Alexander Soifer is a Russian-born American mathematician and mathematics author. A Professor of Mathematics and a Professor of Cinematic Art. The story of the shortest math paper published. Interesting discussion with Professor Alexander Soifer about the nature of Mathematics, his time at Princeton, working with the likes of John Conway, Andrew Wiles and Nobel Laurate John Nash, among other topics.
For more info about Alexander Soifer’s works (including The Mathematical Coloring Book), visit this page.
Avidnote Podcast: In this episode we explore the intriguing story of a mathematician at the center of the crossroads between Mathematics and the Arts. An extraordinary professor who worked together with some of the great mathematicians of this century, we talk about topics such as the nature of Mathematics itself, why it should be viewed as one of the arts, what it feels like to work with famous Nobel Prize winners, and then we end with the curious story of the shortest math paper ever to be published.
Avidnote Intro: You’re listening to the Avidnote Podcast. This is the podcast where we speak with interesting researchers and find out the stories behind the research paper. The podcast is produced by Avidnote – a free web based app that makes it easier to take notes on research papers and organize your documents.
Avidnote Podcast: Ever wondered how short a research paper can be and still become accepted? Today we take a look at the interesting story behind the shortest math paper to ever be published in a serious journal. But more importantly, we also shed light on the facinating life of one of its authors, a man who bridged the arts and the sciences, a Professor of Mathematics and a Professor of Film theory, Alexander Soifer.
Dr. Alexander Soifer: I had to have straight A’s because it was easier to get an A than to explain to my mother why not an A
Avidnote Podcast: That’s Alexander Soifer speaking. Currently a Professor at the University of Colorado and the President of the World Federation of National Mathematics Competitions, Dr Sofier is an accomplished mathematician, artist, filmaker, author, you name it, he did it. Born in the former USSR, he would later emigrate to the U.S where he worked at various prestigious universities including Princeton alongside some rather famous mathematicians.
From “The Beautiful Mind”: So now that I know that you’re real, who are you and what can I do for you? Professor! My name is Thomas King and I’m here to tell you that you’re being considered for the Nobel Prize.
Avidnote Podcast: That’s a scene from the Oscar winning film, a Beautiful Mind, where Russel Crowe plays the role of John Nash, who has just been told that he won the Nobel Prize in Economics. John Nash was made famous for his work on Game Theory and his battles with schizophrenia. Alexander worked at Princeton during the time that John Nash was working there. He recalls his first encounter with the Nobel Prize Winner as follows:
Dr. Alexander Soifer: We knew each other pretty well. You know, at first I didn’t want to even say hello in the elevator because I thought he gets too many hello’s , such a celebrity! But then one time I just told him, hello, my name is Alexander Soifer, and he immediately asked how to spell it.
And we got to know each other. I can’t say I was a friend of John Nash but he inscribed, essential John Nash book to me, as well as the two editors, which is a Harold Kune, double Professor at Princeton of Mathematics Department and Economics Department and Sylvia Nasar who wrote the book A Beautiful Mind.
Avidnote Podcast: Alexander went on to tell a funny incident that occured between himself and John Nash. regarding a book that he was writing at the time which is now available in print entitled “The Mathematical Coloring Book” .
Dr. Alexander Soifer: And so we knew each other, and I was writing my mathematical coloring book, If I were a scientist, I would call this book, Foundations of Ramsey Theory, but I am an artist. And so it’s the Mathematical Coloring Book because Ramsey Theory title would be so boring.
So I asked two people to write about economics One of them was John Nash and another Harold Kuhn, now Harold and John became PhD students at Princeton at the same time in 1948. Ever since that time, they were friends and it was Harold Kune who nominated John Nash for [a] Nobel prize.
And Nash got it in 1994 and the Swedish academy invited Harold Kune and his wife to come so that Harold will preside over noble seminar for John Nash . So I asked them both to write about the economics of Frank Ramsey, because I knew nothing about economics. They both agreed. And a week later, John Nash, you know, we had coffee hour every week day, three to four.
