Math and Reality. What is the deep connection?

In summary: It threw away the concrete notions of flat space, limited dimensionality, simple rotations, that are our most direct impression of the world about us, to explore what happens when such localized constraints are relaxed.Physics later found that these more generalized models of symmetry were actually fundamental in explaining the world.In summary, the conversation discusses the role of mathematics in modeling reality and the connection between mathematical equations and the physical world. The Dirac Equation is mentioned as one way to predict spins of particles and antimatter, and the question of why math is so effective in modeling reality is raised. The conversation also delves into the concept of symmetry and its role in nature, and how it is represented in mathematical language. Ultimately,
  • #1
rogerl
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First of all. We must not have any a priori or preconceived notions of what is reality but instead follow the evidence of what is the true structure (if there is) of reality.

The Dirac Equation is one way to start. It predicts spins of particles and antimatter. How come the mathematics of it correspond to reality? Are their bolts and nuts in our physical space and time wherein they can use the mathematics of the Dirac Equation. If not. What's the reason why math is so effective in modelling reality? You may say that math just describe the world and one must note of the difference between the model and the world being modeled. But why does the model correspond to reality. You can say that as one throws a ball. Trajectory can be calculated using math. That's why math is important. I'm not speaking of such Newtonian model. But more of why our modern physics that is so far from Newtonian models correspond to reality.

You may answer it is because of symmetry. For example, gauge symmetry produce our gauge bosons like the photons. But why does our physical world use symmetry. I mean. What kind of structure of nature can make use of symmetry in the first place?

Perhaps we can start by removing all together any notions of matter, space and time. What is it that can use math to project spacetime and matter into the "world". Maybe it' sall a computer program? What is your theory? Right now. I think it's like there is a pure energy in the Big Bang that uses math to project a world. But what is that energy? Most important. Why does it obey Dirac Equation for example. What is the mechanism, or causal mechanism??

It is not enough to wonder how space and time can morph into each other in Special Relativity. It still doesn't explain why Quantum Mechanics work. Dirac Equation is a good start becaus it already encompass SR and QM.

So what makes the Dirac Equation tick? What mechanisms in nature cause it to follow the math of the Dirac Equation. Anyone got an idea? I'm serious I'm really perplexed by all of this. I know physics just tell you about results of measurements and some models and nothing more about our reality. That is why we must contemplate it... because the physicists won't tell us. They are so compartmentalized that they just don't care what is the structure of reality... as long as they can produce measurements and got paid by the institutions... they are already happy and contented. So let's deal with stuff where they left off and won't touch.
 
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  • #2
This kind of question has been raised often as a thread, so perhaps that is why there are no takers for your question so far. Plus, it is not really that focused.

You can say for instance that all maths derives from concepts formed from experience, so it is less surprising that what is inspired by reality also has a connection with it. Secondly, a lot of maths seems just elaboration without a deep connection to the reality that inspired it at the start.

But if you focus on the meat of your question - the role of symmetry principles in nature - then you might get a more enlightening answer.

Symmetries are about changes that make no change. So they are what "have to be" in nature because nothing can prevent them from being so. No matter how you twist, the symmetry cannot be erased. So symmetries define causal closure (as in Noether's theorem. Symmetries define equilibrium states, as in dynamical modelling.

Thus symmetries are naturally fundamental. If there is change, it is where change "stops" even when change is still "going on". Symmetries are a description of natural limits where otherwise "things are going on".

Maths is then a formal language for describing patterns and describing symmetries is one of the things that maths has learned to do.

So symmetry principles are in the first instance natural (nature couldn't be any other way) and then they become compactly modeled in the language of maths.

Maths has then appeared to run ahead of physics because it was quicker to generalise symmetry principles. It threw away the concrete notions of flat space, limited dimensionality, simple rotations, that are our most direct impression of the world about us, to explore what happens when such localised constraints are relaxed.

Physics later found that these more generalised models of symmetry were actually fundamental in explaining the world.
 
  • #3
apeiron said:
This kind of question has been raised often as a thread, so perhaps that is why there are no takers for your question so far. Plus, it is not really that focused.

You can say for instance that all maths derives from concepts formed from experience, so it is less surprising that what is inspired by reality also has a connection with it. Secondly, a lot of maths seems just elaboration without a deep connection to the reality that inspired it at the start.

But if you focus on the meat of your question - the role of symmetry principles in nature - then you might get a more enlightening answer.

Symmetries are about changes that make no change. So they are what "have to be" in nature because nothing can prevent them from being so. No matter how you twist, the symmetry cannot be erased. So symmetries define causal closure (as in Noether's theorem. Symmetries define equilibrium states, as in dynamical modelling.

Thus symmetries are naturally fundamental. If there is change, it is where change "stops" even when change is still "going on". Symmetries are a description of natural limits where otherwise "things are going on".

Maths is then a formal language for describing patterns and describing symmetries is one of the things that maths has learned to do.

So symmetry principles are in the first instance natural (nature couldn't be any other way) and then they become compactly modeled in the language of maths.

Maths has then appeared to run ahead of physics because it was quicker to generalise symmetry principles. It threw away the concrete notions of flat space, limited dimensionality, simple rotations, that are our most direct impression of the world about us, to explore what happens when such localised constraints are relaxed.

Physics later found that these more generalised models of symmetry were actually fundamental in explaining the world.


But what's the mechanism for the symmetry. Like what structure in reality make it possible
at all? I mean, could actual calculations occur in some 2D surface in the Holographic Paradigm where our 3D world is just a projections of it. How then do the calculations in the 2D turn into projections of actual 3D objects. Or could nature be simply output of computer
programs. Or what other mechanisms could make it possible at all? A purely Newtonian world can't be symmetric in space and time, isn't it. Does this mean this world doesn't really exist as physical object? Because symmetry seems to imply that everything is dynamic.. even spacetime and matter... or else symmetry can't work... and the implications is that these could just be projections.
 
