Fractals (look them up online, for example the Wikipedia and Wolfram/Weisstein articles or summaries on them) are very irregular Boundaries and do have different dimensions from ordinary dimensions (the latter have whole number dimensions like 0, 1, 2, 3, 4, and so on, while the former have dimensions like 1.5, 3.2, and so on).
The Holographic Principle of G. 't Hooft of U. Utrecht (Netherlands) in Quantum Theory maintains that the BOUNDARY of an object contains all its information or knowledge under somewhat (but not completely) general conditions, especially in 3-dimensional space or 4-dimensional spacetime. It is quite valuable in Quantum Physics. Look it up under keywords "Holographic Principle," "Holography," "Gerard 't Hooft" or "Gerhard 't Hooft" or " 't Hooft" (I forget which) online.
My main concern here, however, is to discuss why Boundaries of objects or entities or even perhaps ideas should be regarded as separate Dimensions, which will also perhaps motivate regarding Grice's Intentions for example as separate Dimensions. My Causation blog has some description of what Dimensions are.
The "proof" is extremely simple, which is my "philosophy" of life provided that I stay above the gorillas and alligators/crocodiles and some of their human imitators. Recall that:
1) P(A-->B) = 1 + P(AB) - P(A)
2) P ' (A-->B) = 1 + P(B) - P(A) where P(B) < = P(A).
For Probabilities of continuous random variables on Euclidean or Euclidean-like space (technically taken as proportional to normalized Lebesgue measure, but you can ignore this for rough understanding), Boundaries have Probability 0, which is also true of their Lebesgue measure - it is also 0. Therefore, if A is the Boundary of B, then (1) and (2) become:
3) P(A-->B) = 1 + 0 - 0 = 1, and P ' (A-->B) = 1 + 0 - 0 = 1.
Q.E.D.
So Boundaries maximize Probable Causation/influence of both types (1) and (2) at the value 1 on the usual scale from 0 to 1. They are thus of critical importance as "fundamental" quantities, which roughly is a criterion for something to be a Dimension.
For those who actually want to completely (with every step written) derive (3) mathematically, notice that if P(A) is 0 and if P(B) < = P(A), meaning the probability of B is less than or equal to that of A, then since probabilities are always between 0 and 1, we must have P(B) = 0. Also, if P(A) is 0, then P(AB) is 0, because AB is the intersection of A with some set B, and the intersection of A with B cannot be "bigger" than A or B, so its probability is 0 if that if P(A) is 0. (Technically, this is an axiom of mainstream Probability-Statistics called Monotonicity - if B is a subset of A, then P(A) < = P(B) if defined.)
Osher Doctorow
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I like the ideas. I think 'dimenson' AND 'boundary' requires a good philosophical analysis, even an etymological one!
ReplyDeleteIt first reminded me of ... you know who. Grice. Late in life he felt as if a discipline was lacking in philosophy. Surely, he started by naming it:
He called it 'philosophical eschatology'.
The 'eschato-' bit refers, I trust, to the 'boundary'.
Eschatology IS a discipline in theology; so Grice felt the need to qualify his brand as 'philosophical'. In any, case just some input as to the philosophical desire, on occasion to go _beyond_ boundaries. "Do not trespass" lacks a prosecutor when it comes to what Grice calls 'theory-theory'.
Yes, even an etymological analysis, and even a logical analysis, which brings me to one of the reasons why I remain interested in physics besides psychology, biology, social sciences, philosophy relationships of physics: astronomy and astrophysics, or what in physics is called "cosmology" which studies the physical Universe including galaxies and intergalactic space(time) and clouds or nebulae and so on, especially how they change over long periods of time.
ReplyDeleteOne of the strangest paradoxes in cosmological physics is related to the question of whether the Universe has a BOUNDARY or not. It is considered to be EXPANDING in physics, and presently even accelerating its expansion, but this is considered to be GEOMETRIC rather than MATERIAL although matter is somehow affected by it - and nobody has the faintest clue how this is possible! If the Universe is expanding even geometrically, what is it expanding INTO? If it has no boundary, then how can it be expanding into "NOTHING"? There are even theories of "unparticles", "nothingness", "phantom matter", "ghost particles", taken quite seriously in physics, and "tachyons" and magnetic monopoles (look them up online under those keywords) which are considered to have existed or possibly still exist on a "pre-geometric" level before our physical Universe "started", whatever that means in rough English.
If the Universe DOES have a boundary, then what does it mean for it to be expanding? The same question then arises - what is on "the other side" of the boundary, if it is not the Universe? Is it another Universe, as with the idea of the Multiverse if taken to be partly spatially related to our Universe? Nobody knows.
Osher Doctorow
I like your conclusion. Has an Odysseic ring to it:
ReplyDelete"Nobody knows".
====
From Homer's Odyssey:
"Asking for Odysseus' name,
the hero reply that it
was "Nobody". Polyphemus
promised to eat "Nobody" last.
"Drunk, Polyphemus went wearily
to bed."
"Odysseus hardened the huge
stake point in the fire pit, before
driving the stake into Polyphemus' single eye, blinding him."
"Furious with what Odysseus had done, he proceded to call all the other Cyclops
living in the vicinity to ask for their assistance."
"When they asked Polyphemus what
happened to his eye, Polyphemus answered."
"'Nobody' did this to me."
-- which left Polyphemus's friends at some wonder as to what the implicature of his remark could have been.
A good point here is to discuss the pre-socratics. I thought it was rather unimaginative of some of them to call the Kosmos the apeiron. The indefinite, the infinity. I think I prefer multiverse. "Cosmic" and "cosmetic" are used with still different 'usages' by Joan Rivers.
ReplyDelete