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This is a very good and thought-provoking essay for a short blog post, and I have already shared it in a Facebook community heavily populated by professional ma

This is a very good and thought-provoking essay for a short blog post, and I have already shared it in a Facebook community heavily populated by professional mathematicians (where the moderator, with a Ph. D. in math from Berkeley, has given it a thumbs up). Thanks for sharing. I really like the overall point of the post that mathematics once known can be forgotten or neglected, and mathematics written up for mathematics journals can be difficult to understand. Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998): "What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics. ". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants. "The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety." Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems. http://www.amazon.com/gp/product/0387982892/ http://www.amazon.com/gp/product/0387982892/

tel 11y ago
I think this is an interesting perspective to think about. I certainly remember a long time where I felt this way—that calculus was the entire reason for learning mathematics and the rest are a little silly or outdated. It took a long time for me to catch on to why the simple, silly stuff is where all of the fun is. Today, calculus feels boring and dead to me. Obviously useful, but a mere tool instead of something greater. I spend my time thinking about things like topology where I work really hard to think about what it means for things to be close to one another and nothing more. A younger me would not have understood. Which is a little scary.
- GFK_of_xmaspast 11y ago
  Are you excluding things like 'continuity' from calculus because sure, you get topologies on discrete spaces and such, but on the other hand: manifolds.
  - tel 11y ago
    I'm being a little poetic and talking a little limitedly about calculus. Obviously it's got big connections elsewhere, but my perspective on it has shifted a lot.
LimpWristed 11y ago
> Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. I doubt one's "outgrowth of certain branches of math" is the reason math as taught to non-math majors is a castrated mess it is. It's probably the market forces that reject real analysis, abstract algebra or anything at that level or higher. It's the same reason "some programming language du jour > fundamentals of CS, IRL".
eli_gottlieb 11y ago
>"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety." While I do agree, we have to remember why most math classes actually exist: to teach calculus to physicists and engineers, and, as my stepfather's undergraduate advisor once said, "to keep the children from running in the halls". (For the mathematician's extremely self-centered view of "children" as "anyone who has yet to ace two semesters of real analysis".) I've been starting into real analysis myself via Pugh's textbook[1] after not taking a serious math class since multivariable calculus, and found that, once I get past the applied stuff, I really like the approach of building up calculus from its foundations in real numbers (taken as Dedekind cuts), limits (Cauchy-convergent sequences), the set-theoretic construction of functions, and the construction of topological and metric spaces "from scratch". But I can tell that I like it because, deep down, I have the mind of a theoretical computer scientist (which is what I like to be when I'm not writing firmware), which is a kind of mathematician. I appreciate that someone has to teach the applied classes to the people who aren't going to kvetch about "how can I trust that works!?" and who demand to just get their math over with as quickly as possible. [1] -- http://www.amazon.com/Mathematical-Analysis-Undergraduate-Texts-Mathematics/dp/144192941X http://www.amazon.com/Mathematical-Analysis-Undergraduate-Te...
- arh68 11y ago
  > I've been starting into real analysis myself via Pugh's textbook[1] after not taking a serious math class since multivariable calculus Do you have recommendations for other books? I stopped at multivariable calculus as well. For what it's worth those yellow Graduate Texts in Maths books feel like reading TaoCP or CLRS; I'm looking for more approachable textbooks. I feel like I'm not even up to the 1800s, math-wise, not even up to Gauss.

tel 11y ago

I think this is an interesting perspective to think about. I certainly remember a long time where I felt this way—that calculus was the entire reason for learning mathematics and the rest are a little silly or outdated. It took a long time for me to catch on to why the simple, silly stuff is where all of the fun is. Today, calculus feels boring and dead to me. Obviously useful, but a mere tool instead of something greater. I spend my time thinking about things like topology where I work really hard to think about what it means for things to be close to one another and nothing more. A younger me would not have understood. Which is a little scary.

GFK_of_xmaspast 11y ago
Are you excluding things like 'continuity' from calculus because sure, you get topologies on discrete spaces and such, but on the other hand: manifolds.
- tel 11y ago
  I'm being a little poetic and talking a little limitedly about calculus. Obviously it's got big connections elsewhere, but my perspective on it has shifted a lot.

LimpWristed 11y ago

> Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. I doubt one's "outgrowth of certain branches of math" is the reason math as taught to non-math majors is a castrated mess it is. It's probably the market forces that reject real analysis, abstract algebra or anything at that level or higher. It's the same reason "some programming language du jour > fundamentals of CS, IRL".

eli_gottlieb 11y ago

>"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety." While I do agree, we have to remember why most math classes actually exist: to teach calculus to physicists and engineers, and, as my stepfather's undergraduate advisor once said, "to keep the children from running in the halls". (For the mathematician's extremely self-centered view of "children" as "anyone who has yet to ace two semesters of real analysis".) I've been starting into real analysis myself via Pugh's textbook[1] after not taking a serious math class since multivariable calculus, and found that, once I get past the applied stuff, I really like the approach of building up calculus from its foundations in real numbers (taken as Dedekind cuts), limits (Cauchy-convergent sequences), the set-theoretic construction of functions, and the construction of topological and metric spaces "from scratch". But I can tell that I like it because, deep down, I have the mind of a theoretical computer scientist (which is what I like to be when I'm not writing firmware), which is a kind of mathematician. I appreciate that someone has to teach the applied classes to the people who aren't going to kvetch about "how can I trust that works!?" and who demand to just get their math over with as quickly as possible. [1] -- http://www.amazon.com/Mathematical-Analysis-Undergraduate-Texts-Mathematics/dp/144192941X http://www.amazon.com/Mathematical-Analysis-Undergraduate-Te...

arh68 11y ago
> I've been starting into real analysis myself via Pugh's textbook[1] after not taking a serious math class since multivariable calculus Do you have recommendations for other books? I stopped at multivariable calculus as well. For what it's worth those yellow Graduate Texts in Maths books feel like reading TaoCP or CLRS; I'm looking for more approachable textbooks. I feel like I'm not even up to the 1800s, math-wise, not even up to Gauss.