Why is Hilbert not the last universalist?

[Demystifier]

It is often said that Poincare was the last universalist, i.e. the last mathematician who understood more-or-less all mathematics of his time. But Hilbert's knowledge of math was also quite universal, and he came slightly after Poincare. So why was Hilbert not the last universalist? What branch of math he didn't understood sufficiently well to deserve this title?

 

[fresh_42]

What about Erdös?

 

[Demystifier]

What about Erdös?

If we can prove that Hilbert was not a universalist, then, by induction, it is trivial to prove the same for von Neumann, Erdos, or anybody else who came later.

 

[fresh_42]

If we can prove that Hilbert was not a universalist, then, by induction, it is trivial to prove the same for von Neumann, Erdos, or anybody else who came later.

Yes, but what if I can prove, that Hilbert was a universalist?
http://www.sciencedirect.com/science/article/pii/S0166864103003377

 

[Demystifier]

Yes, but what if I can prove, that Hilbert was a universalist?

First prove it, and then I will tell you what then!
Or perhaps you are trying a reductio ad absurdum? Let as assume that Hilbert was a universalist and derive a contradiction!

 

[fresh_42]

To get serious again. I think the most crucial part of the question is, what really has been known at the time. I've never met Hilbert's name along with the algebraic part of Lie Theory, and similar with algebra (ring or group theory) in general. On the other hand, I have few doubts, that he was aware and knowing of the stuff, which makes it even harder to tell. And what about Russell's work? Hilbert's program is somehow the opposite of what became Gödel's theorems and Russell and others had already pointed out the right direction. It always reminds me on Descartes and on determinism. (On the other hand, I just saw Laplace and Planck on a Wiki list of representatives of determinism - two names we associate with probability nowadays. What an irony!)

So maybe Hilbert just doesn't count as universalist, because all the modern aspects of what revolutionized physics and mathematics in the 20th century were already founded to some extend, and authors, who characterize scientists in such a way, simply didn't took the effort of a closer look.

 

[Demystifier]

Maybe Hilbert's problem was that he lived too long, while Poincare died relatively young:
Poincare: 1854-1912
Hilbert: 1862-1943
A lot of new mathematics happened from 1913-1943, which Poincare didn't need to bother with.

 

[fresh_42]

Maybe Hilbert's problem was that he lived too long, while Poincare died relatively young:
Poincare: 1854-1912
Hilbert: 1862-1943
A lot of new mathematics happened from 1913-1943, which Poincare didn't need to bother with.

Good point.

And I have to adjust my previous comment a little bit. Hilbert did come up in the context of Lie Theory, even though rather indirect. However, Hilbert actively supported his assistant on various occasions and against all odds of the main political stands of his time. This assistant has been definitely well aware of Lie Theory and also Ring Theory, so it's quite unreasonable to assume Hilbert was not. Her name: Emmy Noether. This supports your opinion of a universal mathematician. And even his program, although driven by an impossible aim, influenced others like von Neumann and Gödel, which has to be taken into account, too.

This let's me ask: Where have you read, that Poincaré had been the last universalist? Was it a French author?

 

[Demystifier]

This let's me ask: Where have you read, that Poincaré had been the last universalist? Was it a French author?

I don't know, I have seen it at a dozen of places, including wikipedia.

 

[Ssnow]

Hi,

A lot of new mathematics happened from 1913-1943, which Poincare didn't need to bother with.

yes this can be a remarkable reason but I think there is another, Poincaré refuses to see mathematics as a branch of logic (as happened from Russel and Hilbert ) he belived that intuition was the life of mathematics.
I believe that this happroach much "malleable" allowed him to create a big personal knowledge in the fields where he contributed (these goes from numerical analysis to the group theory and all mathematics), in this sense it is considered the last Universalist...

Ssnow

 

[Demystifier]

yes this can be a remarkable reason but I think there is another, Poincaré refuses to see mathematics as a branch of logic (as happened from Russel and Hilbert ) he belived that intuition was the life of mathematics.
I believe that this happroach much "malleable" allowed him to create a big personal knowledge in the fields where he contributed (these goes from numerical analysis to the group theory and all mathematics), in this sense it is considered the last Universalist...

Hmm, that's an interesting argument. Essentially, you are saying that if you base your reasoning on intuition rather than strict logic and axiomatics, then it is easier to comprehend the whole of mathematics. That makes sense (even though my avatar might not agree ), but how about logic itself? As a universalist, did Poincare understood well the mathematical logic of his time and did he contributed to it? And if he didn't, then was he really a universalist?

