Geometry As A Unifying Idea

[Dubbelosix]

Ah but the experienced mathematician, when dealing with the world, really do believe the equations are out there to be discovered. The old question ''is mathematics invented or discovered'' goes both ways - it is invented, but it is also a discovery. The discovery of the world around us obeying what appears to be rigid laws, being reduced to mathematics, is either a coincidence of reality, or, the universe does obey mathematical laws.

[Moronium]

Ah but the experienced mathematician, when dealing with the world, really do believe the equations are out there to be discovered. The old question ''is mathematics invented or discovered'' goes both ways - it is invented, but it is also a discovery. The discovery of the world around us obeying what appears to be rigid laws, being reduced to mathematics, is either a coincidence of reality, or, the universe does obey mathematical laws.

 

Well, mathematicians can believe what they want.  Math is just another way of conceptualizing.  Neither the greeks nor romans had a place-holding numeral for zero until around 500 A.D. they say.  This lack made mathematical manipulation much more awkward.  Math really began to flourish after the concept of zero was invented.  Somebody didn't just stumble across a zero to discover this.  It was a conceptual feat--a creation of mind, not matter.

Edited March 25, 2019 by Moronium

[Dubbelosix]

What about the Mayans? A later civilization which also came across the concept of zero. How many times does a civilization independently discover something?

 

Did two independent civilizations, come to a concept of zero, simply by imagining it? No ... the concept of zero is that it is an absence of something... real world examples exist of zero, coming across it in my mind, was a natural occurrence.

[Moronium]

Did two independent civilizations, come to a concept of zero, simply by imagining it? No ... the concept of zero is that it is an absence of something... real world examples exist of zero, coming across it in my mind, was a natural occurrence.

 

Sure, the concept of zero was invented multiple times in multiple countries,  China, India, Arabia, etc.  It didn't really hit Europe until the 1100's.  Many north american tribes had no systematic or comprehensive grasp of numbers at all.

 

It is the platonic belief that numbers are "real things" that leads to a lot of the confusion between math and physics, if you ask me.  Many start thinking that math is physics.  It aint.

 

Mathematics are well and good but Nature keeps dragging us around by the nose.

 

======

 

I don't believe in mathematics....Physics is essentially an intuitive and concrete science. Mathematics is only a means for expressing the laws that govern phenomena.  (Albert Einstein).

 

Edited March 25, 2019 by Moronium

[Dubbelosix]

We could quote great scientists all day long... such as Dirac who believed in mathematics and the beauty of mathematics, believing the equations that describe reality (have to be) beautiful in a simple and creative way.

 

I am not saying numbers exist ''out there,'' but the concepts they describe have real world attachment. If there was not such a strong case for the success of mathematical reduction of the physical world, then I probably would never entertain the idea that perhaps the universe is mathematical.

 

Back in the old days when budding scientists where creating new theories, discovered that the world could be potentially made up of zero's and ones. It's not so much that we should believe it is one's and zero's, but it could be explainable through a binary language, not so dissimilar to the one we use in computer logic.

[Moronium]

Back in the old days when budding scientists where creating new theories, discovered that the world could be potentially made up of zero's and ones. It's not so much that we should believe it is one's and zero's, but it could be explainable through a binary language, not so dissimilar to the one we use in computer logic.

 

 

OK, I can buy that.  That seems to summarize the point I am trying to make.

 

Language does not "enable" thinking.  You have to be able to think before you can talk.

 

Likewise, numbers don't enable understanding of the universe.  You have to create math before you can perceive the universe in such conceptual terms.

 

Reification and hypostatization are common category errors.  Numbers and words are merely abstract concepts.  You can see five cows, five horses, five cats, or 5 anything in the world, but you can't see a 5 in the world.

 

In the abstract 2 + 3 = 5.  But two cats and three dogs is not the "same thing as" two chickens and three horses.

Edited March 25, 2019 by Moronium

[Moronium]

Before mathematicians discovered zeros, I bet when some asked how many kids have you got, the answer would have been none, if they had none. 

 

Naw, they would just say "I have no kids," I figure.

 

And then, if it was me, I would add:  "You got the wrong guy, pal.  Don't try to pin no kids on me."

 

It's a damn leading question, can't ya see?  Kinda like askin:  "Will you ever stop beating your wife--yes or no?"

