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Author
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Topic: Fermat
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chimp
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Member # 333
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posted 19. April 2004 05:24
It is clearly self evident, that, if x = something, then x cannot equal ...nothing.
Let x = something
Let not-x = nothing
[ x or not-x] is a tautology
Let [x or not-x] = T(x)
If T(x) then A
If T(not-x) then B
[A or B] = T(T(x))
etc...
A set is a entity with an identity that distributes over its elements.
The set of all chickens is predicated by the "chickeness" of its members.
C[a,b,c,...,n] = [Ca, Cb, Cc,...Cn]
The set becomes a nonparadoxical member of itself due to its distributive identity, an informational structure called abstract containment. The abstract contains the concrete as the concrete contains the abstract.
The identity relates to all members of the set. Einstein said that space and time become modes by which we think, not conditions in which we live, if memory serves.
DNA is an algorithm, a finite set of instructions, which can construct a carbon based life form.
The life form physically contains the DNA and the DNA contains the life form in an "abstract" sense.
So a metric space with distance function r(x,y) involes the real numbers R, allowing the metric space to be embedded in the full structure of manifold, M.
We humans need a quantum field theory of curved spacetime, where the interlocking, discrete, yet causally connected structure of spacetime can be described as a manifold, M, with a metric tensor g_ab. Ergo, the difficulty of formulating a classical background metric is taken care of by the background independence of GR.
Of course the breakdown of the "theory" occurs when spacetime curvature approaches the "Planck" scale. Thus the exact criteria necessary for a valid quantum field theory of curved spacetime requires a quantum theory of gravity. Some potential candidates are loop quantum gravity and string theory...
It appears that a quantum theory of a field system differs from a quantum theory of a particulate system due to the fact that a field system has infinitely many degrees of freedom. Yet, for a system with finitely many degrees of freedom, the kinematic structure of the spacetime in question can be determined by the canonical commutation relations for the position and momentum operators. So the "square matrix" hermitian operators determine up to unitary equivalence the position and momentum observables,
...BUT, it appears there could be problems with formulating a spacetime structure that has infinitely many degrees of freedom.
In general curved spacetime, there does not appear to be any preferred notion of "particles". So for a noncompact space, where the natural notions of particles are available in the asymtotic past and the asymptotic future, the canonical commutation relations corresponding to the two, will, be generally unitarily inequivalent, analogously to the phenomenon of the
ultraviolet catastrophie of quantum electrodynamics.
So if my interpretation is correct, it is necessary to take an algebraic approach, which allows one to consider all states arising in all of the unitarily inequivalent Hilbert space constructions on an equal basis.
So spacetime is locally Euclidean[flat] which allows a universal representation from flat spacetime to curved spacetime, the necessary elements of Poincare symmetry to logically define the canonical commutation relations.
Getting back to the question, "What is a set?", we realize that sets are by definition generalized entities, corresponding to a non-condradictory state of affairs, allowing for meaningful interpretation.
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chimp
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Member # 333
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posted 20. April 2004 14:18
x^2 + 1 = 0
x = sqrt[-1] = i
i^i = e^[-pi/2]
[1+i^i]*[1-i^i] = [e^pi - 1]/[e^pi]
A physical system is described by a normalized vector[state vector] in Hilbert space. All possible information can be known about the system, since, for every physical observable there corresponds a self adjoint operator in Hilbert space.
The only allowed physical results of measurements of some obervable U, are the elements of the spectrum of the operator which corresponds to U.
So all properties of a number may not be completely known, but that which is known, must be specifiable on logical or analytic grounds.
Now if one is trying to say that mathematics is inherently random at its foundations, you must define what randomness is ...exactly.
Take a coin toss for example, as the number of flips of the coin increase
HTTTHTHTHHTHT.....HTHTHTHTHT...
The probability becomes an "exact" number at an infinite limit. 1/2
So a number becomes an identity in the Platonic aeon.
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Claire
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Member # 725
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posted 20. April 2004 22:59
All properties of the value of a physical system may not be completely known because all numbers representing them are just that, a representation, of a part of a value, were the vaule in question is fundamentally called physics. Why say a physical system? Because when physics is a value that is partially represented by number, that number that is doing so might appear to assure us we know the system in full because the number is in full, regardless of whether or not what is being represented by it is full or not. More possible information could be known about a system when we know what and how we can represent all the newer possible physical values about it by new representation. When we use a self adjoint operator we are doing so because we observe the space that is co incorp into the operator. Properties of numbers may not be completely known because there is more stress or importance between known observable technique we are allready familiar with and its expected results, not what new observables could be about and their unexpected outcomes in comparison to allready used ones. Logic on analytical grounds too early could restrict how we create new representations about a physical sytem. Logic later on could round up what new results we create. But we need not stop there because there could be more to know.
