Is the Universe a Quantum Computer Algorithm?

[PFanalog57]

A simple[trivial?] postulate that gives a "Universal Set" and resolves the "set of all sets" paradox[in the geometric sense]:

A circle of radius R, is isomorphic to a circle of radius 1/R.

[1/R]<--->[R]

For any arbitrarily large circle of radius R, there is an exact correspondence with a circle of radius 1/R, such, that the product R*[1/R] = 1

 

[matt grime]

and this resolves russell's paradox? so where is the set of all sets that do not contain themselves in this construction? and in what sense are you using isomorphism? in what category are your morphisms?

 

[PFanalog57]

and this resolves russell's paradox? so where is the set of all sets that do not contain themselves in this construction? and in what sense are you using isomorphism? in what category are your morphisms?

I don't think it resolves "russell's" paradox without some more work.

All circles are isomorphic to each other because they have the same shape. Likewise, all squares are isomorphic to each other. Now if sets can be transformed into geometric shapes, more specifically, circles, or "geometric shape-equivalents", the largest possible set with a geometric radius R, has a corresponding twin with radius 1/R.

 

[matt grime]

well, when you've figured out what it is you're trying to prove let us know.

 

[PFanalog57]

well, when you've figured out what it is you're trying to prove let us know.

Here is a definition of the "Euler characteristic":

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


Graph Theory:

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


If a polyhedron has V vertices, F faces, E edges, and is topologically equivalent to the sphere, the equation is:

V + F - E = 2

2 is the "Euler characteristic" of the polyhedron.

Sets that are members of themselves correspond to a geometric form. Sets that are not members of themselves correspond to a different? geometric form.

Interesting.

 

[Gokul43201]

I don't think it resolves "russell's" paradox without some more work.

All circles are isomorphic to each other because they have the same shape. Likewise, all squares are isomorphic to each other.

I thought circles were isomorphic with squares - they don't have to have the same shape.

Now if sets can be transformed into geometric shapes,

And the elements of the set transform into...?

more specifically, circles, or "geometric shape-equivalents", the largest possible set with a geometric radius R, has a corresponding twin with radius 1/R.

And I thought the paradox involved the cardinality of the Power Set being bigger than the cardinality of the Set (of all sets). I may be wrong...but what does this have to do with isomorphisms ?

 

[PFanalog57]

I thought circles were isomorphic with squares - they don't have to have the same shape.

Circles are homeomorphic to squares, not isomorphic...?

http://www.rdrop.com/~half/Creations/Puzzles/TriangleShapes/

And the elements of the set transform into...?

Elements of a set can be characterized as sets. All sets can be associated to geometric forms...?

And I thought the paradox involved the cardinality of the Power Set being bigger than the cardinality of the Set (of all sets). I may be wrong...but what does this have to do with isomorphisms ?

Any circle of arbitrarily large radius R, is isomorphic to a circle of radius 1/R.

The magnitude of R corresponds to the cardinality of the powerset.

Is the set of all geometric forms, a geometric form?

Can Venn diagrams correspond to light cone cross sections?

 

[Gokul43201]

Can I answer your questions with more questions ?

PS : Yes that should have been homeomorphic. But I'm still not getting the point. What is the resolution of the paradox ?

 

[PFanalog57]

Can I answer your questions with more questions ?

PS : Yes that should have been homeomorphic. But I'm still not getting the point. What is the resolution of the paradox ?

Set intersection is a type of multiplication of sets.

The intersection of two circles of radius R and 1/R, respectively:

R*[1/R] = 1

R[<-[->[<-[1/R]->]<-]->]


The "Universal Set"

For the continual expansion of power set circle R, there corresponds circle[infinitesimal?] 1/R.

 

[Hurkyl]

All sets can be associated to geometric forms...?

Not as far as I know.

 

[Gokul43201]

Set intersection is a type of multiplication of sets.

No, it's not ! It's just a process of picking the common elements.

You can have a set A containing millions of even numbers, and a set B containing thousands of odd numbers and you "multiply" them to get a null set ?

