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Topic: Matheology § 161
Replies: 3   Last Post: Nov 25, 2012 3:43 PM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Matheology § 161
Posted: Nov 25, 2012 3:37 AM

Matheology § 161

{{Yet another application of set theory?}} I propose here, then, first
to illustrate, and then to discuss theoretically, the nature and ideal
outcome of any recurrent operation of thought, and to develope, in
this connection, what one may call the positive nature of the concept
of Infinite Multitude.

Prominent among the later authors who have dealt with our problem from
the mathematical side, is George Cantor. [...] With this theory of the
Mächtigkeiten I shall have no space to deal in this paper, but it is
of great importance for forming the conception of the determinate
Infinite.

A map of England, contained within England, is to represent, down to
the minutest detail, every contour and marking, natural or artificial,
that occurs upon the surface of England.

Our map of England, contained in a portion of the surface of England,
involves, however, a peculiar and infinite development of a special
type of diversity within our map. For the map, in order to be
complete, according to the rule given, will have to contain, as a part
of itself, a representation of its own contour and contents. In order
that this representation should be constructed, the representation
itself will have to contain once more, as a part of itself, a
representation of its own contour and contents; and this
representation, in order to be exact, will have once more to contain
an image of itself; and so on without limit. We should now, indeed,
have to suppose the space occupied by our perfect map to be infinitely
divisible, even if not a continuum.

That such an endless variety of maps within maps could not physically
be constructed by men, and that ideally such a map, if viewed as a
finished construction, would involve us in all the problems about the
infinite divisibility of matter and of space, I freely recognize.

Suppose that, for an instance, we had accepted this assertion as true.
Suppose that we then attempted to discover the meaning implied in this
one assertion. We should at once observe that in this one assertion,
"A part of England perfectly maps all England, on a smaller scale,"
there would be implied the assertion, not now of a process of trying
to draw maps, but of the contemporaneous presence, in England, of an
infinite number of maps, of the type just described. The whole
infinite series, possessing no last member, would be asserted as a
fact of existence.

We should, moreover, see how and why the one and the infinitely many
are here, at least within thought's realm, conceptually linked. Our
map and England, taken as mere physical existences, would indeed
belong to that realm of "bare external conjunctions." Yet the one
thing not externally given, but internally self-evident, would be that
the one plan or purpose in question, namely, the plan fulfilled by the
perfect map of England, drawn within the limits of England, and upon a
part of its surface, would, if really expressed, involve, in its
necessary structure, the series of maps within maps such that no one
of the maps was the last in the series.

This way of viewing the case suggests that, as a mere matter of
definition, we are not obliged to deal solely with processes of
construction as successive, in order to define endless series. A
recurrent operation of thought can be characterized as one that, if
once finally expressed, would involve, in the region where it had
received expression, an infinite variety of serially arranged facts,
corresponding to the purpose in question.

[Josiah Royce: "The world and the individual", MacMillan, London
(1900) p. 500ff]
http://www.archive.org/stream/worldindividual00royciala#page/n0/mode/2up
http://www.archive.org/stream/worldindividual00royciala/worldindividual00royciala_djvu.txt

The repeated application of the fotocopier has been proposed as a
cheap replacement for expensive electron microscopes. Unfortunately I
have forgotten the name of the inventor of this idea.