Rules for the Direction of the Mind; Torrey’s The Philosophy of Descartes (New York, 1892), p. 104.
[1414]. Mathematicians have, in many cases, proved some things to be possible and others to be impossible, which, without demonstration, would not have been believed.... Mathematics afford many instances of impossibilities in the nature of things, which no man would have believed, if they had not been strictly demonstrated. Perhaps, if we were able to reason demonstratively in other subjects, to as great extent as in mathematics, we might find many things to be impossible, which we conclude, without hesitation, to be possible.—Reid, Thomas.
Essay on the Intellectual Powers of Man, Essay 4, chap. 3.
[1415]. If philosophers understood mathematics, they would know that indefinite speech, which permits each one to think what he pleases and produces a constantly increasing difference of opinion, is utterly unable, in spite of all fine words and even in spite of the magnitude of the objects which are under contemplation, to maintain a balance against a science which instructs and advances through every word which it utters and which at the same time wins for itself endless astonishment, not through its survey of immense spaces, but through the exhibition of the most prodigious human ingenuity which surpasses all power of description.—Herbart, J. F.
Werke Kehrbach (Langensalza, 1890), Bd. 5, p. 105.
[1416]. German intellect is an excellent thing, but when a German product is presented it must be analysed. Most probably it is a combination of intellect (I) and tobacco-smoke (T). Certainly I3T1, and I2T1, occur; but I1T3 is more common, and I2T15 and I1T20 occur. In many cases metaphysics (M) occurs and I hold that IaTbMc never occurs without b + c > 2a.
N. B.—Be careful, in analysing the compounds of the three, not to confound T and M, which are strongly suspected to be isomorphic. Thus, I1T3M3 may easily be confounded with I1T6. As far as I dare say anything, those who have placed Hegel, Fichte, etc., in the rank of the extenders of Kant have imagined T and M to be identical.—De Morgan, A.
Graves’ Life of W. R. Hamilton (New York, 1882-1889), Vol. 13, p. 446.
[1417]. The discovery [of Ceres] was made by G. Piazzi of Palermo; and it was the more interesting as its announcement occurred simultaneously with a publication by Hegel in which he severely criticized astronomers for not paying more attention to philosophy, a science, said he, which would at once have shown them that there could not possibly be more than seven planets, and a study of which would therefore have prevented an absurd waste of time in looking for what in the nature of things could never be found.—Ball, W. W. R.
History of Mathematics (London, 1901), p. 458.