 

[PFanalog57]

No, it's not ! It's just a process of picking the common elements.

You can have a set A containing millions of even numbers, and a set B containing thousands of odd numbers and you "multiply" them to get a null set ?

Set intersection obeys the distributive law, which is a multiplicative law:

http://www.jgsee.kmutt.ac.th/exell/Logic/Logic31.htm#13

Two sets without common elements are disjoint.

 

[PFanalog57]

Not as far as I know.

Venn diagrams are circles...

Light cone cross sections are circles, ellipses, etc.

 

[Hurkyl]

Set intersection obeys the distributive law, which is a multiplicative law:

That doesn't mean set intersection has anything to do with arithmetic multiplication.

Venn diagrams are circles...

Light cone cross sections are circles, ellipses, etc.

And what does this have to do with associating all sets to geometric forms?

 

[PFanalog57]

That doesn't mean set intersection has anything to do with arithmetic multiplication.

Since the circle of radius R is isomorphic to the circle of radius 1/R, the cardinality of Circle with radius R is on the same line[radius] as the infinitesimal 1/R

1/R 0--------0 R

Since they are on the same line, they intersect. But perhaps a new type of set multiplicative identity needs to be derived?

And what does this have to do with associating all sets to geometric forms?

When two light cones intersect, they become "phase entangled". The intersection is much like a "set" intersection.


In ordinary quantum mechanics, configuration space is space itself
{i.e.,to describe the configuration of a particle, location in space
is specified}. In general relativity, there is a more general kind of
configuration space: taken to be the space of 3-metrics {"superspace",
not to be confused with supersymmetric space} in the geometrodynamics formulation. The wavefunctions[Venn diagrams-light cones] will be
functions over the abstract spaces, not space itself-- the
wavefunction defines "space itself".


The resultant 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/forms"
intersect/entangle, and are "in phase", they are at "resonance",
giving what is called the "wave-function collapse" of the Schrodinger
equation. the action principle is a necessary consequence of the
resonance principle.

 

[Hurkyl]

Although you're using the words, you don't seem to be doing mathematics, so I'll move this thread over here.

 

[matt grime]

Venn diagrams are circles...

no they aren't. what idiot told you that?

 

[PFanalog57]

no they aren't. what idiot told you that?

You appear to be acting like an ignorant troll.

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

In general, an order-n Venn diagram is a collection of n simple closed curves in the plane such that

1. The curves partition the plane into 2^n connected regions, and

2. Each subset S of {1,2,3,...,n} corresponds to a unique region formed by the intersection of the interiors of the curves in S (Ruskey).

Actually, spacetime does not really need to be "sliced up" in that it can proceed in discrete steps, yet, still be continuous.

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


[<-[->[<-[-><-]->]<-]->]
Intersecting wavefronts = increasing density of spacelike slices

As the wavefronts[circles/Venn diagrams] intersect, it becomes a mathematical computation:

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


If the universe includes everything that is real and excludes that which is not real, then the universe is the "Universal" set.

You cannot refute the above logic...

 

[matt grime]

Since when is a simple closed curve necessarily a circle? As you aren't the ignorant one you must surely know that in order to demonstrate all the possible intersections of 4 sets in a venn diagram you cannot use circles. Moreover, surely you, still not being the ignorant one, must also recognise that a venn diagram is not an element of itself, and thus to take the definition you give, and then deduce that a venn diagram is a circle is most definitely not a logical conclusion?


They, circles and closed curves in the plane, certainly aren't even isomorphic, using your particular definition of isomorphic which appears to mean related be some affine transformation when embedded in the plane.

But I am the ignorant troll, so what do I know about affine transformations? Proved Fermat's last theorem yet?

 

[PFanalog57]

Since when is a simple closed curve necessarily a circle?

