Mathematics, physics and theory of everything?

[Strange]

Just came across these two articles on the relationship between mathematics and physics. Well, we obviously use mathematics to describe physics, but is the nature of physics actually defined or created by mathematical structures? (Beginning to sound like this should be in Philosophy ...)

Anyway, an interesting read:

https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/

And if you follow the links in the "Related" section (near the bottom of each article) you will get sucked into a rabbit's hole of interesting articles.

@neuerwind: You were talking about quaternions recently, so you might like this!

[StringJunky]

2 hours ago, Strange said:

Just came across these two articles on the relationship between mathematics and physics. Well, we obviously use mathematics to describe physics, but is the nature of physics actually defined or created by mathematical structures? (Beginning to sound like this should be in Philosophy ...)

Anyway, an interesting read:

https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/

And if you follow the links in the "Related" section (near the bottom of each article) you will get sucked into a rabbit's hole of interesting articles.

@neuerwind: You were talking about quaternions recently, so you might like this!

I understand maths to be a language which one uses to describe quantitatively ones thoughts. I don't think it is the source of answers. I think sometimes its complexity gives the impression that it can be. The answer has to come from intuition or making a connection in the first instance.

Edited September 11, 2018 by StringJunky

[Markus Hanke]

13 hours ago, Strange said:

Well, we obviously use mathematics to describe physics, but is the nature of physics actually defined or created by mathematical structures?

I think the two mutually depend on each other - physics stimulates new discoveries in maths, and maths has a huge influence on how we think about the physical world. 
One might consider the relationship to be somewhat analogues to the Sapir-Whorf hypothesis for natural languages, albei probably more the weak version of it.

[Conjurer]

On 9/11/2018 at 10:25 AM, Strange said:

Well, we obviously use mathematics to describe physics, but is the nature of physics actually defined or created by mathematical structures? (Beginning to sound like this should be in Philosophy ...)

I don't think it would in an ultimate theory of everything, because mathematics is fundamentally different than physics on the Plank Scale.  It would take an infinite amount of energy in order to detect or even experience something happening below the Plank Scale, so then our physical reality doesn't obey the same principals as an infinitely divisible coordinate plane system.  It could just be a happy coincidence that math tells us as much as it does about physics, and the Plank Scale is just too small to really have any real discernible effect on showing a difference between the math and our physical reality.

[steveupson]

I’ve wanted to reply to several of the ongoing discussions here, but I’ve been a little overwhelmed by events.  (For one thing, I’m going to school again, and I’m trying to study algebra – I’m still no good at it, but I think that I’m learning exactly why I suck at it so bad.)  I’ve also taken an interest in this question recently, and it seems to me as if our mathematics have always been developed in response to questions raised by observations of physical phenomena.  Newton’s calculus used for explaining his laws of motion would probably be the most obvious example of this. 

For those more critical members here that may be offended by me “reintroducing” my opinion at this forum, please do take a look at the interactive video on quaternions that is linked at the end of this post.  It seems to be transformative for this type of learning.   Also, Cohl Furey from the first link in the OP has an interesting set of more traditional video lectures that concludes with octonions. 

 

There’s a fairly recent paper (2015) where the author explains more of the history of how we got here.  It reaches back to Grassmann’s time, and it sort of charts the path that was developed for transitioning between what used to be an understanding of Euclidean 3-space based on spherical trigonometry, and the modern version that is mostly based on vector manipulation (as expressed in the Cartesian coordinate system and some of the higher-dimensional geometries) that is currently popular in physics.  The paper is “QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS by Johannes C. Familton” and can be viewed here:

https://arxiv.org/ftp/arxiv/papers/1504/1504.04885.pdf
 
It’s a pretty long read, so I’ll try and condense the parts that were the most important for me.  My understanding of how this all fits together is still being improved, but I was drawn to the parts in the paper where certain persons spoke about the conceptual (or mathematical) distinction between scalars and vectors.   Here are just a few examples of what I’m talking about:
 

Quote

“It should be noted that when Hamilton discussed ‘vectors’, he was not using them in the way that they are understood today. Hamilton understood ‘vectors’ to mean the non-scalar part of the quaternion equation….”

