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C H A P T E R 7 THE ESSENCE OF MATHEMATICS* 1. RELATION TO SCIENCE Mathematics is frequently encountered in association and interaction with Astronomy, Physics, and other branches of Natural Science, and it also has deep-rooted affinities to what are called Humanities nowadays. Actually, it is a realm of knowledge entirely by itself, and one of consider­ able scope too; the word "mathematics" stems from a root which means Learnable Knowledge as such. Mathematical knowledge is commonly deemed to have a high degree of validity, binding on Homo sapiens irrespective of cultural conditioning and predilection, although it can be argued that in the past cultural settings have affected its develop­ ment noticeably. Even as far as the scientist is concerned, mathematics is not a branch of natural science itself. It does not deal with phenomena and objects of the external world and their relations to each other, but, strictly speaking, only with objects and relations of its own imagery. Mathematical figures in two- or three-dimensional geometry are largely idealizations of objects occurring in the physical world, but figures in «-dimensional space for general η no longer are. Integer numbers 1, 2, 3, . . . and even real numbers in general can be claimed to be abstractions from quantities occurring in the physical world, but the "imaginary" number i = V— 1 has received its name, still current, from the very fact that it no longer so is, even though the use of complex numbers a + bi is indispensable to science nowadays. It a laboratory experiment idealizes a physical * Reprinted, with adjustments, from the McGraw-Hill Encyclo­ pedia of Science and Technology, Copyright © 1960 by McGrawHill Inc. Originally entitled "Mathematics." [255] T H E E S S E N C E OF M A T H E M A T I C S system in order to eliminate secondary features not essential to the study at hand, its ultimate objective is an understand­ ing of the un-idealized physical system nevertheless. But even if it were true that the five regular solids as investi­ gated in the 13th book of Euclid's Elements (cube, tetra­ hedron, octahedron, icosahedron, dodecahedron) had been found due to the occurrence of approximating crystals in nature, nevertheless, once found, the idealized geometric figures become primary objects of mathematics which are definitive as such. Mathematics is not subordinate to natural science by being a handmaiden of it, and one can practice compe­ tently meaningful mathematics without being concerned with science at all. Especially, philosophical attempts to reduce all origin of mathematics to utilitarian motives are wholly unconvincing. But it is fair to say that mathematics is the language of science in a deep sense. Mathematics is an indispensable medium by which and within which science expresses, formulates, continues, and communi­ cates itself. And just as language of true literacy not only specifies and expresses thoughts and processes of thinking but also creates them in turn, so does mathematics not only specify, clarify and make rigorously workable concepts and laws of science which perhaps, partially at least, could be put forward without it; but at certain crucial instances it is an indispensable constituent of their creation and emer­ gence as well. In Newton's formula for the motion of a particle on a straight line the mass m and the force F are non-mathematical objects perhaps. But the instantaneous velocity ν = dx/dt and the instantaneous acceleration a = dv/dt = drx/df are wholly mathematical, and without a mathematical theory of the infinitesimal calculus not conceivable. Newton the physicist [256] THE ESSENCE OF MATHEMATICS was driven to creating his version of the calculus because of this. Also, Newton had to have not only the process of differentiation but also the concept of a mathematical func­ tion, because only a function can be differentiated. He re­ quired not only the path function χ = x(t) but, for the second derivative, he had to envisage the velocity ν = dx/dt itself again as a function depending on t, even though in the definition of ν this dependence had been reduced to "instantaneousness ." This concept of function was given to Newton by the then new theory of analytic geometry of Descartes, and it is a fact that after Archimedes' work on the motion of a lever, Theoretical Mechanics stood virtually still for almost 2,000 years until the twin mathematical concepts of function and derivative were ready to emerge. 2. CREATIVE FORMULAS A formula is a string of mathematical symbols subject only to certain...


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