Circles are simple closed curves.


http://www.combinatorics.org/Surveys/ds5/VennGraphEJC.html

 

[matt grime]

Gee, are circles really closed simple curves? you'd have thought they'd have told me that at university. especially after i had to prove the jordan curve theorem...


a closed simple curve is not necessarily a circle a square being a simple closed curve that isn't a circle, an ellipse also being one, which is what you claimed, and what i pointed out was incorrect, which led you to call me ignorant... hmm, i always take preverse pleasure in being insulted by someone who can't understand a implies b is not equivalent to b implies a.

moreover your statement that a venn diagram is a circle is still incorrect, and now we've seen even more things you don't understand.

 

[PFanalog57]

a closed simple curve is not necessarily a circle

You mean to say, not all simple closed curves are circles.

Wake up.

moreover your statement that a venn diagram is a circle is still incorrect, and now we've seen even more things you don't understand.

:zzz: :zzz: :zzz:

 

[quantumdude]

Matt: You mean to say, not all simple closed curves are circles.

Russell: You mean to say, not all simple closed curves are circles.

Not only did he mean to say it, he did say it. Wake up, yourself, Russell.

He's right, Venn diagrams aren't circles. The definition of a Venn diagram that you quoted doesn't imply that they are, either. The definition of a Venn diagram refers only to the topology of the curves. The definition of a circle, on the other hand, is the locus of all points (x,y) that are equidistant from a fixed point (h,k). They don't mean the same thing.

Why can't you just accept that bit of correction?

 

[PFanalog57]

Not only did he mean to say it, he did say it. Wake up, yourself, Russell.

He's right, Venn diagrams aren't circles.

Why can't you just accept that bit of correction?

Here is what matt ...said:

a closed simple curve is not necessarily a circle

Yes, it is almost equivalent to: "not all simple closed curves are circles"

The definition of a Venn diagram refers only to the topology of the curves

The Venn diagrams have the property of logical inclusion/exclusion.

In nature, a sphere is the most energy efficient configuration. A 2D slice of that sphere is is a circle.

Yes, I accept correction. But what is the point of arguing and pedantic "nit-picking" over definitions?

 

[PFanalog57]

http://www.math.ohio-state.edu/~fiedorow/math655/Jordan.html

Jordan Curve Theorem: Any continuous simple closed curve in the plane, separates the plane into two disjoint regions, the inside and the outside.

Interesting...

Jordan-Schönflies Curve Theorem For any simple closed curve in the plane, there is a homeomorphism of the plane which takes that curve into the standard circle.

If the physical universe includes all that exists and excludes that which does not exist, then by definition, the universe is self containing.

A dynamic process.

 

[quantumdude]

Here is what matt ...said:

a closed simple curve is not necessarily a circle

I know what Matt said.

Yes, it is almost equivalent to: "not all simple closed curves are circles"

There's no "almost". The two statements are equivalent.

The Venn diagrams have the property of logical inclusion/exclusion.

No, Venn diagrams have certain connectivity properties, as your Wikipedia definition states. It is the properties of a specific set, together with the set operations, that have logical inclusion/exclusion properties. Those are what determine how the Venn diagram are populated with elements.

In nature, a sphere is the most energy efficient configuration. A 2D slice of that sphere is is a circle.

So? Physics has no bearing on set theory, Venn diagrams, or circles.

Yes, I accept correction. But what is the point of arguing and pedantic "nit-picking" over definitions?

Because in mathematics, definitions are everything.

edit: fixed a quote bracket

 

[matt grime]

You are subjectively claiming I am nitpicking; perhaps there is another interpretation? Seeing as you managed to misunderstand almost everything that has been written, including failure to understand the important mathematical usage of the word 'necessarily', I'm not going to overly worry about your opinion about what constitutes a 'nit'. Add into that the fact that most of your own posts are off topic in your own thread...

 

[PFanalog57]

I know what Matt said.



There's no "almost". The two statements are equivalent.

I disagree.


[1.] "A simple closed curve is not necessarily a circle"


[2.] "Not all simple closed curves are circles"


[1.] and [2.] are different. Not exactly equivalent.

[2.] better fits the context of THIS thread.

