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 E book,” he would declare when he wished to bestow his highest reward on a gorgeous proof.
By no means thoughts that Erdős doubted God’s very existence. “You don’t should imagine in God, however it’s best to imagine in The E book,” 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 E book—or at the least an earthly shadow of it? Aigner enlisted fellow mathematician Günter Ziegler, and the 2 began accumulating examples of exceptionally lovely 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 instantly again to him, or have been initiated by his supreme perception in asking the suitable query or in making the suitable 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 exquisite relies upon not solely on the proof but additionally on the reader.
The e-book, which has been known as “a glimpse of mathematical heaven,” presents proofs of dozens of theorems from quantity idea, geometry, evaluation, combinatorics and graph idea. Over the 20 years because it first appeared, it has gone by 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 awfully excessive,” the prize quotation reads.
Quanta Journal sat down with Ziegler on the assembly to debate lovely (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 all the time shied away from attempting to outline what is an ideal proof. And I believe that’s not solely shyness, however really, there is no such thing as a definition and no uniform criterion. After all, there are all these parts of a gorgeous proof. It may possibly’t be too lengthy; it needs to be clear; there needs to be a particular concept; it would join issues that normally one wouldn’t consider as having any connection.
For some theorems, there are totally different excellent proofs for various kinds of readers. I imply, what’s a proof? A proof, ultimately, is one thing that convinces the reader of issues being true. And whether or not the proof is comprehensible and exquisite relies upon not solely on the proof but additionally on the reader: What have you learnt? What do you want? What do you discover apparent?
You famous within the fifth version that mathematicians have provide you with at the least 196 totally different proofs of the “quadratic reciprocity” theorem (regarding which numbers in “clock” arithmetics are excellent squares) and practically 100 proofs of the basic theorem of algebra (regarding options to polynomial equations). Why do you assume mathematicians maintain 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 grasp 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 in regards to the theorem and the buildings. So, it’s actually invaluable to discover these proofs to grasp how one can transcend the unique assertion of the concept.
An instance involves thoughts—which isn’t in our e-book however could be very elementary—Steinitz’s theorem for polyhedra. This says that when you have a planar graph (a community of vertices and edges within the aircraft) that stays linked if you happen to take away one or two vertices, then there’s a convex polyhedron that has precisely the identical connectivity sample. This can be a theorem that has three fully various kinds 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 let you know not solely that there’s a polyhedron but additionally 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 means round—the circle packing proof is not going to 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 grasp the state of affairs past the unique primary theorem.
You’ve talked about the ingredient 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 provide you with this?” However there are different proofs which have a sense of inevitability.
I believe it all the time is determined by what you understand 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 few sure kind of graph you’ll be able to assemble from the okay-element subsets of an n-element set—you assemble this graph the place the okay-element subsets are the vertices, and two okay-element units are linked by an edge in the event that they don’t have any parts 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 e-book. God—if he exists—has extra persistence.
It’s fairly simple to indicate you could coloration this graph with n – okay + 2 colours, however the issue was to indicate that fewer colours gained’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 become an entire business of utilizing topological instruments to show discrete arithmetic theorems. And now it appears inevitable that you just use these, and really pure and simple. It’s turn into routine, in a sure sense. However then, I believe, it’s nonetheless invaluable to not neglect the unique shock.
Brevity is one in all your different standards for a BOOK proof. Might there be a hundred-page proof in God’s E book?
I believe there might be, however no human will ever discover it.
We’ve got 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 E book that goes over 100 pages, and on every of those hundred pages, makes an excellent new statement—and so, in that sense, it’s actually a proof from The E book?
Alternatively, we’re all the time 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 more durable to seek 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 consultants decide Andrew Wiles’ proof of Fermat’s Final Theorem. This can be a hundred pages, or many hundred pages, relying on how a lot quantity idea you assume while you begin. And my understanding is that there are many lovely observations and concepts in there. Maybe Wiles’ proof, with just a few simplifications, is God’s proof for Fermat’s Final Theorem.
However it’s not a proof for the readers of our e-book, as a result of it’s simply past the scope, each in technical issue and layers of idea. By definition, a proof that eats greater than 10 pages can’t be a proof for our e-book. God—if he exists—has extra persistence.
Paul Erdős has been known as a “priest of arithmetic.” He traveled throughout the globe—typically with no settled tackle—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 known 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 e-book was a part of his storytelling routine.
While you expertise a gorgeous proof, does it really feel someway religious?
The ugly proofs have their function.
It’s a robust feeling. I keep in mind these moments of magnificence and pleasure. And there’s a really highly effective kind of happiness that comes from it.
If I have been a spiritual individual, I’d thank God for all this inspiration that I’m blessed to expertise. As I’m not non secular, for me, this God’s E book factor is a robust story.
There’s a well-known quote from the mathematician G. H. Hardy that claims, “There is no such thing as a everlasting place on the earth for ugly arithmetic.” However ugly arithmetic nonetheless has a job, proper?
You already know, step one is to ascertain the concept, with the intention to say, “I labored exhausting. I obtained the proof. It’s 20 pages. It’s ugly. It’s a number of calculations, however it’s appropriate and it’s full and I’m pleased with it.”
If the result’s attention-grabbing, then come the individuals who simplify it and put in further concepts and make it increasingly elegant and exquisite. And ultimately you will have, in some sense, the E book proof.
For those who have a look at Lovász’s proof for the Kneser conjecture, folks don’t learn his paper anymore. It’s fairly ugly, as a result of Lovász didn’t know the topological instruments on the time, so he needed to reinvent plenty 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 plenty of confidence. And one approach to get the boldness is that if you understand the factor is true. If you understand that one thing is true as a result of so-and-so proved it, then you may additionally dare to say, “What can be the very nice and brief and stylish approach 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, in fact, 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 e-book, the thought was that this may be a pleasant undertaking, and we’d get finished with it, and that’s it. And with, I don’t understand how you translate it into English, jugendlicher Leichtsinn—that’s kind of the foolery of somebody being younger—you assume you’ll be able to simply do that e-book after which it’s finished.
However it’s saved 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 is not going to be time and power and alternative to do extra. The sixth version would be the ultimate one.
Authentic story reprinted with permission from Quanta Journal, an editorially unbiased publication of the Simons Basis whose mission is to reinforce public understanding of science by overlaying analysis developments and tendencies in arithmetic and the bodily and life sciences.