From a comment by Tamas Gabal:

“This division into 'pure' and 'applied' mathematics is real, as it is understood and awkwardly enforced by the math departments in the US. How is algebraic geometry not 'applied' when so much of its development is motivated by theoretical physics?”

Of course, the division into the pure and applied mathematics is real. They are two rather different types of human activity in every respect (including the role of the “problems”). Contrary to what you think, it is hardly reflected in the structure of US universities. Both pure and applied mathematics belong to the same department (with few exceptions). This allows the university administrators to freely convert positions in the pure mathematics into positions in applied mathematics. They never do the opposite conversion.

Algebraic geometry is not applied. You will be not able to fool by such statement any dean or provost. I am surprised that it is, apparently, not obvious anymore. Here are some reasons.

1. First of all, the part of theoretical physics in which algebraic geometry is relevant is itself as pure as pure mathematics. It deals mostly with theories which cannot be tested experimentally: the required conditions existed only in the first 3 second after the Big Bang and, probably, only much earlier. The motivation for these theories is more or less purely esthetical, like in pure mathematics. Clearly, these theories are of no use in the real life.

2. Being motivated by outside questions does not turn any branch of mathematics into an applied branch. Almost all branches of mathematics started from some questions outside of it. To qualify as applied, a theory should be really applied to some outside problems. By the way, this is the main problem with what administrators call “applied mathematics”. While all “applied mathematicians” refer to applications as a motivation of their work, their results are nearly always useless. Moreover, usually they are predictably useless. In contrast, pure mathematicians cannot justify their research by applications, but their results eventually turn out to be very useful.

3. Algebraic geometry was developed as a part of pure mathematics with no outside motivation. What happens when it interacts with theoretical physics? The standard pattern over the last 30-40 years is the following. Physicists use they standard mode of reasoning to state, usually not precisely, some mathematical conjectures. The main tool of physicists not available to mathematicians is the Feynman integral. Then mathematicians prove these conjectures using already available tools from pure mathematics, and they do this surprisingly fast. Sometimes a proof is obtained before the conjecture is published. About 25 years ago I.M. Singer (of the Atiyah-Singer theorem fame) wrote an outline of what, he hoped, will result from the interaction of mathematics with the theoretical physics in the near future. In one phrase, one may say that he hoped for infinitely-dimensional geometry as nice and efficient as the finitely-dimensional geometry is. This would be a sort of replacement for the Feynman integral. Well, his hopes did not materialize. The conjectures suggested by physicists are still being proved by finitely-dimensional means; physics did not suggested any way even to make precise what kind of such infinitely-dimensional geometry is desired, and there is no interesting or useful genuinely infinitely-dimensional geometry. By “genuinely” I mean “not being essentially/morally equivalent to a unified sequence of finitely dimensional theories or theorems”.

To sum up, nothing dramatic resulted from the interaction of algebraic geometry and theoretical physics. I don not mean that nothing good resulted. In mathematics this interaction resulted in some quite interesting theorems and theories. It did not change the landscape completely, as Grothendieck’s ideas did, but it made it richer. As of physics, the question is still open. More and more people are taking the position that these untestable theories are completely irrelevant to the real world (and hence are not physics at all). There are no applications, and hence the whole activity cannot be considered as an applied one.

Next post: The role of the problems.

Dear Sowa,

ReplyDeleteYes, a similar pattern of hiring happens in my department too. They keep hiring people who do some sort of computations, but rarely, if ever, prove any theorems or come up with new ideas.

The rest of your post was a revelation to me. From talking to my physicist friends I knew how pessimistic most of them are about string theory nowadays, but I always thought that mathematicians working in related areas are very proud of this connection. I was very pleased to read your account of the situations, as it does make more sense.

On the physics side, this obsession with string theory distorted the whole field, pulling attention and resources away from everyone else. For example, recent Fundamental Physics Prizes going (mostly) to string theorists pretend as if string theory was nothing but a huge success. I wonder if in the areas of mathematics connected to string theory a similar effect occurred to any extent, or did mathematicians manage to stay as realistic as you are?

Dear Tamas Gabal,

ReplyDeleteThis happens everywhere, it seems. Initially, I was surprised by the fact that my colleagues comply with such demands very easily, but by now I get used to it. Some get various perks from supporting such hires. It is impossible to know exactly what they get for a particular action supporting the administration, but a rough guess would be that few best pure mathematicians at a given department may expect a additional salary increase about $10,000.00 for support of hiring a chemist positioning himself as an “applied mathematician” (even if the candidate is not hired).

