Note 953 — Chapter VII

As we see in the pretensions set forth by mathematicians, who often suppose that an amount of certainty can be attained in their own pursuits not to be found in any other. This error has probably arisen, as Locke suggests, from confusing clearness with certainty. Essay on Human Understanding, book iv. chap. ii. secs. 9 and 10, in Works, vol. ii. pp. 73, 74. See also Comte, Philos. Pos. vol. i. p. 103, where it is justly observed, that all branches of knowledge capable of being generalized into sciences admit of equal certainty, but not of equal precision: ‘si d'après l'explication précédente, les diverses sciences doivent nécessairement présenter une précision très-inégale, il n'en est nullement ainsi de leur certitude.’ This is handled unsatisfactorily by Montucla (Hist. des Mathémat. vol. i. p. 33), who says, that the principal cause of the peculiar certainty reached by the mathematician is, that ‘d'une idée claire il ne déduit que des conséquences claires et incontestables.’ Similarly, Cudworth (Intellect. System, vol. iii. p. 377): ‘nay the very essence of truth here is this clear perceptibility, or intelligibility.’ On the other hand, Kant, a far deeper thinker, avoided this confusion, by making mathematical clearness the mark of a kind of certainty rather than of a degree of it: ‘Die mathematische Gewissheit heisst auch Evidenz, weil ein intuitives Erkenntniss klarer ist, als ein discursives. Obgleich also beides, das mathematische und das philosophische Vernunfterkenntniss an sich gleich gewiss ist, so ist doch die Art der Gewissheit in beiden verschieden.’ Logik, Einleitung, sec. 9, in Kant's Werke, vol. i. p. 399. On the opinions of the ancients respecting certainty, compare Matter, Hist. de l'Ecole d'Alexandrie, vol. i. p. 195, with Ritter's Hist. of Ancient Philos. vol. ii. p. 46, vol. iii. pp. 74, 426, 427, 484, 614.