No, Venn diagrams have certain connectivity properties, as your Wikipedia definition states. It is the properties of a specific set, together with the set operations, that have logical inclusion/exclusion properties. Those are what determine how the Venn diagram are populated with elements.

A member of the set is included in the "simple closed curve".

That which is not a member of the set is excluded[outside] of the simple closed curve, i.e. a curve that is not necessarily a circle but it does have the property of closure. ...I hope you understand.

So? Physics has no bearing on set theory, Venn diagrams, or circles.

I disagree.

Didn't Ed Witten receive the Fields medal of mathematics for work he did in mathematical physics?


Physics would not exist without mathematics. Geometry can be expressed in terms of algebra. Einstein was very close to a "unified field theory" explained in terms of geometry.

Here is the relevant quote:

https://www.physicsforums.com/showthread.php?t=23034&page=1&highlight=einstein+quantum+gravity

[...]

Since you raised the topic with the subject header, it's both
instructive and revealing to see what Einstein, himself, had to
say on the subject of quantum gravity at the end of his life:

"One can give good reasons why reality cannot at all be represented
by a continuous field. From the quantum phenomena it appears to
follow with certainty that a finite system of finite energy can be
completely described by a finite set of numbers (quantum numbers).
This does not seem to be in accordance with a continuum theory,
and must lead to an attempt to find a purely algebraic theory for
the description of reality. But [sic] nobody knows how to find
the basis of such a theory."

Because in mathematics, definitions are everything.

You refuse to let the horse out of the starting gate.

 

[PFanalog57]

You are subjectively claiming I am nitpicking; perhaps there is another interpretation? Seeing as you managed to misunderstand almost everything that has been written, including failure to understand the important mathematical usage of the word 'necessarily', I'm not going to overly worry about your opinion about what constitutes a 'nit'. Add into that the fact that most of your own posts are off topic in your own thread...

So you promise to go harrass someone else?

Thanks.

 

[quantumdude]

I disagree.


[1.] "A simple closed curve is not necessarily a circle"


[2.] "Not all simple closed curves are circles"


[1.] and [2.] are different. Not exactly equivalent.

[2.] better fits the context of THIS thread.

The two statements are logically equivalent. They needn't have the same wording to be so.

A member of the set is included in the "simple closed curve".

That which is not a member of the set is excluded[outside] of the simple closed curve, i.e. a curve that is not necessarily a circle but it does have the property of closure. ...I hope you understand.

I do understand, and I stick with what I said before: It's not the Venn diagram that has the property of inclusion or exclusion, it's the description of the set, together with binary operators. The Venn diagram by itself can't exclude or include any element from anything.

I disagree.

Didn't Ed Witten receive the Fields medal of mathematics for work he did in mathematical physics?

What's that supposed to prove?

Physics would not exist without mathematics. Geometry can be expressed in terms of algebra. Einstein was very close to a "unified field theory" explained in terms of geometry.

That's not true at all. Experimental physics is not mathematical, but it's still physics. Of course, doing physics would be very difficult without math, but it certainly could exist without it.

You refuse to let the horse out of the starting gate.

Maybe it's time for you to consider that you really don't understand mathematics all that well. The objects of mathematics, and the rules of inference, are all based on definitions. Get those wrong, and you've got math wrong.

 

[PFanalog57]

What's that supposed to prove?

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Witten.html

Basically Witten is a mathematical physicist and he has a wealth of important publications which are properly in physics. However, as Atiyah writes in [3]:-

Although he is definitely a physicist (as his list of publications clearly shows) his command of mathematics is rivalled by few mathematicians, and his ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by his brilliant application of physical insight leading to new and deep mathematical theorems.

Maybe it's time for you to consider that you really don't understand mathematics all that well. The objects of mathematics, and the rules of inference, are all based on definitions. Get those wrong, and you've got math wrong.

Yes, there is much to learn about mathematics.

 

[matt grime]

I promise to harass you whilst you are spouting inaccurate garbage, Russell, don't worry. Why on Earth you chose to cite Ed Witten is a mystery, but then you seem to be beyond the pale of reasonable logical thought and into the realms of the crackpot, so frankly who cares?

