VIII. The majority of the equations whereby we interpret the physical phenomena have certain properties of continuity. The solutions vary in a continuous manner, at least during a certain interval greater or less in length, when the given quantities vary in a continuous manner. Besides, this property is not absolutely general, and it might happen that the theories of the quanta of emission or absorption may lead to attaching more importance than has been done heretofore to exceptional cases; but to-day I do not wish to begin this discussion. I rest content with the general property, verified in a very great number of cases. When we seek to interpret this property in the theory of the potential and of the monogenic functions, we should expect, if for simplification we confine ourselves to the real functions of a single variable, to find a sort of continuous passage between such of these functions as are analytical in the Weierstrassian sense and those which are entirely discontinuous. But it is this which does not occur unless we consider nonanalytical monogenic functions. From the moment when a function ceases to be analytical, it no longer possesses any of the essential properties of analytical functions; the discontinuity is sudden. The new monogenic functions permit one to define functions of real variables which might be called quasianalytical and which constitute in some way a zone of transition between the classical analytical functions and the functions which are not determined by the knowledge of their derivatives in a point. This transition zone deserves to be studied; it is often the study of hybrid forms which best teaches in reference to certain properties of clearly determined species. We see that the points of contact between molecular physics and mathematics are numerous. I have been able only to rapidly point out the principal ones among them. I am not competent to ask whether the physicists will be able to derive an immediate profit from these analogies; but I am convinced that the mathematicians can only gain by going into them thoroughly. It is always by a contact with nature that mathematical analysis is revived. It is only because of this permanent contact that it has been able to escape the danger of becoming a pure symbolism, revolving in a circle about itself; it is owing to molecular physics that the speculations on discontinuity are to take their complete signification and be developed in a way really fruitful. And, for lack of exact applications impossible to foresee, it is sufficiently probable that the mental habits created by these studies will not be without advantage to those who shall desire to undertake the task, which will soon be imposed, of creating an analysis adapted to theoretical researches in the physics of discontinuity. MODERN MATHEMATICAL RESEARCH. By Prof. G. A. MILLER, Mathematics has a large household and there are always rumors of prospective additions despite her age and her supposed austerity. Without aiming to give a complete list of the names of the members of this household we may recall here a few of the most prominent ones. Among those which antedate the beginning of the Christian era are surveying, spherical astronomy, general mechanics, and mathematical optics. Among the most thriving younger members are celestial mechanics, thermodynamics, mathematical electricity, and molecular physics. Usually a large household serves as one of the strongest incentives to activity, and mathematics has always responded heartily to this incentive. As the most efficient continued service calls for unusual force and ingenuity, mathematics has had to provide for her own development and proper nourishment in addition to providing as liberally as possible for her household. This double object must be kept prominently before our eyes if we would comprehend the present mathematical activities and tendencies. There is another important incentive to mathematical activity which should be mentioned in this connection. Mathematics has been very hospitable to a large number of other sciences and as a consequence some of these sciences have become such frequent visitors that it is often difficult to distinguish them from the regular members of the household. Among these visitors are economics, dynamical geology, dynamical meteorology, and the statistical parts of various biological sciences. Visitors usually expect the best that can be provided for them, and the efforts to please them frequently lead to a more careful study of available resources than those which are put forth in providing for the regular household. We have thus far spoken only of what might be called the materialistic incentives for mathematical development. While these have always been very significant, it is doubtful whether they have been the most powerful. Symmetry, harmony, and elegance of form have 1 Read before the Illinois Chapter of the Society of the Sigma Xi, April, 1912, and reprinted by permission from Science, June 7, 1912. always appealed powerfully to dame mathematics; and a keen curiosity, fanned into an intense flame by little bits of apparently incoherent information, has inspired some of the most arduous and prolonged researches. Incentives of this kind have led to the mathematics of the invisible, relating to refinements which are essentially foreign to counting and measuring. The first important refinement of this type relates to the concept of the irrational, introduced by the ancient Greeks. As an instance of a comparatively recent development along this line we may mention the work based upon Dedekind's definition of an infinite aggregate as one in which a part is similar or equivalent to the whole.1 Mathematics is commonly divided into two parts called pure and applied, respectively. It should be observed that there are various degrees of purity, and it is very difficult to say where mathematics becomes sufficiently impure to be called applied. The engineer or the physicist may reduce his problem to a differential equation, the student of differential equations may reduce his troubles to a question of function theory or geometry, and the workers in the latter fields find that many of their difficulties reduce themselves to questions in number theory or in higher algebra. Just as the student of applied mathematics can not have too thorough a training in the pure mathematics upon which the applications are based so the student of some parts of the so-called pure mathematics can not get too thorough a training in the basic subjects of this field. 