“What is called a tensor today is essentially a generalization of a scalar and vector, where a scalar is a tensor of rank zero, and a vector is a tensor of rank one. The rank of a tensor defines the number of directions it has…”

“In order to develop quaternions further as rotations, the concept of ‘pure quaternions’ is needed. Recall that the pure quaternions are the vector part of quaternions without the scalar part. These vectors are in 1-1 correspondence with R 3 . Thus a vector in R 3 corresponds to the quaternion ‘pure vector’ Q = 0 +q1i+q2j+q3k. In order for quaternions to rotate in 3-dimensionsl space the rotation operator LQ is established, where LQ: R 3 −> R 3 . If v = q1i+q2j+q3k then LQ (v) = QvQ* where v is an element R 3 and Q is a unit quaternion, then v is also an element of H0 where H0 is the set of all ‘pure quaternions’…”

 
As many of you are aware, we (a loosely affiliated group of volunteers) are currently working to  develop a new geometry based on newly discovered geometric relationships.  In all of my attempts to explain the math that underlies this new geometry, most of the difficulty is over this specific issue.  We've never really developed a fully comprehensive understanding of how these two types of numbers differ, and this has led to a very imprecise way of speaking about those differences.  
 
In my opinion, the most accurate way to define the difference between these two types of numbers is to recognize that one type of number is a ratio, and the other type of number is a quantity.  We’ve sort of muddled this all together into pabulum like “a vector has magnitude and direction while a scalar has only magnitude.”  Although this may be true, it leads to a very anemic understanding of what’s going on mathematically.  Grassmann seems to be the first to understand that there is a mathematical difference between things that have a direction and things that don’t, and to try and make a study of it in a way that produced results that are still in use today. 
  
Our new understanding (in our informal group) is that there are two distinctly different quantities that make up space or volume.  These two quantities are distance and direction.  We think that we have developed some math that shows how these two quantities are NOT asymmetrically inseparable from one another.   What this last statement means is simple and intuitive, but it also “seems” as if it is in direct conflict with millennia of accepted math and science.   In actuality, there isn’t really any conflict at all, the old and the new fit together perfectly.  What is new is that we can now talk about the direction part of the “vector” separately, in the same way as we have always talked about the scalar portion as being separable.  We can assign a meaningful number to express direction as a quantity, instead of as a ratio.   When we do this correctly, we should be able to express distance as a ratio between directions.  This is a perfect symmetry to the way that we now treat direction as a ratio between distances.

We're catching spherical trigonometry up to include 21st century observations in physics.  All mathematical inquiry into this subject ended fairly abruptly a century and a half ago when quaternions and vector analysis became all the rage.  There are some geometric relationships that are not captured using these traditional methods.  We have stumbled across some of them and we are attempting to use them in order to define a geometry that is different.  While the observation of physical phenomena remains identical, they can be looked at using different mathematical models.  It's the same as reading Don Quixote in two different languages.
 
For more about quaternions, check out:

https://eater.net/quaternions

Edited November 11, 2018 by steveupson
added link

[Endy0816]

When I can rub a few more brain cells together I'll take a look at the above.

 

On 11/5/2018 at 11:04 PM, Conjurer said:

I don't think it would in an ultimate theory of everything, because mathematics is fundamentally different than physics on the Plank Scale.  It would take an infinite amount of energy in order to detect or even experience something happening below the Plank Scale, so then our physical reality doesn't obey the same principals as an infinitely divisible coordinate plane system.  It could just be a happy coincidence that math tells us as much as it does about physics, and the Plank Scale is just too small to really have any real discernible effect on showing a difference between the math and our physical reality.

Discrete mathematics could possibly cover this. You see 'jumps' between numbers rather than what we're used to.

https://en.wikipedia.org/wiki/Discrete_mathematics

Edited November 11, 2018 by Endy0816

[steveupson]

12 hours ago, Endy0816 said:

When I can rub a few more brain cells together I'll take a look at the above.