John Nash told me, you know, I can’t write about economics of Ramsey for you. I’m so sorry, but Harold will do a great job. I’m sure. And I asked him, why wouldn’t you write John? And he said, well, because I’m not an economist. Oh, I told him, I see you are not an economist. You’re only Nobel prize Laureate for Economics and he said I really am not an Economist. I’m a Mathematician. And if you want, I can study economics of Frank Ramsey, but I’m not sure I will be able to write something meaningful for your book. So I said, okay. Okay
Avidnote Podcast: The Princeton Mathematics department in the 90’s and early 2000’s could only be described as stacked. Nothing but famous mathematicians, Nobel Prize winners, Fields medalist, what have you, as far as the eye could see. Besides John Nash, you also had the likes of Andrew Wiles – the man who solved Fermats last theorem, one of the most difficult math problems ever solved. How difficult you ask? Well it stod unsolved for over 350 years, until Andrew came along that is. But it also housed mathematicians with much more varied interests, intellectuals who became famous for their works in numerous fields, not just one field. One of these special rare individuals was a professor by the name of John Conway. Alexander explains how his approach to math differed from others a follows:
Dr. Alexander Soifer: It was science for Andrew Wiles but it was art for John Conway. Two British imports to Princeton. And I spent three years at Princeton about dozen years ago. Couldn’t be more opposite to each other. Andrew Wiles was a narrow specialist, a brilliant one who with the help of Richard Taylor proved Fermat’s last theorem.
That is a huge achievement. But, uh, John Conway was an intellectual.
Dr. John Conway: I started to wonder, you know, whether it was the all nonsense, whether I was not a good mathematician after all and so on, and then I made a certain discovery and was shot into international prominence as a mathematician. When you become a prominent mathematician, in that sense, it doesn’t mean that many people know your name.
It means that many mathematicians know your name and there aren’t many mathematicians in the world anyway, you know, so it doesn’t count very much.
Avidnote Podcast: That’s the late John Conway speaking in an interview with Numberphile, he recently passed away from Covid-19.
Dr. Alexander Soifer: He had very broad interests and he knew a lot about a lot of things. He was interested in history and philosophy, in quantum mechanics, in various branches of mathematics from mathematical logic and set theory, from foundations up to combinatorics and analysis,
So he was an artist, an intellectual. After a short time, we became friends. He would come to my office and litter my Blackboard. Our offices were on the same floor at Princeton. And he would litter my Blackboard with calculations of polyhedra.
And I was fascinated, you know, in my three years at Princeton, I have never seen John Conway with a calculator! I don’t think he owned any! He would brilliantly simplify calculations to the point that they are true, and I was witnessing that.
Avidnote Podcast: Now John Conway was also the co-author of the paper at the center of today’s episode. The world’s shortest math paper. He wrote that paper alongside his collegue and friend, our guest for today, Professor Alexander Soifer whose voice you’ve been hearing so far.
Now we haven’t really talked about who Alexander really is. To know the research, we need to know the man, and to know the man, we need to go back to where it all started, in Soviet Russia, where stories of Math so often begin.
Dr. Alexander Soifer: I was born in Moscow, in the family of an artist, fine artist who was also a set and costume designer for stage. And my mother was a stage actress at Moscow Theater for Youth.
From the age of about 12, I participated in Moscow University’s Mathematical Olympiads, and I won something as a sixth grader, seventh grader, you know, I had to have straight A’s because it was easier to get an A than to explain to my mother why not an A. And Mathematics was nothing special for me until I attended this Mathematical Olympiads because they opened a beauty and elegance and surprises to me that I didn’t see in school.
Avidnote Podcast: Professor Soifer’s experience with Math mirrored the experience that millions of children have with the subject. Some might say, a failure of the schooling system to teach Math in a way that creates curiosity, instead of stifling it. Or as Albert Einstein put it, to regard pure mathematics as the poetry of logical ideas. To express math as something poetic and beautiful. That requires forethought and effort, but to limit it to mere memorization and monotonous instruction, requires less of the instructor and even lesser by the recipient.
Dr. Alexander Soifer: Well the typical school problem sounds like. Given a right triangle with legs three and four by using Pythagoras theorem, calculate hypotenuse. There is no room to think you are told what to prove and what means to use and whether I will have computers and calculators, the problem changed now it’s: uh, right triangle’s legs are 3.1 and 4.2 and by using Pythagoras Theorem and your calculator, figure out the hypotenuse.
So what is important, what attracted me as a freshmen at the university is that my professors would give me not only open-ended problems. Given A, and you don’t know what B is. You have to, you know, experiment and discover it. And they gave me an opportunity to work on open problems as well. So that is important, to feel that you are trying to achieve something new, to walk a path that no one traveled.
Avidnote Podcast: A path that no one travelled. So to truly learn mathematics, to understand it, to grasp it,you need to walk the path that no one travelled, take the road less travelled.
Robert Frost: Two roads diverged in a yellow wood, And sorry I could not travel both. And be one traveler, long I stood And looked down one as far as I could To where it bent in the undergrowth;
Avidnote Podcast: That’s Robert Frost in his older age, reading from his famous poem on the road not taken.
