JEAN TAVERNIER
THE THINKING ABOUT
OF A PHYSICIST
MATHEMATICS
I would like to tell you what is my thinking about the use of mathematics in physics and its teaching. It is to be well understood that what I am going to say comes from my own judgment but I am not sure that it represents the opinion of all my colleagues. First of all i must mention that I am a solid-state physicist who is very attracted by mathematics. For these reasons I teach mathematics on one hand and physics on the other. Accordingly, i could be the right man to bridge the gap between mathematics and physics. Unfortunately, the twenty-years-old students being what they are or more precisely what secondary education made them, I do not succeed entirely in this way. The students having been used to an axiomatic mathematics for which every property is to be proved in detail, they believe that mathematics cannot be taught otherwise. I do not mean that mathematics can be taught without rigour; I just think that the proof of some theorems could be dropped but the enunciation has to be very precise and the physics student must be able to apply any theorem inside its field of validity. So, when I teach mathematics I have to consider it as an abstract science in order to get the students satisfied. I am fully aware of the problem raised by this situation but I would like to ask you the following question: 'Must the students learn to be confident in the results given to them before or after being twenty years old?' In my opinion, it should be beneficent to give them this spirit during secondary education. On the other hand, when I teach physics, I have to give up the few powerful results of mathematics I know. In fact the students have learned so many things in physics during their secondary education that they think they know everything about physics. Then, it is very tedious and too often unsuccessful to explain to them that they have to get a deeper insight into the physical mechanisms responsible for the well-known phenomena and to use a more elaborated mathematical formalism. Consequently it needs several months and sometimes one year in order to get from the students what we expect of them. It is to be remarked that I talk in the same time about physics and mathematics although this meeting is just concerned by the teaching of mathe55
Educational Studies in Mathematics
1 (I968) 55-60; 0 D. Reidel, Dordreeht- Holland
JEAN TAVERNIER
matics. But I think that these two sciences are closely connected as far as the actual gap could be bridged. Furthermore, if I have well understood, the mathematics teacher would be fond of proving that mathematics is useful, what I am aware of, by giving to the students some motivations derived from other sciences. Conversely, I think it would be desirable that the physics teachers use a better and more modern mathematical formalism than they do generally. Then in order to get some very interesting results i wish we had very soon a common meeting enlivened by all the goodwill concerned with physics and mathematics teaching. Now, I would like to give some remarks about the teaching of useful mathematics. First I shall try to answer the question: 'Are the modern mathematics usefuI?' But, what is called modern mathematics? I remind you that Professor Revuz said that: "Modern mathematics are those which have just been discovered". But when the physicist speaks about modern mathematics he means subjects as linear algebra, group representation, Lie algebra, involutive algebra, distributions, etc., which are well known for a long time, by the mathematicians. From this point of view, most of the physicists will agree to usefulness of the modern mathematics. But at this point you will affirm that it is impossible to teach the whole of these subjects during the very short time reserved to mathematics. I am so aware of this fact that I shall remind you that a large part of the mathematical results should be given without proof as I said at the beginning of this paper. For instance it is not very useful to spend a long time to demonstrate the existence and uniqueness theorems relatively to such or such differential equation, important as they are from a mathematical point of view. But it should be more interesting to give a short scheme of the proof showing clearly the reasons why you have to make such or such an assumption. I would specify that the students anxious to make themselves acquainted with the proof must be informed of the textbook where they can find it. Therefore, to teach mathematics to the physicists we have to keep in mind that a lot of matter has to be presented during a small number of hours and for this reason we have to respect the preceding rule which, in my opinion, is the best one to dispense rigorously the most important part of useful mathematics. I know, as Professor Revuz remarked, that it is more difficult to use a theorem given without proof than one which has been demonstrated in details. But the physicist cannot do otherwise and in regard to this point I would mention the very good results obtained by Professor L. Schwartz 56
THE THINKING
OF A P H Y S I C I S T
ABOUT MATHEMATICS
when he taught the mathematical methods in physics at the University of Paris. In order to go a little bit farther I shall ask the question: 'How useful is a good course of mathematics if the lecturer in physics does not apply the mathematical results as a rigorous and powerful tool?' i mean that too many physics teachers use a very poor mathematics and incorrect arguments. How many times does the physicist work with symbols the meaning of which is not precisely specified ! At this point, a good example would be the second quantization which is employed by the physicist specialized in the study of problems connected with quantum mechanics. Most of them know very well the physical meaning and the main results allowing them to handle second quantization. But very few know the mathematical significance of what they do. in order to avoid having to specify the mathematical frame-work they use the Dirac notation which is a systematic way to write down what they do not understand clearly. Working in this way most of the physicists have a better knowledge of the methods of calculus than a clear idea of the mathematical framing. This conception of what is called applied mathematics is realiy unfruitful. As far as teaching is concerned, I think it does not require a greater effort to keep in mind, by memorizing, some results than to learn in a rational manner the definitions and the theorems to be used. To emphasize