Concerning Landau, I am sure he understood the use of existence theorems for science quite well. What he claimed in this letter is that it is useless to teach them to physicists in full amount. Scientists belonging to his circle witnessed that Landau always felt himself to be responsible for the whole Soviet theoretical physics, not only when he was mature and famous, but also when he was young and unknown. This, I believe, is the reason for Landau to have written the letter I translated. Other his letters have been also published in the book of M. Bessarab, where Landau expressed the same ideas, though not with that much details. For example, a lot of people have written him to ask his advices of what and how to learn to become a physicist, or a theorist. He always answered and often repeated the advice to skip theorems, but to focus on solving integrals and differential equations when learning mathematics. As much as I understand, it is because of the shortness of our life, that one needs to only go to the aim taking the shortest possible way. "In view of shortness of life..." is the Landau's true expression.
I do not agree with your last statement, Andrzej, though: "since the great majority of physicists are not Landaus, Wittens, Penroses etc., I don't think it would be very wise for them to take these people as their role models".
I do not think about models, but it is very useful to see such a person as a personal goal, understanding in the same time the distance between myself and Landau, or Penrose, etc.
What I see in the ideas you expressed few times, Andrzej, is the same I have heard many times from mathematicians from whom I learned mathematics in my university times. The idea is: "You do not understand it yet, but all these theorems are necessary for you in your future everyday scientific activity. So, do learn them". The same I have heard from professors in philosophy. In that case the words "all these theorems" were replaced by "these philosophical ideas", the rest was the same.
Now that I am more close to the end of my scientific career, than to its beginning, I can testify that the both played the same role, that is, nearly zero. I cannot tell that I used no theorems in my life. But I for sure used only the constructive ones, and only statements, not proofs. That is, of course, finding a maximum of a function by equating its derivative to zero is a content of a theorem. Of course, all of us use it. But very rarely one needs to recall its proof. It must be a very special case for physicist to be in such a need, and in this case one takes a textbook and learns the proof anew, rather than recalls it.
Of course, it is useful to know the Cauchy's theorem, but a physicist most often only needs is to know, how many boundary conditions he has to account for a given ODE. Does he really need to recall the proof?
The life is short, and the time I spent learning about 1000 theorems with their proofs during my university course of mathematics is the time lost for nothing.
Again, please understand me correctly, all this only concerns education of some classes of people. It does not apply to professional mathematicians. In no case! It applies to physicists, and (probably) some other classes of people, such as chemists, engineers etc. All I wrote here does not mean at all that I think these theorems to be useless for science. No, they are only useless for people who use mathematics as a tool, in the area, where correctness of its application has been proofed already centuries ago.
When you buy a new TV set in a shop, you only need to know how to switch it on, so that you may watch the Olympic games right away. Imagine, however, that during buying it the seller tells you: "Now, before you start watching it, we first teach you the theoretical basis of electronics, which you will surely need all the time you use our TV device, and then you absolutely need to learn making electronic circuits yourself". How would you react? If you agree with the guy, you miss these Olympic games, and probably the next games also. In view of shortness of the life...
-----Original Message----- From: Andrzej Kozlowski [mailto:email@example.com] Sent: Wednesday, August 08, 2012 11:34 AM Cc: Alexei Boulbitch Subject: Re: Landau letter, Re: Mathematica as a New Approach...
On 8 Aug 2012, at 09:16, "djmpark" <firstname.lastname@example.org> wrote:
> Landau just expresses the frustration that non-mathematicians have > when confronted with mathematical derivations and proofs that strive for rigor. > They may need the results and material but, darn it, it's hard. When > they finish reading a proof they might say: "Well, I suppose so." > (Somehow I can't imagine Landau saying that, but maybe he was speaking > for others as well, or perhaps he was truly frustrated with the presentations."
Actually, it was exactly the other way around. Landau was one of those theoretical physicists that were also immensely talented mathematicians (like Edward Witten who has got both a Fields medal and The Fundamental Physics Prize). Such physicists when they need to use certain mathematics often develop it themselves rather than learn from mathematicians (sometimes because nothing of the kind they need exist and sometimes because for them this the quicker route). For example, Landau did this with Catastrophe Theory, as described by Arnold in his little book on the subject. Physicists like this are very exceptional. When they take a look at mathematics as taught by mathematicians they often find that all things that seem to them important they already know and the rest is just formalisation, which they don't care about. But since the great majority of physicists are not Landaus, Wittens, Penroses etc., I don't think it would be very wise for them to take these people as their role models (in their approach to mathematics, of course).