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

[PFanalog57]

x^n + y^n = z^n

x^3 + y^3 = (x+y)[x^2-xy+y^2]

A more general identity for primes p, and numbers, n:

A*B means A times B

A/B means A divided by B

+ and - , we all know...

follow the order of operations:


x^p + y^p

equals

(x+y)*[(x+y)^(p-1) + [{(x^p+y^p)/(x+y)} - (x+y)^(p-1)] ]

for all p and n >= 1

Question:

Let T be a metric space with distance function r(x,y) expressing the definitive predication that involves T with the real numbers, R. Therefore the juxtaposition of left and right hemispheres resonates in perfect accordance with the proposition that T and R are embedded simultaneously in the full structure of manifold M. Ergo we pass on to an enlargement *M of M, whereby the non-standard metric space is diffeomorphism invariant.

So if f(x) is a homeomorphism from T onto S, then for every point p in T, does f(u(p) = u(f(p)) ?


A metric space is a set of points such that for every pair of points, there is a nonnegative real number called their distance that is symmetric, and satisfies the triangle inequality, which states that the sum of the measures of any two sides of any triangle is greater than the measure of the third side. Space is then a tranformation[invariant]. Two objects with relative velocity will have a relative measure that transforms into the other. In effect, the separation does not exist in an extrinsic sense. ABC = BCA = CAB

Utilizing the generalized equation:

x^3 + y^3 = (x+y)*[(x+y)^2 - 3xy]

x^5 + y^5 = (x+y)*[(x+y)^4 -5x^3 y -5(xy)^2 -5x y^3 ]

x^7 + y^7 = (x+y)*[(x+y)^6 -7yx^5 -14x^4 y^2 -21(xy)^3 -14x^2 y^4 - 7xy^5 ]

In general, x^p + y^p = (x+y)*[(x+y)^(p-1) - p*f(x,y) ]

 

[HallsofIvy]

So if f(x) is a homeomorphism from T onto S, then for every point p in T, does f(u(p) = u(f(p)) ?

Well, I don't know. You didn't tell us what "u" is.

Two objects with relative velocity will have a relative measure that transforms into the other. In effect, the separation does not exist in an extrinsic sense. ABC = BCA = CAB

"relative velocity"? How did "velocity" get into a discussion of metric spaces? In any case, I have no idea what you mean by "ABC = BCA= CAB" because you have not defined ABC, BCA, or CAB.

x^3 + y^3 = (x+y)*[(x+y)^2 - 3xy]

What are x and y? They are not points in the metric space you mentioned before because such operations are not defined in a general metric space. They certainly aren't numbers that equation simply isn't true for numbers.

 

[matt grime]

Erm, what the hell are half the words you use meant to mean?

Definitive predication that involves T with the real numbers?


What's S?

Why does your metric space have a ring structure?

Where did M come from? why must R embed into it? How does a general metric space embed into some manifold?

 

[modmans2ndcoming]

what precisly are you trying to make a point of here?

your title has Fermat in it however, no where do you use Fermat's last theorem in anything, you only use the function form.

 

[PFanalog57]

Very interesting:

http://www.space.com/scienceastronomy/time_theory_030806.html
quote:
--------------------------------------------------------------------------------
"There isn't a precise instant underlying an object's motion," he said. "And as its position is constantly changing over time -- and as such, never determined -- it also doesn't have a determined position at any time."
--------------------------------------------------------------------------------
Heisenberg uncertainty: DxDp >= hbar/2 As the observed expansion of the universe continues, and, as the mean temperature of the universe continues to approach absolute zero, could the universe transform into a condition analogous to a "Bose Einstein condensate"?
At the foundation of classical logic is the law of excluded middle,
A|~A|A V ~A
T_|F||__T
F_|T||__T
This forms an invariance principle which is a symmetry
T|F = F|T = T