what I mean, I would like to give you one more example. Most of the solid-state physicists use the theory of finite groups in connection with the determination of band structures. Here there is a continual mixing between the abstract groups and their representations and how tedious are the proofs of the most easy theorems to demonstrate. For instance the well-known theorem according to: 'Every representation of a finite group is equivalent ~o a unitary group of operators' can be demonstrated very simply by constructing an hermitian form for which the operators are unitary. Unlike, in the physics books the p r o o f is very wearisome. One more example would be the Kronecker product of two representations. When this product is defined in a physics book it is given from the matrix elements but it is rarely emphasized what is the intrinsic meaning of this product. This failing is the outcome of the poor definitions given for the group representations. Most of the time a group representation is given as a matrix representation. At this point we could ask ourselves the question: 'Why most of the physicists use the mathematics through weakened form?' I think that the answer could be the following reason. Most of the physicists are not used to think about the mathematical meaning of what they 57
JEAN TAVERNIER
write. In fact they never identify precisely the spaces to which belong the elements they are working with and you know very well that in order to employ the mathematics as a powerful tool you have to keep in mind the nature of the spaces put forward. This is a consequence of the fact that the physicist does not apply the golden rule according to which: 'You must not write the following line before to make yourself sure that the preceding one is meaningful.' We could ask ourselves the question: 'Why doesn't the physicist appropriate this golden rule?' Or more precisely: 'Did he hear about it once during his life?' 'In the affirmative why doesn't he use it?' I think that when he studied mathematics he was used to respect this rule but at the same time during the lectures of physics he did not do so. Obviously when physics is taught, the teacher must try to emphasize the physical phenomena and then to describe them by using mathematical formalism. But I hope I do not betray my colleagues by saying that the teacher hardly succeeds in explaining qualitatively the phenomena although this part should be the most important in physics except for teaching at a high level. Accordingly the teaching of physics is too often reduced to write down a sufficient number of equations and to find out a solution whatever the way used to get it. In fact it is useless to teach rigorously mathematics so as to be useful if the teacher of physics does not apply it along the best way. In order to picture more clearly this point I would like to give you two examples based on variable transformation and the use of complex notation in electricity science. In a schematic form [ consider three spaces E, F and G; two applications f : F ~ G and g :E~F. For a given x e E it corresponds one Y=g (x) belonging to F and one z = f ( y ) belonging to G. Then we can write: z = f (y) = f og (x) = f [ g (x)] = f (x) w h e r e f = f o g is an application from E into G. This correct manner of writing is used in mathematics but too many times, for my taste, the physicist writes: f (y) = f (x). Consequently, most of the students get a confused mind about the variable transformations and the computation of the derivatives of composed functions. As far as the use of the complex notation in electricity is concerned, I 58
THE THINKING
OF A P H Y S I C I S T A B O U T M A T H E M A T I C S
shall remind you what are respectively the mathematical problem and the physicist point of view. Very often we have to solve linear differential equations with constant coefficients, the right-hand member being a sine function. You know that there is a simplification to extend this equation to the complex functions of a real variable making use of the fact that there is an isomorphism of vectorial spaces between the space of the sine function of a given frequency and the complex numbers space: a (t) = A cos (cot + ~o) ~ A e io (ce is given) the operator d / d t being represented by a multiplication by ice. And conversely: A ei~' ~ Real part of ( A e i~~ e i~ = a (t) The isomorphism between the complex numbers space and R 2 gives Fresnel's rule. in a course of physics these properties are rarely invoked and the possibility of use of the complex notation and Fresnel's rule is given as a recipe. Who has to teach these kinds of things? Perhaps, the mathematics teacher could avoid giving too many details concerning the construction of the complex number field and to show clearly the basic ideas of the preceding example which is tightly connected with the complex extension of a real vector space. To summarize what I have told I shall say: the great deal of mathematics subjects needed by the physicist is so huge that we have to keep in mind that it is impossible to teach in details the demonstration of the theorems. Only the simplest ones have to be taught in a precise form so as to show the right lines of thought from the definitions which must be emphasized when they are given, to the results to be obtained. This does not mean that the axiomatic presentation of mathematics is quite desirable but it is important to point out that such or such definition could be modified. If we want the mathematics taught by the mathematician be of some interest it is necessary that the teacher of physics applies it in the right way. I know that this is not our problem, but I think it is a very important one because most of the students are unable to recognize that there are some bonds between the mathematics which they have learned and that used by the professor of physics. In conclusion, although the physicist does not know so much mathematics as the mathematician, he must be able to argue with the same rigour as the mathematician does. It is not very important and it is impossible for him to know the demonstration of all the theorems he has to use, but it is neces59
JEAN TAVERNIER
sary that the physicist was able to perceive what is to be proved and must be educated in this frame of mind. In order to reach this end the mathematician teacher must repeat to himself very often: 'I don't teach what I know to exist, but rather what exists in practice', and the students must become conscious of the fact: 'Mathematics is to be made up and not to be learned'.
University of Paris
60