  • #4
rogerl said:
But what's the mechanism for the symmetry. Like what structure in reality make it possible at all?

That is the issue. And I would turn it round to say that no mechanism is necessary because it is precisely the lack of mechanism, the lack of structure, which makes symmetry possible.

Symmetry is what happens when nothing is detectably happening. So it is the natural default fundamental state. It is instead symmetry breaking which is the issue. This is what we are looking to explain via a mechanism, a structure, a cause.

You can choose to ask how a symmetry exists. But really it is the only thing that in the end cannot be got rid of - made to unexist. Eradicate every other source of difference and the fundamental symmetry is what is left as that which cannot be eradicated.

You can spin an irregular object and see all the time that something is happening. The orientation is changing. The symmetry is broken. You can even see if the spinning is fast or slow.

But spin a circle and you can't see any change. It may as well be stood still. Or it could be being spun at any speed. Or jiggling back and forth, making discrete quantum jumps, whatever you like. It is the natural limit of local rotations in a plane that you reach when every possible distinction has been removed.

It is the symmetry that contains all those possible kinds of actions I just mentioned as potential states - potential directions of symmetry breaking. But it is defined by its lack of any actual such breaking.

How do you then get things as concrete as spacetime or energy out of symmetry notions? Well that's when you need a theory of symmetry breaking.

At this point, I would usually start talking about Anaximander's theory about dichotomies and CS Peirce's semiotics and logic of vagueness. But it is enough now just to say that the challenge is not to explain why symmetries exist (because what would seem more natural than an absence of detectable change?) but why in fact we can see that a broken symmetry exists.

Somehow an expanding universe did arise as a symmetry breaking. So you should now probably ask of maths, where is your model of that? And maybe here, physics has pulled ahead of the race? Or at least you might want to consider condensed matter physics, the literature on phase transitions and self-organising criticality.

If maths is "unreasonably effective", we would be able to find a story on symmetry breaking already extant within maths, as well as all those deep descriptions of the principles of symmetry.
 
  • #5
apeiron said:
That is the issue. And I would turn it round to say that no mechanism is necessary because it is precisely the lack of mechanism, the lack of structure, which makes symmetry possible.

Symmetry is what happens when nothing is detectably happening. So it is the natural default fundamental state. It is instead symmetry breaking which is the issue. This is what we are looking to explain via a mechanism, a structure, a cause.

You can choose to ask how a symmetry exists. But really it is the only thing that in the end cannot be got rid of - made to unexist. Eradicate every other source of difference and the fundamental symmetry is what is left as that which cannot be eradicated.

You can spin an irregular object and see all the time that something is happening. The orientation is changing. The symmetry is broken. You can even see if the spinning is fast or slow.

But spin a circle and you can't see any change. It may as well be stood still. Or it could be being spun at any speed. Or jiggling back and forth, making discrete quantum jumps, whatever you like. It is the natural limit of local rotations in a plane that you reach when every possible distinction has been removed.

It is the symmetry that contains all those possible kinds of actions I just mentioned as potential states - potential directions of symmetry breaking. But it is defined by its lack of any actual such breaking.

How do you then get things as concrete as spacetime or energy out of symmetry notions? Well that's when you need a theory of symmetry breaking.

At this point, I would usually start talking about Anaximander's theory about dichotomies and CS Peirce's semiotics and logic of vagueness. But it is enough now just to say that the challenge is not to explain why symmetries exist (because what would seem more natural than an absence of detectable change?) but why in fact we can see that a broken symmetry exists.

Somehow an expanding universe did arise as a symmetry breaking. So you should now probably ask of maths, where is your model of that? And maybe here, physics has pulled ahead of the race? Or at least you might want to consider condensed matter physics, the literature on phase transitions and self-organising criticality.

If maths is "unreasonably effective", we would be able to find a story on symmetry breaking already extant within maths, as well as all those deep descriptions of the principles of symmetry.


I wonder what part of nature or reality decide what symmetry will work and which won't. For example. The symmetry SU(5) doesn't seem to work.. we still haven't detected protons decay. How come it didn't work and the symmetry of Gauge Theory work? Any idea? This would be important to decide what symmetry combinations can predict new experimental results and the right one.
 
  • #6
rogerl said:
I wonder what part of nature or reality decide what symmetry will work and which won't. For example. The symmetry SU(5) doesn't seem to work.. we still haven't detected protons decay. How come it didn't work and the symmetry of Gauge Theory work? Any idea? This would be important to decide what symmetry combinations can predict new experimental results and the right one.

The symmetry group is the gauge group. So SU(5) might not break down in ways that match observation, but E8 or something else could.

The issue is that we observe particles that fit the gauge group SU(3) × SU(2) × U(1). So the theory works at that level. But the belief is that all these must be fragments of one larger unified gauge group.

Personally I am not so sure there must be a single higher symmetry group. It could be that everything was breaking towards zero gauge - so no other forces represented, just a flat cooling and spreading gravity field - and SU(3) × SU(2) × U(1) represent the three simplest symmetries at the end of the line which trapped some of the energetic slop of the big bang as massive particles.

So one view would say for some reason a particular higher symmetry (like E8) existed, then that got broken into a collection of bits. The other says that there was a general run through every possible symmetry state, but most could not produce any stable structure, and finally a last few very simple and reduced states could.

It would only be in the first view that you would expect to have to find one perfect higher symmetry that explains everything. In the second view, all symmetry states "exist", but most just aren't productive.

And then either way, there is the further need for an energy potential to make anything actually happen. There must be a field that cools/expands, so allowing the simpler symmetry states to crystalise out as particles and their particular kinds of interactions.

So symmetry principles explain the only possible forms that particles could take. But the theory also needs a story of the substance which could become those forms.
 
