The philosophy of mathematics · Auguste Comte
IMAGINARY QUANTITIES.
The difficulties connected with several peculiar symbols to which algebraic calculations sometimes lead, and especially to the expressions called _imaginary_, have been, I think, much exaggerated through purely metaphysical considerations, which have been forced upon them, in the place of regarding these abnormal results in their true point of view as simple analytical facts. Viewing them thus, we readily see that, since the spirit of mathematical analysis consists in considering magnitudes in reference to their relations only, and without any regard to their determinate value, analysts are obliged to admit indifferently every kind of expression which can be engendered by algebraic combinations. The interdiction of even one expression because of its apparent singularity would destroy the generality of their conceptions. The common embarrassment on this subject seems to me to proceed essentially from an unconscious confusion between the idea of _function_ and the idea of _value_, or, what comes to the same thing, between the _algebraic_ and the _arithmetical_ point of view. A thorough examination would show mathematical analysis to be much more clear in its nature than even mathematicians commonly suppose.
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