Mathematical axioms, Engels says, can be reduced to two, borrowed from logic. First, the whole is greater than the part, and second that, if two magnitudes are equal to a third, they are equal to each other, i.e. if A = C, and B = C, A is also equal to B.
“The remaining axioms relating to equality and inequality are merely logical extensions of this conclusion.
These meagre principles could not cut much ice, either in mathematics or anywhere else. In order to get any further, we are obliged to import real relations, relations and spatial forms which are taken from real bodies. The ideas of lines, planes, angles, polygons, cubes, spheres, etc., are all taken from reality, and it requires a pretty good portion of naïve ideology to believe the mathematicians - that the first line came into existence through the movement of a point in space, the first plane through the movement of a line, the first solid through the movement of a plane, and so on. Even language rebels against such a conception. A mathematical figure of three dimensions is called a solid body, corpus solidum, hence, even in Latin, a tangible object; it therefore has a name derived from sturdy reality and not at all from the free imagination of the mind.” (p 49-50)
Duhring notes that, although mathematical elements such as number, magnitude, time, space and motion are empirically observed, in relation to things in the real world, for example, I might count 10 sheep, or 10 Rolls Royces, the concept of number, i.e. of 10, exists on its own, abstracted from whether it is sheep or Rolls Royces. But, as Engels says, this is true, more or less of every abstraction. Marx makes the point in The Poverty of Philosophy.
“In the world schematism pure mathematics arose out of pure thought — in the philosophy of nature it is something completely empirical, taken from the external world and then divorced from it. Which are we to believe?” (p 50)
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