  • #7
To throw my two cents in, aside that rather penetrating analysis by apeiron, I agree that the OP question is a very deep one, one that Einstein himself pondered without making much headway. It might help to reframe the question a little. If we frame the question in a way that sounds like "why is reality like that", we are asking too much of ourselves, we are asking what we just don't get to know. But if we frame it like "why do we get power over reality by thinking about it in such-and-such a way", then we have a question that is as much about ourselves as it is about reality. We are asking a question about the goals that we have set out for ourselves, and are asking why we accomplish those goals, but we can also recognize the goals we have not set, and recognize how completely we have failed to achieve those other goals.

Which brings me to my point about symmetries-- perhaps it is natural that we gain benefits from imagining that reality is described by a set of idealized symmetries, because symmetries are the ultimate simplifiers. Take translation symmetry-- imagine a universe where the laws of physics were different at every point. What kind of physics could we do there? Well, certainly we'd have no reason to even use the concept of "space" at all, if the laws were different at every location. Somehow, the whole reason we imagine there is any such thing as space must arise from the (nearly unbroken) translational symmetry it exhibits.

Is there really any such thing as either space, or translational symmetry? I highly doubt it. It simply isn't true that any experiment we do in one laboratory must be the same in any other-- what is true is that they must be different. However, they are so incredibly close, to the precision we are interested in, that we find more value in noticing the similarities than in noticing the differences, and that is why we generate a concept of space. But we don't want complete translational invariance in reality, because a universe where every point is exactly like every other point is a universe of one point. We only want translational invariance in the idealizations we are using to understand reality, it is necessary that the reality itself not exhibit that invariance. We need a universe of enough complexity to allow interesting things to happen, but then we also need to find simplifying idealizations, like symmetries, that allow us to gain understanding and power in that universe.

In short, we see what we understand (for what we do not understand is just a ? that we have learned to get by without thinking about too much), and what we understand involves nearly unbroken symmetries, so what we see is a universe of nearly unbroken symmetries. Then we turn around and say "I wonder why the universe is like that?", not realizing that we are really looking in the mirror. In this light, we are really asking "why does thought work this way?", and that is the fundamental question. Then we face the daunting conundrum, how can thought figure out what thought is? Personally, I think we just don't get to know the answer to this question, all we get to do is find out how far we can get imagining the universe has the properties we attribute to it.
 
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  • #8
The simple fact is that some mathematics is so "applicable" to "the real world" is similar to the reason why some science fiction correctly "predicts" many future things- there is so much, some of it happens to be right!

More specifically, a mathematical theory is a "template"- in the same way you can have a number of different forms that you then can fit to a specific business application. All mathematical theories are "undefined terms" (like "point" and "line" in geometry). To apply a mathematical theory to a specific situation, physics or other, you assign meanings from those "undefined terms", choosing the meanings and the theory by "best fit"- that theory and those meanings that make the axioms "true" in that particular application. Of course, they are never exactly true because measurements are never exactly correct. But we can choose, out of the infinite number of possible mathematical theories those that fit best.
 
  • #9
But that cannot be enough of an explanation, because it could only explain why math theories work to predict observations that have already occured. It doesn't explain why they can at times predict observations that have not yet occured, like Einstein's prediction of bending light, or Dirac's prediction of positrons. It's true that "winners write the history", so we just don't hear about all the failed attempts, but were there really theories of gravity that successfully explained Mercury's orbit, which had been observed, but failed to get the right deflection of starlight, which had not? There is a sense of inevitability about Einstein's approach to both, so much so that Einstein himself said that he never really paid too much attention to the observations, so certain was he that his theory had to be right. He might have been lucky, certainly, but still, I don't think we selected from a thousand different predictions for the deflection of starlight the one that got it right-- when Eddington did his famed expedition, he really only had one mathematical model in mind that he was testing, and it worked.

So there has to be something about the nearly unbroken symmetries that we have identified, that we tend to get out from them something more than we put in. There really has to be something going on that brings out these special types of symmetries, and we recognize their specialness, but we ignore all the possible ones that clearly are just not working. So perhaps you are saying that we automatically reject so many theories without even thinking about it, that the ones we don't reject appear to take on an undue specialness. I think that's true-- we are doing so much sorting "beneath the radar," it is part of what went into our intelligence, part of why intelligence evolved the way it did.
 
  • #10
I will refer, as I have many times, to Wigner's paper: the unreasonable effectiveness of mathematics in the natural sciences
 
  • #11
Ken G said:
Which brings me to my point about symmetries-- perhaps it is natural that we gain benefits from imagining that reality is described by a set of idealized symmetries, because symmetries are the ultimate simplifiers. Take translation symmetry-- imagine a universe where the laws of physics were different at every point. What kind of physics could we do there? Well, certainly we'd have no reason to even use the concept of "space" at all, if the laws were different at every location. Somehow, the whole reason we imagine there is any such thing as space must arise from the (nearly unbroken) translational symmetry it exhibits.

The key question may be whether we are imagining the symmetries as something we are breaking away from, or headed towards. Are they in the universe's past, or in its future?

I think we can say the universe is a highly broken symmetry from the point of view that it is a long way from the infinite dimensional space it could be. Being 3D is a highly reduced state of affairs - very asymmetric when you compare 3 to infinity. But then a complete breaking of that infinite symmetry would be to arrive at a 0D point.

However, a point or singularity is again a highly symmetric state itself - infinity of another kind as you point out.

So I see a curious resemblance here to the notion of maximum entropy in thermodynamics.

There are in fact two extremes of order in an ideal gas. One pole of order is where all the particles are trapped in the same small corner (and so will want to spread out randomly). The other is where all the particles are trapped in an exact lattice configuration, completely regular in their placing (the situation covered by the third law of thermodynamics). Let free, the particles will again spread out randomly (though will be able to scramble their positions much more quickly of course).

So there could be a good reason why reality ends up poised between infinite dimensionality and zero dimensionality here. There could be a logical reason for the nearly broken state (as it is in fact a fully broken or entropic state once you recognise that the extremes of symmetry or order are dichotomous and being maximally broken falls somewhere between the two states as an equilibrium balance).

Another quick point is that your translational symmetry argument holds true I believe only for flat space (of any number of dimensions).
 