Robert Frost: I shall be telling this with a sigh somewhere ages and ages hence: Two roads diverged in a wood, and I— I took the one less traveled by, And that has made all the difference.
Avidnote Podcast: Taking the road less travelled by is what makes mathematics interesting, to carve out a new path and discover something novel. It is precisely what makes the subject so capitativing to those who are encaptured by its snare, and for those of you don’t like mathematics, it may very well be, the monotonous and boring instructions that you so detest, more so than the math it self. And if you unlike the protagonist in Robert Frost’s poem, choose the side of the path less travelled, perhaps the intrigue of Mathematics would be able to lure you as well.
Alexander then gives us another piece of useful wisdom, this time, regarding the research process it self, and how to produce interesting research around a counter example. He relays the advice given to him by a friend when he was a PhD student in Russia.
Dr. Alexander Soifer: And he said, why are you so dead set on this conjecture? I believe that you feel they’re correct. But I, if I were you, I would try to disprove them, to construct a counter example. And then he added something very nice. There is a Russian proverb: a spoon of tar can spoil a barrel of honey. And so this professor tells me, you can make a barrel of honey out of spoon of tar. In other words, you can create really great research results out of a counterexample. I in the matter of couple of days, constructed counter examples and things went on and the academy of sciences didn’t believe that it was true.
They invited me to travel to Siberia to give a talk about it. I essentially finished my PhD thesis during just the first semester of PhD program. So, you know, that is in fact was a delightful thing that the truth contradicted intuition, it was nice. It’s nice to predict the future, but it’s also nice to be wrong and prove it.
Avidnote Podcast: Yea you heard that correctly, Alexander essentially finished his PhD program in one semester. Astonishing to say the least. Technically speaking, in Soviet Russia at the time, you still had to do some course work, you know, your standard courses in Marxism and Leninism, and then also wait for the results to be published. All in all, at the end of his second year, Alexander Soifer had defended his dissertation.
As I alluded to in the beginning of today’s episode, Professor Soifer is not only a Mathematician but an artist as well. Although these two subject areas may be perceived as separate by many, and perhaps contradictory by some, Alexander views them very much through the same light.
Dr. Alexander Soifer: The difference between me and most mathematicians, is they grew up in the family of engineers, scientists, or people who don’t have any connection to science and engineering. And I grew up in the family of artists, actress and painter. And so mathematics for me was one of the arts. That was the difference.
If I couldn’t perceive mathematics as an art, I would have switched right from the beginning. And you know, for me, the difference between art and science, roughly speaking is this: science studies what is outside of us. Physics, biology, chemistry, study nature, things outside of us, and art expresses what is within.
I enjoyed what is within more. Consequently, great majority of mathematicians, maybe over 90% are Platonists. They believe that Mathematics simply discovers what exists outside, out there. And I think nature is, uh, one of [the] inspirations for doing Mathematics but not the only one. We not only discover, we invent. Invent Mathematics and our criteria are not correspondence to the nature, but beauty, elegance, suprised you know, so, We like results that are not trivial, for Platonists, Mathematics, I guess, is a science.
And for me it’s an art
Avidnote Podcast: and that’s the beauty with having studied more than one subject, it gives you a different lense by which to view things. What Alexander describes here is profound because of the implications involved in viewing math as one of the arts, but also because it shatters an often stereotypical view of Mathematics that is conveyed in pop culture. The idea of the Mathematician as a sort of reclusive, cold-calculating empiricist who becomes so entrenched in their work that they go mad. This stereotype, although perhaps true for a small number of mathematicans, is certainly not true for all. Perhaps no greater example between the disparity between this unfair stereotype and reality is the case of Grigori Perelman. Hailed for his mathematical brilliance, Perelman remains the only person to have solved one of the so called Millenium problems. These are 7 extraordinary difficult math problems, the prize for solving each problem stands at 1 million dollars. Perelman having been awarded the prize money famously refused to accept it, as he had refused to accept the Fields medal before, the highest honor in Mathematics many would argue. The media incorrectlty potrayed him as unstable, as crazy, due to his decision to refuse all awards and recognitions, and for choosing to stop doing Mathematics altogheter. Alexander goes on to explain:
Dr. Alexander Soifer: Crazy are the rest of us.. Not him. He was simply very sensitive toward hypocrisy and lies. For him, integrity was more important than all of science.