 

Discrete mathematics could possibly cover this. You see 'jumps' between numbers rather than what we're used to.

https://en.wikipedia.org/wiki/Discrete_mathematics

At the Planck scale, direction and distance don't have the same relationship with one another as what they have at the human scale or astronomical scale.  Think about a car that needs a certain width of road in order to make a u-turn.  When the road narrows to below the Planck length it is no longer possible for the car to drive in a circle.  It is very likely that a turn in space occupies some amount of direction that cannot be made into a different, smaller amount.  This could be caused by the necessity that the turn in space has to be relative to something.  I won't say it's an aether, because admitting to a belief in an aether is right up there with denying that you're an alcoholic (denial is the first sign, by the way, and that's why we should never deny it.)  Mach's principle may be in play here.  It does look as if there is some room here in this particular model for some discrete math.

Edited November 11, 2018 by steveupson

[Rajiv Naik]

our physics is only one of the solution mathematics provide . Our physics  is  all about pythagoras theorem or slope.

 

[Strange]

8 minutes ago, Rajiv Naik said:

Our physics  is  all about pythagoras theorem or slope.

So use Pythagoras’ theorem to calculate the height of a geostationary orbit. You can’t can you. 

[studiot]

16 minutes ago, Rajiv Naik said:

our physics is only one of the solution mathematics provide . Our physics  is  all about pythagoras theorem or slope.

 

What are you studying?

[Rajiv Naik]

I suppose geostationary orbit is about gluon forces

there are almost consenses now that gluon force and gravity force is one and the same.

solutions would requires quantum physics ,complex vectors,

not classical physics.

[Strange]

2 hours ago, Rajiv Naik said:

I suppose geostationary orbit is about gluon forces

Not at all. 

2 hours ago, Rajiv Naik said:

there are almost consenses now that gluon force and gravity force is one and the same.

No there isn’t. 

2 hours ago, Rajiv Naik said:

solutions would requires quantum physics ,complex vectors,

not classical physics.

Just Newtonian physics. 

[Phi for All]

2 hours ago, Rajiv Naik said:

I suppose geostationary orbit is about gluon forces

there are almost consenses now that gluon force and gravity force is one and the same.

solutions would requires quantum physics ,complex vectors,

not classical physics.

!

Moderator Note

In order to keep the thread on topic, you need to check your facts before stating them so declaratively. Please ask questions if you're unsure. If you're suggesting changes to mainstream science, you need to start your own thread in Speculations.

 

[Rajiv Naik]

On 22/11/2018 at 5:22 PM, Strange said:

So use Pythagoras’ theorem to calculate the height of a geostationary orbit. You can’t can you. 

- my point was-

differential calculus invented by newton is about slope and hence about Pythagoras theorem, isnt it?

I have not read much about geostationary orbits - sorry makiy sweeping comment.

but long back I had read about Keplers eliptical motion and newtons view about it.

Newtons gravity is local based upon Galaleos  work.- doesnt apply to space. maybe that is. The reason I did not  read it in  detail,

however fact remains that Calculus is based upon pythagoras theorem. in fact its a device to get rid of infinity problem ,  giving good aproximation and hence used  in classical Physics.

but calculus is not really classical,

 

 

[Strange]

56 minutes ago, Rajiv Naik said:

differential calculus invented by newton is about slope and hence about Pythagoras theorem, isnt it?

No. 

56 minutes ago, Rajiv Naik said:

Newtons gravity is local based upon Galaleos  work.- doesnt apply to space.

The reason it is called "Newton's Law of Universal Gravitation" us because it isn't local and works for everything from apples to planets. 

58 minutes ago, Rajiv Naik said:

however fact remains that Calculus is based upon pythagoras theorem.

Please provide a reference for this claim. 

 

[studiot]

34 minutes ago, Strange said:

1 hour ago, Rajiv Naik said:

however fact remains that Calculus is based upon pythagoras theorem.

