Math and Reality. What is the deep connection?

[ThomasT]

Wrt the thread title, what do you mean by "deep connection", and how would you know if there was one?

What's the reason why math is so effective in modelling reality?

Modelling is a form of communication. Math is effective in modelling reality because numbers are unambiguous.

Wrt reality, all we have is our private and shared sensory experience. Physical models all ultimately reduce to statements about the qualitative behaviors of objects in our sensory experience. Which can be quantified. We count things, and relate the quantities via various models of 'reality'.

It would be quite strange if math 'wasn't' effective in modelling reality, imho.

 

[rogerl]

Math working with measuring length and volume makes sense.. for example. the pythagorian theories work it's just trigometry. Here it is intuitive math is part or reality because you can obviously see how length add up to the total when measured. Math is also intuitive in calculating trajectories because you can use calculus. But when it comes to Gauge Theory where the gauge bosons arise from the symmetry inherent in the theory. It's no longer about length and volume. It acts as though length and volume don't even exist as SR shows us. But then, SR is simple to visualize, just treat space and time as dynamic and not fixed. About QM, it's just about objects just existing probabilistics. Now when you combine them in the Dirac Equation. It predicts the existence of antimatter for example. Any familiar with the derivation of the Dirac Equation. How does the equation give rise to the positron? Does it use the simple fact that space and time are dynamic and the quantum is probabilistic? Combined, why does the equation work at all. Is reality with dynamical spacetime and probabilistic quantum enough to make it tally with the equation??

What I'm saying or inquiring is whether dynamic spacetime and probabilistic quantum is enough to pull off those Dirac Equation stunts. Or whether all of this has to be processed and calculated in some kind of processor in the 2D surface in Beckenstein Holographic Principle where our 3D is just projection.. or whether all of our reality is just output from a computer program.

We can use deduction and elimination to at least get an idea what is behind this all. If dynamic spacetime and probabilistic quantum is enough to model reality. Then so be it. Projections from 2D surface and Matrix-like virtual reality is not needed.

 

[Ken G]

So maths is an "unreasonably effective" approach to modelling because it does something unreasonable - freezes the global constraints of a system and pushes questions about their development, their reasonableness, right out of the frame.

I get a similar flavor from Wigner's remarks, though he isn't pinpointing the source of the nonuniqueness. It reminds me of a dad playing a game of checkers against their young child, losing on purpose-- at first the child imagines they must be very good at checkers, but as they win game after game, they begin to realize it is unreasonable they should be winning this much just because they are so good, somehow the system must be rigged. It's an interesting conclusion-- the vogue is to conclude that "God must be a mathematician", a la string theorists anticipating the mind of god, so it is refreshing to see a more balanced "maybe the game is rigged" attitude.

At the same time, maths is also very ineffective when it comes to the modelling of global constraints as self-organising, downward causal, developing and evolving, parts of the story.

There are new areas of maths that seem to be tackling this problem now. Hierarchy theory, infodynamics, the various other tools being used by systems scientists. And new more suitable brands of logic, like Peircean semiotics based on a logic of vagueness.

This is interesting. Of course the $64,000 question is then, if this new maths helps to make mathematics better at including both the local dynamics and the downward causality of semi-frozen global constraints, will the success of mathematics once again seem unreasonable, again a victim of its own success? This is why I think it is fundamentally just hubris that makes us believe we can understand our own condition-- if we come up with a mathematical description that explains both the local dynamics and how interaction with global constraints gives rise to rich systemic behaviors, we will still have to wonder why that simple paradigm is so successful at answering these questions-- we will again be the people with keys that fit the doors too often, the child that is too often winning over a more worldly opponent.

 

[rogerl]

Wrt the thread title, what do you mean by "deep connection", and how would you know if there was one?

Modelling is a form of communication. Math is effective in modelling reality because numbers are unambiguous.