Here is the situation, Sylvia Nasar, who wrote ‘A Beautiful Mind’ flew to St. Petersburg, He typically was hiding from the press, but he met with Sylvia and she wrote a fabulous, article in the Atlantic. Right then I think it was 2006, when he refused the Fields Medal He said, and no one listened. It’s so easy to say, he’s crazy because we know better.
He said the majority of mathematicians are honest people but they tolerate the dishonest ones, in their mi dst. And I don’t want to be a poster boy for mathematics that tolerates dishonesty. That’s what he said. And that makes sense. You know, in my coloring book, I describe my interview with Dimitri, who as a high schooler proved a great result that I reproduce in my book, but he left mathematics. He said, mathematicians, always accuse each other and defend themselves. And I was tired of it. I joined those who play Go instead and worked on computers you know, you can say that, Gregory Pe relman is maybe too sensitive, toward integrity and violations of moral norms, but, uh, that’s okay.
I prefer people more sensitive toward morality than people who are not sensitive, like my former president, Donald Trump, who had absolutely no sensitivity toward integrity. He was very practical man, about getting money and fame and the moral issues didn’t interest him at all. So Grigori Perelman was polar opposite of Trump.
Avidnote Podcast: And that brings us to the topic at hand. The story behind how Alexander Soifer together with Jon Conway wrote the shortest math paper ever published in a serious math journal. The story starts, like many stories in academia, over a coffee break.
Dr. Alexander Soifer: So we had coffee hour every day, 3 to 4:00 PM. And the attendees ranged from undergraduate math majors up to John Nash and John Conway, everyone came, for discussions of anything and there was complimentary coffee and tea and cookies that John Conway called biscuits in a British way.
And, once I came and told John that I came up with a problem, take an equilateral triangle network of equally spaced parallel lines. Parallel to each of three sides, partition the triangle into N squared unit triangles. So side of the equilateral triangle is N and we split it into N squared unit triangles. Obviously we can tile this triangle of area say, N squared times a constant with N squared, unit equilateral triangle but what would happen if we enlarge large triangle by a tiny value, by Epsilon, which John in a Brittish way called “Epsilon”
And so, John, you know, said a nice problem and left, left for a conference. Then he came back and showed me this proof that n^2 + 2 unit equilateral triangles suffices. And then it was my turn to leave for also a conference and on board of the plane. I like to work on board of the plane because you’re surrounded by strangers.
So it’s a similar to a solitude. You can’t talk to strangers so you can concentrate on something. And I found a totally different way to tile with n^2 + 2 unit equilateral triangles. I came back and at coffee hour, I showed it to John who said, oh, nice. And we decided let’s publish it together and set a world record of the shortest paper!
You know, I’ll tell you when I was upper undergraduate student and a PhD student, I wanted to publish the longest possible paper that will be accepted. Imagine one paper that was accepted was 54 pages. Whereas the journal had the normal limit 20 and as exception with special decision of editorial board up to 40. They published my 54 because the referees wrote that abridging would make it hard to understand.
So here it was the opposite!. John and I were not interested in long paper. We wanted the shortest. And so we created the title: Can n^2 + 1 unit equilateral triangles cover an equilateral triangle of side > n, say n + ε? And then the whole text n^2 + 2 can, and two pictures, his and mine. And so I sent it to, as John suggested American Mathematical Monthly.
The editorial assistant, Mrs. Margaret Combs wrote to me that Monthly publishes all kinds of papers, short and long, but she says, and I quote: your article, however, is a bit too short to be good, to be a good monthly article, a line or two of explanation would really help. So at the next coffee hour, I asked John, what do you think, they want us to spoil our record.
John said, don’t give up too easily. So I wrote to her, why is the length of a paper a measure of the paper being good or bad? We posed an interest ing open problem. And, showed a result that we achieved, what else is there? Imagine a couple of days later, I received a mail from editor in chief of the American Mathematical Monthly, who says: we publish long papers and we published short papers, but I don’t think the editor of even short papers would accept it.
But here is my offer to you. When we publish papers, sometime paper just starts the last page and we have a blank space, so we can publish your paper in a frame where we have an opening. If you agree. And, you know, I asked John, you know, we’re, co-authors, we can only decide things unanimously. And John said, okay.
And so the paper appeared, but without asking us, they took the title and put it in the paper.
Avidnote Podcast: And finally, having had a long illustrious career as a mathematician, I asked Professor Alexander Soifer about his views on the future of mathematics. Like a true artist, he shifted the discussion to the arts before he answered my question.
