[390] Anal. Post. i. 34.
[391] It may be suggested that there is a similar confusion on this question: when history is called a science, it is often forgotten that its data are essentially such that they can only occur once, while the material of the other sciences is such that cases of ‘the same’ may always be found in it. But neither need it be denied on this account that history can, and should, be written in a scientific spirit.
[392] Science et Méthode, ch. iii, L’Invention mathématique.
[393] Republic, 511 c.
[394] Anal. Post. i. 1.
[395] Diogenes Laertius, ix. 51.
[397] Or more difficult, if the indetermination is conceived as limited.
[398] This we saw ([§ 4]) is really a mistake: mathematical proofs are really hypothetical, and deduced from the initial postulates and definitions. They hold of the ideal objects of mathematics, but that they can be advantageously applied to reality is merely an empirical fact, and it is not inconceivable that the world should grow more recalcitrant to mathematical treatment, though actually it has grown less so.
[399] In Republic vi his whole argument for the existence of metaphysical truth, culminating in a supreme ‘Idea of the Good’, depends on the assumption that the ‘hypotheses’ of the sciences, being insecure originally, remain so until they are deduced from a (self-proving) ‘unhypothetical principle’. This assumes, of course, that they cannot be confirmed empirically by the results of their working, and exhibits the lacuna of logic in a typical way.