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...
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, phasespaces (Stewart and Cohen, 1997) and psuedophase spaces (Edmunds), approximate entropy (New Scientist).
From a sociological viewpoint, 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.1011) 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)
(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