Please provide a reference for this claim. 

 

And there's me thinking 'calculus' is based on something called

"The fundamental Theorem of Calculus"

 

Reference Spivak : Calculus.

 

Silly me.

[Rajiv Naik]

what is calculus basically  ?

its  finding a third unknow variable (slope   ) c by using two other variables x and y of rt. angled triangle

Is it not pythagoras theorem?

this is  my way of looking at  it.

but

there is a proof by John Molokash dated 20/12/2010. One can google it and see.

[Strange]

3 hours ago, Rajiv Naik said:

Is it not pythagoras theorem?

No. Have you studied calculus?

3 hours ago, Rajiv Naik said:

there is a proof by John Molokash dated 20/12/2010. One can google it and see.

it is up to you to support your claims. So provide a link to it. 

[Rajiv Naik]

39 minutes ago, Strange said:

No. Have you studied calculus?

it is up to you to support your claims. So provide a link to it. 

not much in detail , but up to graduation level.

I can understand that its about relative position of particles or any entities in   various dimentions or in various coordinate systems.

its about exploitig infinity and infinitely small distances .insignificant in calculations but giving  a relative position of particles

I am now trying, to understand its linkage to unit vector- planc, constant and tensors- in geometrical way  ,3 dimensional or more.

prima facie personally feel it has reached its limitation and new theories are required.

I would like to get enlightened from you on any other meaning of calcules, So that I can improve on my aproach.

https://www.google.co.in/url?sa=t&source=web&rct=j&url=http://www.cut-the-knot.org/pythagoras/CalculusProof.shtml&ved=2ahUKEwi72Pv4nu_eAhUUaI8KHTZlArcQFjAAegQIBRAB&usg=AOvVaw3J5cjynOqfgHqmBJDuD2wh&cshid=1543138530959

[Strange]

41 minutes ago, Rajiv Naik said:

https://www.google.co.in/url?sa=t&source=web&rct=j&url=http://www.cut-the-knot.org/pythagoras/CalculusProof.shtml&ved=2ahUKEwi72Pv4nu_eAhUUaI8KHTZlArcQFjAAegQIBRAB&usg=AOvVaw3J5cjynOqfgHqmBJDuD2wh&cshid=1543138530959

This link (to here: http://www.cut-the-knot.org/pythagoras/CalculusProof.shtml) is a proof of the Pythagorean theorem using calculus, not the other way round.

It does NOT say that calculus is based on the Pythagorean theorem, it says that you can use calculus to prove the Pythagorean theorem. That is obvious, just from the title.

 

 

[studiot]

2 hours ago, Rajiv Naik said:

not much in detail , but up to graduation level.

 

I would like to get enlightened from you on any other meaning of calcules, So that I can improve on my aproach.

 

So did you pass yout graduation?

I would have thought that anyone who wants to get enlightened about a subject would find out about it when told that there is a Fundamental Theorem of that subject.

(I have been doing just exactly that after I read the thread on magnetostatics recently posted by beecee)

The Fundamental Theorem of Calculus asserts (in one dimension)

[math]\int_a^b {df}  = f(b) - f(a)[/math]

Clearly Pythagoras is not involved since there are no squares or square roots involved.

Does this mean anything to you?

(It should tell you that there is so much more to 'calculus' than differentiation)

Wikipedia has a reasonable discussion of this theorem.

https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

Since you appear to think calculus is only about derivatives here is a definition of the derivative

[math]f'\left( x \right) = \mathop {\lim }\limits_{h \to 0}  = \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}[/math]

 

Once again neither squares nor square roots are involved.

 

2 hours ago, Rajiv Naik said:

I can understand that its about relative position of particles or any entities in   various dimentions or in various coordinate systems.

its about exploitig infinity and infinitely small distances .insignificant in calculations but giving  a relative position of particles

I am now trying, to understand its linkage to unit vector- planc, constant and tensors- in geometrical way  ,3 dimensional or more.

prima facie personally feel it has reached its limitation and new theories are required.

