Note 42 — Chapter VIII · PLATONIC COMPOSITIONS GENERALLY.

Plato, Epist. vii. 343 D. The difficulties which Plato had here in his eye, and which he required to be solved as conditions indispensable to real knowledge — are jumped over in geometrical and other scientific expositions, as belonging not to geometry, &c., but to logic. M. Jouffroy remarks, in the Preface to his translation of Reid’s works (p. clxxiv.):—“Toute science particulière qui, au lieu de prendre pour accordées les données à priori qu’elle implique, discute l’autorité de ces données — ajoute à son objet propre celui de la logique, confond une autre mission avec la sienne, et par cela même compromet la sienne: car nous verrons tout à-l’heure, et l’histoire de la philosophic montre, quelles difficultés présentent ces problèmes qui sont l’objet propre de la logique; et nous demeurerons convaincus que, si les différentes sciences avaient eu la prétention de les éclaircir avant de passer outre, toutes peut-être en seraient encore à cette préface, et aucune n’aurait entamé sa véritable tâche.”

Remarks of a similar bearing will be found in the second paragraph of Mr. John Stuart Mill’s Essay on Utilitarianism. It has been found convenient to distinguish the logic of a science from the expository march of the same science. But Plato would not have acknowledged ἐπιστήμη, except as including both. Hence his view about the uselessness of written expository treatises.

Aristotle, in a remarkable passage of the Metaphysica (Γ. p. 1005, a. 20 seqq.) takes pains to distinguish the Logic of Mathematics from Mathematics themselves — as a separate province and matter of study. He claims the former as belonging to Philosophia Prima or Ontology. Those principles which mathematicians called Axioms were not peculiar to Mathematics (he says), but were affirmations respecting Ens quatenus Ens: the mathematician was entitled to assume them so far as concerned his own department, and his students must take them for granted: but if he attempted to explain or appreciate them in their full bearing, he overstepped his proper limits, through want of proper schooling in Analytica (ὅσα δ’ ἐγχειροῦσι τῶν λεγόντων τινὲς περὶ τῆς ἀληθείας, ὃν τρόπον δεῖ ἀποδέχεσθαι, δι’ ἀπαιδευσίαν τῶν ἀναλυτικῶν τοῦτο δρῶσιν· δεῖ γὰρ περὶ τούτων ἥκειν προεπισταμένους, ἀλλὰ μὴ ἀκούοντας ζητεῖν — p. 1005, b. 2.) We see from the words of Aristotle that many mathematical enquirers of his time did not recognise (any more than Plato recognised) the distinction upon which he here insists: we see also that the term Axioms had become a technical one for the principia of mathematical demonstration (περὶ τῶν ἐν τοῖς μαθήμασι καλουμένων ἀξιωμάτων — p. 1005, a. 20); I do not concur in Sir William Hamilton’s doubts on this point. (Dissertations on Reid’s Works, note A. p. 764.)

The distinction which Aristotle thus brings to notice, seemingly for the first time, is one of considerable importance.