2 As mathematics is such an old science and as there is such a close relation between various fields, it might be supposed that fields of research would lie in remote and almost inaccessible parts of this subject. It must be confessed that this view is not without some foundation, but these are days of rapid transportation and the student starts early on his mathematical journey. The question as regards the extent of explored country which should be studied before entering unexplored regions is a very perplexing one. A lifetime would not suffice to become acquainted with all the known fields, and there are those who are so much attracted by the explored regions that they do not find time or courage to enter into the unknown. In 1840 C. G. J. Jacobi used an illustration, in a letter to his brother, which may serve to emphasize an important point. He states that at various times he had tried to persuade a young man to begin research in mathematics, but this young man always excused himself on the ground that he did not yet know enough. In answer to this statement Jacobi asked this man the following question: "Suppose your family would wish you to marry would you then also 1"Encyclopédie des sciences mathématiques," vol. i., pt. 1, 1904, p. 2. 2"Der Urquell aller Mathematik sind die ganzen Zahlen," Minkowski, Diophantische Approximation, 1907, preface. "Briefwechsel zwischen C. G. J. Jacobi und M. H. Jacobi," 1907, p. 64. reply that you did not see how you could marry now, as you had not yet become acquainted with all the young ladies?" In connection with this remark by Jacobi we may recall a remark by another prominent German mathematician who also compared the choice of a subject of research with marriage. In the "Festschrift zur Feier des 100 Geburstages Eduard Kummer," 1910, page 17, Prof. Hensel states that Kummer declined, as a matter of principle, to assign to students a subject for a doctor's thesis, saying that this would seem as if a young man would ask him to recommend a pretty young lady whom this young man should marry. While it may not be profitable to follow these analogies into details, it should be stated that the extent to which a subject has been developed does not necessarily affect adversely its desirability as a field of research. The greater the extent of the development the more frontier regions will become exposed. The main question is whether the new regions which lie just beyond the frontier are fertile or barren. This question is much more important than the one which relates to the distance that must be traveled to reach these new fields. Moreover, it should be remembered that mathematics is n-dimensional, n being an arbitrary positive integer, and hence she is not limited in her progress to the directions suggested by our experiences. If we agree with Minkowski that the integers are the source of all mathematics, we should remember that the numbers which have gained a place among the integers of the mathematician have increased wonderfully during recent times. According to the views of the people who preceded Gauss, and according to the elementary mathematics of the present day, the integers may be represented by points situated on a straight line and separated by definite fixed distance. On the other hand, the modern mathematician does not only fill up the straight line with algebraic integers, placing them so closely together that between any two of them there is another, but he fills up the whole plane equally closely with these integers. If our knowledge of mathematics had increased during the last two centuries as greatly as the number of integers of the mathematician we should be much beyond our present stage. The astronomers may be led to the conclusion that the universe is probably finite, from the study of the number of stars revealed by telescopes of various powers, but the mathematician finds nothing which seems to contradict the view that his sphere of action is infinite. From what precedes one would expect that the number of fields of mathematical research appears unlimited, and this may serve to 1 This view was expressed earlier by Kronecker, who was the main founder of the school of mathematicians who aim to make the concept of the positive integers the only foundation of mathematics. Cf. Klein und Schimmack, "Der mathematische Unterricht an den höheren Schulen," 1907, p. 175. furnish a partial explanation of the fact that it seems impossible to give a complete definition of the term mathematics. If the above view is correct, we have no reason to expect that a complete definition of this term will ever be possible, although it seems possible that a satisfactory definition of the developed parts may be forthcoming.1 Among the various fields of research those which surround a standing problem are perhaps most suitable for a popular exposition, but it should not be inferred that these are necessarily the most important points of attack for the young investigator. On the contrary, one of the chief differences between the great mathematician and the poor one is that the former can direct his students into fields which are likely to become well known in the near future, while the latter can only direct them to the well-known standing problems of the past, whose approaches have been tramped down solid by the feet of the mediocre, who are often even too stupid to realize their limitations. The best students can work their way through this hard crust, but the paddle of the weaker ones will only serve to increase its thickness if it happens to make any impression whatever. It would not be difficult to furnish a long list of standing mathematical problems of more or less historic interest. Probably all would agree that the most popular one at the present time is Fermat's greater theorem. In fact, this theorem has become so popular that it takes courage to mention it before a strictly mathematical audience, but it does not appear to be out of place before a more general audience like this. The ancient Egyptians knew that 32+42-52, and the Hindus knew several other such triplets of integers at least as early as the fourth century before the Christian era. These triplets constitute positive integral solutions of the equation x2 + y2 = 22. Pythagoras gave a general rule by means of which one can find any desired number of such solutions, and hence these triplets are often called Pythagorean numbers. Another such rule was given by Plato, while Euclid and Diophantus generalized and extended these rules. Fermat, a noted French mathematician of the seventeenth century, wrote on the margin of a page of his copy of Diophantus the theorem that it is impossible to find any positive integral solution of the equation xn + yn = zn (n>2). 1 Bocher discussed some of the proposed definitions in the Bulletin of the American Mathematical Society, vol. 2 (1904), p. 115. * Lietzmann, "Der Pythagoreische Lehrsatz," 1912, p. 52. |