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Marx [#!Mar94!#] and Trotsky were both interested in mathematics but were unable to pursue it to any degree because of other, more pressing issues they faced. It was Engels who developed the materialist view of science furthest. Here we can find the beginnings of a materialist understanding of mathematics [#!Eng!#,#!Eng76!#].

To start with it must be said that many of the mathematical problems addressed by Engels in Anti-Dühring have been superseded by later developments in the subject. For example Engels' discussion of the infinite is only marginally less confused than that of Dühring and it was not until the work of Cantor that a rigorous treatment of infinite arithmetic was given. For example, Engels argues ``... beginning and end necessarily belong together, like the North Pole and the South Pole, and that if the end is left out, the beginning just becomes the end -- the one end which the series has; and vice versa. ... all mathematical series positive or negative, must start with 1, or they cannot be used for calculation. " ([#!Eng76!#] page 63) However the negative integers form an infinite series with an end but no beginning, the positive integers have a beginning but no end, the integers have neither and in fact there is a vast collection of different order types of infinite sequences.

Infinitesimals, similarly, are of great interest to Engels, but cause much confusion. Consider the following quote ([#!Eng76!#] p 175) ``Therefore, , the ratio between the differentials of x and y, is equal to , but taken as the expression of ". This doesn't make much sense mathematically and the problem of infinitesimals had been fully and rigorously reduced to the treatment of finite numbers, some years earlier, by Weierstrass. Indeed Engels argues that until the development of the theory of infinitesimals in calculus, mathematics and logic were the only areas which could claim to deal with absolute and final truth. (Though we have already seen that even this part of science has an evolution, is conditional and needs to be considered dialectically.) But once this step beyond the finite was taken mathematics, too, fell from grace. However, this is not a helpful way of analysing the subject and is not in accordance with the state of mathematics today. These criticisms do not detract from the central points that Engels makes, which are sketched, in outline, as follows.