Famous exercise from Lang's Algebra - MathOverflow

Famous exercise from Lang's Algebra

2010-01-06T07:33:36Z

There's a famous story about an exercise from Lang's Algebra that says something along the lines of "pick up a homological algebra book and prove all of the theorems yourself". I cannot find it in the third revised edition, and I'm wondering if it's still in the third revised edition, if it's only in the older editions, or if it's an urban legend.

Answer by Jonas Meyer for Famous exercise from Lang's Algebra

2010-01-06T08:01:38Z

It's real, but only in the first and second editions. ~~(I don't have any electronic proof, but I've seen it in my copy of the second edition and someone else's copy of the first edition.)~~ It's the only exercise in the chapter.

The full quote in the second edition is:

Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book.

Homological algebra was invented by Eilenberg-MacLane. General category theory (i. e. the theory of arrow-theoretic results) is generally known as abstract nonsense (the terminology is due to Steenrod).

If I'm not mistaken, the quote is the same in the first edition. First edition: page 105; Second edition: page 175; so you can look in your library to see if I messed up the quote!

And I do have an electronic copy of the third edition, which I've searched to confirm it is not there. The historical remarks were expanded, written less dismissively and put at the intro to Part Four.

Update: I've scanned the evidence. First, Second.

Answer by José Figueroa-O'Farrill for Famous exercise from Lang's Algebra

2010-01-06T08:02:05Z

It is not an urban legend at all. It's certainly in the second edition, which is the one I studied from when I was a grad student. However it seems to be missing from the Springer 2002 edition. Certainly a loss, in my opinion.

Answer by Leonid Positselski for Famous exercise from Lang's Algebra

2010-01-06T12:28:21Z

In the Russian 1968 translation of Lang's Algebra (which is done from the Addison-Wesley 1965 edition), this exercise is there on page 126, exactly as quoted in Jonas' answer. It is complemented by two footnotes by the editor of the tranlation.

The footnotes say: "1) We suggest to skip these exercises on the first reading." and "2) It should be pointed out that the term `abstract nonsense' in this book has a positive character and is used below in the serious sense."

Answer by David L Johnson for Famous exercise from Lang's Algebra

2010-08-04T20:29:45Z

I got to this list since I was trying to remember whether the quote from Lang was true or not. Indeed, my copy, first edition I guess, does have that quote on p. 105. I don't know exactly how "positive"(ly) the term "abstract nonsense" is meant in that quote, though. I would imagine that Lang would have been annoyed had someone suggested they do that with any of his books.

Re: Spanier. Yes, the book is terse, and hard to read. I sincerely doubt that the intent was primarily to make a teaching text for beginning grad students; this was, and is, the standard reference book for basic algebraic topology. What makes it hard to read is the fact that every statement is made is as great a generality as he could. This is an advantage for a reference text. I sincerely doubt, however, that there was anything in that book that Spanier could not prove completely.

That being said, I certainly struggled through a course using it. I chose to take algebraic topology from someone else, to avoid taking it from Spanier himself, since I thought he would teach it as generally as was in the book. It turns out, though, that he did back down from that level of abstraction when teaching the course, and perhaps I missed something not taking the course from him.