The word element is a term which frequently occurs in philosophy. It s — Abstraction (mathematics)

"The chemist, in describing some mineral, may present all its attributes, color mass, density, volume, molecular construction, and its properties exhibited in relation to heat, light, sound, and electric waves, and the resulting conception will not be mathematical. On the other hand, in describing its form as cubical, he relies upon the purely mathematical abstraction involved in the definition of a cube, which includes that of the mathematical plane, which involves the geometric line, which, in turn, resolves into an assemblage of points. Not one of these conceptions finds its realization in physical phenomena, but at best an approximation. The types conceived by the mind remain, however, definite and unchangeable. The idea of the ultimate element, the "point," best illustrates the nature of these abstractions, and involves the first difficulty that lies in the way of the understanding. The point is defined as having but one positive attribute, and that is, position in space. ...The mathematical point cannot be realized by this process, for the resulting principle is this: all material things are indefinitely divisible, and, no matter how small, possess magnitude and occupy space. The negative of this thought is the point, which does not have magnitude, but position only. It is a negative thought concept, a mathematical abstraction..."

"In solving a problem, be it one in the calculus, in algebra, or in the second year of arithmetic, we begin by substituting for the actual things certain abstractions represented by symbols; we think in terms of these abstractions, aided by symbols, and finally from our result we pass back to the concrete and say that we have solved the problem. It is all a matter of "one to one correspondence," it being easier for us to work with the abstract numbers and their corresponding figures than to work with the actual objects. Fundamentally the process is something like this: 1. By abstraction we pass to numbers. 2. Thence we pass to symbols, and we make an equation, either openly, as in algebra, or concealed, as in many forms of arithmetic. This equation we solve, the result being a symbol. 3. We find the number corresponding to this symbol, and say that the problem is solved. All this does not mean that primary number is to be merely a matter of symbols. It means that in mathematics we find it more convenient to work purely with symbols, translating back to the corresponding concrete form as may be desired. And so those teachers who fear lest the child shall drift into thinking in symbols instead of in number, are really fearing that the child shall drift into mathematics."

"Change of state involves what is meant by the word "time," which, like space, is a necessary condition for thought, or the formation of concepts. These occur in succession, as do all phenomena involving a change, and whenever motion or change is involved, time is required, and the universal knowledge of this fact is an elementary abstraction. Unity apart from any concrete thing... is the primary abstraction in arithmetic, and number and the theory of numbers grow from this concept to one of great importance. Since time and space are necessary to the realization of sensible objects and the phenomena of motion, the attributes of time and space adhere of necessity to the theory of these subjects, and hence the principles demonstrated in the abstract concerning time and space relations, are applicable to such phenomena. Herein lies the secret of the universal application of pure mathematical deductions to the many sciences that sum up human knowledge of the universe."

"It was only in the nineteenth century that the winds of change started to blow. First, the introduction of abstract geometric spaces and of the notion of infinity (in both geometry and the theory of sets) had blurred the meaning of "quantity" and of "measurement" beyond recoginition. Second, the rapidly multiplying of mathematical abstractions helped to distance mathematics even further from physical reality, while breathing life and "existence" into the abstractions themselves."

"[W]hile we put number into objects, on the other hand we derive our idea of number only from the presence of the world external to the mind. We see a group of people, and we begin by making an abstraction ("people"), and we say, "Here are ten people"—thus calling them all by the one abstract name, even though the individuals be very different. "A careful observation shows us, however, that there are no objects exactly alike; but by a mental operation of which we are quite unconscious, although it holds within itself the entire secret of mathematical abstraction, we take in objects which seem to be alike, rejecting for the time being their differences. Here is to be found the source of calculation." So the idea of number is generated in the mind by the sense perception of a group of things supposed to be alike."

"Abstraction is the immediate ulterior result of analysis. We may speak of the analysis of the mathematical whole, and so of the abstraction of any of its parts. Wherever analysis may take place, abstraction likewise, is possible. ...The reason on account of which the analysis and abstraction of the mind are directed to the parts of the metaphysical whole as such, lies in the fact that the mental division of an object into its mathematical, or separable, parts, is not sufficient even for the ends of ordinary thought. We cannot, from such a division, adequately understand and express the nature of things. This purpose requires that we should consider and designate inseparable parts, such as powers, shapes, magnitudes, and attributes generally."