John Nash gives a most excellent equation for spacetime:


http://www.math.princeton.edu/jfnj/texts_and_graphics/Equation_an_Interesting/note2
http://www.math.princeton.edu/jfnj/texts_and_graphics/Equation_an_Interesting/equation.gif
http://www.math.princeton.edu/jfnj/texts_and_graphics/

Remarks added 22 May 2002: The remarks below are given as they were in a (memo) note that wasn't generally accessible. Now I am not really updating it, but since the equation (vacuum) itself is now included on my "web page" it is time also to include these remarks. At the present time I think the "input" of the gravitational action of matter, etc. might be studied in terms of boundary value problems. Then on one side of a boundary there could be the vacuum equation to be satisfied. And there are some ideas that relate to this. But these are ideas that call for further study.
3 memo of May 31, 2001: The equation is tensor equation which has a parallel or similarity to "wave" equations and can be described in terms of a d'Alembertian operator. It is thought of as of interest as an alternative description of the general relativistic space-time continuum that allows for "compressional" waves rather than allowing only for "transverse" waves. At the present time I am still seeking to find a good "input relation" for matter as the source of gravitation (analogous to the relation found by Einstein and Hilbert for the 2nd order tensor equation of standard GR). The vacuum equation can be described as having (on a LHS side that is equated to zero) a fourth order term formed by the covariant d'Alembertian operating on the G-tensor of Einstein plus an additive portion of second order (as to the differentiation) formed by quadratic combinations of curvature tensor elements. The precise additive portion or set of terms is defined by the condition that the total LHS is so structured so as to be formally divergence free (like the G-tensor is intrinsically divergence free). The plan is to put into this directory, ultimately, files of graphic type including the tensor equations in handwritten form.

Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness[qualia]. Properties, or "attributes" like red are numbers in the sense that they interact algebraically according to the laws of Boolean algebra. Take one object away from the set of red objects and the distributive number "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. The "infinite" Cantorian alephs are really distributive[qualia].

Yet, if we have a finite set of 7 objects, the cardinal number 7 does not really distribute over its individual subsets. Take anything away from the set and the number 7 ceases to describe it[wave function collapse-condensation into specific localization].

Symmetry is analogous to a generalized form of self evident truth, and it is a distributive attribute via the laws of nature, being distributed over the entire system called universe. A stratification of Cantorian alephs with varying degrees of complexity. Less complexity = greater symmetry = higher infinity-alephs. So the highest aleph, the "absolute-infinity" distributes over the entire set called Universe and gives it "identity".

The highest symmetry is a distributive mathematical identity[also a total unknown but possibly analogous to a state of "nothingness"]. This fact is reflected in part, by the conservation laws.

So an unbound-infinite-potentia and a constrained-finite-bound-actuality, are somehow different yet the same. The difference and sameness relation is a duality. Freedom(higher symmetry) and constraint-complexity-organizational structure(lesser symmetry) form a relation that can be described by an invariance principle.

On a flat Euclidean surface, the three angles of a triangle sum to 180 degrees. On the curved surface of a sphere, the three angles add up to more than 180 degrees. On the hyperbolic surface of a saddle they sum to less than 180 degrees. The three types of surface are not equivalent.

There is a rotational invariance for a triangle, that seems to hold for the three types of surface though.

ABC = BCA = CAB

CBA = BAC = ACB


[ 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. Topologically it is equivalent to a "point". The abstract description contains the concrete topology. Likewise, the concrete contains the abstract.

A duality.

A point contains an infinite expanse of space and time?

Could it be, that the "absolute" infinity, is actually a dimensionless point? Or more correctly, an "infinitesimal"?

Universe? = Zero?


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."

A point can be defined as an "infinitesimal". 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.

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

Quantum mechanics leads us to the realization that all matter-energy can be explained in terms of "waves". In a confined region(i.e. a closed universe or a black hole) the waves exists as STANDING WAVES In a closed system, the entropy never decreases.

