What is the significance of Fermat's Last Theorem in modern mathematics?

[matt grime]

hyper surface? do you even know what one of those is? And therefore it is an operation not on the topological space but.. well, assuming it even has a meaningful notion of hypersurface, then an operation on something of dimension 'one fewer' anyone got an inner product lying around? Or a ring of algebraic functions to help define this hyper surface?

Here is a topological space (with a metric)

R^n ; n>17

Define a multiplication on R^n which allows you to do this specious nonsense with Fermat's LAst theorem? Make sure that you have at least an integral domain. Field would be nice... Heck, even a division algebra would be a start.

Wow, defining a manifold structure on the set of all sets, you're a genius, my cap is doffed.

 

[PFanalog57]

Originally posted by matt grime
hyper surface? do you even know what one of those is? And therefore it is an operation not on the topological space but.. well, assuming it even has a meaningful notion of hypersurface, then an operation on something of dimension 'one fewer' anyone got an inner product lying around? Or a ring of algebraic functions to help define this hyper surface?

Here is a topological space (with a metric)

R^n ; n>17

Define a multiplication on R^n which allows you to do this specious nonsense with Fermat's LAst theorem? Make sure that you have at least an integral domain. Field would be nice... Heck, even a division algebra would be a start.

Wow, defining a manifold structure on the set of all sets, you're a genius, my cap is doffed.

Let's definitely define, in accordance, with that which we have most excellently learned:

A topological space is a set X along with a happy family of subsets of X, called the open sets, requred to satisfy certain conditions, like the empty set and X itself are both open, if the subsets of X, U and V are open, so is the intersection of U and V, of course! And if the sets U_a of X are open, then so is the union of U_a. The collection of sets taken to be open is called the topology of X. An open set containing a point x, which is an element of X, is called a neighborhood of x. The complement of an open set is called "closed".

So with the use of topology it becomes possible to define continuous functions, and roughly speaking, a function is continuous if it sends nearby points to nearby points, of course. The conceptual notion of "nearby" can be made precise using open sets. Ergo a function f: X-->Y from one topological space to another is defined to be continuous because if we are given any open set U subset of Y, then the inverse image f^-1 U subset of X, is open. So the concept of manifold can be likened to that of a globe, whereby it may be covered with patches that look just like R^n.

A collection, U_a of open sets "covers" a topological space X if their union is all of X.

Well allrighty then, so, given topological spaces X and Y, there is a product X x Y, i.e. the product topology in which a set is open iff it is a union of sets of the form U x V, where U is open in X and V is open in Y .
If M is an m-dimensional manifold and N is an n-dimensional manifold, then M x N is an (m+n) dimensional manifold.

So simultanaety "S" is a spacelike hypersurface or "slice" through spacetime that cuts through event P, with a set of observers having worldlines crossing the simultanaety "simultaneously-orthogonally" having clocks that all read the same "proper" time at the instant of crossing.


The metric spaces are thus defined as being diffeomorphism invariant. Intersecting cotangent bundles[manifolds] are the set of all possible configurations of a system, i.e. they describe the phase space of the system. When the "wave-functions" intersect, and are "in phase", they are at "resonance", giving what is called the "wave-function collapse" of the Schrodinger equation.


Yes, is it possible to derive Einstein's field equation strictly in terms of quantum mechanical operators? using n-dimensional cross sections of cotangent vector spaces? Near a massive object M, the *isobar* cross sections increase in density as wavefunction density gradients, a possible solution? to Hartle and Hawking's "wavefunction of the universe"?

There is the Schrodinger equation: H(psi) = E(psi), where H is the Hamiltonian operator, the sum of potential and kinetic energies, and "psi" is the wavefunction. E is the energy of the system. The square of the wavefunction, is the probability of the position and momentum for the system.

The Wheeler DeWitt equation is the Schrodinger equation applied to the whole universe. Since the total energy of the universe is postulated to be zero[but is still not defined], the Wheeler DeWitt equation is: H(psi) = 0

There is a complementary path integral approach for this equation. The brilliant physicist Stephen Hawking, derived the wavefunction of the universe as a path integral, for a complex function of the classical configuration space:

psi(q) = integral exp(-S(g)/hbar) dg

The problem is that "dg" is not well defined either.

"exp" is the base of the natural logarithm "e" raised to a power.

The power in this case, is the quantity -S(g)/hbar, where S(g) is the Einstein Hilbert action. The Einstein Hilbert Action, is defined via the Lagrangian, which is the difference of kinetic and potential energies, and it has a formulation in general relativity:

Lagrangian = R vol

R is the Ricci scalar curvature of the metric g, derived by contracting the Ricci tensor and "vol" is the volume form associated to g.

The Einstein Hilbert action then becomes: S(g) = integral R vol

A topological group G, is a topological space such that the group multiplication ab = c is a continuous function from G x G into G and the operation inversion a^-1 = b is a continuous function from G into G, so one of the characteristic properties of a continuous function in a topological space, and if ab = c, with W being an open neighborhood of c, then there exist open neighborhoods U and V of a and b respectively, such that UV is a subset of W. UV is therefore defined as the set of all products a'b' where a' is an element of U and b' is an element of V.


