Math and Reality. What is the deep connection?

rogerl

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 modelled. 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 wont 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.

apeiron

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 learnt to do.

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

rogerl

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.

apeiron

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.

rogerl

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.

apeiron

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.

Ken G

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.

HallsofIvy

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.

Ken G

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.

Pythagorean

I will refer, as I have many times, to Wigner's paper: the unreasonable effectiveness of mathematics in the natural sciences

apeiron

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).

rogerl

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 paralel 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.

nismaratwork

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.

rogerl

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 maintainance 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.

apeiron

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.

apeiron

You realise 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.

rogerl

So what you are saying is that if any one 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?