(26) Dugald Stewart in his Elements of the Philosophy of the Human Mind, (P. II., C. II., Section 3, §2,) cites a passage from a dissertation printed at Berlin in 1764, which does not appear so unreasonable as he pretends. We subjoin it, because the German philosopher's opinion seems to us the same that we gave in the text.
"Omnes mathematicorum propositiones sunt identicæ et representantur hac formula, A=A. Sunt veritates identicæ, sub varia forma expressæ, imo ipsum quod dicitur contradictionis principium vario modo enuntiatum et involutum; si quidem omnes hujus generis propositiones revera in eo continentur. Secundum nostram autem intelligendi facultatem ea est propositionum differentia, quod quædam longa ratiociniorum serie, alia autem breviore via, ad primum omnium principiorum reducantur, et in illud resolvantur. Sic. v. g. propositio 2+2=4 statim huc cedit: 1+1+1+1=1+1+1+1; id est, idem est idem; et, proprie loquendo, hoc modo enuntiari debet,—si contingat adesse vel existere quatuor entia, tum existunt quatuor entia; nam de existentia non agunt geometræ, sed ea hypothetice tantum subintelligitur. Inde summa oritur certitudo ratiocinia perspicienti; observat nempe idearum identitatem; et hæc est evidentia assensum immediate cogens, quam mathematicam aut geometricam vocamus. Mathesi tamen sua natura priva non est et propria; oritur etenim ex identitatis perceptione, quæ locum habere potest, etiamsi ideæ non repræsentent extensum."
ON CHAPTERS XXX AND XXXI.
(27) We have shown that Dugald Stewart had perhaps in view Vico's doctrine; but without wishing to bring against him the charge he brought against his master, Reid, that of resuscitating the doctrines of the Jesuit Buffier, we would, in order that the reader may judge with full knowledge of the cause, subjoin a remarkable extract from the Scotch philosopher, which will show coincidence between some of his doctrines and those of the Neapolitan. Had Stewart read Vico, we are inclined to believe that he would not have complained of the confusion with which various ancient and modern authors have explained this doctrine.
"The peculiarity of that species of evidence which is called demonstrative, and which so remarkably distinguishes our mathematical conclusions from those to which we are led in other branches of science, is a fact which must have arrested the attention of every person who possesses the slightest acquaintance with the elements of geometry. And yet, I am doubtful if a satisfactory account has hitherto been given of the circumstance in which it arises. Mr. Locke tells us, that 'what constitutes a demonstration is intuitive evidence of every step;' and I readily grant, that if in a single step such evidence should fail, the other parts of the demonstration would be of no value. It does not, however, seem to me that it is on this consideration that the demonstrative evidence of the conclusion depends, not even when we add to it another which is much insisted on by Dr. Reid,—that 'in demonstrative evidence our first principles must be intuitively certain.' The inaccuracy of this remark I formerly pointed out when treating of the evidence of axioms; on which occasion I also observed, that the first principles of our reasonings in mathematics are not axioms, but definitions. It is in this last circumstance (I mean the peculiarity of reasoning from definitions) that the true theory of mathematical demonstration is to be found; and I shall accordingly endeavor to explain it at considerable length, and to state some of the more important consequences to which it leads.
"That I may not, however, have the appearance of claiming, in behalf of the following discussion, an undue share of originality, it is necessary for me to remark that the leading idea which it contains has been repeatedly started, and even to a certain length prosecuted, by different writers, ancient as well as modern; but that, in all of them, it has been so blended with collateral considerations, although foreign to the point in question, as to divert the attention both of writer and reader, from that single principle on which the solution of the problem hinges. * * * * * * *
"It was already remarked, in the first chapter of this part, that whereas, in all other sciences, the propositions which we attempt to establish, express facts real or supposed,—in mathematics, the propositions which we demonstrate only assert a connection between certain suppositions and certain consequences. Our reasonings, therefore, in mathematics, are directed to an object essentially different from what we have in view, in any other employment of our intellectual faculties;—not to ascertain truths with respect to actual existences, but to trace the logical filiation of consequences which follow from an assumed hypothesis. If, from this hypothesis, we reason with correctness, nothing, it is manifest, can be wanting to complete the evidence of the result; as this result only asserts a necessary connection between the supposition and the conclusion. In the other sciences, admitting that every ambiguity of language were removed, and that every step of our deductions were rigorously accurate, our conclusions would still be attended with more or less of uncertainty, being ultimately founded on principles which may, or may not, correspond exactly with the fact." (Elements of the Philosophy of the Human Mind. P. II., C. II., S. 3, § 1.)
This is exactly Vico's doctrine of the cause of the difference in the degrees of evidence and certainty; although he makes a general system, in order to explain the problem of intelligence, what the Scotchman only assigns as a fact to show the reason of mathematical evidence. Père Buffier (Trait. des prem. Vérités, P. I., C. II.) explains the same thing with great clearness.