Most definitely the standard operator T is bounded iff T transforms every point of the monad of the origin, 0, into a point in the monad of 0. Therefore a point is in the monad iff its norm is infinitesimal. The necessity of the condition is abundantly clear, and obviously so, because, T is bounded iff T transforms every point a in *B into a finite point.

||Ta|| =< ||T|| ||a||

If a is finite then Ta is also finite. Most definitely the standard operator T, is bounded iff T transforms every near standard point into a near standard point.

Hence a topological space is compact iff all points of *T are near standard.

Yes, a point is defined as "near standard" if there exists a standard point q such that p is an element of mu(q) , where mu(q) is the intersection of the set of open sets in T containing q.

The closed unit interval [0,1] is compact. But this finite unit is comprized of an infinite number of fractions... more later


Question:

Can light cone cross sections represent the operations of union and intersection analogously to Venn diagrams?

 

[ranyart]

Russ, when you state:
According to conventional theories, the surface area of the horizon surrounding a black hole, measures its entropy, where entropy is defined as a measure of the number of internal states that the black hole can be in without looking different to an outside observer, who must measure only mass, rotation, and charge. Another theory states that the maximum entropy of any closed region of space can never exceed one quarter of the area of the circumscribing surface, with the entropy being the measure of the total information contained by the system.

S' = S_m + A/4

So the "black hole" theorists came to realize that the information associated with all phenomena in the three dimensional world, can be stored on a two dimensional boundary, analogous to the storing of a holographic image.

Plus:Since entropy can also be defined as the number of states, that particles can be in within within a region of space, and the entropy of the universe must always increase, the next logical step is to realize that the spacetime density, i.e. the information encoded within a circumscribed region of space, must be increasing in the thermodynamic direction of time.

Of course, thermodynamic entropy is popularly described as the disorder or "randomness" in a physical system. In 1877, the physicist Ludwig Boltzmann defined entropy more precisely. He defined it in terms of the number of distinct microscopic states that the particles in a system can be configured, while still looking like the same macroscopic system. For example, a system such as a gas cloud, one would count the ways that the individual gas molecules could be distributed, and moving.

In1948, mathematician Claude E. Shannon, introduced today's most widely used measure of information content: entropy. The Shannon entropy of a message is the number of binary digits, i.e. "bits" needed to encode it. While the structure, quality, or value, of the information in Shannon entropy may be an unknown, the quantity of information can be known. Shannon entropy and thermodynamic entropy are equivalent. END-QUOTES.


I see a problem? starting here:Another theory states that the maximum entropy of any closed region of space can never exceed one quarter of the area of the circumscribing surface, with the entropy being the measure of the total information contained by the system.

S' = S_m + A/4

If Entropy(MAX) cannot exceed a 1/4 the Area, and Entropy being the shannon information of that area(being eqivilent to thermo) then ONE bit of anything needs THREE bits of nothing in order to be measured?

The ZERO-DIMENSIONAL TRI-QUARKs cannot be isolated thus! if one pulls hard enough on a Single Quark, there will be 3 times the Energy needed initally..but then the further one takes it from its Coupled partners..the increase in ENERGY IS EXPONENTIAL!

This is the Negative Probability of Feynmans doing, since you are trying to convey path integrals, I SUGGEST YOU GO AND SEEK OUT Negative Probability Functions.

Like I said here:
https://www.physicsforums.com/showthread.php?threadid=14123
You cannot class a 'rotation' as an added dimension?

The minimum 1 of:S' = S_m + A/4 relates to the positve function expressed as Entropy Arrows, the Negative ARROW tends towards a Singularity

 

[matt grime]

Thanks for the idiot's guide to basic point set topology. It is almost correct - I wouldn't say that continuity is about closeness in an arbitrary topological space - just take the discrete topology.


But you didn't define a reasonable algebraic structure on R^17 and do your fermat's last theorem thing there did you? Wonder why?

You didn't indicate you know what hypersurface means either.

 

[PFanalog57]

Originally posted by matt grime
Thanks for the idiot's guide to basic point set topology. It is almost correct - I wouldn't say that continuity is about closeness in an arbitrary topological space - just take the discrete topology.


But you didn't define a reasonable algebraic structure on R^17 and do your fermat's last theorem thing there did you? Wonder why?

You didn't indicate you know what hypersurface means either.

In simplest terms, a hypersurface is an [ n-1] dimensional space. For example, 3 dimensional space can be generalized to a 2-dimensional surface.

What the h**l is the relevance of R^17 ? 17 legs to the right triangle? All that is required is R^3 + 1 Then the [n-1] hypersurface is 2D.

x^n + y^n = z^n

 

[PFanalog57]

Hawking writes:

Information about the quantum states in a region of spacetime may be somehow coded on the boundary of the region, which has two dimensions less. This is like the way that a hologram carries a three dimensional image on a two dimensional surface.

