MS 842 (1908) - A Neglected Argument - Early Drafts

G42

therein all but to a line or point.

The apeiry (τὁ άπειρον; Listing was not guilty of this word,) of any place, L, is the least number of places in it of the dimensionality less by 3 than L's, that would suffice, if all were impenetrably occupied, to leave no room for a solid that could not continuously shrink therein to indefinitely close approach to becoming a surface, line, or point.

For each of the four numbers it is true that its for a group

G42

of places its value is the sum of those for the separate parts. The homoge[n]eity of the concepts is excellent. Once Listing's analysis had shown that these four concepts are the ideas upon which the extended theorem had to be con mainly considered (after those of continuity and dimensionality,) in the investigation of the extended theorem it was became a very simple thing to trace out its formula; much simpler and more facile than the ordinary theorematic demonstrations of the mathematicians. It was more like

G43

the reasoning of a corollary, in that the way was plainly mapped out and no invention was called for. Now when we recall what trouble the narrower proposition gave to such demonstrators as Euler and Legendre, the Reader, I think, will be inclined to assent to the division of Deduction into the two branches of Logical Analysis which clears up definitions and Demonstration which invents ne apodictic proofs.

Hurrying on to Induction, since Ratiocinative Deduction

G44

can be of little aid in natural theology, I would like to call a moment's attention to a point which seems to afford an important aid in understanding toward attaining a right comprehension of the peculiar function of this third stage of inquiry. The point I mean is that Retroduction and Induction face opposite ways. The function of retroduction is not unlike those fortuitous variations in reproduction which played so important a rôle in Darwin's original theory. In point of fact, according to him every step in the long history of the development of the moner

G45

into the man was first taken in that arbitrary and lawless phenomena mode. Whatever truth or error there may be in that, it is quite indubitable, as it appears to me, that every step in the development of primitive notions into modern science was in the first instance mere guess-work, or at least mere conjecture. But the stimulus to guessing, the hint of the conjecture was derived from experience. The order of the progression march of suggestion in retroduction is from experience to hypothesis. A great many people who may be admirably trained in