  • #12
There are consequences or emergence by knowing how math is related to reality.

1. For example, if all the abstract math calculations occur in the 2D surface in the Bekestein's Holographic Principle and our 3D world is just a projection, how is it projected, there will be additional forces or bosons that do the projections much like the screen projector at the movie theater sending all those photons to the screen.

2. If our reality is just output of a computer program, there will be consequence in that we can understand how the output occurs and even predict new *phenomenon* esp when the program is interactive with the source. Note all Relativity, quantum mechanics, and the second law of thermodynamics (entropy) are all limitations on the speed, quantity, or quality of information transfer. This may be so in order they can be processed by some kind of computer the finite discrete information in the 3. Do you think this is possible?

3. If the Big Bang started with first an Idea from a Mind, then the Idea become matter and energy first thru Symmetries, then symmetries is spontaneously broken to give rise to the universe. If this were true, then all the constants of nature were fine tune that way to cause the existence of living matter and mind. This is as likely as Anthropic Principles where infinite parallel universes were created and life occurs here because we live right in the one where life is possible. But believing constants of nature were fined tune is equally likely and even simpler via Occam's Razor. Now if there were true, the consequence is that the Will that main the constant of nature can be tapped by the Will of our Mind itself. Is this the explanations for all the miracles done by Saints because they can sometime tap the Mind and create/uncreate reality?

4. What else? What kind of structure of reality (whether mechanical or fluid-like process) is versatile enough to use the math of the Dirac Equations for example to produce antimatter. What else beside Holographic Principle, Matrix-like computer program virtual reality and Universe as output of Mind? Which of these 3 are more likely?

I think nailing the Final Theory may involve knowing at least how this occur.
 
  • #13
This strikes me as a question that could only ever be answered AFTER a final TOE is discovered and understood... i.e hindsight... if ever.
 
  • #14
I think we can never arrive at the TOE if we don't have answers to it first. For example. SU(5) is very elegant and should work. How come SU(3)xSU(2)xU(1) works and not SU(5).. that is.. Protons don't decay. In the case of #1. Maybe we can to integrate the
3D projection scalar field to explain a sub-broken symmetry or sora. In case #2 where we are inside a computer program, perhaps SU(5) doesn't work because of certain programming algorithm use. In the case of #3. Perhaps the will aspect that hold in maintenance the values of the constants of nature are not included in the calculations. So it's possible we will never arrive at any Final Unification without taking this into account. Just for example Superstring Theory. Lee Smolin said we don't even know what is the basic principle of it and zero experimental confirmation of any kinds of its dozens of predictions. A phycisist even describe our current physics as "Recreational Mathematical Theology". Now knowing the mechanisms of how it all occur whether one of the 3 cases or others can give us guiding principle that can make us nail the TOE.
 
  • #15
rogerl said:
What else beside Holographic Principle, Matrix-like computer program virtual reality and Universe as output of Mind? Which of these 3 are more likely?

I think nailing the Final Theory may involve knowing at least how this occur.

But all your thoughts here depend on a particular presumption about maths - that it is all about "someone" doing calculations. Constructing outcomes. Computing results.

I would instead argue that the fundamental part of maths is identifying the global constraints - generalising to discover axiomatic truths. After that comes the building, constructing, calculating, computing.

Axioms can't themselves be calculated or computed. The ability to calculate and compute is instead what they create.

Physics cannot have a final theory which has to posit mechanism that constructs more mechanism. That is simply an endless regress. Instead it must have a theory about how mechanism in general arises. So the ground of reality must be viewed non-mechanistically.

The holography principle, for instance, is firstly a theory about global constraints - an exact relationship between the surface area of a sphere and the information that can be contained within the volume of that sphere.

A projection mechanism would then be a secondary use of the basic idea. And not a plausible one IMO.

So a creating god fails as an explanation due to infinite regress (creators need creating). Matrix style simulations fail for the same reason (if there is a computer constructing reality, someone has to construct the computer).

And holography is a general constraints-based view. But it smuggles in a symmetry-breaking because there is a Planck scale that produces the concrete result by introducing a local cut-off. And attempts to tell a projection story based on dualities such as ads/CFT is I think mistaking the natural reason for such dualities of description.

String theory suggest extra local degrees of freedom to quantum field theory. Now those degrees could be found "inside" a point in the field. Or you can instead map them to the outside of the field and treat them as a projection back on to the locale. The description is symmetric.

This in fact fits my systems POV. What exists locally is the product of global constraints. So there is a kind of projection happening from the global to the local. But it is, once again, a matter of constraint, not construction. There is no top-down calculation happening, no extra party needed to do some work and ensure an outcome occurs.
 
  • #16
rogerl said:
Now knowing the mechanisms of how it all occur whether one of the 3 cases or others can give us guiding principle that can make us nail the TOE.

You realize that in talking about SU(5), you are talking about GUTs not TOEs?

GUTs - and SO(10) would be a more popular candidate these days - unify the three forces, but not gravity.

A TOE would be a unification that includes gravity - like string theory hopes to be.

Although E8xE8 is a candidate gauge symmetry that might be large enough to make a connection with string symmetries and so incorporate gravitons as gauge particles.

So there is the gauge stuff that seems robust. The SU(3) + SU(2)/U(1) works. Then there is a lot that is extremely sketchy. There is no clear candidate for a single GUT unification symmetry. And a TOE unification is a very sketchy idea still.

It seems like gauge symmetry is a basic way reality organises. But there is much less that can be said about any symmetry breaking mechanism that gets us to what we can observe.

There could be one higher symmetry that gets broken. Or there could be quite a different way that "breakings" occur, and so quite a different view on how to expect a TOE to look.
 
  • #17
apeiron said:
But all your thoughts here depend on a particular presumption about maths - that it is all about "someone" doing calculations. Constructing outcomes. Computing results.

I would instead argue that the fundamental part of maths is identifying the global constraints - generalising to discover axiomatic truths. After that comes the building, constructing, calculating, computing.