 

[fresh_42]

As a universalist, did Poincare understood well the mathematical logic of his time and did he contributed to it? And if he didn't, then was he really a universalist?

When I read about Poincaré in Jean Dieudonné's book on the history of mathematics (1700-1900), I cannot avoid the impression, that mathematics at his time hasn't been developed sufficiently, to speak of the kind of rigor and therewith logic, we demand today - a point of view which is hardly to accept, considering all the other geniuses until then (Euler, Gauss, Kummer, Legendre, Lagrange, etc.). Or the other possibility would be, that he gave a da.. about precision but had brilliant ideas, others worked out later on. That's commonly a good start to be called a "universalist".

E.g. Poincaré and the fundamental group:
"The composition of loops doesn't appear by P. (He called them routes, because to him loops have been routes which were successively walked through in both directions.) The group is defined by substitutions of certain values into (not uniquely defined) functions on the manifolds, inspired by the automorphisms in function theory, especially those which P. called Fuchs' functions, which we call automorphic functions today. The homotopy wasn't described by P. ... The role of a basis point wasn't mentioned at all ..."

Not very trustful, and Dieudonné was French, too! The short biography in this book also says: "Poincaré was a full mining engineer and also practiced this profession during his dissertation." - Maybe he thought he has dug deep enough.

 

[lavinia]

When I read about Poincaré in Jean Dieudonné's book on the history of mathematics (1700-1900), I cannot avoid the impression, that mathematics at his time hasn't been developed sufficiently, to speak of the kind of rigor and therewith logic, we demand today - a point of view which is hardly to accept, considering all the other geniuses until then (Euler, Gauss, Kummer, Legendre, Lagrange, etc.). Or the other possibility would be, that he gave a da.. about precision but had brilliant ideas, others worked out later on. That's commonly a good start to be called a "universalist".

E.g. Poincaré and the fundamental group:
"The composition of loops doesn't appear by P. (He called them routes, because to him loops have been routes which were successively walked through in both directions.) The group is defined by substitutions of certain values into (not uniquely defined) functions on the manifolds, inspired by the automorphisms in function theory, especially those which P. called Fuchs' functions, which we call automorphic functions today. The homotopy wasn't described by P. ... The role of a basis point wasn't mentioned at all ..."

Not very trustful, and Dieudonné was French, too! The short biography in this book also says: "Poincaré was a full mining engineer and also practiced this profession during his dissertation." - Maybe he thought he has dug deep enough.

While mathematics today demands rigor in proofs, mathematicians still believe that rigor is an afterthought and that ideas come first - proofs later. So Poincare's intuition is still considered the well spring of mathematical ideas.

To illustrate, I once sat in on a course in "Elementary Topology" taught by Dennis Sullivan in which he would ask students to give proofs at the blackboard. If the student would try to present a rigorous deduction, Sullivan would get angry and say "That's not a proof!" and he was not satisfied until the student explained his intuition for what was going on.

Science still does not understand the process by which new ideas and insights arise in consciousness. It may be that intuition and insight can be reduced to an unconscious computation. But it is certainly false that we experience new ideas as paths of deduction. Deduction is always an afterthought not a source.

- In most sciences e.g. biology or astronomy, the 20'th century saw a proliferation of new knowledge that made it impossible for any single person to understand everything in his field. The same is true in modern mathematics. Today mathematicians are relative specialists. On the other hand, the amount of knowledge that a modern geometer or algebraist has at his fingertips probably dwarfs the entire mathematical knowledge of Poincare or Hilbert.

 

[Ssnow]

but how about logic itself? As a universalist, did Poincare understood well the mathematical logic of his time and did he contributed to it? And if he didn't, then was he really a universalist?

These are good questions. I don't have answer for each of these questions but we can analyze the situation at the Poincaré time. First the branch of mathematics called mathematical-logic at the Poincaré time was just at the beginning, so still impossible to contribute in this field of mathematics (if are not the creator)... I think that the Logic considered until Russel was properly confined in the philosophy with no much connection with the world of mathematics (with this I don't want to say that mathematician didn't know the logic obviously but only that its connection with mathematics was not so enthusiastic ).
I mean to say that the rigorous treatment and organization of mathematics only started in this time so the major part of mathematicians were not only mathematicians but they did a lot of works as engineers,legislators, and so on ... the mathematics was a large collection of results (sometimes caotical) obtained by thinking on real or ideal problems. I think that Poincaré was the last Universalist because without the rigorous organization of the mathematics that we have at recent days he was able to give contributions in all fields of mathematics (of its time) only using the intuition (and observation) as guideline ...