So information is encoded on the 2-dimensional boundary of spacetime.

http://www.encyclopedia4u.com/t/topological-space.html



Any metric space turns into a topological space if define the open sets to be generated by the set of all open balls. This includes useful infinite-dimensional spaces like Banach spaces and Hilbert spaces studied in functional analysis.

http://www.encyclopedia4u.com/m/metric-space.html

Metric space

A metric space is a space where a distance between points is defined. It is a topological space.

Formally, a metric space is a set of points M with an associated distance function (also called a metric) d : M × M -> R (where R is the set of real numbers). For all x, y, z in M, this function is required to satisfy the following conditions:


d(x, y) ≥ 0
d(x, x) = 0 (reflexivity)
if d(x, y) = 0 then x = y (identity of indiscernibles)
d(x, y) = d(y, x) (symmetry)
d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).

These axioms express intuitive notions about the concept of "distance": distances between different spots are positive and the distance between x and y is the same as the distance between y and x. The triangle inequality means that if you go from x to z directly, that is no longer than going first from x to y, and then from y to z. In Euclidean geometry, this is easy to see. Metric spaces allow this concept to be extended to a more abstract setting.
A metric space in which every Cauchy sequence has a limit is said to be complete.

Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly normal). An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.

Two metric spaces (M1, d1) and (M2, d2) are called isometrically isomorphic iff there exists a bijective function f : M1 → M2 with the property d2(f(x), f(y)) = d1(x, y) for all x, y in M1. In this case, the two spaces are essentially identical. An isometry is a function f with the stated property, which is then necessarily injective but may fail to be surjective.

Every metric space is isometrically isomorphic to a closed subset of some normed vector space. Every complete metric space is isometrically isomorphic to a closed subset of some Banach space.


Distance between points and sets
If (M,d) is a metric space, S is a subset of M and x is a point of M, we define the distance from x to S as

d(x,S) = inf {d(x,s) : s ∈ S}
Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following generalization of the triangle inequality:
d(x,S) ≤ d(x,y) + d(y,S)
which in particular shows that the map x |-> d(x,S) is continuous.

 

[phoenixthoth]

so is it that the known universe is four dimensional with a 2 dimensional boundary? why are there 11 dimensions?

 

[PFanalog57]

Originally posted by phoenixthoth
so is it that the known universe is four dimensional with a 2 dimensional boundary? why are there 11 dimensions?

[zz)] [zz)] [zz)]

http://superstringtheory.com/basics/basic6a.html

T duality
The duality symmetry that obscures our ability to distinguish between large and small distance scales is called T-duality, and comes about from the compactification of extra space dimensions in a ten dimensional superstring theory. Let's take the X9 direction in flat ten-dimensional spacetime, and compactify it into a circle of radius R, so that



A particle traveling around this circle will have its momentum quantized in integer multiples of 1/R, and a particle in the nth quantized momentum state will contribute to the total mass squared of the particle as



A string can travel around the circle, too, and the contribution to the string mass squared is the same as above.
But a closed string can also wrap around the circle, something a particle cannot do. The number of times the string winds around the circle is called the winding number, denoted as w below, and w is also quantized in integer units. Tension is energy per unit length, and the the wrapped string has energy from being stretched around the circular dimension. The winding contribution Ew to the string energy is equal to the string tension Tstring times the total length of the wrapped string, which is the circumference of the circle multiplied by the number of times w that the string is wrapped around the circle.



where



tells us the length scale Ls of string theory.
The total mass squared for each mode of the closed string is



The integers N and Ñ are the number of oscillation modes excited on a closed string in the right-moving and left-moving directions around the string.
The above formula is invariant under the exchange



In other words, we can exchange compactification radius R with radius a'/R if we exchange the winding modes with the quantized momentum modes.
This mode exchange is the basis of the duality known as T-duality. Notice that if the compactification radius R is much smaller than the string scale Ls, then the compactification radius after the winding and momentum modes are exchanged is much larger than the string scale Ls. So T-duality obscures the difference between compactified dimensions that are much bigger than the string scale, and those that are much smaller than the string scale.
T-duality relates type IIA superstring theory to type IIB superstring theory, and it relates heterotic SO(32) superstring theory to heterotic E8XE8 superstring theory. Notice that a duality relationship between IIA and IIB theory is very unexpected, because type IIA theory has massless fermions of both chiralities, making it a non-chiral theory, whereas type IIB theory is a chiral theory and has massless fermions with only a single chirality.
T-duality is something unique to string physics. It's something point particles cannot do, because they don't have winding modes. If string theory is a correct theory of Nature, then this implies that on some deep level, the separation between large vs. small distance scales in physics is not a fixed separation but a fluid one, dependent upon the type of probe we use to measure distance, and how we count the states of the probe.
This sounds like it goes against all traditional physics, but this is indeed a reasonable outcome for a quantum theory of gravity, because gravity comes from the metric tensor field that tells us the distances between events in spacetime.

 

[PFanalog57]

Take two opposing mirrors as an overly simplistic example.

The number of self contained images increase, while the size of the images decrease.