 

Which kinda make me wonder...what kind of geometry would correspond to beating your wife?  I mean like good, long, and hard, ya know?  A good mathematician could probably also impose some structure on the action which would encapsulate it all in a sequence of numbers, too.  An Italian could probably tell me how to say "I just beat the hell out of my wife" in the Italian language, too.  But he would have to also speak English, know what I'm sayin?

Edited March 25, 2019 by Moronium

[Moronium]

 How many times does a civilization independently discover something?

 

Did two independent civilizations, come to a concept of zero, simply by imagining it? 

 

They don't "discover" it, they invent it.

 

I don't know how many different languages and dialects there are in the world, probably thousands.

 

But, ya know what?  I bet they all have all invented a word for "ugly."  But that's a little different, I suppose.  Ugly is really out there.

Edited March 25, 2019 by Moronium

[Dubbelosix]

 

 

 

With any computer model, if garbage goes in then garbage comes out. The model must be correct, to get good results. In your OP what geometry would you use, or suggest using.

 

Just the good old geometry found in general relativity - I want people to consider that we will have ''gravity without gravity.''

 

What do I mean?

 

Einstein's equations crop up, without gravity all the time. I have been learning more and more just how generic the construction of geometry is. Ok, so I have demonstrated the electromagnetic field in terms of geometric algebra

 

[math]\gamma_0 \rho^2 = ( \epsilon_0 \mathbf{E} \cdot \mathbf{E} + \frac{1}{\mu_0}\mathbf{B} \cdot \mathbf{B})^2\gamma_0 + 2i \vec{\sigma}\ (\epsilon_0 \mathbf{E} \cdot \mathbf{E} \times \frac{1}{\mu_0}\mathbf{B} \cdot \mathbf{B})^k \gamma_k[/math]

 

The Berry curvature is given as

 

[math]F_{ij} = [\partial_i, A_j] - [\partial_j, A_i][/math]

 

Where [A] is the Berry connection. In comparison, take a look at the curvature tensor with zero torsion

 

[math]R_{ij} = -[\partial_i, \Gamma_j] + [\partial_j, \Gamma_i][/math]

 

Now take a look at the gauge invariant Berry curvature

 

[math]F_{ij} = [\partial_i, A_j] - [\partial_j, A_i] + [A_i,A_j][/math]

 

It has a structure identical to the non-zero torsion formulation of the field equations

 

[math]R_{ij} = -[\partial_i, \Gamma_j] + [\partial_j, \Gamma_i] + [\Gamma_i, \Gamma_j][/math]

 

The only formal difference between the Berry curvature and the Einstein curvature appears to be the definition of the connection. And as was stated to me, this was no accident, but both identities come from the same mathematical beauty underlying the premise behind them. Let's take a look at another example, notice the density of an electric part of the field is:

 

[math]\rho = \epsilon_0 \int \mathbf{E} \cdot \mathbf{E} = \epsilon_0 \int (\nabla \phi + \frac{\partial \mathbf{A}}{\partial t})(\nabla \phi + \frac{\partial \mathbf{A}}{\partial t})[/math]

 

As you can see, I have rewritten this so we can expand the right hand side. Doing so yields:

 

[math]\nabla \phi \nabla \phi + \nabla \phi \cdot \frac{\partial \mathbf{A}}{\partial t} + \frac{\partial \mathbf{A}}{\partial t} \cdot \nabla \phi + \frac{\partial \mathbf{A}}{\partial t}\frac{\partial \mathbf{A}}{\partial t}[/math]

 

This is a geometry that arises... aside from the first term [math]\nabla \phi \nabla \phi[/math] which may not resemble general relativity, certainly the last part does, namely

 

[math]\nabla \phi \cdot \frac{\partial \mathbf{A}}{\partial t} + \frac{\partial \mathbf{A}}{\partial t} \cdot \nabla \phi + \frac{\partial \mathbf{A}}{\partial t}\frac{\partial \mathbf{A}}{\partial t}[/math]

 

This is gravity, without said gravitational field. Gravity is already encoded into the electric definition of the field. While I cannot identify the first term, it does appear in other formula's from geometric algebra, for instance, it appears in the derivation of my bivector representation, which too is expanded:

 

[math]\nabla \gamma_0 \mathbf{D} = \nabla^k \gamma_k \gamma_0\gamma_0 \nabla^j \gamma_j \gamma_0  [/math]

 

[math]- \nabla^k \gamma_k \gamma_1 \gamma_0 \gamma_0 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3 [/math]

 

[math]- \mathbf{D}^k\gamma_k \gamma_1 \gamma_2 \gamma_3 \gamma_0 \nabla^j\gamma_j \gamma_0 [/math]

 

[math]+ \mathbf{D}^k \gamma_k \gamma_1 \gamma_2 \gamma_3 \gamma_0 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3[/math]

 

We notice that the first term

 

 

[math] = \nabla^k \gamma_k \gamma_0\gamma_0 \nabla^j \gamma_j \gamma_0  [/math]

 

has strong similarities to

 

[math]\nabla \phi \nabla \phi[/math]

 

But anyway, the whole point is that geometry (the thing we name gravity) is already encoded in the physical fundamental systems in the universe. This ''geometry'' could create the phenomenon we call gravity.

 

Edited March 25, 2019 by Dubbelosix

[Moronium]

Suppose I got a wild hair up my azz and started thinking:  There's not really 3 dimensions, there's only two in actuality.   The appearance of three dimensions is just a holograph.  Ya think some mathematician could reduce that proposition to a set of formulas?

Edited March 25, 2019 by Moronium

[Dubbelosix]

Well, I don't believe that there are only two dimensions, and that the fundamental world encodes far too much to be taken seriously as some mere holograph, however, if you want to follow those idea's, follow the work of Verlinde.

[Moronium]

Well, I don't believe that there are only two dimensions, and that the fundamental world encodes far too much to be taken seriously as some mere holograph, however, if you want to follow those idea's, follow the work of Verlinde.

 

I wonder how much "believing" affects the math?

[Dubbelosix]

I don't believe there are two dimensions, simply for the fact that we can measure three. That is not enough to disparage holography, but again, how would you prove this?

[Moronium]

I don't believe there are two dimensions, simply for the fact that we can measure three. That is not enough to disparage holography, but again, how would you prove this?

 

I don't think it's possible to "prove" it.  You could probably "prove" it mathematically, I figure, but I wouldn't know how.  That's why I ask.  Then I can act like my pet belief is "scientific," see?

Edited March 25, 2019 by Moronium

[Dubbelosix]

Well.. fair enough. However, I do know some of the math behind it and that knowledge did not sway me either way. I try and think of things intuitively, as all should do. A physicist friend of mine said the other day, ''I don't follow cosmology any more, because if you ask ten of the top physicists what big bang was or whether anything happened before it, you'll get a different answer each time.''

[Dubbelosix]

See, I have come to find something important about, the treatment of gravity. it is well-known gravity is a pseudo-force from the first principles of relativity, (even though physicists have wasted their time trying to quantize it, and I cannot stress this latter part enough).

 

1. You cannot fit the laws of electromagnetic field theory, into that of gravity, or any other force. But gravity on the other hand, can describe electromagnetism including the strong force. The weak force is a bit different concerning decays, but if phases related to those decays are geometrically-related, then there might be some sort of Berry-like phase describing those transitions. It remains, a matter of fact, that gravity is a universal application to the geometry behind real fields.

 

2. You can deduce the laws of curvature, without a notion of gravity, from electromagnetism, including strong interactions. The latter is known as ''strong gravity theory.''

 

These are important facts, which highlights how dynamic, and yet, generic Einstein's description of curvature was. The big thing realized, was that you can have curvature of space, from massive objects to the small - from contributions of the energy associated to those fields. Suppose there was a universe, with only electromagnetic, weak and strong forces, the first and latter are entirely capable of curving space, without a priori of an independent gravitational field.

[Dubbelosix]

Flummoxed, you said, after second thoughts...

 

''You put garbage in, you get garbage out.''

 

What do you know of chaos theory? Some say out of chaos comes an order. For this to happen, there must be something fundamental in the chaos. It only appears chaotic, but follows the same rules of cause and effect.

 

Suppose you put into a computer what would appear, garbage to a scientist on Earth, but what if it was so complicated, and deep, with meaning, that it had a function bringing about an ordered world?