Date: Nov 25, 2012 3:55 AM
Author: Graham Cooper
Subject: Re: Matheology § 161

On Nov 25, 6:37 pm, WM <> wrote:
> 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]
> 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.
> Regrads, WM

there are no mathematicians reading.

logic is a study of literature only, no different to knowing vast
journals of legal precedents, the entire art history of the Incubus
period, the works of Shakespear or illustrated Medical Journals.

You are talking to Clayton's Arts Graduates who wish to further the
'study' of logic, which opposes the _application_ of logic.

if( if(t(S),f(R)) , if(t(R),f(S)) ).
if it's sunny then it's not raining
if it's raining then it's not sunny

S: if stops(S) gosub S
G. GREENE: this proves stops() must be un-computable!