[<-[->[<-->]<-]->]


A self including set.

 

[ranyart]

Originally posted by Russell E. Rierson
Take two opposing mirrors as an overly simplistic example.

The number of self contained images increase, while the size of the images decrease.

[<-[->[<-->]<-]->]


A self including set.

You are correct in this:Information about the quantum states in a region of spacetime may be somehow coded on the boundary of the region, which has two dimensions less. This is like the way that a hologram carries a three dimensional image on a two dimensional surface.


The thing is this:Take two opposing mirrors as an overly simplistic example.

The number of self contained images increase, while the size of the images decrease.

[<-[->[<-->]<-]->]


A self including set.

Has allready been contemplated here:http://groups.msn.com/Youcanseehomefromhere

And the Universe has 2-Dimensional locations where there is an intersection of dimensional projection.

We are used to our conceptions that our 3-D world is continuous and surrounds the micro-quantum fields, the question is which surrounds what and where. The space between Galaxies is made from 2-Dimensional fields, which surround a 3-D space containing our Galaxy, the action of Dimensionality constricts our Galaxy, the Galaxy 3-D is Embedded in a 2-D lattice, quite a Quantum Leap in perceptional thinking is needed to see this (dont you think!).

The Flat Geometric space between Galaxies is not the same generic make up as the Field-Space that is in and around Matter(Atoms).

When we look inwards from our 3-D space down to Matter, 3-Dimensions allways Surround the (QUANTUM-2-D and inner Vaccum/fields) matter we are observing. When we turn our observation around and look outwards into the Universe at Large, we are SURROUNDED by a 2-Dimensional Field(E-M-Vaccum), quite the reverse of our pre-conceptual Spacetime idealistic Galactic Notions.

 

[PFanalog57]

Take two opposing mirrors as an overly simplistic example.

The number of self contained images increase, while the size of the images decrease.

[<-[->[<-->]<-]->]


A self including set, all relations are "intrinsic". The "outside" = first iteration = alpha = 0. The final "last" iteration = omega = 0.

alpha = omega



http://superstringtheory.com/basics/basic6a.html

T duality

The duality symmetry that obscures our ability to distinguish between large and small distance scales is called T-duality,

If space is *quantized* yet also continuous, then it too, has the property called "wave-particle" duality. If space consists of indivisible units, then a measurement of space means that Fermat's last theorem holds, for it.

According to the Pythagorean theorem:

x^2 + y^2 = z^2

All possible integer solutions are then rerpresented as:

[a^2 - b^2]^2 + [2ab]^2 = [a^2 + b^2]^2

a^4 -2(ab)^2 + b^4 + 4(ab)^2 =

a^4 + 2(ab)^2 + b^4 = [a^2 + b^2]^2




all odd numbers can be represented as:

[a^2 - b^2] or Z^p - Y^p

if Y is an "even" natural n and Z is odd, same for a and b .

Fermat's last theorem, for integers a,b,Z,Y,p:

[a^2 - b^2]^p + Y^p = Z^p

[a^2 - b^2]^p = Z^p - Y^p

a^2 - b^2 = [Z^p - Y^p]^[1/p]

When Z^p - Y^p is a prime number, it cannot have an integer root.

a^2 - b^2 is not an integer, for [Z^p - Y^p]^[1/p] , for a,b,Z,Y,p, unless p = 2.


To every set A and every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds. This is the axiom that leads to Russell's paradox. For if we let the condition S(x) be: not(x element of x), then B = {x in A such that x is not in x}. Is B a member of B? If it is, then it isn't; and if it isn't, then it is. Therefore B cannot be in A, meaning that nothing contains everything.

This means that relativity holds in the "topological" sense and T-duality is correct.



Vector fields can be defined on manifolds in a concise "algebraic" way. Given an arbitrary vector field on R^n, the directional derivative of function f may be formed in the direction of vector v, denoted by vf.

x^1, ... , x^n can represent coordinates on R^ n[though eventually coordinate independence is preferred].

The "Einstein summation convention"

vf = v^1 @f/@x^1 + ... +v^n @f/@x^n

where @ denotes the partial derivative symbol.

Infinitely many differentiable functions may then be defined on manifold M


f+g = g+f

f+ [g+h] = [f+g]+h

f[gh] = [fg]h

f[g+h] = fg+fh

[f+g]h = fh+gh

1f = f

Vector Fields:

v[f+g] = v[f] + v[g]

v[af] = av[f]

v[fg] = v[f]g + fv[g]

The tangent vector space is infinitesimally distrubuted over M.


If memory serves, the distance formula for two dimensions:

ds^2 = g_11 dx^2 + 2g_12 dydx + g_22 dy^2

 

[PFanalog57]

A + B = C

A^2 + B^2 = C + A^2 - A + B^2 - B

A^2 + B^2 = [A+B] + A[A-1] + B[B-1]

A^p + B^p = [A+B] + A*[A^(p-1)-1] + B*[B^(p-1)-1]



3^2 + 4^2 = 5^2

3^3 + 4^3 = 5^3 - 34

34 is a Fibonacci number.