Claire
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chimp
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Member # 333
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posted 21. April 2004 02:43
An infinite number of coin flips gives an equal amount of heads and an equal amount of tails.
[1/2 H and 1/2 T]*n, for n--->oo
A radioactive nucleus decays in accordance with probability P within time t_0 to time t_1
Probability P becomes a timeless mathematical entity governing the future iterations of events at time t. There exists a spectrum of possibilities for the observed quantities. Certain deterministic factors become contingent with respect to uncertainty, DxDp >= h .
An infinite number of observations of the radioactive decay, converges to an exact number for t?
Wave function probability density =
|psi (r, t)|^2
The physical meaning of the expectation value, is the value that would be found by taking the average of many measurements of the observables in question, on a large collection of systems all in the state psi. the individual results are weighted by the probability.
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chimp
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Member # 333
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posted 21. April 2004 06:14
It seems that many valued logic must be formulated in terms of a stable 2-valued logic background.
Suppose the limit as n-->oo, s_n = s, in the classical sense. It must then be demonstrated that s_n - s is infinitesimal for all infinite n. That is to say, for any epsilon > 0 and for any infinite natural number n , it must be proved that |s - s_n| < epsilon.
For any given epsilon > 0 in R there exists a natural number v in N such that
|s_n - s| < epsilon for n > v, n is an element of N.
For all x, if x is an element of N and x > v then |s_x - s| < epsilon.
Since any infinite natural number is greater than v it can be deduced that |s_n - s| < epsilon for all infinite n.
so if I flip a coin, it will be Heads H, or not-Heads, ~H
So if the coin lands on its side, it is still H or ~H, this being the case that it is ~H. Absolutely true.
So if I go for a more specific multivalued logic it becomes H, ~H, S.
H or ~H is still true.
H or ~H or S is just adding more specification...?
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chimp
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Member # 333
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posted 25. April 2004 00:18
quote:
Evan wrote:
Is there anything special about the seuence 0, 10, 40, 155, 624? other than the fact that they are the numbers that make things work?
X^2 :
3^2 = 2*(1+2)+3
4^2 = 2*(1+2+3)+4
5^2 = 2*(1+2+3+4)+5
6^2 = 2*(1+2+3+4+5)+6
X^3 :
3^3 = 3*(1*2 + 2*3)+3
4^3 = 3*(1*2+2*3+3*4)+4
5^3 = 3*(1*2+2*3+3*4+4*5)+5
6^3 = 3*(1*2+2*3+3*4+4*5+5*6)+6
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chimp
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Member # 333
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posted 26. April 2004 04:18
5^2 = 4*5/2 + 5*6/2
13^2 = 12*13/2 + 13*14/2
17^2 = 16*17/2 + 17*18/2
25^2 = 24*25/2 + 25*26/2
5^2 = 10+15
13^2 = 78+91
17^2 = 136+153
25^2 = 300+325
1+2+3+...+N = N*[N+1]/2
X^2 = X*[X-1]/2 + X*[X+1]/2
X^3 = X*[X^2-1]/2 + X*[X^2+1]/2
X^4 = X*[X^3-1]/2 + X*[X^3+1]/2
X^n = X*[X^(n-1) -1]/2 + X*[X^(n-1) + 1]/2
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chimp
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Member # 333
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posted 16. May 2004 20:16
The General Equation?
3*(3+4) + 4*|4-3| = 5^2
3*(3+4) + 4*1 = 5^2
3*(3+4) + 4*(1 + 5^2) = 5^3
3*(3+4) + 4*(1 + 5^2 + 5^3) = 5^4
3*(3+4) + 4*(1 + 5^2 + 5^3 + 5^4) = 5^5
etc...
5*(5+12) + 12*|12 - 5| = 13^2
5*(5+12) + 12*(7 + 13^2) = 13^3
5*(5+12) + 12*(7 + 13^2 + 13^3) = 13^4
etc...
The equation? :
p is a prime number > 2.
z^p = x*(x+y) + y*( |y-x| +...+ z^(p-1) )
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