Axioms can't themselves be calculated or computed. The ability to calculate and compute is instead what they create.

Physics cannot have a final theory which has to posit mechanism that constructs more mechanism. That is simply an endless regress. Instead it must have a theory about how mechanism in general arises. So the ground of reality must be viewed non-mechanistically.

The holography principle, for instance, is firstly a theory about global constraints - an exact relationship between the surface area of a sphere and the information that can be contained within the volume of that sphere.

A projection mechanism would then be a secondary use of the basic idea. And not a plausible one IMO.

So a creating god fails as an explanation due to infinite regress (creators need creating). Matrix style simulations fail for the same reason (if there is a computer constructing reality, someone has to construct the computer).

And holography is a general constraints-based view. But it smuggles in a symmetry-breaking because there is a Planck scale that produces the concrete result by introducing a local cut-off. And attempts to tell a projection story based on dualities such as ads/CFT is I think mistaking the natural reason for such dualities of description.

String theory suggest extra local degrees of freedom to quantum field theory. Now those degrees could be found "inside" a point in the field. Or you can instead map them to the outside of the field and treat them as a projection back on to the locale. The description is symmetric.

This in fact fits my systems POV. What exists locally is the product of global constraints. So there is a kind of projection happening from the global to the local. But it is, once again, a matter of constraint, not construction. There is no top-down calculation happening, no extra party needed to do some work and ensure an outcome occurs.


So what you are saying is that if anyone of the 3 were what was behind the scene of our relaity. Then it is possible we will never have a TOE.. because then we can't know what is say behind the Mind that wills the values of the constants of nature? Or maybe we can only reach the unification of the 4 fundamental force and physics end and we can say we have physical TOE for the forces relevant to measurements only and not beyond. Of course, this assumes that we can unite GR and QM in the first place by Superstrings. If GR, QM can only be united by treating the Planck scale as the pixel of reality, then perhaps there comes a point when we can no longer go beyond it and call it quits. Right?
 
  • #18
apeiron said:
You realize that in talking about SU(5), you are talking about GUTs not TOEs?

GUTs - and SO(10) would be a more popular candidate these days - unify the three forces, but not gravity.

A TOE would be a unification that includes gravity - like string theory hopes to be.

Although E8xE8 is a candidate gauge symmetry that might be large enough to make a connection with string symmetries and so incorporate gravitons as gauge particles.

So there is the gauge stuff that seems robust. The SU(3) + SU(2)/U(1) works. Then there is a lot that is extremely sketchy. There is no clear candidate for a single GUT unification symmetry. And a TOE unification is a very sketchy idea still.

It seems like gauge symmetry is a basic way reality organises. But there is much less that can be said about any symmetry breaking mechanism that gets us to what we can observe.

There could be one higher symmetry that gets broken. Or there could be quite a different way that "breakings" occur, and so quite a different view on how to expect a TOE to look.

Yes i know SU(5) is the GUT. I just rereading Lee Smolin book Trouble with Physics and he said the SU(5) is elegant and should work but didn't, so we mustn't be mesmerized by the beauty myth hence Superstrings is very like not true.

You are saying it is possible that the 4 forces can't be unified at all? What motivate us to believe they can in the first place? Maybe electroweak and strong force and gravity are really different and not mergeable. Do we believe electroweak and the strong force can be unified because when we run and penetrate beyond the virtual particles haze, we can see them coincide at a certain Gev? Can't it be just coincident and they are really not unifiable?
 
  • #19
rogerl said:
I think we can never arrive at the TOE if we don't have answers to it first. For example. SU(5) is very elegant and should work. How come SU(3)xSU(2)xU(1) works and not SU(5).. that is.. Protons don't decay. In the case of #1. Maybe we can to integrate the
3D projection scalar field to explain a sub-broken symmetry or sora. In case #2 where we are inside a computer program, perhaps SU(5) doesn't work because of certain programming algorithm use. In the case of #3. Perhaps the will aspect that hold in maintenance the values of the constants of nature are not included in the calculations. So it's possible we will never arrive at any Final Unification without taking this into account. Just for example Superstring Theory. Lee Smolin said we don't even know what is the basic principle of it and zero experimental confirmation of any kinds of its dozens of predictions. A phycisist even describe our current physics as "Recreational Mathematical Theology". Now knowing the mechanisms of how it all occur whether one of the 3 cases or others can give us guiding principle that can make us nail the TOE.

I'm reminded of the 'Hitchiker's Guide To The Galaxy', That you want the answer before you know the question. I don't think that this is a place where the cart can be placed before the horse. I doubt we'll ever find a TOE, but certainly we won't have spontaneous answers before we have questions... or rather, when we do we require even more time and effort to formulate the questions that make the answers sensible.
 
  • #20
rogerl said:
If GR, QM can only be united by treating the Planck scale as the pixel of reality, then perhaps there comes a point when we can no longer go beyond it and call it quits. Right?

It could be that some aspects of a TOE remain arbitrary - such as a handful of constants needed to define coupling strengths, the Planck scale, or other essential parameters.

In that case, either some aspects of reality are irreduciably random, a matter of chance which way the symmetry broke. Or it could instead be anthropic. Every choice was made "somewhere" and so there is an infinity of universes out there.

But what would be really nice is if even the constants are fixed by some kind of self-organising logic. Like they could be emergent ratios in the way that pi, e and other mathematical constants are.

So perhaps answering my own question earlier in the thread, maths models the symmetries, and what maths has to say about constants could be central to the story of how symmetries get broken.

Constants arise in maths as the limits of processes. Limits of complex processes in fact.

Anyway, getting back to your going beyond the final pixel, there is a way to do this based on a logic of vagueness. Because instead of just dissolving smallness, you are also dissolving any largeness. As you zero in on the pixel, it dissolves, and you dissolve, so nothing definite now can lie beyond (as finding anything smaller would demand that you as the observer are still around to see it).