The analogy with black holes is an interesting one but if there is nothing outside the universe, then it cannot be radiating energy outside itself as black holes are explained to be. So the amount of information i.e. "quantum states" in the universe is increasing. We see it as entropy, but to an information processor with huge computational capabilities, it is compressible information.

Quantum field theory calculations where imaginary time is periodic, with period 1/T are equivalent to statistical mechanics calculations where the temperature is T. The periodic waveforms that are opposed yet "in phase" would be at standing wave resonance, giving the action.

Periodicity is a symmetry. Rotate into the complex plane and we have
real numbers on the horizonal axis and imaginary numbers on the
vertical axis. So a periodic function could exist with periodicity
along both the imaginary AND the real axis. Such functions would have
amazing symmetries. Functions that remain unchanged, when the complex
variable "z" is changed.

f(z)---->f(az+b/cz+d)

Where the elements a,b,c,d, are arranged as a matrix, forming an
algebraic group. An infinite number of possible variations that
commute with each other as the function f, is invariant under group
transformations. These functions are known as "automorphic forms".

Topologically speaking, the wormhole transformations must be
invariant with regards to time travel. In other words, by traveling
backwards in time, we "complete" the future, and no paradoxes are
created.

So when spacetime tears and a wormhole is created, it must obey
certain transformative rules, which probably appear to be
discontinuities from a "3-D" perspective, but really, these
transformations are continuous!

So the number of holes[genus] on the surface of space, determine
whether there exist an infinite, or finite, number of solutions to
the universal equations?

 

[modmans2ndcoming]

ok, now I get it.

since X,Y, and Z are irrational numbers according to Fermat's Theorem, then at any given time, you can not be measured at a single point in space.

that makes sense.

 

[matt grime]

wow,russell, that must count as the longest and most off topic reply in the history of everything ever in forums.

Nash, quantum, qualia, GR, infinity, completeness.

Shame none of them bears any relation to the criticisms of your post.

 

[PFanalog57]

Originally posted by HallsofIvy
What are x and y? They are not points in the metric space you mentioned before because such operations are not defined in a general metric space.

They certainly aren't numbers that equation simply isn't true for numbers.

[?] The "equation simply isn't true for numbers"?

x^3 + y^3 = (x+y)*[(x+y)^2 - 3*x*y ]

3^3 + 4^3 = (3+4)*[(3+4)^2 - 3*3*4 ]

91 = 7*[49 - 36]

91 = 7*13

x^3 + y^3 = (x+y)*[(x+y)^2 -3xy] is true for numbers

The metric space has distance function r(x,y), definitively characterized by involvement with the real numbers, R, such that the metric space and R are embedded simultaneously in the full structure of manifold M. A topological space consists of sets of points which are defined[in this case] to be the intersections of cotangent bundles.

We move on to functions of a complex variable in the 2-D Euclidean space Z, utilizing the algebraic structure of complex numbers, where points of Z may be regarded as pairs of real numbers, R. Formal statements concerning Z, are also expressable as statements about R.

Complex-valued functions e.g. w = f(z), can also be represented as binary, or quaternary relations, these sets of complex valued polynomials then have elements that map Z into Z. Every complex valued polynomial subset p(z), has natural number coefficients a_k, or the "kth" coefficient of p(z)..

So, for non-zero polynomials, we take into account the number of zeros OF the polynomial O[p(z)] with O ranging over the natural numbers such that
O[p(z), R ] = v, where v is a finite natural number.

Thus if G is a metric group and E is a topological group, such, that an open neighborhood U, of the identity is a metric space, with a compliant metric, specifiable to the topology of U, then the distance r(a,b) between any two points (a,b) in U, then in terms of the distance u(E) = J, consists of the points a of U with the fact that r(a,E) is an infinitesimal.

 

[modmans2ndcoming]

fermat's theorem is true

it is true that there are no rational solutions for x and y and z.

that means that there are irrational solutions for x or y or z. (aint logic grand)

any way, there are real solutions, however at least one must be irrational.