 

[quantumdude]

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Witten.html

Again, what is that supposed to prove?

 

[PFanalog57]

Again, what is that supposed to prove?

You wrote this:

Physics has no bearing on set theory, Venn diagrams, or circles.

Sets "contain" elements, members, etc. Venn diagrams can be represented as conic sections.


A______B

____P____

C______D

A, B, C, and P are "co-moving" i.e. they are at rest with respect to each other. The radius[hypotenuse] from P to the other points{A, B, C, D} is the same length. An expanding circle of light[from point P] reaches A, B, C, and D, "simultaneously". The invariance of "c".

Experimental physics is not mathematical, but it's still physics. Of course, doing physics would be very difficult without math, but it certainly could exist without it.

:zzz: :zzz: :zzz:


There is no experiment unless "numbers" can be attached to the quantity being observed.

Your statement that "experimental physics is not mathematical" is total hog-wash.

Any measurement uses numbers.

Light cones are cutting edge stuff :

http://www.mpi-hd.mpg.de/ilcac/98SPeter_prop/node4.html

It has been known for some time that light-cone field theory is uniquely suited for to address problems in string theory. In addition recently new developments in formal field theory associated with string theory, matrix models and M-theory have appeared which also seem particularly well suited to the light-cone approach

 

[PFanalog57]

I promise to harass you whilst you are spouting inaccurate garbage, Russell, don't worry. Why on Earth you chose to cite Ed Witten is a mystery, but then you seem to be beyond the pale of reasonable logical thought and into the realms of the crackpot, so frankly who cares?

Yes, I suppose it is psychologically healthfull [for you at least] to get in touch withn your inner troll

 

[quantumdude]

You wrote this:

Physics has no bearing on set theory, Venn diagrams, or circles.

Yes, I know what I wrote.

Sets "contain" elements, members, etc. Venn diagrams can be represented as conic sections.

LOL, thanks for the lesson.

A______B

____P____

C______D

A, B, C, and P are "co-moving" i.e. they are at rest with respect to each other. The radius[hypotenuse] from P to the other points{A, B, C, D} is the same length. An expanding circle of light[from point P] reaches A, B, C, and D, "simultaneously". The invariance of "c".

This has nothing whatsoever to do with the question I asked. In fact. your entire post looks as though it were written by a random word generator.

There is no experiment unless "numbers" can be attached to the quantity being observed.

False. You can do an experiment without attaching any numbers to the results, and it would still be called "physics".

Your statement that "experimental physics is not mathematical" is total hog-wash.

That's not what an experimental physicist would say.

Any measurement uses numbers.

No kidding. That doesn't mean that physics wouldn't exist without mathematics.

Light cones are cutting edge stuff :

http://www.mpi-hd.mpg.de/ilcac/98SPeter_prop/node4.html

?

And what does this have to do with anything being discussed here?

 

[PFanalog57]

?

False. You can do an experiment without attaching any numbers to the results, and it would still be called "physics".

An experiment is something that can be repeated over and over again. The repetition leads to "equations".

I flip a coin for a large number of repetitions, I notice[observe] certain regularities.

No kidding. That doesn't mean that physics wouldn't exist without mathematics.

Yes it does. Numbers are part of "mathematics".

And what does this have to do with anything being discussed here?

You also appear to act like an ignorant troll.

 

[quantumdude]

An experiment is something that can be repeated over and over again. The repitition leads to "equations".

I flip a coin for a large number of repetitions, I notice[observe] certain regularities.

Observation is not the same as doing mathematics.

Yes it does. Numbers are part of "mathematics".

Again, simply attaching numbers to measurements is not the same thing as doing mathematics.

You also appear to act like an ignorant troll.

Guess what? So do you.

Seriously, so what if I appear that way to you? You said the same thing to Matt Grime. He is a PhD student in mathematics, and I am a PhD student in physics. Given that you seem to be struggling to understand both of those two disciplines, it hardly seems unsettling that you should say that.

