# Serge Lang´s remarks on the superiority of algebra. What it actually means? [closed]

I read two comments of Lang that basically places algebra over other math subjects. One of this comments is on his calculus book preface (see Remark 1 below); I am not finding his other comment, but it was an interaction he had with someone at Yale´s math department coffee break and is written somewhere. It basically says that algebra is superior to any other math subject, if I recall it correctly. My problem with his comments is that I have no idea what he is talking about. They baffle me. I suppose I stand in the exact opposite from his viewpoints. For me, you can´t compare the applicability and importance of analysis and differential equations to that of modern algebra. Hence:

1) Is there an article of Lang explaining in detail his viewpoints? 2) or, do you know what is his point?

Remark 1) On the preface of his Calculus book, Serge Lang basically says that he thinks bright students may benefit more in studying abstract algebra before or at the same time they learn calculus. My book is in portuguese, so I give a rough translation: " when I was a student I didn´t like calculus nor analysis. I probably woundn´t like this book either... [today I think] that calculus and analysis are overestimated, with a loss to algebra, mainly because of historical accidents." He goes on to say that a beginner course in algebra should consist in a study of vector spaces and groups, that this is independent of calculus and has important applications to other fields, and that some people prefer this material over calculus. He also says that there is no reason someone should be forced to study calculus before algebra. This being true specially for the most talented students.

Remrak 2) "I remember one time when I was a grad student, I was standing next to him at tea while he was explaining to a first-year that analysis is just “number theory at infinity”. I said Come on, that’s not true. He immediately turned up the volume, challenging me to stop bullshitting and give an example. I said OK, p-adic analysis, and then walked away. But I’ve always wished I had stayed to see what his reaction would have been. We need more trouble makers like him. ", from the site Not Even Wrong.

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## closed as not constructive by Pete L. Clark, lhf, Benjamin Lim, Asaf Karagila, Willie Wong♦Mar 14 '12 at 13:39

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It's an interesting question. But nothing you have quoted by Lang gets anywhere near claiming the superiority of algebra. Rather, he is questioning the primacy of analysis in the mathematical curriculum and the fact that one must wade through it to get to other branches of mathematics. From my pure mathematician's perspective, I agree that this seems to be largely due to historical inertia: one could equally well learn algebra first, or some topology, or number theory, or... (Like many other people, I did learn some number theory before I studied university-level mathematics.) –  Pete L. Clark Mar 9 '12 at 6:37
What I wrote is perhaps very long and you missed that the "superiority" part was expressed in his Yale common room exchange. Unfortunately, I did not find it. –  Espinho da Flor Mar 9 '12 at 6:42
$@$Espinho: I am responding to what you directly quoted. If you are going to ask a question about what someone (especially no someone no longer living) said, it seems only reasonable to supply a direct reference. Otherwise we're just trafficking in rumors and hearsay. –  Pete L. Clark Mar 9 '12 at 6:44
I would advise extreme caution about "philosophical" comments by celebrated mathematicians. Some of them, like Lang and Arnold, delight in controversies and provocative statements. I've heard both make outrageous claims that didn't strike me as particularly convincing, to put it mildly. Fortunately other mathematicians show remarkable restraint and totally abstain from pontificating, Deligne being the ultimate example of such wise behaviour. –  Georges Elencwajg Mar 9 '12 at 8:31
@Espinho: Your title is Serge Lang's remarks on the superiority of algebra. But you haven't given any documentation of this whatsoever. You've only alluded to a coffee break conversation at which you were not present: that's pretty much the definition of hearsay. Further, it is not really clear to me that insight into a deceased mathematician's mental math map is on topic for this site: how are the rest of us to judge the correctness of such an answer? Finally, as to what Lang thought...he had (and documented) a lot stranger thoughts than topology before analysis, to be sure. –  Pete L. Clark Mar 13 '12 at 5:22

I don't know about the remark at Yale, but as for his comment in the preface I agree 100%. I think you are misunderstanding his remark though, as his point seem to be mostly pedagogical and not asserting the superiority of algebra over analysis. It is true that in many departments, the first year of an undergraduate program is largely calculus, and serious algebra is left until later. For example at the University of Sydney, in first year the focus is on Differential and Integral Calculus being done with more rigor and topics than in high school, and other than that there is Statistics, Discrete Math and "Linear Algebra" which is actually a course in basic matrix algebra. The axioms of a vector space never appear.

This is not a dig at the department here, it is simply a statement of the reality of things - universities often leave serious algebra until after calculus because the majority of first year students could not handle the new level of abstraction so quickly, and also many of them never intend to - most people in the class intend to be majors in other scientific fields.

This comes at a disadvantage to the students that could handle it, and indeed would thrive. Algebra has a distinctly different flavor to analysis (not always true, but certainly calculus vs group theory). Some students are more inclined to thinking algebraically, and some students are adept at thinking in both mindsets. For these students, a beginners course in algebra would go a long way in expanding their mathematical horizons and may increase their interest in mathematics. Lang's remark is simply noting that it is a pity that these students don't receive this.

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Serge Lang is simply expressing his disappointment with universities pushing for calculus courses as the first mathematics courses available for first year students. In fact at my university the first serious proof based algebra course is not till the second semester of second year.

Linear algebra is taught in first year but alas - computing eigenvalues, taking determinants and row reducing (though important techniques to know) I believe do not show students the power of linear algebra. It was not till I did a reading course in algebra (of which required a lot of linear algebra) that I absolutely loved it. I realised that with an operator on say a complex vector space $V$, one can decompose $V$ as a direct sum of invariant subspaces. If these are cyclic, you have Jordan Canonical Form. This for me was true power. However I still had to go through classes where we learned 10 different methods to evaluate integrals and another 15 methods on solving differential equations.

I think Lang's remark can equally be applied to the case where students are exposed to cookbook clases( matrix algebra or calculus). As Ragib has already mentioned, Serge Lang is just simply saddened that the more talented students have to endure these classes (calculus or matrix algebra) as their first ones at university. Their is not mention of the superiority of Algebra over Analysis. I don't think from what he provided did he ever claim that say Commutative Algebra was more powerful than Functional Analysis.

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