Paul Erdős, the famously eccentric, peripatetic and prolific 20th-century mathematician, was keen on the concept that God has a celestial quantity containing the proper proof of each mathematical theorem. “This one is from The Ebook,” he would declare when he needed to bestow his highest reward on an exquisite proof.
By no means thoughts that Erdős doubted God’s very existence. “You don’t need to imagine in God, however it is best to imagine in The Ebook,” Erdős defined to different mathematicians.
In 1994, throughout conversations with Erdős on the Oberwolfach Analysis Institute for Arithmetic in Germany, the mathematician Martin Aigner got here up with an concept: Why not really attempt to make God’s Ebook—or a minimum of an earthly shadow of it? Aigner enlisted fellow mathematician Günter Ziegler, and the 2 began gathering examples of exceptionally stunning proofs, with enthusiastic contributions from Erdős himself. The ensuing quantity, Proofs From THE BOOK, was printed in 1998, sadly too late for Erdős to see it—he had died about two years after the undertaking commenced, at age 83.
“Lots of the proofs hint straight again to him, or had been initiated by his supreme perception in asking the fitting query or in making the fitting conjecture,” Aigner and Ziegler, who are actually each professors on the Free College of Berlin, write within the preface.
Whether or not the proof is comprehensible and delightful relies upon not solely on the proof but in addition on the reader.
The guide, which has been referred to as “a glimpse of mathematical heaven,” presents proofs of dozens of theorems from quantity concept, geometry, evaluation, combinatorics and graph concept. Over the twenty years because it first appeared, it has gone via 5 editions, every with new proofs added, and has been translated into 13 languages.
In January, Ziegler traveled to San Diego for the Joint Arithmetic Conferences, the place he obtained (on his and Aigner’s behalf) the 2018 Steele Prize for Mathematical Exposition. “The density of chic concepts per web page [in the book] is very excessive,” the prize quotation reads.
Quanta Journal sat down with Ziegler on the assembly to debate stunning (and ugly) arithmetic. The interview has been edited and condensed for readability.
You’ve stated that you just and Martin Aigner have the same sense of which proofs are worthy of inclusion in THE BOOK. What goes into your aesthetic?
We’ve at all times shied away from making an attempt to outline what is an ideal proof. And I believe that’s not solely shyness, however really, there isn’t any definition and no uniform criterion. After all, there are all these parts of an exquisite proof. It might probably’t be too lengthy; it needs to be clear; there needs to be a particular concept; it would join issues that often one wouldn’t consider as having any connection.
For some theorems, there are totally different good proofs for several types of readers. I imply, what’s a proof? A proof, in the long run, is one thing that convinces the reader of issues being true. And whether or not the proof is comprehensible and delightful relies upon not solely on the proof but in addition on the reader: What are you aware? What do you want? What do you discover apparent?
You famous within the fifth version that mathematicians have give you a minimum of 196 totally different proofs of the “quadratic reciprocity” theorem (regarding which numbers in “clock” arithmetics are good squares) and almost 100 proofs of the basic theorem of algebra (regarding options to polynomial equations). Why do you suppose mathematicians hold devising new proofs for sure theorems, after they already know the theorems are true?
These are issues which are central in arithmetic, so it’s essential to know them from many alternative angles. There are theorems which have a number of genuinely totally different proofs, and every proof tells you one thing totally different concerning the theorem and the buildings. So, it’s actually beneficial to discover these proofs to know how one can transcend the unique assertion of the theory.
An instance involves thoughts—which isn’t in our guide however may be very basic—Steinitz’s theorem for polyhedra. This says that you probably have a planar graph (a community of vertices and edges within the aircraft) that stays linked in case you take away one or two vertices, then there’s a convex polyhedron that has precisely the identical connectivity sample. It is a theorem that has three fully several types of proof—the “Steinitz-type” proof, the “rubber band” proof and the “circle packing” proof. And every of those three has variations.
Any of the Steinitz-type proofs will inform you not solely that there’s a polyhedron but in addition that there’s a polyhedron with integers for the coordinates of the vertices. And the circle packing proof tells you that there’s a polyhedron that has all its edges tangent to a sphere. You don’t get that from the Steinitz-type proof, or the opposite approach round—the circle packing proof won’t show that you are able to do it with integer coordinates. So, having a number of proofs leads you to a number of methods to know the state of affairs past the unique fundamental theorem.
You’ve talked about the aspect of shock as one function you search for in a BOOK proof. And a few nice proofs do depart one questioning, “How did anybody ever give you this?” However there are different proofs which have a sense of inevitability.
I believe it at all times depends upon what and the place you come from.
An instance is László Lovász’s proof for the Kneser conjecture, which I believe we put within the fourth version. The Kneser conjecture was a couple of sure kind of graph you may assemble from the ok-element subsets of an n-element set—you assemble this graph the place the ok-element subsets are the vertices, and two ok-element units are linked by an edge in the event that they don’t have any components in frequent. And Kneser had requested, in 1955 or ’56, what number of colours are required to paint all of the vertices if vertices which are linked should be totally different colours.
A proof that eats greater than 10 pages can’t be a proof for our guide. God—if he exists—has extra persistence.