 

[PFanalog57]

Seriously, so what if I appear that way to you? You said the same thing to Matt Grime. He is a PhD student in mathematics, and I am a PhD student in physics. Given that you seem to be struggling to understand both of those two disciplines, it hardly seems unsettling that you should say that.

edited after second thoughts:

Thank you for your perspective Tom Mattson

 

[PFanalog57]

False. You can do an experiment without attaching any numbers to the results, and it would still be called "physics".

A measurement is "numerical": e.g. distance, length, time, force, counting repititions, counting quantities, weight, mass, colour-wavelength, symmetric/asymmetric/anti-symmetric patterns, volume, area, etc.


Distinctions or differentiating between quantities is inherently LOGICAL/MATHEMATICAL.

My[Phd student] interlocutors only response has been "that aint true".

What can I say...

College aint what it used to be?

 

[matt grime]

A measurement is "numerical": e.g. distance, length, time, force, counting repititions, counting quantities, weight, mass, colour-wavelength, symmetric/asymmetric/anti-symmetric patterns, volume, area, etc.


Distinctions or differentiating between quantities is inherently LOGICAL/MATHEMATICAL.

My[Phd student] interlocutors only response has been "that aint true".

What can I say...

College aint what it used to be?

that isn't what you've been saying before, and our replies do not reflect what you claim we have said.

In fact I believe we are pointing out that you do not konw what a venn diagram is, nor that you understand the usage and meaning (if they are different) of 'necessary' and that you stated all simple closed curves were circles, which is obviously wrong.

I don't recall ever using or seeing the word 'distinction' before in this thread and I certainly can't imagine I needed to use 'measurement' at any point other than in this post.

You are distinguishing between things, and you are inherently unmathematical. At least you are consistent in your inconsistency.

 

[PFanalog57]

In fact I believe we are pointing out that you do not konw what a venn diagram is, nor that you understand the usage and meaning (if they are different) of 'necessary' and that you stated all simple closed curves were circles, which is obviously wrong.

Nope. I never said simple closed curves were "only" circles.

But your continual berating is very distracting. Hopefully you can put that education to good use and say something constructive.

 

[quantumdude]

A measurement is "numerical": e.g. distance, length, time, force, counting repititions, counting quantities, weight, mass, colour-wavelength, symmetric/asymmetric/anti-symmetric patterns, volume, area, etc.

Distinctions or differentiating between quantities is inherently LOGICAL/MATHEMATICAL.

No one denies that numbers are attached to measurements, but as I said, measuring and doing mathematics are two different things. Also, one could do experimentation qualitatively, without numbers. Did you know that there is not a single equation in any of Michael Faraday's lab notebooks? And yet who would say that he did not do physics?

My[Phd student] interlocutors only response has been "that aint true".

Well, that's not really fair. We have explained ourselves.

 

[PFanalog57]

Did you know that there is not a single equation in any of Michael Faraday's lab notebooks? And yet who would say that he did not do physics?

Didn't Michael Faraday discover somethiong like "field" lines of force? If his notebooks contained diagrams, then he was thinking in terms of vectors.

Vectors are mathematical.

 

[quantumdude]

Didn't Michael Faraday discover somethiong like "field" lines of force? If his notebooks contained diagrams, then he was thinking in terms of vectors.

Vectors are mathematical.

You keep stating the obvious, as if it proves what your are saying, but it doesn't. No one denies that physical forces can be described by vectors, and quite well at that. But it is not the case that physical forces are vectors. One is the map, and the other is the territory: they aren't the same thing.

 

[PFanalog57]

You keep stating the obvious, as if it proves what your are saying, but it doesn't. No one denies that physical forces can be described by vectors, and quite well at that. But it is not the case that physical forces are vectors. One is the map, and the other is the territory: they aren't the same thing.

Faraday was an experimentalist who thought in "mathematical" terms, even though he was not formally trained in physics or mathematics.

You concede then?