Dr. Alexander Soifer: Let me start with the future of cinema, may I? Well I teach film regularly in 19… I think 76, I was sitting in my Moscow apartment watching my portable black and white television. And it was an interview with the Japanese genius film director, Akira Kurosawa who just finished filming a film in Russian Siberia and he was asked the question: What is the future of cinema and he just shocked me and maybe everyone else, he said, it’s in the silent cinema. You know? And he didn’t elaborate. And I live my life with this statement and here’s how I understood him. Cinema will become so expressive that dialogue would be superfluous, unnecessary. It will be so expressive, visual. And, you know, I can tell you that the number of my favorite films has almost no dialogue.
You know, so now future of Mathematics…
Some people don’t accept proofs where computer was used. I think it’s silly. I asked, Paul Erdős around 1991, what do you think about Appel-Haken proof of the four-color theorem? And fortunately it wasn’t in-person conversation. It was in the le tters so I have a letter where Paul Erdős writes: I accept Ap pel-Haken proof Beggars are not choosers! But I would prefer a computer -free proof. Beggers are not choosers.
Avidnote Podcast: By the way, that’s the late Paul Erdős that is mentioned casually there by Alexander. Erdős aside from being a friend of Alexander, also had a rather noteworthy background, but before I can talk about this, I will let him introduce himself
Dr. Paul Erdős: I introduced myself as a joke like this. Now it takes a little bit of explaining PGOM means Poor Great Old Man. And the first LD means living dead. You get that title when you are a 60. AD, the second means archeological discovery, when you’re 65. And this means legally dead when you are 70.
Avidnote Podcast: Paul Erdos would later become one of the most prolific mathematician perhaps of all time. In terms of his extreme proclivity to publish papers. He is credited as an author for more papers than any other mathematician in history. Atleast 1250 academic publications with over 500 collaborators all around the globe. In fact, it became so common to collaborate with him that a specific number, the Erdős number, was deviced to denote how close you were to having worked with him. Those who worked with him directly had an Erdős number of 1, that includes our esteemed guest Alexander Soifer. Those who had worked with someone who had worked with Erdős had an Erdős number of 2, which includes the likes of Albert Einstein and many Nobel Prize winners. Any way, I digress. Let’s get back to the story.
Dr. Alexander Soifer: It’s inevitable that computers are used in new results that I’m including in the coloring book. Most of new results were achieved with the aid of computers Now, what is good and bad about it? Well, first of all, if you look at four color theory, there are infinitely many maps.
You can’t just plug in computer and check infinitely many cases, computer will melt before it finishes infinitely many tasks.
So computer cannot do everything. What is sometime draw back from computer-aided proof is that we’re a curious people: we want to know, not just answer ” 4 hours”, we want to understand why it’s four. We want to understand why things are the way they are. Not just know the answer.
So we like proofs without computer if they answer why things are the way they are. But it’s inevitable. Computers are here to stay. So reject use of computers, my friend, Paul Kainen from Georgetown University wrote: it’s like denying discoveries that astronomers achieved with telescope.
Let me quote my other friend, Yuri Norstein is the world’s greatest animator He is in Moscow, but he’s animations won the title Greatest Animation of all time in Los Angeles, in Hollywood, in Tokyo, in Japan. I attended his lecture in Tokyo.
He gave a lecture for 200-300 professional animators and it so happened, at the year 2000 that I was in Tokyo for mathematical congress. So I came and one Japanese animation director, asked him: everyone uses computers in animation today, and you don’t, why not? And Yuri Norstein just shocked 300 Japanese present by saying: because computers don’t make mistakes! But then he elaborated. He said, computers do everything right. Everything predictable. Now, when I do everything by hand, I can make a mistake and that mistake may open a new direction. A new means, new horizons. Remember, like in my case, from a spoon of tar, I managed to make a barrel of honey.
Same here from a mistake, animator can create something unpredictable and computer cannot. So that this art and science and computers.
And that’s my long answer to your short question.
Avidnote Podcast: I think that’s a lovely place to end today’s episode. A long answer to a short question given by a man who gave the shortest answer to a long math problem. Be sure to subscribe to get access to upcoming episodes. A quick thanks to our guest, Professor Alexander Soifer. If you’re interesting in learning more about the Mathematical Coloring Book or any other publication written by Alexander, I’ve left a link to to his publications in the episode shownotes.
Until next time, all the best!
Avidnote Intro: Thanks for listening and stay tuned for more episodes. This podcast has been produced by Avidnote. The app that enables you to organize your research and documents. Find out more at: avidnote.com
Avidnote is the new digital tool for writing and organizing your research notes. Become a more effective researcher.