I wonder if the Fibonacci series can lead to a "proof" of Fermat's Last Theorem?

1,2,3,5,8,13,21,34,55,89,144,233,377,610,...


5^2 - 3^2 = 2^4

[2^2 - 1^2]^2 + [2*2*1]^2 = [2^2 + 1^2]^2

5^2 + [2*2*3]^2 = 13^2



Squares of fibonacci numbers give fibonacci numbers or multiples of them:


1^2 + 2^2 = 5

2^2 + 3^2 = 13

3^2 + 5^2 = 34

5^2 + 8^2 = 89

8^2 + 13^2 = 233

But Fibonacci cubes aren't so predictable?

1^3 + 2^3 = 9 = 8 + 1

2^3 + 3^3 = 35 = 34 + 1

3^3 + 5^3 = 152 = 144 + 8

5^3 + 8^3 = 637 = 1 + 5 + 21 + 610


These Fermat characters are HILARIOUS!


http://www.fermatproof.com/

Is the "proof" valid?

He appears to be using the rule of Pythagorean triples, trying to extend it out to all values of n > 2. Then using reducto ad absurdum to establish the absurdity of n > 2



Interesting diagrams with the right triangles

...2
...22
...222
..1111

= 4^2

2+2+2+2+2+2+1+1+1+1 = 4^2

6+4 = 10

1+2+3+4

10 + 5 = 15

10*2 + 5*1 = 5^2

15 + 6 = 21

15*2 + 6*1 = 6^2


N(N+1)/2 = 1+2+3+...+ N

Interesting...


Z = [X^p + Y^p]/[Z^(p-1)]

For p > 2, Z cannot be an integer...



That is the big question, why do two cubes not result in another cube?

3^2 + 4^2 = 6[4]+1

5^2 + 12^2 = 14[12]+1

7^2 + 24^2 = 26[24]+1

and

3^3 + 4^3 + 5^3 = 43[5]+1 = 6^3

3^3 + 4^3 = [45/2]*[4] +1






http://www.umcs.maine.edu/~chaitin/unknowable/ch1.html

At that same time that Turing shows that any formalism for reasoning is incomplete, he exhibits a universal formalism for computing: the machine language of Turing machines. At the same time that he gives us a better proof of Gödel's incompleteness theorem, he gives us a way out. Hilbert's mistake was to advocate artificial languages for carrying out proofs. This doesn't work because of incompleteness, because of the fact that every formal axiomatic system is limited in power. But that's not the case with artificial languages for expressing algorithms. Because computational universality, the fact that almost any computer programming language can express all possible algorithms, is actually a very important form of completeness! It's the theoretical basis for the entire computer industry!


[...]


And I begin to get the idea that maybe I can borrow a mysterious idea from physics and use it in metamathematics, the idea of randomness! I begin to suspect that perhaps sometimes the reason that mathematicians can't figure out what's going on is because nothing is going on, because there is no structure, there is no mathematical pattern to be discovered. Randomness is where reason stops, it's a statement that things are accidental, meaningless, unpredictable, & happen for no reason.

Interesting...


The "ABC conjecture" :

http://www.maa.org/mathland/mathtrek_12_8.html

 

[matt grime]

whenever people raise questions you can't answer or don't understand you post 'bored' smilies. Perhaps you are a genius. Perhaps you're an idiotic crank. I know where my money is

 

[PFanalog57]

Originally posted by matt grime
whenever people raise questions you can't answer or don't understand you post 'bored' smilies. Perhaps you are a genius. Perhaps you're an idiotic crank. I know where my money is

That appears to be a "bi-valent" logic statement.

What if I am not a genius and I am not a crank?




I could say that you have no sense of humor or that you are

But I would probably be wrong[I hope] [?]

 

[matt grime]

Originally posted by Russell E. Rierson
In simplest terms, a hypersurface is an [ n-1] dimensional space. For example, 3 dimensional space can be generalized to a 2-dimensional surface.

What the h**l is the relevance of R^17 ? 17 legs to the right triangle? All that is required is R^3 + 1 Then the [n-1] hypersurface is 2D.

x^n + y^n = z^n

You were attempting to define a well-defined algebraic operation on an arbitrary manifold, ie turning it into field or division algebra, at least that was one interpretation and you didn't present any explanation of what you wre doing. No such structure exists fo R^{17}, as most mathematicians know (division algebra structures exist for 2,4,8,16 degree extensions of R and no others), so where does your Fermat Last Theorem argument get you here? Your last 'sentence' is numerically challenged.

If you are neither an idiot nor a crank I'd lose that money. But then I just dislike people posting huge quoted bits of text for neither rhyme nor reason and not justifying any of their bizarre assertions, so I might be letting it affect my judgement.

 

[PFanalog57]

as most mathematicians know (division algebra structures exist for 2,4,8,16 degree extensions of R and no others)

n-1 = 2


Information may be stored on the two dimensional boundary of space, much like a holographic image.

Space is a perception of separation between points, perception being cognitive processing of the "mind".

