Hegel on direct ratio
48. Again, were Hegel’s mathematics correct ? Was he right about the mathematics of his own time, and, if so, would he be right about the mathematics of the present day? To answer these questions requires a knowledge of mathematics which I am very far from possessing.
Mr Bertrand Russell — one of the few philosophers who are also mathematicians — says: “ In Hegel’s day, the procedure of mathematicians was full of errors, which Hegel did not condemn as errors but welcomed as antinomies; the mathematicians, more patient than the philosophers, have removed the errors by careful detailed work on every doubtful point.
A criticism of mathematics based on Hegel can, therefore, no longer be regarded as applicable to the existing state of the subject.”
But the value of Hegel’s treatment of Quantity would only be slightly affected by the fact that his criticisms of mathematics were based on ignorance or by the fact that they had been invalidated by the progress of that science.
The main object of the dialectic, after all, is to reach the Absolute Idea, and so to demonstrate what is the true nature of reality. Thus the principal function of the lower categories is to lead on to the Absolute Idea. And for this it is only requisite that each of them should logically follow from the one before it, and lead on to the one after it.
Now the question whether Hegel’s various categories of Quantity do perform this function is not affected by any mathematical mistakes which he may have made, nor can it be settled in the negative by any mathematical criticisms.
The only question is whether Hegel was justified in starting the dialectic with the category of Pure Being, and whether the validity of the Hegelian categories of Quantity can be shown to be involved in the validity of the category of Pure Being. And this is a question for metaphysics and not for mathematics.
John McTaggart A Commentary on Hegel’s Logic 1910