Duhring believed that he could conjure out of his head the whole of pure mathematics, with no prior reference to the real world, simply on the basis of abstraction,
“In pure mathematics, in his view, the mind deals “with its own free creations and imaginations” ; the concepts of number and form are “its adequate object which it itself creates”, hence mathematics has “a validity which is independent of particular experience and of the real content of the world” (p 46-7)
But, the very concept of abstraction, requires a starting point, something that is abstracted from, as Marx also set out in The Poverty of Philosophy.
What is more, there are certain abstract concepts, necessary in mathematics, in order to perform mathematical operations, whilst there is no real world existence of these concepts. For example, the abstract concept of zero took a long time for mathematicians to develop, and yet, without it, modern maths and much of the science that depends on it, would be impossible. Why did it take time to develop the concept of zero? Because, in the material world, there is no equivalent of it. If I am counting beans or sheep, I start with 1 not zero.
The experiments with some animals, showing that they might be able to grasp the concept of zero, does not change that. For example, chimpanzees can be trained to carry out additions using digits, and parrots are often used to perform similar functions, which includes a zero-like concept. But, in all these cases, it is humans that have developed the mathematical concepts as abstractions, and which they then train the animals to perform. In none of the cases did the animals themselves develop the concepts as abstractions or as symbolic language.
Do animals have intelligence, in the sense that they can take experiences of the real world, and, on the basis of it, usually by some conditioned reflex, adopt a purposive behaviour? Absolutely. Birds, learn that they can drop snails from a height, so as to crack their shells and so on, but this amounts again to learned behaviour, from direct experience of the real world, not birds sitting and pondering the question of how to crack shells, let alone the concept of gravity and its nature.
The same is true with mathematical forms. Without the concept of a point, geometry is impossible, and, indeed, many of the other mathematical concepts become impossible too. But, a point is a zero in space. It is something, which has no dimension. Yet, as Trotsky set out, arguing against Burnham, there is, in the real world, no such thing as a point. Everything that exists, exists in space-time, it has dimension and duration.
“How should we really conceive the word “moment”? If it is an infinitesimal interval of time, then a pound of sugar is subjected during the course of that “moment” to inevitable changes. Or is the “moment” a purely mathematical abstraction, that is, a zero of time? But everything exists in time; and existence itself is an uninterrupted process of transformation; time is consequently a fundamental element of existence. Thus the axiom “A” is equal to “A” signifies that a thing is equal to itself if it does not change, that is, if it does not exist.”
Without the abstract concept of a point, other mathematical forms and concepts become impossible, too. For example, a line is a point that has extension in one dimension. But, again, that is something that does not exist in the real world. No matter how thin a line may be, it always has some thickness. The fact that these mathematical concepts, however, exist, purely as abstractions, does not mean that, in order for the mind to develop them, the starting point is not things that actually do exist in the real world.
“There must have been things which had shape and whose shapes were compared before anyone could arrive at the concept of form. Pure mathematics deals with the spatial forms and quantitative relations of the real world — that is, with material which is very real indeed. The fact that this material appears in an extremely abstract form can only superficially conceal its origin in the external world. But in order to make it possible to investigate these forms and relations in their pure state, it is necessary to separate them entirely from their content, to put the content aside as irrelevant; hence we get points without dimensions, lines without breadth and thickness, a's and b's and x's and y's, constants and variables; and only at the very end do we reach the mind's own free creations and imaginations, that is to say, imaginary magnitudes.” (p 47-8)
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