Now we come to the "Universal Set", which is analogous to a Universal Algorithm.

Algorithms contain information.

The abstract[algorithm] contains the concrete[units-bits of information]

Quite simple really, the "Platonic form" is an abstract description of a concrete object or space[topology].

The Platonic form is isomorphic to the concrete instantiation.

P = Platonic form

C = Concrete object or space

P<---->C

P[C] = C[P]

 

[phoenixthoth]

while it may be correct, can you give like some information about the isomorphism to further prove that the two realms are isomorphic? like in what context are they isomorphic (as sets, as groups, as vector spaces, etc.)?

 

[PFanalog57]

Originally posted by phoenixthoth
while it may be correct, can you give like some information about the isomorphism to further prove that the two realms are isomorphic? like in what context are they isomorphic (as sets, as groups, as vector spaces, etc.)?

Well phoenix, I have been trying to explain, much to the chagrin
of

An algorithm contains the information with regards to the necessary and sufficient topologies-geometries, RELATIONS, and other "bijections" that are required. Algorithms correspond to configurations.

I recall you posted the equation f[x] = x

Say F = U , where F is a particular calculation that feeds back into itself.

F = U

G[F] = U

U[...H[G[F]]] = U

A Platonic form is an abstract "thought" form, or configuration, where if it has the necessary internally self consistent "logics" to sustain its own existence, then it actually "exists". The Platonic thought form is also known as the Schrodinger wave function. A damped oscillation.

As I explained a couple of posts back, while being rudely interrupted and insulted, the isomorphism is analogous to two opposing mirrors, where the images are intersecting.

[<-[->[<-[-><-]->]<-]->]

The real physics of the above diagram corresponds as a one to one and onto correspondence, which creates a condition of standing-wave resonance. The optimal configuration is in accordance with the "action principle". The axiom of choice is governed by an operator that forces it to "choose" the most optimal configuration.

So I know that you are strictly a mathematician, but my advice is to grapple with the physics also, if you are going to "prove" the Absolute Infinity...

The information density of the universal system must be increasing. The increase of information density is analogous to a pressure gradient.

[density 1]--->[density 2]--->[density 3]---> ... --->[density n]

The universal laws of nature are explained in terms of symmetry identities. The completed infinities, mathematician Georg Cantor's infinite sets, could be explained as cardinal identities, akin to "qualia" [Universally distributed attributes] distributing over their universes in accordance with a guiding principle.

Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness. Properties, or "attributes" like red are numbers in the sense that they interact algebraically according to algebraic laws. Take one object away from the set of red objects and the distributive identity "red" still describes the set. The distributive identity[attribute] "natural number" or "real number" describes an entire collection of individual objects.

The alephs can be set into a one to one correspondence with a proper subset of of themselves.

On the other hand, a finite set of objects becomes less than its previous self, if an element is removed from it.


[abstract representation]-->[semantic mapping]-->[represented system]


An abstract representation is exactly that, "abstract". It is not a space, or time, but is instead a product of consciousness, or a mental construct.

On one level of stratification, two photons are separate. On another level, of stratification, the photons have zero separation.

Instantaneous communication between two objects, separated by a distance interval, is equivalent to zero separation[zero boundary] between the two objects.

According to the book "Gravitation", chapter 15, geometry of spacetime gives instructions to matter telling matter to follow the straightest path, which is a geodesic. Matter in turn, tells spacetime geometry how to curve in such a way, as to guarantee the conservation of momentum and energy. The Einstein tensor[geometric feature-description] is also conserved in this relationship between matter and the spacetime geometry. Eli Cartan's "boundary of a boundary equals zero."

The Topological spaces are defined as being diffeomorphism invariant. Intersecting cotangent bundles[manifolds] are the set of all possible configurations of a system, i.e. they describe the phase space of the system. They have "zero" boundary...

Waves are then abstract distributions and particles are convergent "concrete" localizations.


Here is what Max Tegmark says:

http://www.sciam.com/article.cfm?articleID=000F1EDD-B48A-1E90-8EA5809EC5880000&pageNumber=7&catID=2

As a way out of this conundrum, I have suggested that complete mathematical symmetry holds: that all mathematical structures exist physically as well. Every mathematical structure corresponds to a parallel universe. The elements of this multiverse do not reside in the same space but exist outside of space and time. Most of them are probably devoid of observers. This hypothesis can be viewed as a form of radical Platonism, asserting that the mathematical structures in Plato's realm of ideas or the "mindscape" of mathematician Rudy Rucker of San Jose State University exist in a physical sense. It is akin to what cosmologist John D. Barrow of the University of Cambridge refers to as "ð in the sky," what the late Harvard University philosopher Robert Nozick called the principle of fecundity and what the late Princeton philosopher David K. Lewis called modal realism. Level IV brings closure to the hierarchy of multiverses, because any self-consistent fundamental physical theory can be phrased as some kind of mathematical structure.

I certainly hope that you understand the "relevance" of the above quote...







