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Bounded infinities


20 Sept 1998

Notes about the an idea of a bounded infinity (BI). Something that has infinite forms within the bound part of another space. An extremely simple BI is a line. It is constrained to a particular part of a nD space (n>1) but has infinite points within that line.

Are there more complicated forms that can describe some of the interactions between two rival theories. For example, the difference between Value Added and Labour theories of value in economics.

These are very rough notes...


2nd-3rd March 1998

Bits about Injective and Surjective:

"Injective: associating two sets in such a way that different members of the domain are paired with diferent members of the codomain, although not all of the latter need be members of the specified range." Borowski & Borwein (1989)

i.e with sets,
S {A,B,C,D,E}, T{A',B',C',D',E'} and T* (A',B',C',D',E',F',G',H')

the mapping from S to T* is injective (as is the mapping from T to S)

A' - A
B' - B
C' - C
D' - D
E' - E
F'
G'
H'

"Surjective: associating two sets in such a way that every member of the codomain is the image of at least one member of the domain, although There may be some members of the domain that are not mapped on to any element of the codomain." Borowski & Borwein (1989)

with the example before:

the mapping from T* to S is surjective (as well as from T to S, and from S to T)

Bijective: the mapping is both surjective and injective, i.e. from S to T, and from T to S)

This is alright for finite sets. What happens if you have infinite sets?

one - 1
two - 2
three - 3
four - 4
five - 5
... - ...
ten - 10
eleven - 11
... - ...

infinite - ?
irrational -
pi - 3.141 592 653 589 79...
e - 2.718 281 828 ...
... - ...

65 - A
66 - B
67 - C
68 - D
69 - E
... - ...
111,110,101 - one
116,119,111 - two
... - ...

The sets of letters, L, and digits, D, are finite (for a given alphabet or base). However, the sets of possible words, W and numbers, N, are infinite (for a given alphabet or base).

Are the mappings from given W to given N injective or surjective, or are they bijective?

I think that they are injective and surjective with out being bijective. (Or finitely bijective, if there is such a thing)

This has implications for numerical computation. Links to Turing machines (Penrose), Godel numbers, phase-spaces (Stewart and Cohen, 1997) and psuedo-phase spaces (Edmunds), approximate entropy (New Scientist).

From a sociological view-point, this has implications for signification.

"Signification is a term from semiotics, the study of signs (Hawkes, 1977). Saussure defines signification as the process that links the signifier (e.g. the word "night") to the signified (a division of a day). This is a abstract link, the word "Nacht" or "potato" would do equally well as long as the signification is understood by a communicating group." D318 TMA 01 - Representation - (1998)

How much reality can be contained within a language (signification system)?

Ferdinand de Saussure distinguished langue and parole. One was the set of possible words, the other was the set of actual words (or something like that).

Does this have anything to do with the hypothesis that:

If
Infinite subsets of W, W' are injective on N.
and
Infinite subsets of N, N' are injective on W.
Can
The mappings between W and N be bijective
While
The mappings between W' and N' are in some way none of these relations.

There's lots of mathematical paradoxes, Cantors paradox, the Lawyers paradox, etc.. (Borowski & Borwein, 1989) that deal with these ideas but I don't understand much of it.

I've got ideas of bound infinities, sets whose elements are bound in certain ways but have infinite members. Are signifier and signified similar to injective and surjective in some ways?

Does this also have links to ideas of reflexivity (Beck ).

"Hammersley (1993, pp.10-11) describes the progress of social science as the application of rational, scientific methodology to the humanities. Has cultural studies arisen because this approach fails to describe social meanings? Beck (1986: 1992) argues that rational methodology fails because society is reflexive: that culture is an artefact in which meanings are dependent on themselves. du Gay et. al. (1997, p.13) highlight the importance of signification and language in definitions of culture. Are there any other artefacts in which internal signification occurs? Mies and Shiva (1993) suggest that biology is one area where signification is more relevant than rational methodology."
D318 TMA 01 - Culture - (1998)

(Note by rational methodology I mean finite represention, or the idea of a universal Platonic reality. That is a larger set that the set of all possible universes with our set of physical laws. That there is a perfect language both explains and predicts all actions and all future actions. Maybe these aren't the same things, but they have something of the same essence)


Non-linearity -
Separable functions -
Analytic approximation -
Base functions -
Rational numbers -
Irrational numbers - pi, e, - shortest algorithm - approximate entropy - Chaos - psuedo phase spaces - phase spaces - Figments -
Imaginary numbers - i - differential geometry - co-ordinate geometry - Wave equations - Physics -

(BACK) - Analytic approximation - computation - statistical analysis - bell curve - Disney - astrophysics - selection effects - measurement -

Statistics in social science - complexity - abstraction processes - Figments -
Semiotics - perfect language - Cantor's paradox - Chaotic representation -
Godel - Disney - (BACK) Godel numbers - finite representation -
Power series - Chaos - network analysis - wave equations - (BACK)

(BACK) - Separable equations - models of stars - analytic approximations - Virial equation -
Thermal descriptions - ideal gas - gravitation - degeneracy pressure - quantum descriptions - solution forms - massive analytic computation - Ockhams Razor & Plato's Lens - significance - signification & index -

Ecological description - energy flow and nutrient cycling - Biological description - bioenergetics - chemical gradients - photon gradients - thermal gradients - -
DNA - chemical description - quantum description - -
Chemical identity - gibbs energy - difference & similarity - -

Thermal descriptions - periodicity - periodic table - (Chemical) phase transistions - quantum descriptions - -
Life - dynamic equilibrium - chaotic representations -
Nomenclature - spectrum of chlorophyll - colour - Figments - semiotics - (mathematical) phase space - quantum descriptions -

Statistical analysis - analytical approximation -

Index - bound infinity - bound infinity
unit - e - pi
equality - imaginary index
index - signifier
integers - real - complex
Logical analysis -

Statistical effect of being translated through two irrationals - bell curve?

Index - bound infinity - bound infinity - index
unit - e - pi - i

Analytic representation of irrationals

Separable indices



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Created 20/9/98