The late great Paul Erdös was not a religious man (his take on religion seems to have been fairly ironic, referring for example to God as “The Supreme Fascist”), except of course when it came to mathematics. Ever the Platonist, he considered that when he died, he might finally get a chance to gaze upon “The Book” which, as if written by God, contains the most beautiful and enlightening proofs of all theorems. The highest form of praise from Erdös for a proof was, “It’s straight from The Book!” He also said, “You don’t have to believe in God, but you should believe in The Book!”

Do you believe in The Book? I’m not sure I do!

In fact, there is this book by Aigner and Ziegler, “Proofs from The Book”. In it they include the following one-sentence proof by Don Zagier on Fermat’s two square theorem (that a prime congruent to is a sum of two squares):

A One-Sentence Proof That Every Prime p congruent to 1 modulo 4 Is a Sum of Two Squares D. Zagier Department of Mathematics, University of Maryland, College Park, MD 20742 The involution on a finite set S = {(x,y,z) \in N^3 : x^2 +4yz = p } defined by ( x+2z, z, y-x-z ) if x < y-z (x,y,z) ---> { ( 2y-x, y, x-y+z ) if y-z < x < 2y ( x-2y, x-y+z, y ) if x > 2y has exactly one fixed point, so |S| is odd and the involution defined by (x,y,z) ---> (x,z,y) also has a fixed point.

I plucked this off the Web from here; the author of the page prefaces it with a comment:

The following constitutes the essential text of a complete research article; I have omitted only some comments at the end concerning the history of this type of argument. The author reproves a famous result. He builds his proof into a single sentence as simply a tour-de-force. In fact, he has left many straightforward steps for the reader to verify. 1. As an exercise in critical reading, list all the implicit claims that the reader must verify in order to accept this argument as a proof. 2. As an exercise in logic and algebra, supply all the details necessary to support these claims. Package all this as a long-winded rewrite of Zagier's article written so that any high school algebra student could easily read it with comprehension. You should expect to expand Zagier's single sentence to a full page or more.

Um, yeah.

My own reaction to this proof: it is surely dazzling in its compression, although one’s first reaction is likely to be “WTF?!?” — what just happened here? The underlying idea is that the number of fixed points of an involution on a finite set (i.e., a function equal to its own inverse) has the same parity as itself; it follows that if has odd parity, then any involution on has at least one fixed point; such a fixed point of the involution on Zagier’s set yields a solution to , whence the theorem. So the bulk of the proof is in showing that has odd parity, by showing that his nontrivial involution has exactly one fixed point.

And I guess you can see, by staring at his casewise-defined involution for a while, that its only fixed point is where . It then remains to check that this really is a well-defined function from to , and it really is an involution. The full verification probably does take up at least a page.

It truly is a jaw-dropping proof. My problem though is that it looks like black magic. I mean, I can construct a line-by-line verification that the proof does what it purports to do, but in a deeper sense I *still don’t get it*. How Zagier cooked up this involution is a mystery to me, and unless I made a concerted effort to memorize it, it would remain unmemorable to me (that is, unless someone were to reveal the underlying mystery to me — I suspect that *that* would take a few sentences or more! Can anyone help me?).

What do you think — does it qualify as a Book proof? Me personally, I prefer proofs which are *enlightening* — arguments that I can really understand, proofs that stick, proofs I can take with me to the grave. Put it this way: if God were to write a proof which consumed an absolute minimum number of bytes in some optimal language, it still wouldn’t be much of a Book proof to me unless I (a limited human) could *really* understand it, and if it were really better in that sense than its closest competitors.

I don’t think I believe too strongly in the reality of “Book proofs”, or at least I’m skeptical that every theorem can be said to have a Book proof. Every mathematical statement and proof is embodied in some larger context or matrix of ideas, many requiring patient assimilation before a light suddenly flashes on. I tend to believe that’s the rule rather than the exception, and the idea that we should believe in a Book proof for every theorem, possessing a snappy immediacy which cries “Behold!”, is based on a dangerous and even crazy fallacy concerning the nature of mathematics.