 

[matt grime]

"The metric space has distance function r(x,y), definitively characterized by involvement with the real numbers, R, such that the metric space and R are embedded simultaneously in the full structure of manifold M. A topological space consists of sets of points which are defined[in this case] to be the intersections of cotangent bundles."

Would you care to elaborate? Let X be some abstract space (possibly even containing a proper subclass) with the trivial metric (r(x,y) =0 if x=y 1 otherwise.) How does one embed this in a manifold? which manifold, what do you mean by embed? I mean as X is not a manifold, you don't mean the usual embedding (or immersion or submersions) of differential geometry. As the metric only takes exactly two values, how do you relate it to all of R? Must every manifold contain an *embedded* copy of R? I can't remember the explicit definititions for embeddings, you see - I was never very good at memorizing all those things, you know, an immersion that's not this is not the other.

 

[phoenixthoth]

this is pure insanity . or is it pure genius ?? i leave the details of the proof of either to the reader .

 

[PFanalog57]

If f(x) is a homeomorphism from T onto S, and for every point p in T, f(U(p)) = U(f(p)), and the monad is invariant under standard topological transformations, with the caveat that the definition also comprizes a type of dynamic situation sematics, where concepts, such as "proper set", "ordinal" and "cardinal" are relativised to context, taking care of paradox at all levels via symmetry, or an invariant many-valued logic, and the "top[set of all sets]", would naturally not exist, of course, since there is nothing outside the universe. So it becomes an infinite chain or composition of ever more inclusive situated sets expressing an interesting informational -topological dynamic.


Outside of "Total Existence"[TE] there is nothing. This is an irrefutable fact. Or we could say that there IS not an "outside" of Total Existence.

Therefore the largest possible set does not exist, where "does not exist" is equivalent to "nothing".

Nothing contains everything, as Russell's paradox so eloquently puts it.

Before the beginning there is nothing. After the end, there is nothing.

ERGO

Alpha = Omega.

 

[phoenixthoth]

if alpha is 0 and U is omega, where U is the universal set, then
{0}=1~U where ~ is 'hemitopy'.

ergo
alpha = unity = omega (depending on what you mean by =)

 

[matt grime]

I hope your last post was tongue in cheek, Phoenixtoth.


And, Russell, are you going to explain anything ever or just keep posting wildly unrealistic statements that are lacking any meaningful content?

 

[phoenixthoth]

yeah, it was. there are about 50 million things wrong with what i wrote but i was mimicking russel's style for a second.

 

[PFanalog57]

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.

Quantum entities are described as probability distributions, which are attributes of an underlying phase space, where the properties-attributes such as "spin" and "charge" are not the attributes of individual particles, but they are universally distributive entities, being the attributes of a "coherent wave function". It is this wave-distribution property that then "decoheres" into the ostensible "wave function collapse", as waves become localized particles that are "in phase" creating standing-spherical-wave resonances, which are condensations of space itself. The continual collapse-condensation of space into matter-energy is the continual "change", i.e. the property called "time". The spherical waves, or probability distributions are represented by the Schrodinger wave function, "psi".


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]


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

Intersecting wavefronts = increasing density of spacelike slices

As the wavefronts intersect, it becomes a mathematical computation:

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...2^n


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.

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.

The universal laws of nature are explained in terms of symmetry. The completed infinities, mathematician Georg Cantor's infinite sets, could be explained as cardinal identities, akin to "qualia" [Universally distributed attributes] from which finite subsets, and elements of subsets [quantum decoherence-wave function collapse] can be derived.

Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness[qualia]. Properties, or "attributes" like red are numbers in the sense that they interact algebraically according to the laws of Boolean algebra. Take one object away from the set of red objects and the distributive number "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. The "infinite" Cantorian alephs are really distributive[qualia].