This is the quantum foam story of course. The HUP principle. Except that now we have a better name to describe what "lies beyond" - vague potential.
 
  • #21
rogerl said:
You are saying it is possible that the 4 forces can't be unified at all?

Personally I'm not sure whether they should or shouldn't. But I see gravity as a global symmetry rather than a local or gauge one, so that is one reason to think that it is not going to end up as a "fourth force" unified with a gauge symmetry.

The reason for expecting EW and strong to be unified is definitely because of the way the couplings run and look to become vanilla at the GUT scale. That is a strong motivation. And gravity would also run, suggesting a Planck scale unification of some sort

Or instead, in my view, a Planck scale dissolution into vagueness. That is gravity would have the vanilla force strength, but it would no longer be coherently directed so as to be pulling in a clear direction. The 3D space would have dissolved into foam by that stage.
 
  • #22
Guys. Is there a possibility that the TOE would involve no math at all... since physics involves math.. and TOE is beyond math. Then it's the end of physics? What I mean to say is. What if beyond the Planck is a realm that is no longer describable by math. Is this possible? Or would physics evolve to other systems of modelling that don't use math anymore? If so, Symmetry may be the final concept that is still math connected. Beyond Symmetries may lie mathless facts... for example, what if in the Planck realm where spacetime breaks down and quantum fluctuations ruin any continuous manifold lie something that is no longer modelable by math?
 
  • #23
rogerl said:
Guys. Is there a possibility that the TOE would involve no math at all... since physics involves math.. and TOE is beyond math. Then it's the end of physics? What I mean to say is. What if beyond the Planck is a realm that is no longer describable by math. Is this possible? Or would physics evolve to other systems of modelling that don't use math anymore? If so, Symmetry may be the final concept that is still math connected. Beyond Symmetries may lie mathless facts... for example, what if in the Planck realm where spacetime breaks down and quantum fluctuations ruin any continuous manifold lie something that is no longer modelable by math?

Most would say the problem lies in observable maths - scientific models that are framed as formal statements with measurable consequences (rather than unmeasurable ones).

Science involves both models and measurements in interaction. So for a TOE to be scientific, it has to be testable.

If a TOE is just a formally stated hypothesis that is ultimately untestable, then it becomes metaphysics.

So I would say it is looking like the maths will stretch beyond the eventual reach of science. Instead of mathless fact, it is more likely we will end up with factless maths!

Which is good news for the future career prospects of metaphysicians.
 
  • #24
apeiron said:
Most would say the problem lies in observable maths - scientific models that are framed as formal statements with measurable consequences (rather than unmeasurable ones).

Science involves both models and measurements in interaction. So for a TOE to be scientific, it has to be testable.

If a TOE is just a formally stated hypothesis that is ultimately untestable, then it becomes metaphysics.

So I would say it is looking like the maths will stretch beyond the eventual reach of science. Instead of mathless fact, it is more likely we will end up with factless maths!

Which is good news for the future career prospects of metaphysicians.


So even if it were true that the constants of nature were fine tuned into values compatible with life by a Universal Mind. It still can be modeled by math? It's like this. Big Bang might occur to seed life and mind. Now this can explain how our mind can tap the Mind of the universe and even make us possesses the power to create/uncreate reality (nothing that Miracles occurred down the centuries where Saints can bilocate (be in 2 places at once) and even levitate as well as numerous phenomena documented by many sources). So you mean all these can be modeled by math? If so, then it is part of physics and not metaphysics (which can't be modeled by math). In other words, any measurable effects can be modeled by math, is this what you are saying?

"There are more things in Heaven and Earth, Horatio than are dreamt off in your philosophy".
 
  • #25
rogerl said:
So even if it were true that the constants of nature were fine tuned into values compatible with life by a Universal Mind. It still can be modeled by math? It's like this. Big Bang might occur to seed life and mind. Now this can explain how our mind can tap the Mind of the universe and even make us possesses the power to create/uncreate reality (nothing that Miracles occurred down the centuries where Saints can bilocate (be in 2 places at once) and even levitate as well as numerous phenomena documented by many sources). So you mean all these can be modeled by math? If so, then it is part of physics and not metaphysics (which can't be modeled by math). In other words, any measurable effects can be modeled by math, is this what you are saying?

"There are more things in Heaven and Earth, Horatio than are dreamt off in your philosophy".

I'd tell Billy that he's in the wrong forum for metaphysics.
 
  • #26
rogerl said:
So even if it were true that the constants of nature were fine tuned into values compatible with life by a Universal Mind. It still can be modeled by math? It's like this. Big Bang might occur to seed life and mind. Now this can explain how our mind can tap the Mind of the universe and even make us possesses the power to create/uncreate reality (nothing that Miracles occurred down the centuries where Saints can bilocate (be in 2 places at once) and even levitate as well as numerous phenomena documented by many sources). So you mean all these can be modeled by math? If so, then it is part of physics and not metaphysics (which can't be modeled by math). In other words, any measurable effects can be modeled by math, is this what you are saying?

"There are more things in Heaven and Earth, Horatio than are dreamt off in your philosophy".

OK, so now you want to make me regret taking your questions seriously :smile:.

Miracles would have to count as observables to fall within the realms of scientific discussion. So you say these things happened, but if there is no good evidence so far as the rest of us are concerned, then there is no real interest in debating the implications for theory.

And the universal mind seems to me a vacuous concept unless it can be pinned down as a formalism that is at least potentially measurable. If you can provide a source that does that for you, please do.

Peircean semiotics is for example a kind of "universal mind" approach - but nothing like what you probably mean here.

So time to source your model of the universal mind, citing some mainstream philosophical approach. But it is unlikely I would have much to say about it.
 
  • #27
apeiron said:
OK, so now you want to make me regret taking your questions seriously :smile:.

Miracles would have to count as observables to fall within the realms of scientific discussion. So you say these things happened, but if there is no good evidence so far as the rest of us are concerned, then there is no real interest in debating the implications for theory.