[At the same time: we can all agree that Erdös was an absolute genius at finding Book proofs! 🙂 ]

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May 4, 2008 at 9:31 pm

John ArmstrongI have to agree with you here. “Clear”, “simple”, and “elegant” are all so subjective as to be meaningless. I don’t believe in “The Book”, but I probably believe in “my Book”, and might even be convinced that you have one of your own.

On a side note, it’s beginning to look like the title “Vishal Lama’s Blog” is getting a little inaccurate…

May 4, 2008 at 9:53 pm

Todd Trimble(Yes, Vishal and I have been talking about that.)

Another interesting case is the Fundamental Theorem of Algebra (that every polynomial with complex coefficients possesses a complex root). I’m sure many of us have our pet favorite proofs — actually I have a memory that you (John) may have brought this up on your blog/blath.

I think a lot of them (like one based on elementary homotopy theory, or another on complex variable theory) are simply gorgeous. My own personal favorite may be one which is almost purely algebraic, based on real closed fields and Sylow theorems. In any case, they

allinvolve some background theory and patient build-up of auxiliary concepts; the internal compression that we all come to experience and love as mathematicians comes only after a lot of hard work, and I think that fact somehow tempers this fanciful notion of a “Book proof”.May 4, 2008 at 10:23 pm

Vishal LamaIf there is anyone who is most painfully aware of the inaccuracy of the current blog title, then it is me! 🙂 This has been on my mind for quite some time now. I will be changing the title to something more appropriate after consulting with Todd, some time soon.

Coming back to the topic of this post, I completely agree with the assessment that “

Every mathematical statement and proof is embodied in some larger context or matrix of ideas, many requiring patient assimilation before a light suddenly flashes on.In fact, I would even say it is almost impossible for me to learn any new mathematics without having some larger context that I can always refer to for, say, motivation, somewhere at the back of my mind. And, I believe a lot of people do/learn mathematics that way. It provides “meaning” to mathematics, if I may say so.Also, I couldn’t agree more with the statement that “

…the idea that we should believe in a Book proof for every theorem, possessing a snappy immediacy which cries “Behold!”, is based on a dangerous and even crazy fallacy concerning the nature of mathematics.” I would say such an idea is extremely dangerous to a “beginner”, especially when he/she starts out . Unfortunately, such “Platonic” ideas are reinforced through Hollywood movies (“A Beautiful Mind” is a typical example) as well as in popular culture all the time!January 10, 2010 at 10:22 am

Américo TavaresInside Proofs From The Book is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. Some of the proofsare classics, but many are new and brilliant proofs of classical results. Still others are recent results.

Proofs from THE BOOK

Reviewed by Daniel H. Ullman

NOTICES OF THE AMS, AUGUST 1999, pp. 789-791

May 4, 2008 at 10:25 pm

Urs SchreiberIsn’t it true that book-like proofs have a chance of existing only for questions not too remote from number theory, anyway? Where the concepts involved in the statement to be proven (numbers, functions) are so comparatively elementary?

For instance the Atiyah-Singer index theorem. Could any proof of it feel like a “book proof”?

May 5, 2008 at 12:09 am

Todd TrimbleI think you’re right, Urs, that Erdös’s own interests happened to be in areas where the case for “The Book” may look a little stronger. That is, certain aspects of number theory, graph theory, combinatorics where the problems are susceptible to solutions which require cleverness, but not necessarily much theoretical build-up. It is well-known that Erdös was not terribly interested in theory, and was probably largely impatient with it.

(A fellow Hungarian, Peter Lax, once quoted Courant as saying that there was always a tradition in Hungary to paint exquisite miniatures!)

One could interpret Erdös’s “should” (One should believe in The Book!) somewhat more generously, as in never give up hope that a slick ultra-clever proof of some statement might exist — and he did score some notable successes which might strengthen such faith. But I dunno, the attitude which underlies that sentiment strikes me as having an anti-theory bias — and there’s a lot of it around.

May 5, 2008 at 2:37 am

John ArmstrongI was actually thinking about the various proofs of FToA as an illustrative example, Todd, but decided against including the details of alternatives in the comment. Small minds think alike, eh?

Oh, and Urs has a great point. In fact, in the areas I’m most interested in there could only be a “book proof up to isomorphism”, I suppose.