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22 Insert p 21 and Skip to p 42.
Thus in place of a sheet of assertion, we have a book of separate sheets, tacked together at points, if not otherwise connected. For our alpha sheet, as a whole, represents simply a universe of existent individuals, and the different parts of the sheet represent facts or true assertions made concerning that universe. At the cuts we pass into other areas, area of conceived propositions which are not realized. In these areas there may be cuts where we pass into worlds which the in the imaginary worlds of the outer cuts are themselves represented to be imaginary and false,
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but which may, for all that, be true, and therefore continuous with the sheet of assertion itself, although this is uncertain. You may regard the ordinary blank sheet of assertion as a film upon which there is, as it were, an undeveloped photograph of the facts in the universe. I do not mean a literal picture, because its elements are propositions, and the meaning of a proposition is abstract and altogether of a different nature from a picture. But I ask you to imagine all the true propositions to have been formulated; and since facts blend into one another, it can only be in a continuum that we can
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conceived this to be done. This continuum must clearly have more dimensions than a surface or even than a solid; and we will suppose it to be plastic, so that it can be deformed in all sorts of ways without the continuity and connection of parts being ever ruptured. Of this continuum the blank sheet of assertion may be imagined to be a photograph. When we find out that a proposition is true, we can place it wherever we please on the sheet, because we can imagine the original continuum which is plastic to be so deformed as to bring any number of propositions to any
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places on the sheet we may choose. So far I have called the sheet a photograph, so as not to overwhelm you with all the difficulties of the conception at once. But let us rather call it a map,— a map of such a photograph if you like. A map of the simplest kind represents all the points of one surface by corresponding points of another surface in such a manner as to preserve the continuity unbroken, however great may be the distortion. A Mercator's chart, however, represents all the surface of the earth by a strip, infinitely long, both north and south poles being at infinite distances,
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near the poles are magnified so as to be many times larger than the real surfaces of the earth that they represent, while in longitude the whole equator measures only two or three feet; and you might continue the chart so as to represent the earth over and over again in as many such strips as you pleased. Other kinds of map, such as my Quincuncial Projection which is drawn in the 4th volume of the American Journal of Mathematics, show the whole earth over and over again in checkers, and there is no arrangement you can think of in which the different representations of the same place might not appear on a perfectly correct map. This accounts for our