Scientific objectivity for "Plato":

http://www.mmsysgrp.com/stefanik.htm

Quote:

Thesis: Scientific objectivity is best characterized by the concept of invariance as explicated in category theory than the concept of truth as explicated in mathematical logic.

 

[Deeviant]

You know I was reading through this thread, tepidly amused while I happened to read my coffee cup and Russell's theory began make more and more sense.

What did the coffee cup say?

If you can't dazzle them with brilliance, bury them in bull****.

 

[digiflux]

Fermat's Last Theorem

That solution to Fermat's Last Theorem is my father's work.

My father (photo included) was an electrical engineer who worked at Vought on rocket missile systems. He was really smart and I have total faith in his theorem www.fermatproof.com . Please forward this link to interested parties.

Thanks!

 

[PFanalog57]

Why not put the Fermat equation in terms of one variable, where the polynomial must have positive integer roots, for degree n = prime ?

f[x] + g[x] = h[x]

f[x] = x^p

g[x] = [x+y]^p

h[x] = [x+z]^p

where y and z are arbitrary integer constants > 0.


f[x]+g[x] = h[x]


1 + g[x]/f[x] = h[x]/f[x]


{f[x]g'[x] - g[x]f '[x] }/{f[x]}^2 =

{f[x]h'[x] - h[x]f '[x] }/{f[x]}^2


divide out the {f [x]}^2

rearrange terms

f[x]{g'[x] - h'[x]} = f '[x]{g[x] - h[x]}

f[x]/f '[x] = {g[x] - h[x]}/{g'[x] - h'[x]}

f[x]/f '[x] also equals:

{h[x] - g[x]}/{h'[x] - g'[x]}


[1.] x^p + y^p = z^p

differentiate [1.]

p*x^[p-1] * x' + p*y^[p-1] * y' = p*z^[p-1] * z'

divide both sides by p

[2.] x^[p-1] * x' + y^[p-1] * y' = z^[p-1] * z'

[1.] - [2.] & factor

x^[p-1] *[xy'-yx'] = z^[p-1] *[zy'- yz']

[xy' - yx'] = z^[p-1] *[zy'-yz']/x^[p-1]

Since x,y,z are relatively prime, z^[p-1] does not divide x^[p-1]

interesting...

 

[matt grime]

non-formally differentiating functions of integers is it now?

 

[PFanalog57]

Originally posted by matt grime
non-formally differentiating functions of integers is it now?

No, in the last paragraph, x,y,z are polynomial functions of x.

 

[PFanalog57]

Types are entities built up in a finite number of steps, beginning with an arbitrarily chosen object, 0, also identifiable with the number 0, in accordance to certain rules.

[1.]

0 is a type

[2.]

If n is a positive integer and t_1,...,t_n are types then the sequence [t_1,...,t_n] is a type. Types can be assigned to all individuals, sets, and relations, as needed, with regards to the universe under consideration.

[3.]

Let the set of all types be T.

If, for every type t, B_t, contains all relations of A_t, then the higher order structure, M, which is any set, B_t, that is indexed in T( or basically function defined on T ), such, that for any non-empty set of individuals A_t, where B_t is a subset of A_t, with every element of A_t appearing as a function satisfying certain necessary and sufficient conditions, then the higher order structure M, is full.

If no B_t contains any relation repeatedly, such that the value a function, f, takes on each value at most "once", then the higher order structure is normal. Loosely speaking, a structure is full and normal if B_t = A_t for all t.

At the very rock bottom of the set theory universe, is the singular entity called the "empty set" At first there is nothing at all, with possibly an undefined, unknown, realm in the reverse direction of the empty set, such that the imagination runs amok contemplating the possibilities.

The whole universe of set theory is constructed from the empty set, and since it is the raw idea of set formation, the set theory universe in ever increasing levels of complexity.

The empty set is the simplest set, which is the set that has no elements, represented as { }

The next most elementary set is the set containing the empty set, i.e. { {} }

etc...etc...etc...


A topological group G is a topological space, and also a group, such, that group multiplication ab = c is a continuous function from G x G into G and the inverse operation a^-1 = b is also a continuous function in a topological space. Yes, it is much to the bewilderment and chagrin of certain friends at physics forums. So one of the defining characteristics of a topological space, is, that if ab = c and W is an open neighborhood of c, then there exist open neighborhoods U and V of a and b respectively, such that UV is a subset of W.

 

[PFanalog57]

Pythagorean triples:

http://grail.cba.csuohio.edu/~somos/rtritab.txt

The Fibonacci numbers are also within Pythagorean triples

a^2 + b^2 = c^2 .

15^2 + 8^2 = 17^2

35^2 +12^2 = 37^2

99^2 + 20^2 = 101^2

255^2 + 32^2 = 257^2

675^2 + 52^2 = 677^2

1763^2 + 84^2 = 1765^2

4623^2 + 136^2 = 4625^2

etc.

