by other signs indicating—under a most original guise—a gradual transition of the mathematical sciences to a cosmic orientation. In the seventies of the last century, G. Cantor formulated the basic propositions of the theory of sets, whose main concern is the investigation of infinite sets. The contradictions of mathematical infinity, and certain of its paradoxical properties (for instance, in the arithmetic of transfinite numbers), are a reflection of the actual contradictions and properties involved in the real infinity of the universe. Perhaps the most noteworthy thing about the theory of sets is the fact that it has not only evolved quite rapidly but also impinged upon literally all the main branches of mathematics. Set-theoretical axiomatics gained increased importance in arithmetic, geometry, the theory of probability, and mathematical logic. Many important new mathematical disciplines, such as the theory of functions of a real variable, general topology, to mention but a few, could not be made possible without the theory of sets. Whereas at first mathematics proceeded from the particular and finite to the infinite, now the finite itself is interpreted from the standpoint of the infinite, and thus the general validity of mathematics becomes richer in implications than was the case with the older mathematics.
However, mathematical science was not the only one to exhibit a cosmic slant. At the turn of the 20th century, many branches of natural science started, one after the other, to contemplate their subject of study in its cosmic aspect. The cosmic outlook broke in full force upon physics, when Einstein formulated the special, and later the general, theory of relativity, which nowadays constitutes the theoretical foundation of the principal domains of the new, nonclassical physics. This theory, as is known, is concerned with super-high, truly cosmic velocities, and in general, with many pro cesses which occur mainly (or which are most detectable) not under specifically terrestrial conditions but in the wide expanses of the universe (which include our planet as well). But matters do not end with Einstein's discoveries. Across the span of half a century, there has been the forma tion of atomic and nuclear physics, cosmic-ray physics, plasma physics, vacuum physics, cryophysics, etc. Each of these disciplines investigates either specifically cosmic processes, or phenomena produced by cosmic factors, or else some terrestrial occurrences considered as a special instance of cosmic phenomena. In view of the multiplication of such specialized departments in physics and of the preponderance they are gaining, it becomes clear that physics as a whole is being transformed into a cosmic science.
The same may be observed, though to a lesser extent, in the other branches of natural science of the first half of the century. The cosmic element in one form or another is making its way into chemistry, biology, geology, and geography. Chemistry, for example, now includes radiation chemistry and cryochemistry, and research has been initiated in the chemistry of super-high temperatures (in particular, where this is directly related to the development of rocket fuels and engines) and on the effect of cosmic factors on chemical reactions and the chemical properties of sub stances. Geology and geography (and even meteorology)—these most
terrestrial of all branches of natural science—tend to consider their problems more and more from an extraterrestrial standpoint, or as related with cosmic aspects, and to treat the Earth as one of the planets, i.e. , as a cosmic body. These changes are even more pronounced in geophysics, which was born, to begin with, under the sign of the cosmos.
On the basis of the foregoing we can deduce a second trend toward the cosmization of natural science. This trend involves the expansion in the last century (to say nothing of the present one) of the object of study of many of the old classical sciences, and the emergence from the framework of purely terrestrial phenomena and processes, with an interest in the cosmic aspects of research.
But this is not all. A notable advance occurred in the development of natural science of the last decades, brought about by the creation of new, primarily cosmic, scientific disciplines, chiefly concerned with the study of cosmic objects. After the inception of astrophysics at the end of the 19th century, the beginning of our century was marked by the appearance of theoretical astronautics. This event dates from 1903, when K.E. Tsiolkovskii's celebrated work "Issledovanie mirovykh prostranstv reak tivnymi priborami" (The Exploration of Outer Space by Means of Reaction Devices) was published. Later to develop were the elements of astrobiology and in particular of astrobotany, and the fundamentals of space chemistry. Toward the 50's of this century, the science of aeronomy was founded, as well as astrogeography and astrogeology. Aviation medicine was also developed, the direct forerunner of space medicine.
Admittedly, it is rather arbitrary to call most of these sciences in their initial form cosmic, or "purely" space sciences. Astrogeography and astrogeology, for instance, are as yet ordinary geography and geology darried over into space and constitute to a large extent a generalization from the appropriate terrestrial data to cosmic data (the particular case of astro geology basically takes into account cosmic factors in the geology of the Earth). But such will not always be the case. Sooner or later, quantity will change into quality. The direct geographical and geological study of other planets and planetlike bodies will not only uncover new facts and lead to a deeper understanding of familiar laws but will also call for a reappraisal of our knowledge of ordinary terrestrial geography and geology. We shall have occasion to come back to this point. Obviously, at present astrogeography and astrogeology may be classified as cosmic sciences in potential only. On the other hand, astrophysics, astronautics, or astro chemistry started out as primarily space sciences (the extension of any particular data to space is in the present case of secondary importance); they have a "direct" approach to their cosmic objectives.