And the universal mind seems to me a vacuous concept unless it can be pinned down as a formalism that is at least potentially measurable. If you can provide a source that does that for you, please do.

Peircean semiotics is for example a kind of "universal mind" approach - but nothing like what you probably mean here.

So time to source your model of the universal mind, citing some mainstream philosophical approach. But it is unlikely I would have much to say about it.

Ok. Miracles can be observable too and can be modeled by math. Now let's not discuss further on this lest this thread be locked. Let's go back to GR, SR QM and be within just its small circle. Anyway. I just found out about Brian Greene new book "The Hidden Universe
... I'm listening to MP3 now about chapter 10 called "Computers, Universes and Mathematical Reality". It sounds interesting.
 
  • #28
rogerl said:
Ok. Miracles can be observable too and can be modeled by math. Now let's not discuss further on this lest this thread be locked. Let's go back to GR, SR QM and be within just its small circle. Anyway. I just found out about Brian Greene new book "The Hidden Universe
... I'm listening to MP3 now about chapter 10 called "Computers, Universes and Mathematical Reality". It sounds interesting.

No rogerl, you don't throw out "mathematically modeled miracles" and walk away. You should be worried about your fate, not the thread's.

Please, do explain in accordance with PF guidelines or retract your claim. And roger... I'm in a mood to hurt someone, so let's not make this emotionally painful.
 
  • #29
nismaratwork said:
No rogerl, you don't throw out "mathematically modeled miracles" and walk away. You should be worried about your fate, not the thread's.

Please, do explain in accordance with PF guidelines or retract your claim. And roger... I'm in a mood to hurt someone, so let's not make this emotionally painful.


Ok. I'll retract it. Mind has nothing to do with the Big Bang or the Constants of Nature. General Relativity, Special Relativity and Quantum Mechanics are the final words and we are bound to them. Our brain is just emergence of biochemistry. And the world is just pure physical, with no possibility of anything higher. If I need to delete the threads.. just let me know as one can only delete them within 2 hours.
 
  • #30
rogerl said:
Ok. I'll retract it. Mind has nothing to do with the Big Bang or the Constants of Nature. General Relativity, Special Relativity and Quantum Mechanics are the final words and we are bound to them. Our brain is just emergence of biochemistry. And the world is just pure physical, with no possibility of anything higher. If I need to delete the threads.. just let me know as one can only delete them within 2 hours.

No, this works for me. :smile:
 
  • #31
Brian Green new book is very interesting... so relevant to many of the topics we discuss like Mathematical Reality. I wonder what would happen if superstrings and branes are just figment of the imagination. Lee Smolin believes in "Trouble with Physics" that they may not be real.. instead of 20 constants of nature.. we now have over 250 constants of nature if Superstrings are real.. so maybe some other Ultimate Symmetry can bind all that can explains the constants? He who discovers it can win 2 Nobel at the same time.. lol
 
  • #32
rogerl said:
Brian Green new book is very interesting... so relevant to many of the topics we discuss like Mathematical Reality. I wonder what would happen if superstrings and branes are just figment of the imagination. Lee Smolin believes in "Trouble with Physics" that they may not be real.. instead of 20 constants of nature.. we now have over 250 constants of nature if Superstrings are real.. so maybe some other Ultimate Symmetry can bind all that can explains the constants? He who discovers it can win 2 Nobel at the same time.. lol

So long as you realize these guys are beating each other up over the 2% they don't agree on rather than the 98% that they do.

The 2% is of course where they make a career for themselves.
 
  • #33
apeiron said:
The key question may be whether we are imagining the symmetries as something we are breaking away from, or headed towards. Are they in the universe's past, or in its future?
Another duality. Perhaps it could never be one without the other-- the symmetries from our past that are breaking gives meaning to the concept of past, and the symmetries we are breaking toward give meaning to the concept of future. The duality gives meaning to the concept of time.
I think we can say the universe is a highly broken symmetry from the point of view that it is a long way from the infinite dimensional space it could be. Being 3D is a highly reduced state of affairs - very asymmetric when you compare 3 to infinity. But then a complete breaking of that infinite symmetry would be to arrive at a 0D point.

However, a point or singularity is again a highly symmetric state itself - infinity of another kind as you point out.
Yes, probably the near-symmetry is that the potentially infinite dimensional universe is "modded out" by the four dimensions of spacetime, much like the integers mod 4. The other dimensions are probably not completely gone-- there would be a hierarchy of these nearly unbroken symmetries, the nearly unbroken symmetry between time and space, and the nearly unbroken symmetry of the 11 dimensions of M-theory (if it is a valid theory). So we cannot completely mod out the dimensions above 4, but we nearly can, and most likely we could not completely mod out the dimensions above 11, but we nearly can, etc. ad infinitum. We live in the world of the most broken symmetry, the symmetry between space and time and the mod 4 symmetry of spacetime, with only ghostly hints of the higher-order much more nearly unbroken symmetries. Since we see what we understand, our minds automatically collapse those nearly unbroken symmetries into obscurity, until very careful and high energy observations resurrect them.
There are in fact two extremes of order in an ideal gas. One pole of order is where all the particles are trapped in the same small corner (and so will want to spread out randomly). The other is where all the particles are trapped in an exact lattice configuration, completely regular in their placing (the situation covered by the third law of thermodynamics).
Yes-- delta functions in the two complementary variables of "x" and "k" space. Another duality-- one gives us constructive interference with location, the other constructive interference with motion. Neither by themselves has any meaning-- location means nothing without motion to connect locations, and motion means nothing without locations to connect. Since neither can trigger recognition in us without the other, what we see as reality must involve a blend of both, a blend that is not a pure state of either. Equilibrium between the poles.