At closer inspection:

(4^2-1)^2 + (2*4)^2 = (4^2+1)^2

(6^2-1)^2 + (2*6)^2 = (6^2+1)^2

(10^2-1)^2 + (2*10)^2 = (10^2+1)^2

(16^2-1)^2 + (2*16)^2 = (16^2+1)^2

(26^2-1)^2 + (2*26)^2 = (26^2+1)^2

(42^2-1)^2 + (2*42)^2 = (42^2+1)^2

(68^2-1)^2 + (2*68)^2 = (68^2+1)^2

etc...

Notice:

4 = 2*2

6 = 2*3

10 = 2*5

16 = 2*8

26 = 2*13

42 = 2*21

68 = 2*34

2*Fibonacci

etc...

Interesting...




While researching "squares" of numbers I find that ratios of certain types of squares always involve something like a + (2b)^(1/2).

I found many squares that have successive ratios that tend towards the number
3 + (8)^(1/2)

This is of course one solution to the polynomial x^2 - 6x + 1 = 0.

The challenge is to answer the question "why"
are these squares ratios tending towards
3 + (8)^(1/2) as the numbers are getting larger?

I will give some "lists" of these squares!

3^2 + 4^2 = 5^2

20^2 + 21^2 = 29^2

119^2 + 120^2 = 169^2

696^2 + 697^2 = 985^2

4059^2 + 4060^2 = 5741^2

23660^2 + 23661^2 = 33461^2

137903^2 + 137904^2 = 195025^2

Now the ratios

29/5 = 5.8

169/29 = 5.827586207...

985/169 = 5.828402367...

5741/985 = 5.828426396...

33461/5741 = 5.828427103...

195025/33461 = 5.828427124...

3 + (8)^(1/2) = 5.828427125...

This number also appears for the successive ratios of any of the three numbers
(a, b, c), a2/a1, b2/b1, or c2/c1 of the Pythagorean triples on the above list.

Here are some more...

3 = 2^2 - 1

17 = 4^2 + 1

99 = 10^2 - 1

577 = 24^2 + 1

3363 = 58^2 - 1

19601 = 140^2 + 1

114243 = 338^2 - 1

665857 = 816^2 + 1

The ratios

17/3 = 5.666666667...

99/17 = 5.823529412...

577/99 = 5.828282828...

3363/577 = 5.828422877...

19601/3363 = 5.828427...

114243/19601 = 5.828427121...

665857/114243 = 5.828427125...

Also approaching 3 + (8)^(1/2)




The mathematicians tell me "no-problemo" an easy problem!

This is what one mathematician explained to me...

Hi, this is easy to explain.

Notice that your equation is basically of the form

a^2 + (a+1)^2 + 1 = b^2

Never mind, for the moment, that you are considering consecutive ratios of b. First start by trying to understand which a's and b's can be solutions to this equation.

Well, this equation is the same as:

2a^2 + 2a + 1 - b^2 = 0.

Which is the same as

4a^2 + 4a + 2 - 2b^2 = 0

which is

(2a+1)^2 -2b^2 = -1

This is a special case of Pell's Equation

X^2 - 2Y^2 = -1 (i.e. where X=2a+1 and Y=b).

Now observe that (X,Y) = (1,1) is a solution.

It turns out that all the solutions
(X_k, Y_k) to this equation are given by

X_k + sqrt(2) Y_k = (1+sqrt(2))^k where k is odd.

Now your question relates to those X_k which are odd. Now compute some examples with k = 1,3,5,7,9,11,13... you will see a pattern. This explains your observations.

1+sqrt2)^2 = (3 + sqrt8)

(1+sqrt2)^3 = (7 + sqrt50)

(1+sqrt2)^4 = (17 + sqrt288)

(1+sqrt2)^5 = (41 + sqrt1682)

(1+sqrt2)^6 = (99 + sqrt9800)

(1+sqrt2)^7 = (239 + sqrt57122)

(1+sqrt2)^8 = (577 + sqrt332928)


The Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233...

1+0 = 1

1+1 = 2

1+2 = 3

2+3 = 5

3+5 = 8

5+8 = 13

8+13 = 21

etc...

Fibonacci squares... also generate Fibonacci numbers...

1^2 - 0^2 = 1

1^2 + 1^2 = 2

2^2 - 1^2 = 3

2^2 + 1^2 = 5

3^2 - 1^2 = 8

3^2 + 2^2 = 13

5^2 - 2^2 = 21

5^2 + 3^2 = 34

8^2 - 3^2 = 55

8^2 + 5^2 = 89

13^2 - 5^2 = 144

13^2 + 8^2 = 233

21^2 -8^2 = 377

21^2 + 13^2 = 610

etc...

The consecutive ratios of these numbers... are of course approaching (1 + sqrt(5))/2

The "golden" ratio...

Sum the digits of 2^n and get a repeating pattern:

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16, 1+6 = 7

2^5 = 32, 3+2 = 5

2^6 = 64, 6+4 = 10, 1+0 = 1

2^7 ---> 2

2^8 ---> 4

2^9 ---> 8

2^10 ---> 7

2^11 ---> 5

2^12 ---> 1

2^13 ---> 2

2^14 ---> 4

etc...

etc...

etc...