May 16, 2013
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| Figure1: Local and global optimum |
This post is an extension of my previous post Ordinals of life, where I discussed about Mr. Sanjay who
always wanted to enjoy a normal life and was always considered as extraordinary by rest of the normal
population. In this post again, I try to capture as usual a mathematical purview of human life, Some of my
friends try to put this in to fixed genre(i.e Metaphysics). This post is motivated by Weierstrass’ Theorem
for functions and human interactions. My knowledge in the field of sociology and psychology is limited. So I
tried to surrogate the social exchange theory and related issues to information theory, contributed by Claude E.
Shannon.
We know that a human gets birth and dies one day so this set of life(time), which closed and bounded. And if success(happiness/utility) is function time i.e. f(t) then we can map each success points on to Real line.
Definition:
Success is a real valued continuous function f(s) : s-->ℜ. Where f(s)∈ S, s∈F.
Theorem 1 (Weierstrass’ Theorem for functions) Let f(s) be a continuous real-valued function on the compact non-empty set F ⊂ℜ . Then F contains a point that minimizes(maximizes) f(s) on the set F.
We know that a human gets birth and dies one day so this set of life(time), which closed and bounded. And if success(happiness/utility) is function time i.e. f(t) then we can map each success points on to Real line.
Definition:
Success is a real valued continuous function f(s) : s-->ℜ. Where f(s)∈ S, s∈F.
Theorem 1 (Weierstrass’ Theorem for functions) Let f(s) be a continuous real-valued function on the compact non-empty set F ⊂ℜ . Then F contains a point that minimizes(maximizes) f(s) on the set F.
Above theorem clearly states that if the set is closed and bounded then the function will be having a
maximum or minimum value in that domain. As i have already defined above the life of individual is closed
and bounded set of time. It means that a human life will reach to its maximum value(in terms of
success/happiness) in that compact set. This success(happiness) is defined in the previous post(see figure:2). It can be only considered a success when there will be ordinal comparison or benchmark.
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| Figure2: Ordinals and success as continuous function |
Now I would like to bring the notion of information exchange. This information can be exchange by
different means, if I consider that the natural language was not discovered, our sensory capability could
capture some information from the surrounding. It means neighbour hood. For the
time being, I try to introduce two types of neighbourhoods. First one is temporal and the other is spatial. In
the case of single human with time(temporal neighbourhood) he/she sees his/her own success/happiness and it’s
graph and traverse according to his/her ordinal preference and jumping on other curves(see
Figure2). In this case I have dropped its spatial movement and its implicit nature of comparison. I would like to mention again the person named Sanjay who was
always enjoying his life where he considered his success with his own
compact set of time, and his comparison was always with him.
If I apply Theorem1 to understand the two dimensions, where I include both the temporal and spatial type of neighbourhood and then the whole situation would be like Figure1 where there are successes, continuous function with respect to time and space. So if I take every individual human and his life is a subset of the bigger set which includes the every human and with either a fixed time frame or we can consider the superimposing of different time window where success can be brought in the same comparison platform.
Hypothesis
Every single person is success full and happy if i consider subset d ∈ {space X time} and this subset is assumed as a compact set, so means whatever that has been consider a local optimal success becomes a global optimal success with respect to that person in that compact set. This subset is defined within the constraint of information exchange. The more volatile and large information network will exist the bigger the compact set will be. And than his own global optimal success will become a local optimal success in this bigger compact set.
Hypothesis
Every single person is success full and happy if i consider subset d ∈ {space X time} and this subset is assumed as a compact set, so means whatever that has been consider a local optimal success becomes a global optimal success with respect to that person in that compact set. This subset is defined within the constraint of information exchange. The more volatile and large information network will exist the bigger the compact set will be. And than his own global optimal success will become a local optimal success in this bigger compact set.