 

You really need to know and properly understand more about basics before shooting for the moon trying for tensors and tensor calculus.

Confusingly tensor calculus was called 'the absolute differential calculus' when it was first introduced.
This was first applied to many and varied geometrical situations such as surfaces, shapes, curves and curvature as well as position.

This is now all part of the subject now known as 'differential geometry'.

Here is a typical multidemensional form of the Fundamental Theorem of Calculus, recast in multimensional form.

[math]\int_{\partial \omega } \omega   = \int_\Omega  {d\omega } [/math]

Where it relates the integral over some n dimensiona l differentiable manifold, [math]\Omega [/math] to the integral over the (n-1) dimensional  boundary of that manifold,  [math]{\partial \omega }[/math]

via

the (n-1) differential form,  [math]\omega [/math] with differential   [math]{d\omega }[/math]

this can be directly related to the one dimensional version i gave earlier.

and there is still no Pythagoras in sight.

 

As an aside

What is known as the triangle inequality or the CauchySchwarz inequality also appears as heisenberg's uncertainty principle and in many other fundamental places in Physics and applied Maths.

Perhaps you are confusing this situation with Pythagoras??

 

 

 

 

Edited November 25, 2018 by studiot

[Rajiv Naik]

I got the point What you are trying to say. thanks sir  for taking so much pains for explaining

usually I  type on mobile using dictation or handwriting software..so I usually try to keep it short , it could be little unscientific language but usually based on what I read.' hope III be excused for that.

I think  to. explain why I am saying that pythagoras theorem is the way chosen by all the further mathematics- includig calculus and Tensors in 3d or higher dimensions  requires more elaborate explaination. So will have to start from the beginning-  I "II try to do it by  going through details.. later.

Meanwhile

I feel that the defination of derivatives is about right angle triangle and finding slope of infinitely small portion..  ultimately .its a piece  of information

its property of a thing in a space-

and coordinate systems are just tools. 

I thinlK ,,,vectors  mathematical properties give rise to complex systems which are represented by tensors, its   interesting.

fundamental, its logical probability of an event -

which has two states in classical physics 1 and 0.

I am not really talking about more advanced mathematics or calculus- what I meant was that mathematics. is based upon logic , and also another aspect i.e. rules with wich  information /Light travels or gravity travels.  they  are all functions of this rules of π.

einstine was last person who tried to think differently-others are just exploring that physics- thereby so far only increasing the complexity of mathematics involved, last best thing that we have achieved is particle accelerators and instruments like ligo and advanced space telescopes. its too less since our last invention of  fission reaction.

it has become so complicated that Iwe now need superconputers assist us.

Arent there other ways than to follows sterotype mathematically which is just calculation and calculations - taking us nowhere except some materialistic inventions we have (not even to mathematical singularity like other fields in science are doing)

I am not an expert at advanced mathematical  methods. But I feel they are not difficult either if one spends some time with it.  otherwise as my profession, I have a lawfirm and studied Civil engineering also, at Goa , India.  -so not a scienceman realy.

 

I will like to stress that my entire  Pt. was ,we are heading tonowhere for next 40 years with current math, (except Al which will help us one day) and whatever Petty inventions physisists make after reading and  calculating for their entire life is not helping us practically.

even Trees and nature are still more intelligent  than us and we are like more evolved cavemen.

so we need a change in vision  in Physics and mathematical approach.

this are my pesonal views and   nothing personal against scientists or any person.

I joined the forum as I am expecting same new approach and minds I could learn about

ultimately we should know the truth of universe before dying

I 'll try to place. Some Proper material  after draftig it on computers with proper references as that is what is expected on this forum (which is great )

I liked this forum, even though  occasionally reprennanded- which is intact a good thing - will make me more serious about life and this forum. However If I am not filting in the scheme, and causing inconvinience, I may quit in the interest of others on forums

I must  thank you all for this, who are learned in the subjects involved.