 

[martinbn]

When was that said first? If it was while Poincare was still alive or just after his death, then it may be because of that.

fresh_42, I'd say that Hilbert was quite big in algebra, especially if you consider commutative algebra and algebraic number theory. There quite a few theorems there that are knows as Hilbert's theorem. On the other hand it seems that Poincare wasn't much of an algebraist.

By the way how much knowledge/expertise in an area of mathematics is sufficient for universalism? For example from the list of books authored by Lang, one may argue that he was also an universalist. Or is more depth needed in each subfield?

 

[fresh_42]

While mathematics today demands rigor in proofs, mathematicians still believe that rigor is an afterthought and that ideas come first - proofs later. So Poincare's intuition is still considered the well spring of mathematical ideas.

Oh, I don't want to play down the role of ideas nor their importance. A friend of mine once said: "A genius is not the one with a brilliant idea late at night. It's the one who sits down the next morning and works it out." I think there is much truth to it.

 

[lavinia]

Oh, I don't want to play down the role of ideas nor their importance. A friend of mine once said: "A genius is not the one with a brilliant idea late at night. It's the one who sits down the next morning and works it out." I think there is much truth to it.

Formal rigor as now required does not have much to do with genius IMO. As you said in your post, there were many geniuses before the advent of formal proofs. It is possible to " work it out"with out reducing the problem to a syllogism. Much of "doing the work" is finding new ideas that elucidate the original idea to be proved. It is not finding formal demonstrations per se. For instance, attempts to prove the Poincare conjecture inspired entire new areas of mathematics. The same goes for research on Fermat's last theorem. Often different proofs of the same theorem employ new and entirely different techniques. These techniques are just as important or maybe more important than the theorem itself.

- There is a paper by Witten that did not contain formal proofs but spurred an intense area of research. For years mathematicians did not know whether his claimed theorems were actually true.

- I was once told about a mathematician who hardly every published but was so insightful that other mathematicians wanted to have him around just to talk to.

 

[Demystifier]

For example from the list of books authored by Lang, one may argue that he was also an universalist. Or is more depth needed in each subfield?

Writing textbooks is one thing, making new original contributions is another. Did Lang made original contributions in all these branches of math? And for that matter, did Bourbaki made original contributions in all branches of math?

Which reminds me of a joke:
Why did Bourbaki stopped writing books? Because they realized that Lang is a single person.

 

[martinbn]

That's why I asked about the criteria. So an universalist is not some one who knows "everything" in his subject, but some one who has original contribution to all subfields of his subject. I don't know if that disqualifies Lang (most likely it does), but does Poincare fit in this definition?

 

[lavinia]

http://www.abelprize.no/c53720/binfil/download.php?tid=53562

Here is an incomplete list of the areas of mathematics where Milnor has made stunning contributions..

- Differential Topology - many contributions including the discovery of exotic 7 spheres
- Combinatorial Topology - Counterexample to the Hauptvermutung, invention of micro bundles
- Algebra - group theory, Algebraic K-theory
-Dynamical Systems
- Knot theory - Discovered Milnor invariants.
- Characteristic classes
- Differential Geometry - Total curvature of knots
- Algebraic Geometry - Singularities of Complex hyper surfaces - Milnor fibrations

Milnor is also a great teacher and his books have become bibles for certain subjects.

In this modern world of specialization in Mathematics, Milnor had broad influence in many areas. He has been called "The Mozart of Mathematics"

 

[Demystifier]

I didn't know that Milnor worked on dynamical systems.

 

[Demystifier]

but does Poincare fit in this definition?

That's what I would also like to know.

 

[lavinia]

I didn't know that Milnor worked on dynamical systems.

this is from the Wikipedia article on him

Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarized by Peter Makienko in his review of Topological Methods in Modern Mathematics:

It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems.[5]

 

[atyy]

In this modern world of specialization in Mathematics, Milnor had broad influence in many areas. He has been called "The Mozart of Mathematics"

Why Mozart and not Beethoven?

 

[bhobba]

Maybe he was lacking in some areas of applied math. Poincare made contributions to Mining engineering, and while he never practiced it, to the best of my knowledge anyway, Von-Neumann was a qualified Chemical Engineer.

To me a polymath is a better classification. Both Poincare and Von-Neumann were polymaths of the highest order. Any other great polymaths after Von-Neumann - don't know.

I however have to say Terry-Tao seems to pretty much jump easily from area to area. Is he a polymath? Probably not.

