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I came across this statement in the autobiography by Saunders Mac Lane.

It was the interaction between solenoids and group extension that got our collaboration started, and this first work of collaboration revealed much else to be done, some stimulated by a result of Heinz Hopf. It can be best summarized by a striking proof that 2 is more than 1 plus 1.

Ref: pp 346-347 1

Where can I find the above mentioned proof? I tried google but failed.

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The 3rd paragraph of page 9 of math.uiuc.edu/K-theory/0245/survey.pdf provides references to some related papers. –  KConrad Jun 23 '13 at 6:11
Could you give some context? What is the topic? Non-boolean topoi? –  The User Jun 23 '13 at 12:33
The User: It would be extraordinary indeed if Mac Lane was thinking about non-boolean toposes in 1942. –  Steven Landsburg Jun 23 '13 at 13:44
Oh, come on! Don't you think Saunders meant that two mathematicians working together are better than the disjoint sum of them? –  Todd Trimble Jun 23 '13 at 13:51
As this question is based on an apparent misunderstanding of an offhand little quip by Mac Lane, I'm voting to close as "no longer relevant". –  Todd Trimble Jun 23 '13 at 14:39
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closed as no longer relevant by Todd Trimble, Dan Petersen, Timothy Chow, Henry Cohn, Qiaochu Yuan Jun 23 '13 at 19:51

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1 Answer

up vote 11 down vote accepted

The paper Mac Lane is referring to must be "Group Extensions and Homology" from May, 1942. (This fits the description about "interactions between solenoids and group extensions".) The main result in that paper is a form of the universal coefficient theorem for cohomology. I don't see how this can be interpreted as saying that 2 is more than 1 plus 1.

However, the paragraph in Mac Lane's autobiography from which you've quoted begins:

All these examples of collaboration must yield in size and consequence to my long continued work with Eilenberg. He and I came together on a problem that combined our expert knowledge in topology and algebra. The combination turned out to be very fruitful, leading to our many joint papers and covering discoveries such as the cohomology of groups, homological algebra, Eilenberg-Mac Lane spaces, and category theory ....

I infer that the "1 plus 1" being referred to is "Eilenberg plus Mac Lane" and the "more than 1 plus 1" is "Eilenberg-Mac Lane". And the proof, as they say, is in the pudding.

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This of course would have all been clearer if Mac Lane had written "It can be best be summarized as a striking proof that 2 is more than 1 plus 1". I blame the misuse of "by" on either lackadaisical editing or an editor who didn't get the point. –  Steven Landsburg Jun 23 '13 at 14:13
You didn't quite say it right: the 2 in "2 is more than 1 plus 1" is Eilenberg-Mac Lane. Not the "more than 2". –  Todd Trimble Jun 23 '13 at 14:50
Todd: Fixed; thanks. –  Steven Landsburg Jun 23 '13 at 15:37
The proof of the pudding, as they say, is in the eating: phrases.org.uk/meanings/proof-of-the-pudding.html –  MTS Jun 23 '13 at 15:55
MTS: I've heard both phrases. –  Steven Landsburg Jun 23 '13 at 16:04
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