I am still puzzled, because others support the administration also, contrary to their own interests. I am inclined to think that this is a special case of the extraordinary level of compliance in the US with the authorities and even with persons having no authority, but claiming to have one.

What is the “revelation”? The fact that there is a problem with the string theory is known to everybody, I believe. A physical theory existing without any experimental confirmation for decades is bound to be in a (scientific) trouble. But during these decades string theorists took over all key positions in high energy physics. So, everything is OK for them and they do not hesitate to say (I think E. Witten said something like this) that string theory predicts (!) the general relativity, and hence the general relativity provides an experimental confirmation of string theory. It does not matter for them that the general relativity was developed long before the string theory. Moreover, from the very beginning the requirement to include the general relativity was one of two may requirement imposed on the development of string theory (another one is to include the quantum theory). There is no surprise that it does include the general relativity.

ReplyDeleteI think that you meant my not very exalted position with respect to the contributions of the string theory to mathematics. Surely, this is not a universally accepted position. 10 or even less years ago I would (and I did on some occasions) defend the string theory and (less enthusiastically) its contributions to mathematics. Well, if our expectations do not materialize long enough, we usually abandon them.

I am not sure that may mean “being proud of the connection”. It is out there, like the Everest is. Could one be proud of Everest? But if you managed to climb it to the top, you have all ground to be proud. I think that mathematicians who proved the conjectures (or predictions, as they like to say) of physicists, also have all grounds to be proud. The issue here is to what degree they are proud, and how such proofs are valued by the mathematical community.

My observations long ago lead me to the conclusion that a connection with physics and especially with the string theory substantially increases the appreciation of your work. Suppose that there are two precisely stated and equally difficult conjectures, one stated by a mathematician (not involved with the connection), and the other stated by a physicist/string theorist. The one who proves the second conjecture would be appreciated about one level higher: she or he will get better job, will be invited on more prestigious conferences, etc. If the conjecture was suggested by E. Witten, a proof, theoretically, may catapult you from a speaker at a section of a Congress to a Fields medalist. (I think this happened once.) In fact, any connection with physics adds value to your work, and you do not have to be the discoverer of this connection.

All such “added values” distort the picture and distort the natural development of mathematics. I have no objections against using string theory, biology, a local rock band, whatever, as a source of inspiration. But if the end result is a theorem or a mathematical theory, then it should be judges as such.

Dear Sowa,

ReplyDeleteI did not know until recently that string theory was in such a big trouble within physics, perhaps, because string theorist still occupy key positions and continue to receive many prestigious awards. What was a revelation to me was your point of view as a mathematician. I thought that mathematicians in general were impressed by the connections to string theory and physics and that most (if not all) mathematicians working in related areas were proud of this connection, in the sense that it made their entire areas somehow more exciting and more important within mathematics. In a way, what you wrote at the end supports this view, as it was mathematicians who "added value" to the conjectures related to string theory. Maybe, people like you, who are more realistic and are not easily impressed, are an exception?

By the way, it is not only the connections to string theory but also to statistical physics that resulted in the Fields medals lately. Maybe, it makes sense because you can point to a problem in physics to say why some mathematical result is important. Another result could be more important in the long term, but the decision-making process of the various award committees must be based on information already available to them. Of course, this is also an argument against such awards.

I should add that the connections to physics and other applied disciplines can be very healthy for mathematics. For example, in the words of von Neumann, "... at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas."

Dear Tamas Gabal,

ReplyDeleteMathematicians are usually too immersed in their own work in order to notice even a new trend in the mathematics itself. I always had more broad interests than a typical mathematician. This is not just my own comparison of myself with others. For example, in the graduate school I was reproached more than once for studying “irrelevant” or “useless” things, and even for proving a minor theorem in my area of mathematics, but irrelevant to my Ph.D. thesis problems. Despite having more broad interests than expected, I did not notice some important events in time.