Yet, if we have a finite set of 7 objects, the cardinal number 7 does not really distribute over its individual subsets. Take anything away from the set and the number 7 ceases to describe it[wave function collapse-condensation into specific localization].

Symmetry is analogous to a generalized form of self evident truth, and it is a distributive attribute via the laws of nature, being distributed over the entire system called universe. A stratification of Cantorian alephs with varying degrees of complexity. Less complexity = greater symmetry = higher infinity-alephs. So the highest aleph, the "absolute-infinity" distributes over the entire set called Universe and gives it "identity".

The highest symmetry is a distributive mathematical identity[also a total unknown but possibly analogous to a state of "nothingness"]. This fact is reflected in part, by the conservation laws.

So an unbound-infinite-potentia and a constrained-finite-bound-actuality, are somehow different yet the same. The difference and sameness relation is a duality. Freedom(higher symmetry) and constraint-complexity-organizational structure(lesser symmetry) form a relation that can be described by an invariance principle.

On a flat Euclidean surface, the three angles of a triangle sum to 180 degrees. On the curved surface of a sphere, the three angles add up to more than 180 degrees. On the hyperbolic surface of a saddle they sum to less than 180 degrees. The three types of surface are not equivalent.

There is a rotational invariance for a triangle, that seems to hold for the three types of surface though.

ABC = BCA = CAB

CBA = BAC = ACB

According to Einstein, and the CTMU of Langan, www.ctmu.org , "space and time are modes by which we think, and not conditions in which we live". Space becomes abstract, a relation that is perceptual and "mental", where distance interval between two points becomes a mental perception.

[ 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. Topologically it is equivalent to a "point". The abstract description contains the concrete topology. Likewise, the concrete contains the abstract.

A duality.

A point contains an infinite expanse of space and time?

Could it be, that the "absolute" infinity, is actually a dimensionless point? Or more correctly, an "infinitesimal"?

Universe? = Zero?


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."

A point can be defined as an "infinitesimal". 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.

 

[phoenixthoth]

i have neither the time nor the inclination nor the desire to refute your points. let's just take one:

A point can be defined as an "infinitesimal".

incorrect. an infinitesimal is bigger than a point yet smaller than every nondegenerate circle. it is correct to say it "Can" be defined that way, but no one usually does.

 

[PFanalog57]

Originally posted by phoenixthoth
i have neither the time nor the inclination nor the desire to refute your points. let's just take one:


incorrect. an infinitesimal is bigger than a point yet smaller than every nondegenerate circle. it is correct to say it "Can" be defined that way, but no one usually does.

http://www.fact-index.com/h/hy/hyperreal_number.html

Infinitesimal and infinite numbers

A nonstandard real number e is called infinitesimal if it is smaller than every positive real number and bigger than every negative real number.

Zero is an infinitesimal,

but non-zero infinitesimals also exist: take for instance the class of the sequence (1, 1/2, 1/3, 1/4, 1/5, 1/6, ...) (this works because U contains all index sets whose complement is finite).

 

[phoenixthoth]

it's 'trival' to call 0 an infinitesimal. the hard part is to show that there are nonzero infinitesimals. now what would your quote become if one replaced 'infinitesimal' with 0? what would it say?

 

[PFanalog57]

In Newton's calculus 0/0 is replaced with:

infinitesimal/infinitesimal.

A point CAN be defined as an infinitesimal ;)

 

[matt grime]

An arbitrary topological space has neither a sum or product defined on it. You are talking gibberish. Again.

 

[phoenixthoth]

a topological space's powerset has a boolean ring structure.

 

[matt grime]

Doesn't make a valid ring structure on the topological space though.

But, hey, seeing as the set

0,1,2,3,4,5,6,7

has the discrete topology, and the equation

x^4+y^4=z^4 mod 8

has non-trivial solutions,
maybe he isn't a complete, raving lunatic.

 

[PFanalog57]

Originally posted by matt grime
An arbitrary topological space has neither a sum or product defined on it. You are talking gibberish. Again.

