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."
Let's assume that the mathematical proofs and derivations to be presented actually are useful with regard to the material. Certain techniques, some requiring Mathematica, can be a great help.
Background Knowledge: Does the student know what the objects are that the proof is dealing with? It's one thing to say "Let n be an integer", something else to say "Let f be a p-linear symmetric mapping." Can we see some examples? It is one thing to see the derivation of a surprising relation among objects we thought we knew all about, another thing to end up with a hazy relation about hazy objects that we didn't know that much about to begin with. Mathematica can help because we can generate many examples that should give familiarity with the objects.
Motivation: To really understand most derivations and proofs takes a fair amount of time. Is the student convinced ahead of time that the result will be worth the effort? Can he see the problem that the derivation is to solve? If this is a generalization from an easier problem can he see why one would want to generalize and why it is not so obvious how to do it?
Known Material and Axioms: Does the reader have a clear knowledge of the axioms and theorems that go into the derivation? Does he have a clear distinction between the starting point and what is to be derived? Is he given references to equations in earlier chapters where the matter is stated in a different notation and context? And perhaps hard to find - like the only numbered equation in 25 pages. Does he have to play the role of a real estate lawyer doing title searches? With Mathematica notebooks space is inexpensive and all the required material COULD be gathered together perhaps in a separate window and perhaps in an active form.
Coherent Unit: Is a derivation or proof presented as a coherent unit, or does it spread out over many pages with incoherent typography? Mathematica allows a derivation to be presented in outline form and space to be reused such that it is easy to switch between sections of a derivation or to display several sections together in adjacent windows. It is possible to bring the material to the reader when and where they need it.
Active Calculation: Many derivations and proofs in Mathematica can be done with active calculation using routines and rules provided. People understand actions much better than successions of static expressions. Carrying out a derivation actively, as a series of calculated steps, is much more convincing. Providing the routines for doing this may be a great deal of work, but they also provide a base for further work in the applied area. The student studying the proof not only gets the proof but also useful tools. Rather than a forced purchase she's gotten an extra bonus. Active calculation also carries out text-proofing, eliminating silly errors and misprints and producing a document of higher integrity.
Proofs for Mathematicians vs. Proofs for Users: Proofs for mathematicians carry the burden of correctness and rigor. They can assume quite a bit of advance knowledge from fellow mathematicians. Proofs for users have an EXTRA burden of motivation, elaboration and turning the proof into an active useful tool. If the reader isn't going to use it, then why ask him to read it in the first place?
Proof Presentation Research: The active and dynamic features of Mathematica provide means to present proofs and derivations that are not only correct but easier for readers to follow and use. It is not at all obvious how to do this. It is very new. Some of the techniques are things I mentioned above. One good technique with complicated diagrams is to use Checkboxes to turn various features of the diagram on or off as they pertain to elements of the proof. There are probably many other techniques that would arise from or be adapted to special elements in a proof. There is a matter of judgment and taste - no sense in elaborating the obvious, but what are the sticking points that need elaboration?
There is a video kindly done for me by Roger Williams illustrating some of these proof presentation techniques. There are two 8 minute videos but it is the second that shows the techniques. I've posted these before so some of you may have already seen them.
Dear Community, dear Jon, dear Craig, After my recent post where I mentioned the letter of L.D. Landau concerning mathematical education for physicists I received several requests for the full text. The full (or whatever) text I read years ago in the book of Maiia Bessarab<http://www.amazon.de/s/ref=ntt_athr_dp_sr_1?_encoding=UTF8&field-au thor=Maiia%20Bessarab&search-alias=books-de> "Lev Landau: Novel-biography" which can be found here: http://www.amazon.de/Lev-Landau-Roman-biografiia-Novel-biography/dp/B001MIKD 24/ref=sr_1_2?s=books&ie=UTF8&qid=1344239945&sr=1-2 and in few other places on the web. One can also find sites from where it can be downloaded for free. It is, however, in Russian. Because of your requests, and since Landau's ideas today are as valid as they have been 80 years ago, I translated the letter into English for those of you who are interested to read it, cannot do it in Russian though. Please find it below. I tried to keep the translation as close to Landau original style and expressions as I could. Sometimes this lead to phrases that in English sound cumbersome, some sentences look too long, and special expressions are used, such as "exorcise" and alike. Landau liked strong (often impolite) expressions. The term "lyrics" is not from this letter, as I realized when re-read it now. However, it is a true Landau expression he applied to existence theorems. I simply picked it up somewhere else. Finally I apologize, that my translation is not at all professional. Have fun. Alexei
This letter has been written to be sent to the rector of one of Moscow technical universities as a response to its program in mathematics, that probably had been made public at that time. No more details on the origin and effect of this text is available*.
Taking into account the importance of mathematics for physicists, (as it is of general knowledge, physicists experience a need in a calculating, analytical mathematics), the mathematicians, however, for incomprehensible for me reason just fob us logical exercises off as an involuntary purchase**. In your curriculum this statement is directly expressed in a special note in the beginning of the program. It seems me that it is high time to teach physicists things that they consider necessary for themselves, rather than save their souls against their own wish. I do not want to argue against the scholastic mediaeval idea that one can allegedly learn to think by the way of learning unnecessary things. I strongly think that all existence theorems, too rigorous proofs and so on, and so on... should be completely exorcised from the mathematics studied by physicists. For this reason I will not specially focus on those numerous points of your program that are in a drastic contradiction to this point of view. I will only make few additional notes. Historical introduction makes a strange impression. It is self-understood that communicating interesting historical details may only make the lectures more interesting. It is not clear, however, why this is considered as a program point. I hope that this is not intended to be included into tests. The vector analysis is placed between the multiple integrals. I have nothing against such a combination. I hope, however, that it makes no damage to a very necessary knowledge of formal formulae of the vector analysis. The program concerning series is especially overloaded by unnecessary things, in which the scarce useful information about Fourier series and integrals sink. The course of the so-called, mathematical physics*** I would make optional. It is not possible to require physicists-experimentalists to master such things. One should also say that the program is too much overloaded. The necessity in the course of the probability theory is rather doubtful. Physicists anyway teach (their students, AB) all they need within the courses of statistical physics and quantum mechanics. Anyway the presented program is got flooded with uselessness. For this reason I think that teaching of mathematics needs a most serious reformation.
*I believe that this letter has been sent to the rector, but made no effect at all. Landau was not yet that broadly famous at that time. And if he were, nothing would be changed. **Involuntary purchase - a practice existed in USSR, when a shop customer had to buy an unnecessary good, in order to be able to buy a necessary, but rare one. *** In the USSR this was the subject unifying partial differential equations and complex numbers theory.