Wrt reality, all we have is our private and shared sensory experience. Physical models all ultimately reduce to statements about the qualitative behaviors of objects in our sensory experience. Which can be quantified. We count things, and relate the quantities via various models of 'reality'.

It would be quite strange if math 'wasn't' effective in modelling reality, imho.

I was contemplating on this from time to time. About the deep connection. Someone of you answered earlier in the thread that whenever something has dynamics, it can be described by math because all things dynamical has connections that is modellable by numbers. So if it's true, the question now becomes "What is behind our physical models like GR, QFT, etc.?" No longer the question "math and reality, what is the deep connection?". The answer to what is the deep connection is "dynamics". Right guys?

 

[apeiron]

Math is also intuitive in calculating trajectories because you can use calculus.

Tell that to the philosophers who railed against infinitesimals as the ghosts of departed quantities.

Like Cantor's approach to infinity, what seems patently unreal as ontology has a strange way of becoming instead an ontological fact simply because an epistemological stance proves so effective.

But when it comes to Gauge Theory where the gauge bosons arise from the symmetry inherent in the theory. It's no longer about length and volume. It acts as though length and volume don't even exist

It works the other way round. Gauge talks about local degrees of freedom that are not eliminated by constraining spacetime action to a "point". You can locate a point, but you can't stop it then spinning. Those local symmetries cannot be changed by any amount of global spacetime jiggering about (breakings of translational symmetries).

Any familiar with the derivation of the Dirac Equation. How does the equation give rise to the positron? Does it use the simple fact that space and time are dynamic and the quantum is probabilistic?

It said hey, whoops, there seems to be a symmetry in my equations. So maybe there is a particle to express that? I have been thinking of a breaking of the fundamental symmetry only in the positive direction, but there is one in the negative as well. And what is not forbidden, must exist.

Or whether all of this has to be processed and calculated in some kind of processor in the 2D surface in Beckenstein Holographic Principle where our 3D is just projection.. or whether all of our reality is just output from a computer program.

Fanciful speculation. You appear to be alluding to Ads/CFT which is about a duality of models of reality - mapping a string theory description to a quantum field one. This is not a claim that reality itself is some kind of holographic projection, just that one model can be related to another in this way.

(OK, I admit some physicists do talk as if they think this is an ontologically realistic view, rather than a statement about models, but there are plenty of loopy physicists out there. Some of them will believe absolutely anything.)

There is a good SciAm article that talks about how it is models to models.

http://homepage.mac.com/photomorphose/documents/qpdf.pdf

It winds back from the crazy stuff as you can see...

In particular, does anything similar hold for a universe like ours in place of the
anti–de Sitter space? A crucial aspect of anti–de Sitter space is that it has a
boundary where time is well defi ned. The boundary has existed and will exist
forever. An expanding universe, like ours, that comes from a big bang does
not have such a well-behaved boundary. Consequently, it is not clear how to defi
ne a holographic theory for our universe; there is no convenient place to put
the hologram.

 

[rogerl]

Wrt the thread title, what do you mean by "deep connection", and how would you know if there was one?

Modelling is a form of communication. Math is effective in modelling reality because numbers are unambiguous.

Wrt reality, all we have is our private and shared sensory experience. Physical models all ultimately reduce to statements about the qualitative behaviors of objects in our sensory experience. Which can be quantified. We count things, and relate the quantities via various models of 'reality'.

It would be quite strange if math 'wasn't' effective in modelling reality, imho.

Another thing. I was supposed to title this thread "Why does the Dirac Equation work".. but at the last seconds, I changed the title to "Math and reality, what is the deep connection?" as this is a philosophy forum and didn't want the thread deleted for out of topic so changed it suddenly without thinking. Anyway. After all the answers and reflections. The more appropriate title should be "Physical models and Reality: What is the deep connection?"

 

[Ken G]

Like Cantor's approach to infinity, what seems patently unreal as ontology has a strange way of becoming instead an ontological fact simply because an epistemological stance proves so effective.