? [zz)] [?]

http://en.wikipedia.org/wiki/Direct_product

Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff; the product of connected spaces is connected, and the product of compact spaces is compact. That last one, called Tychonoff's theorem, is yet another equivalence to the axiom of choice.

http://mathworld.wolfram.com/DirectProduct.html

Direct products satisfy the property that, given maps a: S-->A and ,b: S-->B there exists a unique mapS--> A x B given by
(a(s),b(s)). The notion of map is determined by the category, and this definition extends to other categories such as topological spaces. Note that no notion of commutativity is necessary, in contrast to the case for the coproduct. In fact, when A and B are Abelian, as in the cases of modules (e.g., vector spaces) or Abelian groups) (which are modules over the integers), then the direct sum
A(+)B is well-defined and is the same as the direct product. Although the terminology is slightly confusing because of the distinction between the elementary operations of addition and multiplication, the term "direct sum" is used in these cases instead of "direct product" because of the implicit connotation that addition is always commutative.

Product Topology:

http://en.wikipedia.org/wiki/Product_topology

 

[UltraPi1]

This thread is cool.

But is it possible to discuss this without the equations.
I found some occasional gathering together of words that I agree with.

 

[matt grime]

If you believe that product and coproduct of topological spaces defines a sum and multiplication rule *inside* *a* toplogical space you are a deluded idiot. To be fair you were just solving Fermat's last theorem in a METRIC space, but such is a topological space too, unless you're doing something odd.

Or was throwing in Fermat just a red herring?

Do you also realize that invoking Russells paradox in the form you do requires the class of all sets to be a set, which it isn't.
So why that implies anything about relativity in a topological space, whatever that sentence means is a mystery.

You and Organic should get together, the results could be most amusing

 

[phoenixthoth]

"Do you also realize that invoking Russells paradox in the form you do requires the class of all sets to be a set, which it isn't."

indeed it is, or at least it can be. see http://math.boisestate.edu/~holmes/holmes/nf.html

 

[matt grime]

That's one view, not one used much outside of abstract set theory. As it disproves the axiom of choice it isn't very nice is it? However, by axiomatizing in that way you also remove Russell's paradox for some well-foundedness of predicates reasons or something, so even if there were a set of all sets, russell's paradox doesn't come into it either. Seeing as, given any cardinal A, there is a proper subclass of the class of all sets with cardinality strictly greater than A, is an intuitive idea (I have a strange mind) then the class of sets can't be a set, that is we are in ZF territory.

 

[phoenixthoth]

cantor himself argued that consistency is the only requirement for something to be mathematical, whether or not you 'like' dropping the axiom of choice.

in my universal set theory, i have no reason to suspect i need to drop any axioms. i just changed the subsets axiom and the foundation axiom. they are done in such a way that if one did throw out the third truth value, you are left with ZF set theory. however, if you do allow three truth values, then the universal set axiom does not appear to me now to contradict any of the other axioms and russell can be avoided.

 

[PFanalog57]

Originally posted by matt grime
If you believe that product and coproduct of topological spaces defines a sum and multiplication rule *inside* *a* toplogical space you are a deluded idiot. To be fair you were just solving Fermat's last theorem in a METRIC space, but such is a topological space too, unless you're doing something odd.

Or was throwing in Fermat just a red herring?

Do you also realize that invoking Russells paradox in the form you do requires the class of all sets to be a set, which it isn't.
So why that implies anything about relativity in a topological space, whatever that sentence means is a mystery.

You and Organic should get together, the results could be most amusing

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

Products and sums "within?"[more correctly, on the "hyper-surface of"] a particular topological space, are defined by the operations of intersection and union. Manifolds are characterized as intersecting cotangent bundles, which become self embedding. The "set of all sets" is its own powerset and is a dynamic self including-self embedding[it is also its own subset] manifold.

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

Therefore T-duality is correct. Relativistically speaking, of course.

 

[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!