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First on the objectivity of mathematics. Engels attacks Dühring's notion that mathematics is a free creation of the imagination independent of the world and our particular experience.

The concepts of number and form have been derived from no source other than the world of reality. The ten fingers on which men learnt to count, that is, to carry out the first arithmetic operation, are anything but a free creation of the mind. Counting requires not only objects that can be counted, but also the ability to abstract from all properties of the objects being considered except their number -- and this ability is the product of a long historical development based on experience. Like the concept of number, so the concept of form is derived exclusively from the external world and does not arise in the mind as a product of pure thought. There must have been things which had shape and whose shapes were compared before anyone could arrive at the concept of form ([#!Eng76!#] page 47).

It should be added that mathematics does not always derive directly from nature, but can arise out of science and even from mathematics itself. Thus elementary arithmetic is abstracted from the very concrete task of counting and comparing physical objects (like fingers), group theory is abstracted from algebraic equations and geometric symmetries. Tensor theory owed its origin to the attempt by Riemann, in the mid-19th century, at solving the problem of unifying gravity and electromagnetism, a premature attempt which, nevertheless, led ultimately to Einstein's more successful work in relativity. In this way mathematics can become very abstract indeed. Nevertheless mathematics, like all other ideas, ultimately arises from experience.