Or the quintessential example of this, the quantum mechanical wavefunction, complete with imaginary numbers to get the interference.

 

[apeiron]

This is why I think it is fundamentally just hubris that makes us believe we can understand our own condition-- if we come up with a mathematical description that explains both the local dynamics and how interaction with global constraints gives rise to rich systemic behaviors, we will still have to wonder why that simple paradigm is so successful at answering these questions-- we will again be the people with keys that fit the doors too often, the child that is too often winning over a more worldly opponent.

This would be the deep division between our belief systems. I am optimistic (you would say deluded ) because I just cannot believe how deeply we can come to know reality. Everything that seemed pretty impossible to answer when I was a kid has turned out to be amazingly knowable - and indeed, check out the first proper philosopher there ever was (Anaximander) and the basics were understood right away. Most of it hasn't even turned out to be difficult. You push at a locked door and its swings open on oiled hinges.

You on the other hand express the voice of doubt and pessimism (or honest appraisal you would say). We may think we know a lot, but it is all an edifice of invention, and we know that we can never really know "the thing in itself".

I see mathematic's claims of specialness as just a social one-upmanship. Another example of the attitude caught my eye today.

https://www.physicsforums.com/showpost.php?p=3211743&postcount=5

We are so mysteriously clever because we are in touch with the rationalist paradise of Platonia. We have left the logical clay of lesser mortals behind to touch the mind of god. etc. etc.

Maths doesn't like being told it is just standard metaphysical wisdom being worked out as formal computational structure, with the important bit (the axioms, the global constraints) frozen and left to one side for the moment.

 

[apeiron]

Or the quintessential example of this, the quantum mechanical wavefunction, complete with imaginary numbers to get the interference.

Aha, but complex number magic is all about the irreducibility of dichotomies. It is no surprise at all that you need two dimensions to "number" a system. You must have a local variable and a global variable (such as energy and time, or location and momentum, or even spin and charge now).

Irreducibly orthogonal measurements at the fundamental level are exactly what the systems approach predicts.

QM had to be a theory about complementary properties because systems have no choice but to be organised that way.

 

[ThomasT]

I wrote this:

@ rogerl
Ok, so you're not really asking about a deep (reality) connection, which would be meaningless. And you're not asking why mathematics is effective in modelling theoretical/mathematical constructs, which is obvious. Or even why some theoretical/mathematical constructs are difficult to reconcile, which is difficult. What you're asking is how these constructs are translated into the language of instrumental behavior. Is that right?

Then I noticed this:

The more appropriate title should be "Physical models and Reality: What is the deep connection?"

It's a physically meaningless question, since the only thing we have to compare physical models with is the reality of our sensory experience.

 

[rogerl]

I wrote this:

@ rogerl
Ok, so you're not really asking about a deep (reality) connection, which would be meaningless. And you're not asking why mathematics is effective in modelling theoretical/mathematical constructs, which is obvious. Or even why some theoretical/mathematical constructs are difficult to reconcile, which is difficult. What you're asking is how these constructs are translated into the language of instrumental behavior. Is that right?

Then I noticed this:
It's a physically meaningless question, since the only thing we have to compare physical models with is the reality of our sensory experience.

How does physical model like general relativity correspond to reality is not a meaningless question. We have instruments like GPS, fermion detectors that interface with them.. it's not just our sensory experience because we don't do the direct detectors. We just interpret it and the results of the instruments correspond to the data which we input into our brain.

 

[nismaratwork]

How does physical model like general relativity correspond to reality is not a meaningless question. We have instruments like GPS, fermion detectors that interface with them.. it's not just our sensory experience because we don't do the direct detectors. We just interpret it and the results of the instruments correspond to the data which we input into our brain.

What. The. Hell?

I don't think you understood a word that TT said, adn I surely don't understand your point if there is one.