So there could be a good reason why reality ends up poised between infinite dimensionality and zero dimensionality here. There could be a logical reason for the nearly broken state (as it is in fact a fully broken or entropic state once you recognise that the extremes of symmetry or order are dichotomous and being maximally broken falls somewhere between the two states as an equilibrium balance).
Yes, and I think that reason must have to do with what we count as important, worth noting. Our intelligence has evolved to notice an incredibly tiny fraction of "what is really going on", and label it as "important", because that's how we get power over our environment, and that's how we survive. It's as much about us, about how we think about reality, as it is about reality itself (another duality-- neither would mean anything without the other).
Another quick point is that your translational symmetry argument holds true I believe only for flat space (of any number of dimensions).
Yes, symmetries in GR are local, they have differential generators rather than global ones. Not that I'm any GR expert.
 
  • #34
apeiron said:
So long as you realize these guys are beating each other up over the 2% they don't agree on rather than the 98% that they do.

The 2% is of course where they make a career for themselves.

Have you read Smolin's "Trouble with Physics"? He wants us to go back to fundamentals and try to understand for example what made QM ticks. He even thinks the entire Superstring theory is false.. so I wonder what 98% are you talking about. I know he digs for Loop Quantum Gravity.. but he is saying quantum mechanics needs to be reinvestigated because the unification may not be along the superstring programme. Anyway. After rereading the book. I realized that there are two kinds of physicists, the realist and the other one. Einstein, Schroedinger's are realists.. like Smolin, in that they want to look for a physical side to the theory. But the other parties like Feynman, Julian, 't Hoofe... mathematical beauty is enough without regards to any physical basis or mechanism. This is why I wonder what math has to do with reality because it can give us insight if Smolin has basis of the need to look for the physical side of it or just pure mathematical investigations in the case of the others without regards for what form of structure or nature reality have take.
 
  • #35
Lee Smolin wrote about the following in Trouble with Physics (just brief quotes):

"In the approach of particle physics developed and taught by Richard Feynman, Freeman Dyson, and others, reflections on foundational problems had no place in research"

"However, as I will argue in detail in the pages to come, the lesson of the last thirty years is that the problems we're up against today cannot be solved by this pragmatic way of doing science. To continue the progress of science, we have to again confront deep questions about space and time, quantum theory, and cosmology"


Do you guys believe in Smolin approach? If Smolin is right and we don't do re-investigation of foundational problems, we will never have any TOE. So it's NOT like we have to find the TOE first and then contemplate on the insight later. I still can't decide if Smolin is right or wrong.. and this is why I asked the initial questions in the thread.
 
<h2>1. What is the relationship between mathematics and reality?</h2><p>The relationship between mathematics and reality is a complex and debated topic. Some argue that mathematics is a human construct and therefore not inherently connected to reality. Others believe that mathematics is a fundamental part of the universe and that it accurately describes and predicts the workings of reality. Ultimately, the answer may lie somewhere in between, with mathematics being a tool that humans use to understand and manipulate the world around us.</p><h2>2. How does mathematics help us understand reality?</h2><p>Mathematics provides a precise and logical framework for understanding and describing the natural world. It allows us to make predictions and test theories, and has been crucial in many scientific discoveries and advancements. For example, the laws of physics are described using mathematical equations, and these equations have been used to predict and explain phenomena such as the motion of planets and the behavior of particles.</p><h2>3. Can mathematics prove the existence of reality?</h2><p>Mathematics itself cannot prove the existence of reality, as it is a tool created by humans to describe and understand the world. However, the fact that mathematical models and equations accurately predict and describe real-world phenomena can be seen as evidence for the existence of reality.</p><h2>4. How does the study of mathematics impact our perception of reality?</h2><p>The study of mathematics can impact our perception of reality in several ways. It can help us understand the underlying patterns and relationships in the world around us, leading to a deeper appreciation and understanding of reality. It can also challenge our preconceived notions and expand our understanding of what is possible. Additionally, the study of mathematics can lead to technological advancements that shape our perception of reality, such as the development of computers and other technologies.</p><h2>5. Are there any limitations to the connection between mathematics and reality?</h2><p>While mathematics has proven to be a powerful tool for understanding and describing reality, it does have its limitations. For example, there are still many phenomena that cannot be fully explained or predicted using mathematical models. Additionally, our current understanding of mathematics may be limited by our human perspective and may not fully capture the complexity of reality. As our understanding of mathematics and the universe continues to evolve, so too may our understanding of the connection between the two.</p>

1. What is the relationship between mathematics and reality?

The relationship between mathematics and reality is a complex and debated topic. Some argue that mathematics is a human construct and therefore not inherently connected to reality. Others believe that mathematics is a fundamental part of the universe and that it accurately describes and predicts the workings of reality. Ultimately, the answer may lie somewhere in between, with mathematics being a tool that humans use to understand and manipulate the world around us.

2. How does mathematics help us understand reality?

Mathematics provides a precise and logical framework for understanding and describing the natural world. It allows us to make predictions and test theories, and has been crucial in many scientific discoveries and advancements. For example, the laws of physics are described using mathematical equations, and these equations have been used to predict and explain phenomena such as the motion of planets and the behavior of particles.

3. Can mathematics prove the existence of reality?

Mathematics itself cannot prove the existence of reality, as it is a tool created by humans to describe and understand the world. However, the fact that mathematical models and equations accurately predict and describe real-world phenomena can be seen as evidence for the existence of reality.

4. How does the study of mathematics impact our perception of reality?

The study of mathematics can impact our perception of reality in several ways. It can help us understand the underlying patterns and relationships in the world around us, leading to a deeper appreciation and understanding of reality. It can also challenge our preconceived notions and expand our understanding of what is possible. Additionally, the study of mathematics can lead to technological advancements that shape our perception of reality, such as the development of computers and other technologies.

5. Are there any limitations to the connection between mathematics and reality?

While mathematics has proven to be a powerful tool for understanding and describing reality, it does have its limitations. For example, there are still many phenomena that cannot be fully explained or predicted using mathematical models. Additionally, our current understanding of mathematics may be limited by our human perspective and may not fully capture the complexity of reality. As our understanding of mathematics and the universe continues to evolve, so too may our understanding of the connection between the two.

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