Just here we are in position where we have only to look steadily a little in order to per ceive clearly the sharp and ultimate distinc tion between mathematics, on the one hand, and physical or other science, on the other. These are discriminated according to the kind of curiosity whence they spring. The mathe matician is curious about definite abstract relationships, about logically possible modes of order, about varieties of abstract implica tions, about completely determined or deter minable functional relationships considered solely in and of themselves, that is. without the slightest concern about the question whether or no they have external or sensuous or other sort of validity than that of being logically thinkable. It is the aggregate of logically thinkable relationships that consti tutes the mathematician's universe, an in definitely infinite universe, worlds in worlds of worlds in worlds of wonders, inconceivably richer in mathetic content than can be any outer world of sense. Immense indeed and marvelous is our own world of sense with its rolling seas and stellar fields and undulating ether. But compared, one need not say with the entire world of mathesis, but only with the hyperspaces (see HYPERSPACES) explored by the geometrician, the whole vast region of the sensuous universe is literally as a merest point of light in a shining sky.
Now this mere speck of a physical universe, in which the chemist, the physicist, the astrono suer, the biologist, the sociologist, and the rest of nature students, find their great fields, may be, as it somewhat seems to be, a realm of invariant uniformities or laws; it may be a mechanically organic aggregate, connected into an order whole by a tissue of completely definable functional relationships; and it may not. In other words, it may be that the uni verse eternally has been and is a genuine cosmos, that the external sea of things im mersing us, although it is ever changing in finitely, changes only lawfully, in accordance with a system of immutable laws, constituting an invariant (see INVARIANTS AND COVARIANTS) at once underived and indestructible and securing everlasting harmony through and through; and it may not be such. The student of nature assumes, he rightly assumes, that it is, and, moved and sustained by appropriate curiosity, he endeavors to find in the outer world what are the elements and relationships assumed to be valid there. "Natural science,* said Bernhard Riemann, "is the attempt to comprehend nature by means of exact con cepts?) The mathematician, as such, does not make that assumption and does not seek for relationships in the outer world.
Is the assumption correct? Undoubtedly it is admissible, and as a working hypothesis it is undoubtedly very useful or even indis pensable to the student of external nature; but is it true? The mathematician, as man, does not know although he greatly cares. Man, as mathematician, neither knows nor cares. The mathematician does know, how
ever, that, if the assumption be correct, every relationship that is valid in nature is, itt. ab siractu, an element in his domain, a subject for his study. He knows, too, at least he strongly suspects, that, if the assumption be not correct, his domain remains the same abso lutely. The two realms, of mathematics, of natural science, like the two attitudes, the mathematician's and that of the nature student, are fundamentally distinct and disparate. To think logically the logically thinkable—that is the mathematician's aim. To assume that nature is thus thinkable, an embodied rational logos, and to discover the thought supposed in carnate there — these are at once the principle and the hope of the student of nature.
Suppose the latter student is right, suppose the outer universe really is an embodied logos of reason, an infinitely intricate garment ever weaving and ever woven, warp and weft, of logically determinate relationships, does that imply that all of the logically thinkable is incorporated in it? It seems not. A cosmos, a harmoniously ordered universe, one that through and through is self-compatible, can hardly be the whole of reason materialized and objectified. There appears to be many a ration al logos. At any rate the mathematician has delight in the construction and contemplation of divers systems that are inconsistent with one another, though each is composed of consistent relationships. He constructs in thought, as witness the geometry of hyperspaces, ordered worlds, worlds that are possible and logically actual, and he is content not to know if any of them be otherwise actual or actualized. There is, for example, a Euclidean geometry and there are infinitely many kinds of non-Euclidean (see Nox-Eticunenti C-nms-raY). These theories regarded as "applied° mathematics, regarded, that is, as true descriptions of some one actual space, are incompatible. In our universe, to be specific, if it be, as Plato thought and nature science takes for granted, a geometrized or geometrizable affair, then one of these geometries may be objectively valid in it. But in the vaster world of thought, in the world of pure mathesis, all of them are valid; there they coexist, and interlace among them selves and others as differing strains of a hyper cosmic harmony.
A geometry, indeed any mathematical theory, consists of a definite system of deter minate compatible principles or assumptions or hypotheses or postulates (commonly called axioms) together with their implications, their logically deducible consequences. Accordingly, natural science, the term being broadly em ployed to signify knowledge that is ultimately dependent upon "observation and experi ment,* cannot be or become strictly mathe matical. It aspires to the character, and ap proximates and imitates the form, of mathe matics, but it can never really attain either.