Page:Mind (Old Series) Volume 9.djvu/469

 w. WUNDT'S LOGIK, n. 457 an element in calculation, the infinitely great and the infinitely little. Here he has some remarks which will seem rather hard sayings to what may be called old-fashioned mathematicians. Speaking from this latter standpoint, and with an acute con- sciousness that those who have not followed the speculations of later mathematicians cannot judge adequately of the question in dispute, I must confess to feeling considerable hesitation and doubt here. With regard to the contradictions found to attend the notion of anything absolute in infinity, in either direction in which it may be taken, as illustrated alike by mathematical philosophers like Berkeley and by the non-mathematical like Hamilton, these seemed almost entirely avoided by adhering to the conception of a ' limit'. All geometrical difficulties seemed to be smoothed when we assumed that a differential coefficient simply indicated the rate of change of some variable element a perfectly determinate magnitude ; that all magnitudes were to be regarded as essentially finite ; and that all that was meant by calling them infinite was to consider them capable of indefinite extension or diminution : the value ' at infinity ' or ' at nothing ' being the value again perfectly determinate and as a rule finite towards which we found the variable magnitude or its function to continually tend. Now, as against all this, at least as against its completeness, we have the conception introduced of the absolutely infinite number. " By reason of the perfected freedom of conceiving (Begriffsconceptionen) enjoyed by mathematics, it is possible to assume that there is an absolute value w = oo, which not only marks the limit to which the series of numbers approxi- mates without end but in which this limit is actually attained " (p. 127). And various consequences are deduced from this assumption, such as that the ordinary law of commutation may fail for these " transfinite " numbers as they are called in dis- tinction from the commonly recognised infinite numbers so that though 1 + w = w, we may have w + 1 > w. This distinction between transfinite and infinite (which is not, of course, Prof. "Wundt's own introduction, but taken from the speculations of some advanced mathematicians) is repeated in the discussion upon the foundations of the Differential Calculus, and it is maintained that the neglect of it has marred the analysis and criticism of other authors. After the discussion of the more abstract subject of Mathe- matics, we come to what is one degree more concrete, >:>z., Mechanics. This section is mostly devoted to a minute account, partly historical, partly critical, of the various general principles which have been assumed or deduced in order to explain the observed facts of nature, so far as motion is concerned. Thus we have as a starting-point some of the principles assumed by the early Greek investigators, ?.$., their rather crude doctrines of me- chanical causation on the one hand, and of teleology on the other. These are explained and tested, and it is inquired to what extent