Page:Mind (Old Series) Volume 9.djvu/468

 456 w. WUNDT'S LOGIK, n. the priority is naturally assigned to the most abstract, viz., Mathematics. One of the first inquiries meeting us here is the special use and significance of the processes of Analysis and Synthesis as applied to this science. The common but extremely vague employment of the former term, as in ' Analytical Geo- metry,' calls for some determination. Prof. Wundt assigns the leading place in the interpretation to the two following character- istics : It will be remembered that Analysis is always supposed to take a proposition for granted ; it then proceeds to deduce consequences therefrom, and the verification of the proposition is rested upon the acknowledged truth of these consequences. Now, in mathematical analysis, according to the general concep- tion of it introduced by Descartes, there are two reasons why the process is distinguished from the ordinary application. One of these is found in the employment of algebraic symbols. By their use we are enabled to secure, over a vastly enlarged range of cases, the condition that some element shall be considered as known or given ; for all the elements of the problem, whether originally included among the data or not, admit of equally easy representation by means of symbols, and it becomes easy then to detach and isolate the one which we require. The other consists in the fact that we couch our expressions in the form of equa- tions. The defect of the ordinary analytical process is that the deduction of a true consequence is no necessary proof of the truth of the premiss, but only confers upon it a greater or less degree of probability. But if the consequence and the ground are reci- procally connected, i.e., if the former necessarily carries the latter with it, matters are altered. Now this is obviously the case in Mathematics, for the equational form is convertible, and the relation of equality enables us to proceed with certainty from consequence to cause in a way which could not generally else- where be allowed. The more detailed examination into the logical nature of mathe- matical assumptions and pi'ocesses is too intricate to interest, or indeed be intelligible to, any but professed mathematicians ; but a few remarks may be made upon the treatment of such critical questions as the foundations of the Differential Calculus and the explanation of the ' Infinite '. On the former of these two points Prof. Wundt's account is very full and suggestive. He distin- guishes and discusses the characteristic assumptions in each of the three main accounts of the basis of Differentiation : viz., the doctrine of Infinitesimals, that of Limits, and that of Derived Functions or of Expansion ; accounts which may, roughly speaking, be connected with the three great names respectively of Leibniz, Newton, and Laplace. But, though not regarding these as really hostile views, he seems to recognise in them a reality and depth of distinction which many will hardly be able to follow. This point is connected with a view which he has expressed in an earlier chapter (on Number) on the possibility of admitting, as