Thanks
Bill

 

[Auto-Didact]

Hmm, that's an interesting argument. Essentially, you are saying that if you base your reasoning on intuition rather than strict logic and axiomatics, then it is easier to comprehend the whole of mathematics. That makes sense (even though my avatar might not agree ), but how about logic itself? As a universalist, did Poincare understood well the mathematical logic of his time and did he contributed to it? And if he didn't, then was he really a universalist?

From a history of mathematics point of view, the (pre)intuitionist philosophy that Poincaré employed seems indeed to play a great deal in this. Its pretty clear that pretty much none of the universalists (Newton, Bernoulli, Euler, Gauss etc) before Frege et al. worked extensively on logic, with Leibniz being the exception. In this sense the logicist philosophy, and especially formalism championed by Hilbert, are really modern phenomena, i.e. a departure from classical pure mathematics.

Ironically, many modern mathematicians, heavily influenced by being educated in logicist/formalist programmes, seem to view non-rigorous, non-proof driven mathematics in a similar manner as how the classical purists viewed applied mathematics, and that while not necessarily having any disdain for applied mathematics either given that it can be treated rigorously.

It therefore seems that the universalist/specialist categorisation is a more accurate way of categorising mathematicians and their philosophies, especially those who straddle the two eras such as Poincaré and von Neumann, and comparing them with other mathematicians while taking other differences of both classical and modern mathematicians into account (such as being more purist/applied).

In this sense Poincaré clearly was more of a universalist, while Hilbert, despite being from the same era and having worked in many branches, has much more in common with modern mathematicians who are practically all specialists.

 

[mathwonk]

Hilbert is quite famous for his algebraic work on invariant theory. In fact some say his work gave birth to the subject of abstract algebra, as opposed to computational methods in algebra of the 19th century. David Mumford famously resurrected ideas and techniques of Hilbert to develop what he calls Hilbert's criterion for "stability", a crucial technique for constructing moduli spaces in algebraic geometry. A fundmental object used for constructing such moduli spaces developed by Grothendieck is the so called "HIlbert scheme". Most of us encounter first the more elementary algebraic result due to Hilbert that a polynomial ring over a so called "Noetherian" ring is also Noetherian. One of the classic algebraic works on my shelf is (a translation of) Hilbert's theory of algebraic number fields. According to its introduction, this was the key text on the subject for some 30 years at the beginning of the 20th century, and that the famous algebraists Artin, Hasse, Hecke, and Weyl all learned algebraic number theory from it. Some 70 years later Lang is also said to have attributed the structure of his own book to this source. Hilbert's presentation of his famous problems about 1900 clearly demonstrates his large scale grasp of the mathematics of his time. These problems influenced the progress of mathematics for decades afterwards.

Here is a link on invariant theory:

https://en.wikipedia.org/wiki/Invariant_theory

and another on Hilbert's problems, several about algebraic and analytic number theory, and the famous 5th problem on topological versus lie groups.

 

[Chronos]

Alexander Grothendieck, a remarkable man famed nearly as much for insanity as his contributions to mathematics, is a worthy contender, but, John von Neumann stands out in any crowd of mathematicians.

 

[Auto-Didact]

Another point is that all universalists up to and including Poincaré were clearly not only mathematicians but also specifically physicists during practically their entire careers; apart from the indirect mathematical role of Hilbert space in von Neumann's QM and the Einstein-Hilbert action in GR, I am not aware of any other direct contributions to physics proper from Hilbert.

I would even go as far as to say that Hilbert's purist mathematical stance and constant formalist want for axiomatization is as anti-physics as it gets; moreover, I presume Feynman shared this opinion based on his Messenger Lecture on the relation of mathematics to physics.

 

[lavinia]

Another point is that all universalists up to and including Poincaré were clearly not only mathematicians but also specifically physicists during practically their entire careers; apart from the indirect mathematical role of Hilbert space in von Neumann's QM and the Einstein-Hilbert action in GR, I am not aware of any other direct contributions to physics proper from Hilbert.

I would even go as far as to say that Hilbert's purist mathematical stance and constant formalist want for axiomatization is as anti-physics as it gets; moreover, I presume Feynman shared this opinion based on his Messenger Lecture on the relation of mathematics to physics.

Following up on your post I started to listen to the Messenger Lecture. I think Feynman was talking not about mathematics per se but about the application of mathematics to physical problems.

Mathematicians examine the realm of mathematics much as physicists examine nature. As Feymann described for Physics, Mathematics also has need for the "Babylonian method" as he called it.