It’s relatively simple to point out you can colour this graph with n – ok + 2 colours, however the issue was to point out that fewer colours received’t do it. And so, it’s a graph coloring drawback, however Lovász, in 1978, gave a proof that was a technical tour de power, that used a topological theorem, the Borsuk-Ulam theorem. And it was an incredible shock—why ought to this topological software show a graph theoretic factor?
This was a complete business of utilizing topological instruments to show discrete arithmetic theorems. And now it appears inevitable that you just use these, and really pure and easy. It’s grow to be routine, in a sure sense. However then, I believe, it’s nonetheless beneficial to not neglect the unique shock.
Brevity is one among your different standards for a BOOK proof. May there be a hundred-page proof in God’s Ebook?
I believe there might be, however no human will ever discover it.
We’ve these outcomes from logic that say that there are theorems which are true and which have a proof, however they don’t have a brief proof. It’s a logic assertion. And so, why shouldn’t there be a proof in God’s Ebook that goes over 100 pages, and on every of those hundred pages, makes an excellent new commentary—and so, in that sense, it’s actually a proof from The Ebook?
Then again, we’re at all times completely satisfied if we handle to show one thing with one shocking concept, and proofs with two shocking concepts are much more magical however nonetheless tougher to search out. So a proof that could be a hundred pages lengthy and has 100 shocking concepts—how ought to a human ever discover it?
However I don’t understand how the specialists decide Andrew Wiles’ proof of Fermat’s Final Theorem. It is a hundred pages, or many hundred pages, relying on how a lot quantity concept you assume while you begin. And my understanding is that there are many stunning observations and concepts in there. Maybe Wiles’ proof, with a number of simplifications, is God’s proof for Fermat’s Final Theorem.
However it’s not a proof for the readers of our guide, as a result of it’s simply past the scope, each in technical problem and layers of concept. By definition, a proof that eats greater than 10 pages can’t be a proof for our guide. God—if he exists—has extra persistence.
Paul Erdős has been referred to as a “priest of arithmetic.” He traveled throughout the globe—typically with no settled deal with—to unfold the gospel of arithmetic, so to talk. And he used these non secular metaphors to speak about mathematical magnificence.
Paul Erdős referred to his personal lectures as “preaching.” However he was an atheist. He referred to as God the “Supreme Fascist.” I believe it was extra essential to him to be humorous and to inform tales—he didn’t preach something non secular. So, this story of God and his guide was a part of his storytelling routine.
Whenever you expertise an exquisite proof, does it really feel someway non secular?
The ugly proofs have their function.
It’s a robust feeling. I bear in mind these moments of magnificence and pleasure. And there’s a really highly effective kind of happiness that comes from it.
If I had been a non secular particular person, I might thank God for all this inspiration that I’m blessed to expertise. As I’m not non secular, for me, this God’s Ebook factor is a robust story.
There’s a well-known quote from the mathematician G. H. Hardy that claims, “There isn’t any everlasting place on the earth for ugly arithmetic.” However ugly arithmetic nonetheless has a task, proper?
You recognize, step one is to ascertain the theory, so to say, “I labored exhausting. I obtained the proof. It’s 20 pages. It’s ugly. It’s plenty of calculations, however it’s right and it’s full and I’m pleased with it.”
If the result’s fascinating, then come the individuals who simplify it and put in further concepts and make it an increasing number of elegant and delightful. And in the long run you’ve got, in some sense, the Ebook proof.
For those who have a look at Lovász’s proof for the Kneser conjecture, folks don’t learn his paper anymore. It’s relatively ugly, as a result of Lovász didn’t know the topological instruments on the time, so he needed to reinvent quite a lot of issues and put them collectively. And instantly after that, Imre Bárány had a second proof, which additionally used the Borsuk-Ulam theorem, and that was, I believe, extra elegant and extra easy.
To do these brief and shocking proofs, you want quite a lot of confidence. And one option to get the boldness is that if the factor is true. If that one thing is true as a result of so-and-so proved it, then you may also dare to say, “What can be the very nice and brief and chic option to set up this?” So, I believe, in that sense, the ugly proofs have their function.
You’re presently making ready a sixth version of Proofs From THE BOOK. Will there be extra after that?
The third version was maybe the primary time that we claimed that that’s it, that’s the ultimate one. And, after all, we additionally claimed this within the preface of the fifth version, however we’re presently working exhausting to complete the sixth version.
When Martin Aigner talked to me about this plan to do the guide, the concept was that this could be a pleasant undertaking, and we’d get performed with it, and that’s it. And with, I don’t understand how you translate it into English, jugendlicher Leichtsinn—that’s form of the foolery of somebody being younger—you suppose you may simply do that guide after which it’s performed.
However it’s stored us busy from 1994 till now, with new editions and translations. Now Martin has retired, and I’ve simply utilized to be college president, and I believe there won’t be time and power and alternative to do extra. The sixth version would be the last one.
Authentic story reprinted with permission from Quanta Journal, an editorially impartial publication of the Simons Basis whose mission is to reinforce public understanding of science by overlaying analysis developments and developments in arithmetic and the bodily and life sciences.