17th April 1869
My Dear Sir,
I have read attentively your paper on Mill’s Theory of Geometrical Reasoning. I think you have detected some slight flaw in his mode of stating the subject, but not to the extent that you seem to imagine. It has long been remarked (by De Morgan[3]) that the fifth proposition of the First Book [of Euclid] needs an additional postulate to justify the process of allowing a triangle to be turned round on itself. That postulate is omitted in the common demonstration, and Mill simply repeats the omission; in that respect being nowise inferior to Euclid himself. I would not call this assumption an axiom. It seems to me that in this whole demonstration Mill copied the original and has no faults but what belongs to it.
As to the dispute between Intuition and Induction, I think you have disposed, in a too off-hand way, of a large question, not to say a series of questions. I cannot pretend to consider these in a note; but I would remark that you seem to confound the alleged inductive origin of the axioms of Geometry with the inductive formulation of the separate demonstrations. I don not think Mill holds that the proofs of the propositions are inductive; he would say they are deductive. As to the final axioms, ‘the sums of equals are equal’, ‘things equal to the same thing are equal’, I agree with him in calling these inductions, and I see nothing in your remarks to make me depart from that opinion. That ‘two straight lines cannot enclose a space’ is not an induction because it is not an axiom.
Where Mill first handles the subject of Definition in the ‘Logic’ I think he is careful to add that with the definition of a thing, we must postulate its existence, in order to infer true propositions from them. As to the hypothetical nature of the definitions of Geometry, that admits of being easily explained without incurring any absurdity.
When Mill talks of dropping the use of diagrams and carrying on the demonstration algebraically, I do not see that he commits any mistake. Of course he would have to make his formulae express all that is meant by the diagrams, and to suppose that the user of the formulae had conceived the full diagrams for once, in order to know what the symbols stand for.
You seem to find a contradiction in Mill’s using the phrase ‘intuition’ to express the perception that [the triangle] ABC is the difference between ABE and CBE. I see nothing of the sort. It clashes neither with the doctrine that figure is unessential, nor with the doctrine that a general truth cannot flow from a single intuition. Not with the first because he would hold that you may see the figure, or you may not; if you see it, you are at liberty to judge it by its help; if you do not you must dispense with it by the proper & thorough analytic translation. In the present instance, he merely used the first alternative, as Euclid does. It does not contradict the second doctrine, because Mill makes special allowance for the process named ‘parity of reasoning’, whereby a generality may be made to arise from a single instance.
Whether he, or any one else had thoroughly expounded, or properly, or properly hit, this peculiar operation, I do not know; I have seen many attempts at it, and have mentioned it myself. But I do not know the man that has so far triumphed in the matter as to look with pity on Mill’s views respecting it.
It seems to me that in going so far as you have done, you ought to have gone much farther, and made good a theory of the inductive foundations of geometry, free from all the difficulties so often pointed out, from Locke downwards, as attending to innate truth.
You and Mill both make use of Intuition, to mean simply ‘perception’ which I think tends to confuse the whole argument. The ordinary sense of intuition, in which it stands opposed to induction, is ‘innate’ or primordial.
The whole strain of your paper appears to me hypercritical and over-done, while you have not yourself so guarded your language as to avoid a damaging retort by an equally acute opponent.
With best remembrances to your father and mother,
I am,
Yours sincerely,
A. Bain
Mr W. R. Smith
[1] CUL ADD 7449 D029 MS
[2] Written on official paper of the University of Aberdeen with its coat of arms on top of the page.
[3] De Morgan, Augustus (1806–1871): one of the most noted of nineteenth century British arithmeticians and mathematicians, who held the chair of Mathematics at University College London from 1828 until his death. A prolific contributor to the Penny Cyclopaedia, De Morgan was much admired by WRS, who later was to recommend his work to G. .H. Lewes.