Science > Physics > Causation: How It Originates in Algebra, Category Theory, Topology, etc.
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Science > Physics |
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"OsherD" |
| Date: |
01 Jun 2005 03:16:19 PM |
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Causation: How It Originates in Algebra, Category Theory, Topology, etc. |
From Osher Doctorow
Causation: How It Originates in Algebra, Category Theory, Topology,
etc.
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
We have seen in my recent thread that Banach Space's usefulness from
the viewpoint of physics and PI originates largely in its absence in
general of the inner product which forces one to examine dimensionality
instead and replace multiplication by addition.
Causation's usefulness from the viewpoint of physics and PI originates
almost as subtlely or unusually and has some similarities to the
previous paragraph but also some differences.
Here the addition or addition-subtraction versus
multiplication-division is "forced" or derived from more direct
structural relationships.
In Category Theory and much of the remainder of mathematics, sums and
products are in a sense derived from the way in which they operate on
spaces or "large objects" like sets and groups and rings and modules.
In Category Theory, the sum operation in its large scale is the
coproduct, while the product operation in its large scale is also known
as the product, respectively defined by objects C = UXi for i in index
set I and P = PI(Xi) for i in index set i such that respectively:
1) d o ci = di
2) pi o q = qi
in the notation of Wolfram's mathworld.wolfram.com (Coproduct and
Category Product entries), with these qualifications: d and pi are
described by "There exists (is) a unique morphism (d: C->D, q: Q->P)
such that," the respectively equations hold for all i in the index
sets, and where the family of morphism of form ci take Xi to C and the
family of morphisms di take Xi to D for every object D, and
correspondingly for pi: P->Xi and qi: Q->Xi for every object Q.
I'll try to return soon.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation: How It Originates in Algebra, Category Theory, Topology, etc. |
01 Jun 2005 03:40:02 PM |
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From Osher Doctorow
For sets, we can look at the category of sets or just at sets
themselves. For the category of sets, the coproduct is the disjoint
union:
1) U(Xi)
of sets Xi for i in some index set I and the morphisms sending Xi to
this disjoint union.
The product is the Cartesian product:
2) PI(Xi)
and the morphisms send the Cartesian product to Xi and are called
projections onto the factor Xi from PI(Xi).
This disjointness carries over to the general set picture and mostly
crosses groups, vector spaces, rings, Banach spaces, Hilbert spaces,
topological spaces.
The idea is roughly speaking is that, in Probable Influence (PI)
language:
1) x + y or x - y = some important quantity
where x and y differ fundamentally. For example, in PI:
2) P(A-->B) = 1 + y - x
where y = P(AB) and x = P(A), and therefore we have:
3) y - x = P(A-->B) - 1
where the right side is an important quantity in itself and x, y differ
fundamentally since x is the probable cause or probable influence and y
is the probable effect or probably influenced quantity.
In some scenarios the important quantity might be either the universe S
or the null set N or 1 or some "null object" also denoted N or 0:
4) x + y or x - y = S or N
Notice that if N is written 0, we have:
5) x - y = 0
and if S is written 1 we have:
6) x - y = 1
and (6) can be rewritten:
7) y - x + 1 = 0 = P(A-->B)
which would be a special case of 0 Probable Influence. If N is written
0, we have:
8) x - y= 0
and therefore:
9) x - y + 1 = 1 = P(A-->B)
so that A exerts probable influence of 1 (maximal) on B, or since we
can start with y - x rather than x - y in (8), with y < = x, readers
will more accurately recognize:
10) y - x + 1 = 1 = P(A-->B)
so that A exerts maximum probable influence on B.
The fundamental difference here is between probable cause and probable
effect, or probable influencing and probable influenced.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation: How It Originates in Algebra, Category Theory, Topology, etc. |
01 Jun 2005 04:00:37 PM |
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From Osher Doctorow
The direct sum of two Banach spaces is well explained in
www.algebra.com/algebra/about/history/Direct-sum.wikipedia by beginning
with direct sum of two vector spaces V and W over field K, where the
cartesian product V x W is turned into a vector space over field K by
componentwise definition of operations (x1, y1) + (x2, y2) = (x1 + x2,
y1 + y2), k(x, y) = (kx, ky), and the direct sum is typically written V
+ W except that + has a circle around it. Every element of this V +
(circle) W can be uniquely written as the sum of an element of V and an
element of W, and the dimension of the sum is the sum of dimensions of
V and of W. For Banach spaces, the direct sum of X and Y is their
direct sum considered as vector spaces with norm //(x, y)// as //x//_V
+ //y//_W for x in V and y in W. For Hilbert spaces, the only
difference is that in addition to the above an inner product such as
for exampe the dot product is defined and the summands are orthogonal
or in orthogonal spaces.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation: How It Originates in Algebra, Category Theory, Topology, etc. |
01 Jun 2005 04:17:57 PM |
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From Osher Doctorow
One might ask why, if the underlying idea of an expression like 1 + y -
x as a real element of a direct sum is so easy, with y and x
fundamentally different objects via their signs (+ vs -, both y and x
being nonnegative), the idea was missed in algebraic geometry and
algebraic topology. There was nothing deliberate about it - there are
simply too much data and too few theories in algebra.
For example, look at this:
1) x o y = x + y - xy
This is the star product or circle composition product in the Jacobson
Radical which is applicable to rings, modules, and even topological
spaces. If you saw a real expression like x + y - xy, would you be
likely to recognize it? Probably not unless you dealt with Jacobson
Radicals almost every day. Algebraists do occasionally stumble into
Jacobson Radicals or other types of Radicals (yes, there are quite a
few more), but unless they happen to specialize in them (only
occasionally true), they tend to seldom see them.
Now think of all the thousands (or more!) of expressions in different
algebraic "objects", and you have a picture of the enormous data in
algebra, but the theory is relatively meagre. In fact, we've barely
scraped the surface of Jacobson Radicals in algebra, and its physics
applications are harder to find than a needle in a haystack. Keep
looking though, since Rare things are often quite valuable!
Osher Doctorow
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