is easier to work with and show that the larger thing approaches zero. Many of the most important sequence and function spaces rely on famous inequalities to establish the triangle inequality. Many convergence and continuity arguments rely on clever use of inequalities. Theory of Equations, however, is an excellent example. Jul 12 '19 at 19:09
• @DaveLRenfro You can solve a system of linear eqs without knowing any linear algebra but knowing some theory helps a lot not just for solving a simple system but to see the bigger picture. The same goes for inequalities: one can prove some basic inequalities but more than 90% of an inequality course will be new to most students. Jul 12 '19 at 21:06
• That history article you link is interesting. It makes me realize how fortunate I was to meet a few graduates of the golden age of the math major. They showed me higher standards are possible. It seems it is a lesson that society is bound and determined to ignore as we continue to ignore the need to have math literate teachers for lower level math courses in K-12 especially. Also, the failure of the AP program and the travesty of CC education continues to be ignored. I went to a CC and I don't look down on them in principle, but in practice I saw mostly dismal results from transfers... Jul 13 '19 at 2:18
• of an inequality course will be new to most students --- Among undergraduate mathematics courses (in the U.S., and even elsewhere that I am aware of) nothing remotely qualifies as "an inequality course". Are you sure you're using the correct term? The closest I can think of would be Putnam Exam problem solving meetings, which are typically not courses for credit (but sometimes might be). Jul 13 '19 at 6:17
• @Paracosmiste "A 12th grade book nowadays is a piece of cake compared to a 1970 book." - I think you meant to say "piece of crap". Jul 15 '19 at 5:34