Abstraction (mathematics)

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Numbers Abstraction
after Jasper Johns      

Mathematical abstraction is the process of extracting the underlying essence of a mathematical concept.


  • [M]ental Abstraction... is not [only the] Property of Mathematics, but is common to all Sciences. For every Science considers the Nature of its own Subject abstracted from all others; forms its own general Precepts and Theorems; and separates its own Properties from the Properties of others... For example, Physics considers the constitutive Principles, Matter, Form, &c. of Body in general; then the Affections common to all Bodies, viz. Quantity, Place, Motion, Rest, and the like; from whence it descends to the next lower Species, investigating their particular Natures and Properties; but meddles not with particular Bodies or Individuals, as well because they are innumerable and distinguished from one another by innumerable Differences... The same way Geometry proposes Magnitude for the Subject of its Enquiry, not the peculiar Magnitude of this or that Body, but the Magnitude taken universally; together with its general Affections, viz. Divisibility, Congruence, Proportionality, a Capacity of different Situation and Position, Mobility &c. declaring these to be inherent to it, and after what manner they are so: Next it defines the various Species of Magnitude, (viz. a Line, Superficies, and a Body or Solid) and particularly draws forth and demonstrates their distinct Properties; continually dividing these Species into others more contract, and searching and proving their Affections by universal Propositions, Rules and Theorems lawfully demonstrated, till it has wholly exhausted its Subject, and descended to the very lowest Species. And these Theorems however more or less general as to their Matter, may be truly and properly accommodated to Subjects particular to themselves. True Mathematical Abstraction then, is such as agrees with all other Sciences and Disciplines, nothing else being meant (whatsoever some do strangely say of it) than an Abstraction from particular Subjects, or a distinct Consideration of certain things more universal, others less universal being ommitted and as it were neglected.
  • They who are acquainted with the present state of the theory of Symbolical Algebra, are aware that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. ...
    It has happened in every known form of analysis, that the elements to be determined have been conceived as measurable by comparison with some fixed standard. The predominant idea has been that of magnitude, or more strictly of numerical ratio. The expression of magnitude, or of operations upon magnitude, has been the express object for which the symbols of Analysis have been invented, and for which their laws have been investigated. Thus the abstractions of the modern Analysis, not less than the ostensive diagrams of the ancient Geometry, have encouraged the notion that Mathematics are essentially, as well as actually, the Science of Magnitude. ...
    [T]his conclusion is by no means necessary. If every existing interpretation is shewn to involve the idea of magnitude, it is only by induction that we can assert that no other interpretation is possible. ...The history of pure Analysis is, it may be said, too recent to permit us to set limits to the extent of its applications. ...That to the existing forms of Analysis a quantitative interpretation is assigned, is the result of the circumstances by which those forms were determined, and is not to be construed into a universal condition of Analysis. It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical Analysis, regardless that in its object and in its instruments it must at present stand alone.
  • 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...
    • J. Brace Chittenden, "Mathematics", The Encyclopedia Americana: A General Dictionary of the Arts and Sciences (1903)
  • 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.
    • J. Brace Chittenden, "Mathematics", The Encyclopedia Americana: A General Dictionary of the Arts and Sciences (1903)
  • Geometry can in no way be viewed... as a branch of mathematics, instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to those of geometry.
    • Hermann Grassmann, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844) [The Theory of Linear Extension, a New Branch of Mathematics] as quoted by Mario Livio, Is God a Mathematician? (2009)
  • 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.
    • Edward John Hamilton, The Human Mind: A Treatise in Mental Philosophy (1883) p. 292.
  • The word element is a term which frequently occurs in philosophy. It signifies any of those parts of an object into which it is or may be separated by analysis; and which, therefore, may be separately considered by abstraction. ...A notion of a thing may be formed by the composition of mathematical parts, and such a composition in its relation to parts, and such a composition in relation to the object might be spoken of as mathematical conception.
    • Edward John Hamilton, The Human Mind: A Treatise in Mental Philosophy (1883) p. 293.
  • 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.
  • Berkeley Bishop of Cloyne was a man of first-rate talents, distinguished as a metaphysician, a philosopher, and a divine. His geometrical knowledge, however, which, for an attack on the method of fluxions, was more essential than all his other accomplishments, seems to have been little more than elementary. The motive which induced him to enter on discussions so remotely connected with his usual pursuits has been variously represented; but whatever it was, it gave rise to the Analyst, in which the author professes to demonstrate, that the new analysis is inaccurate in its principles, and that, if it ever lead to true conclusions, it is from an accidental compensation of errors that cannot be supposed always to take place. The argument is ingeniously and plausibly conducted, and the author sometimes attempts ridicule with better success than could be expected from the subject; thus when he calls ultimate ratios the ghosts of departed quantities, it is not easy to conceive a witty saying more happily fastened on a mere mathematical abstraction.
    • John Playfair, The Works of John Playfair Esq. Vol. 2, Dissertation, exhibiting a general view of the progress of mathematical and physical science since the revival of letters in Europe (1822) pp. 321-322.
  • Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say.
  • [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.
  • 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.

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