April 6, 1869
Dear Sir,
I am obliged to you for sending me your paper.[2] I am always glad when my speculations are criticized by anyone who understands what he is talking about.
The oversight[3] you have pointed out consists only in not making explicit in the statement of the demonstration, what is implicitly contained in its terms; and this was an oversight, inasmuch as I tacitly undertook to state everything explicitly. You will, however, not fail to observe, that I did not introduce Euc.I.5. In confirmation or illustration of my opinion respecting the foundations of mathematical evidence. I gave it as a specimen of a train of reasoning in illustration of my view of the Syllogism and it is, probably, from an acceptance of it as what it was intended for, and not as anything else, that (as you remark) none of my other critics have noticed the defect.
I do not see on what ground, even on the gravest view of the error, it can be imputed, as you seem to impute it, to ignorance of geometry. To set out only a part of the premises of a reasoning instead of the whole, is a logical, not a mathematical error. And, the premise omitted not being included by geometry among their axioms, it is a logician’s rather than a mathematician’s part to point out that it is tacitly assumed by them.
The whole discussion between us, though important in itself, seems to me to have no bearing on the question respecting mathematical evidence. If you succeed in showing that geometers require, and habitually use, a great many more axioms than they recognise, you will have done a useful service, but you will not have made any progress towards proving that the truth of axioms is a priori and not, as I contend a posteriori.
I shall carefully keep your paper to be referred to when I next revise my Logic, or otherwise write again on the subject; and I shall then be able to give a more thorough examination to the various points raised by you than I have leisure for at present.
I am, Dear Sir,
Yours very faithfully,
J. S. Mill[4]
Wm Robertson Smith Esq.
[1] CUL ADD 7449 D473 MS
[2] “Mr Mill’s theory of geometrical reasoning mathematically tested”: read to the Royal Society of Edinburgh by P. G. Tait on 1 February, 1869, and published in the Proceedings of the RSE (vi, pp. 477–483 (cf.. L&E pp.3-18).
[3] Mill corrected his “oversight” in subsequent editions by making a substantial addition in his System of Logic Ratiocinative and Inductive at ch. iv, § 4, where he attempts to prove Euclid’s fifth proposition “directly from first principles”.
[4] Mill, John Stuart (1806–1873): the world-renowned philosopher, logician and scholar, was a ready target for the young WRS, not least because of his close friendship with Alexander Bain and their joint espousal of Auguste Comte’s secular theories.