 

[ThomasT]

How does physical model like general relativity correspond to reality is not a meaningless question. We have instruments like GPS, fermion detectors that interface with them.. it's not just our sensory experience because we don't do the direct detectors. We just interpret it and the results of the instruments correspond to the data which we input into our brain.

You want to know how a physical model corresponds to a reality that we can't access. It's a meaningless question. We speculate and make theoretical/mathematical models, and test those models via instrumental behavior amenable to our senses. If the model works, it doesn't mean the model corresponds to some vision of a deep reality. It means the model corresponds to the observed instrumental behavior.

 

[rogerl]

You want to know how a physical model corresponds to a reality that we can't access. It's a meaningless question. We speculate and make theoretical/mathematical models, and test those models via instrumental behavior amenable to our senses. If the model works, it doesn't mean the model corresponds to some vision of a deep reality. It means the model corresponds to the observed instrumental behavior.

I wrote that message with the assumption that we could access the reality that physical instruments couldn't. That is.. i was thinking our mind can access it.. and by contemplating how physical models correspond to reality.. we can use deductions and mind probe to know if the physical model is close to the reality. For example. I was meditating a lot about space and time and what it means to warp and curve. What if our pure mind is interactive with space and time.. then we can view spacetime using our mind's and know how the physical model correspond to it... something like that.

 

[Pythagorean]

Yeah sure. I'll entertain myself with your content-free insults while you get round to thinking things through.

At least you have signalled your conclusion. Now you are just working on the argument that gets you there. I can see my patience will be rewarded.

actually, if you weren't so deluded by the victim role, I was asking questions to get me started in understanding your colorful language. I wasn't aware I insulted you...?

and what exactly was my conclusion?

there seemed to be a lot more in there for a non-reply...

 

[nismaratwork]

I wrote that message with the assumption that we could access the reality that physical instruments couldn't. That is.. i was thinking our mind can access it.. and by contemplating how physical models correspond to reality.. we can use deductions and mind probe to know if the physical model is close to the reality. For example. I was meditating a lot about space and time and what it means to warp and curve. What if our pure mind is interactive with space and time.. then we can view spacetime using our mind's and know how the physical model correspond to it... something like that.

Whaaaat, the hell is "mind probe"?

 

[Ken G]

This would be the deep division between our belief systems. I am optimistic (you would say deluded ) because I just cannot believe how deeply we can come to know reality. Everything that seemed pretty impossible to answer when I was a kid has turned out to be amazingly knowable - and indeed, check out the first proper philosopher there ever was (Anaximander) and the basics were understood right away. Most of it hasn't even turned out to be difficult. You push at a locked door and its swings open on oiled hinges.

You on the other hand express the voice of doubt and pessimism (or honest appraisal you would say). We may think we know a lot, but it is all an edifice of invention, and we know that we can never really know "the thing in itself".

Yes, that's a fair characterization of our positions. Perhaps it is not that one or the other of us has to be right, but that we are caught in yet another duality-- neither of our positions would make any sense if the other's did not also hold some truth. If the universe was as fundamentally impenetrable as I imagine, then there would be no point in thinking about it in the first place-- that's exactly the stance of the rock, which has given up the effort altogether and so lives in a universe where there is nothing to understand, so nothing is not understood either. Or, if the universe was really our intellectual oyster, just one or two really big ideas away from giving up its deepest secrets (as you might hope, I would not put the words in your mouth), then we have the disastrously appalling possibility of knowing all there is to know, and returning again, thusly, to the same status of the rock. Having nothing left to understand is a lot like having nothing to understand in the first place-- and so instead we find ourselves constantly balanced on the brink of enlightenment, another dynamical equilibrium, between what we do know and what we can know.

Maths doesn't like being told it is just standard metaphysical wisdom being worked out as formal computational structure, with the important bit (the axioms, the global constraints) frozen and left to one side for the moment.