In contrast, there is a guy named John Horgan, who published in 1996 a book

“The End of Science: Facing the Limits of Knowledge in the Twilight of the Scientific Age”. If I remember correctly, he was a staff writer (may be even the chief staff writer) for the“Scientific American”. One of the chapters of this book is devoted to physics, and contains a lot of criticism of string theory. He is not a physicist, or a scientist, but he did a lot of leg work. For example, he interviewed E. Witten. He worked in the right way, using the leading scientists as his sources (and not any opponents of this or that science). It was the time when everyone was excited about the prospects of string theory, but J. Horgan arrived at different conclusion. It seems that he was right, and most of us were wrong.About the same time he published in

“Scientific American”an article“The end of the proof”. This thesis was based on the notion that relatively soon mathematics will be done by computers or will be assisted by computers to such extent that traditional proofs will be irrelevant. My summary may be not quite correct; I did not consult his paper now. Still, his article looks quite prophetic in view of Gowers’s plans.I was impressed, less than many others, but still impressed. All theorem and theories resulting from this interaction are good anyhow. They are just not as brilliant as claimed.

ReplyDeleteI don’t think that my opinions are very exceptional. Other people do say the same things, but not in public, not even partially anonymously as me.

I don’t think that importance for physics plays a real role in the decisions of who get awards, positions, etc. People would say something like this and include it into the address presenting the work of the winner. I think that the reasons are deeper. There is a parallel phenomenon in the careers of quite a few top class mathematicians. After reaching the top of their mathematical creativity, they decide they cannot go higher in mathematics (probably, that much is correct, but they could continue to work at their extremely high level), and the only way higher is to turn to something else. Speaking of Fields medalists only, S. Smale turned to economics, then to the theory of computations, and then to the “theory” of learning (there is no such theory). S. Novikov turned to physics, D. Mumford to image recognition, M. Freedman moved to Microsoft, T. Gowers to the artificial intelligence and elimination of mathematics. In order for such a strategy to work, other areas should be valued by mathematicians – none of them made a contribution to other science comparable to his contribution to mathematics.

When von Neumann said these words, he wasn't a mathematician for about a decade or longer. He was involved in the development of first computers, in the design of the atomic and nuclear bombs, and eventually turned to purely government work. These words are self-serving, justifying the choices he already made. This quote can be countered by many other quotes, but it is better to use examples. Say, algebraic number theory survives at least since P. Fermat (or even from the ancient Babylon) without any external stimuli. The problem of finding formulas for solutions of polynomial equations of degrees 3,4, and higher lead to the whole field of algebra, which never needed external stimuli. Only analysis experienced significant influences from outside. I am aware of only one area which seems to be unable to live without outside support: PDE, and only on the largest scale. It seems that experts in PDE are able to investigate almost all equations suggested to them. But they depend on physics and geometry as a source of the new equations. Why experts in PDE did not invented the Seiberg-Witten equations as a simpler alternative to the Yang-Mills equations for applications to the topology of 4-manifolds? This is a mystery.

ReplyDeleteI have conjectured that the main problem with string theory is that string theorists fail to realize that Milgrom is the Kepler of contemporary cosmology. The probability that Milgrom is wrong is zero — study the evidence. String theorists ignore Milgrom to their own detriment.

ReplyDelete"About 25 years ago I.M. Singer (of the Atiyah-Singer theorem fame) wrote an outline of what, he hoped, will result from the interaction of mathematics with the theoretical physics in the near future."

ReplyDeleteDear Owl,

Could you please provide a more explicit reference to this text?

Thanks a lot in advance.

Best wishes,

DD

DD: I had in mind the paper

DeleteI.M. Singer, Some problems in the quantization of gauge theories and string theories, In: The mathematical heritage of Hermann Weyl, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988.

About 20 years earlier Singer wrote a similar article about the future of the index theory. It seems that in this field his predictions turned out to be true, and even more than true. Of course, the development did not followed exactly his vision, but it was anything but disappointing.

Dear Owl,

DeleteThanks a lot for your reply. I will have a look on this paper with a lot of interest.

Having read some of your posts, I can conclude that you are clearly a pure mathematician, me being a very applied one (I work in Hydrodynamics and a bit of scientific computing). Nevertheless, I would be delighted to read some of your thoughts about the (modern) Applied (including computational) Mathematics.

Another question: according to you, which parts of the modern mathematics, applied mathematicians should learn?

I understand that it can be difficult to address these points, especially if you didn't think about it before, but I would really appreciate any points of view from the pure side of our profession. Hopefully it will allow to build a better future for all of us :)

Thanks a lot in advance!

Dr. D