Today I attended the funeral of Jürgen Ehlers who died recently and quite unexpectedly for those around him. Here I want to collect some personal thoughts about Jürgen. I was associated with him for twenty years during which time he was my colleague and mentor. I was first a member of his research group in Munich and later at the institute in Potsdam (Max Planck Institute for Gravitational Physics) in whose founding he was the key figure. This is all the more remarkable since his passion was for science and not for scientific politics.

This is not the place to write about the personal qualities which I admired in Jürgen and I will confine myself to talking about some of his scientific qualities. Although he decided to go into physics rather than mathematics at a very early stage of his career, he had a keen sense for mathematical questions and a wide mathematical knowledge. He was not the kind to devote himself to the production of long and complex mathematical proofs. What he did was to identify important physical questions and the challenging mathematical problems which lay hidden beneath them. Once he had identified them he was a master of exposition whose explanations were a pleasure to listen to. He came back again and again to statements which are frequently made more or less uncritically in textbooks on general relativity such as ‘Newtonian gravitational theory is the limit of general relativity as the speed of light tends to infinity’ or ‘It follows from the Einstein equations that small bodies move on geodesics’. He saw that these phrases were in need of not only a proof but of a precise statement. He contributed to the first of these questions by formulating his ‘frame theory’ and also worked hard on the second. His efforts are now beginning to bear mathematical fruits in the form of rigorous theorems on these questions.

In doing science Jürgen emphasised making distinctions, e.g. ‘distinguish between a mathematical model in physics and its physical interpretation’, ‘distinguish between an intuitive argument and a rigorous proof’, ‘insist on precision in terminology where it can bring clarification’. These things may seem obvious but they are often enough neglected. In communicating these things to me he was preaching to the converted but he encouraged me by his example to propagate these same ideas. I remember him once talking about the difference between the concept of proof in physics and mathematics. To establish a mathematical fact one proof suffices. Additional, more or less independent, proofs may bring more insight but they are not necessary. The physicist, on the other hand, often likes to have several proofs. He likes to reach the same endpoint by various routes. Each of the two procedures has its own advantages. Very roughly speaking, the physicist’s approach takes one fast and far. The mathematician’s approach has the advantage that the gains it achieves are lasting. What I have said about distinctions might sound like abstract philosophy but my experience is that these ideas are of great worth in improving communication between physics and mathematics. No doubt similar things are true for the relations between other sciences.

June 16, 2008 at 3:22 pm |

Hello

I think this is a nice and fair description of the philosophy of Jürgen Ehlers. However I would like to make a comment on the distinction between ”mathematical ” and ”physical ” proofs. I think the situation is a little more complicated especially there are situations in which a proof might be more interesting than the theorem which is proven. Let me give you some examples.

1. The first is based on an article of Halmos whose reference I don’t remember right now. It is about the celebrated 4 color problem. The structure of the proof was known for some time, however the details required a huge amount of calculations which finally were done by a computer and so the complete proof seems not to provide any “new” insight.

2. The same holds for a theorem which is even more famous, the n-body problem in Newtonian gravity. The original problem, as formulated in the King Oscar II price, was to find a global solution which could be expanded by a convergent power series. That problem was not solved, but finally Poincare was awarded for his contribution which lead to the discovery of chaos. 20 years later Sundman proved the theorem as demanded by the scientific board. It turned out however that the power series converges so slowly that it does not provide any qualitative nor quantitative insight. So we have here an example of an important theorem but the proof does not provide any insight.

3. The second example concerns Fermats last theorem. I think Gauss himself complained about some conjectures in number theory saying that he would be able to provide hundreds of similar conjectures. However Wiles proof was celebrated for the new techniques and insights it provides but so much for the fact that the theorem was proven or disproven. (As Halmos puts it: I don’t want to know WHETHER a particular conjecture is true, I want to know WHY it is true or not).

4 The last example would concern a theorem with outstanding importance in the sense that many statements would depend on it, such as the Riemann hypothesis or (maybe) P=NP. In that case any proof would be welcome.

October 14, 2009 at 2:10 pm |

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