And it likes Godel's proof even less, the fundamental divide between what can be certain and what can have meaning. But as you can guess from the above, I view Godel's proof that meaning and truth are fundamentally different as the saving grace, the one Wigner key that doesn't fit, giving us hope that all the other ones that did were not just a completely rigged universe, "losing" to our intellectual advances just to humor us.

 

[ThomasT]

I wrote that message with the assumption that we could access the reality that physical instruments couldn't. That is.. i was thinking our mind can access it.. and by contemplating how physical models correspond to reality.. we can use deductions and mind probe to know if the physical model is close to the reality. For example. I was meditating a lot about space and time and what it means to warp and curve. What if our pure mind is interactive with space and time.. then we can view spacetime using our mind's and know how the physical model correspond to it... something like that.

Well, that's a whole other question, and since I don't have those powers I can't answer it.

However, when models do correspond to instrumental behavior, then the visions of deep reality that they might provoke can be preserved, for the time being, and maybe tweaked, added to, combined, etc. by theoretical physicists. Wrt most of them, I think it's possible to get a good idea of their thinking processes via their writings.

As for spacetime curvature, I wouldn't ponder that too much. Think in terms of waves and wave complexes, resonances, hierarchies of particulated wave structures, that sort of stuff.
Just a suggestion.

 

[Ken G]

QM had to be a theory about complementary properties because systems have no choice but to be organised that way.

I think you may be right about this, and I might add that the key thing that the complex humbers are allowing is an undreamed-of escape from the frozen prison of Parmenides' and Zeno's impossibility of change. Complex numbers are what allow a state to be "stationary" (unchanging in regard to a universe of only that state), and yet also be dynamical (bringing about change when considered as part of a larger system), all at the same time. Perhaps we could say they are what allows a particle to participate in a larger system in a way that does not project onto the internal degrees of freedom of that particle itself.

 

[apeiron]

Yes, that's a fair characterization of our positions. Perhaps it is not that one or the other of us has to be right, but that we are caught in yet another duality-- neither of our positions would make any sense if the other's did not also hold some truth.

Agreed. What I would add is that I have had to make some radical changes in POV along the way. So I started out as a naive realist and it was a big breakthrough to then get that "everything is just models". Embracing complete doubt is what allows you to turn around and have complete confidence in the fact "now you can model".

Likewise I was first a reductionist, then became completely disillusioned with reductionism. And then in turn found that reductionism was what I also completely believe in because it was a subset (or more correctly, the dichotomous partner) to a fully worked out systems view.

So going all the way over to one side is only worthwhile if you turn round and see now that there is also other side you left behind. To have moved away successfully means that there was definitely the other thing there in the first place.

You will know one of the favourite symbols of what we are talking about here?
http://en.wikipedia.org/wiki/Drawing_Hands

 

[apeiron]

actually, if you weren't so deluded by the victim role, I was asking questions to get me started in understanding your colorful language.

Really, I have no idea why I thought that you might have been insulting. It would be out of character.

and what exactly was my conclusion?

"Essentially though, what it seems like you're saying is what I've said here before, that mathematics is type of logical clay."

 

[Pythagorean]

So I concluded what you sounded like to me, yes. But it's you who chose to make the story end there.

You never confirmed whether semiotics was synonymous with organic logic either.

So the irony of your accusations is that you're the one jumping to conclusions, while I'm still struggling to understand you.

 

[nismaratwork]

Yeah, let's sail right past the 'crazy' mind probing, and get on with bickering... wait, what?

 

[Pythagorean]

Yeah, let's sail right past the 'crazy' mind probing, and get on with bickering... wait, what?

sometimes, to properly probe each other's minds, we have to bicker past the anticipatory language barriers. (probe sent)

 

[nismaratwork]

sometimes, to properly probe each other's minds, we have to bicker past the anticipatory language barriers. (probe sent)

Sometimes people are just bonkers, and believe in magic, and insist on bringing it up (only to be deleted) over and over in the wrong forum.

 

[Math Is Hard]

That'll do. You can play in GD.