OK...

 

[quantumdude]

Faraday was an experimentalist who thought in "mathematical" terms, even though he was not formally trained in physics or mathematics.

That's just it: He thought in physical terms. It is you who is imposing the mathematical viewpoint onto this.

An experimentalist with no training or interest in mathematics looks at Faraday's apparatus and sees coils, batteries, wires, magnets, capacitors, inductors, etc. He hooks them up and he sees wires being attracted and repelled, ammeter needles deflecting, etc.

You look at those things, and you see vectors, whose line integrals you can compute and whose divergence you can calculate. There's nothing wrong with that (indeed, I see the same thing), but there's also no reason to think that everyone is going to see it that way. Further, there's further no reason to think that such people aren't "doing physics".

The statement "Physics would not exist without mathematics" is simply false. In fact I think a better case could be made for the converse.

You concede then?

OK...

Why should I?

 

[PFanalog57]

That's just it: He thought in physical terms. It is you who is imposing the mathematical viewpoint onto this.

Faraday visualized field lines of force; directed line segments. Vectors in simple form.

In certain respects, physical existence is mathematical. If not, then why does mathematics explain the world so well?

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

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

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."

An arbitrary n-dimensional surface, a curve, can be defined by a parametric equation:

x^a = x^a(u) , (a = 1,2,...,n) where u is the parameter with x^1(u),x^2(u),...,x^n(u) denoting n functions of u. A subspace/surface of m dimensions has (m &lt; n) degrees of freedom, depending on m parameters according to the parametric equations:

x^a = x^a(u^1 , u^2 ,..., u^m) , (a = 1, 2,...,n)

When m = n-1 the subspace is called the hypersurface:

x^a = x^a(u^1 , u^2 ,..., u^n^-^1), (a=1, 2,..., n)


If the manifold with n degrees of freedom is restricted to a hypersurface of an n-1 subspace, its coordinates must satisfy the constraint:

f(x^1 , x^2,..., x^n) = 0

Intersecting "level surfaces" or simple closed curves, of n-1 dimensions with radius R and 1/R, respectively, form in-phase standing waves. A geometric "universal set".

 

[matt grime]

nxn matrices and nxn matrices of nonzero determinant are manifolds, one has one fewer degree of freedom than the other, yet they have the same dimension (that of the dimension of its tangent space at any point).

We'll leave the rest as specious nonsense at this stage

 

[quantumdude]

Faraday visualized field lines of force; directed line segments. Vectors in simple form.

I can see that my comment went in one ear and out the other.

Once again: it is you who is imputing the mathematical interpretation onto this.

In certain respects, physical existence is mathematical. If not, then why does mathematics explain the world so well?

First, a question is not a valid argument. Second, mathematics isn't used "explain" the world at all. It is used to describe the world. And third, your position as to the nature of existence is neither a scientific viewpoint nor a mathematical one. It is a philosophical viewpoint, and a rather bad one at that, as it is a form of idealism that mistakes the ideal forms used to describe the physical reality, for the physical reality itself.

Anyway, the point that led up to this line of discussion was my response to your comment that energy arguments could somehow be invoked in set theory. They can't. Physical arguments are of no use whatsoever in proving mathematical results, because math is not physics, and vice versa.

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

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

I have read the Wigner article more than once. Do you know why he calls the effectiveness of mathematics "unreasonable"? It's because no one can explain it. That is precisely why the article is not of any use in proving your assertions.

Intersecting "level surfaces" or simple closed curves, of n-1 dimensions with radius R and 1/R, respectively, form in-phase standing waves.

Incorrect. First, "standing waves" are physical phenomena. And second, even if you mean that they are "standing wave solutions to the wave equation" (which are bona fide mathematical objects), your analysis here by no means proves such a thing. You haven't even specified the parametric equations, nor a dynamical equation that they are supposed to satisfy.

A geometric "universal set".

What is so "universal" about this set?

I'm afraid Matt is right: What you have written here is specious nonsense, and has nothing to do with either math or physics.