Math and Reality. What is the deep connection?

[nismaratwork]

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.

I think you should look at what QM has correctly predicted, despite detractors and want of a single Interpretation. Then, you draw your own conclusions, I'm not sure there is a right one.

 

[apeiron]

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.

I think Smolin is right (along with Laughlin, Prigogine and other good critics of the reductionist status quo). Something is lacking, and it is a systems perspective.

But equally, that does not obselete all that has been achieved with reductionist approaches - all the varieties of mechanics like quantum mechanics, classical mechanics, relativistic mechanics, statistical mechanics.

These models are good, simple and effective. So whatever comes along as a TOE has to be able to incorporate these various models - and indeed incoporate reductionism as a metaphysics.

My only beef with Smolin is that he has not really understood the systems view (he just seems busy trying to reinvent it eventually).

So it is about re-investigating the foundations of science's metaphysics. And it is something high energy physicists ought to be doing (as biologists, for instance, have already done it).

Replacing particles with strings or loops is still thinking like atomists. But replacing particles with resonances or solitons is thinking like a systems thinker.

Yet what you can guarantee is that a re-investigation is not going to replace atomism with any kind of supernatural ontology, like the mind of god or whatever. Well, given what some physicists will say to sell their books, perhaps this cannot be promised .

 

[nismaratwork]

As if, were there a god, we'd want to know its mind. Sounds to me like a rather frightening proposition, I'd prefer nature red in tooth and claw.

 

[Ivan Seeking]

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.

So, according to Smolin, in order to fine a TOE, you have to put the shoe on the other foot?

 

[rogerl]

I think Smolin is right (along with Laughlin, Prigogine and other good critics of the reductionist status quo). Something is lacking, and it is a systems perspective.

But equally, that does not obselete all that has been achieved with reductionist approaches - all the varieties of mechanics like quantum mechanics, classical mechanics, relativistic mechanics, statistical mechanics.

These models are good, simple and effective. So whatever comes along as a TOE has to be able to incorporate these various models - and indeed incoporate reductionism as a metaphysics.

My only beef with Smolin is that he has not really understood the systems view (he just seems busy trying to reinvent it eventually).

So it is about re-investigating the foundations of science's metaphysics. And it is something high energy physicists ought to be doing (as biologists, for instance, have already done it).

Replacing particles with strings or loops is still thinking like atomists. But replacing particles with resonances or solitons is thinking like a systems thinker.

Yet what you can guarantee is that a re-investigation is not going to replace atomism with any kind of supernatural ontology, like the mind of god or whatever. Well, given what some physicists will say to sell their books, perhaps this cannot be promised .

Where did you hear about replacing particles with resonances or solitons? Any references? Superstrings is the only game in town. If it was false, everything flows down the drain and millions of dollars lost (in institution fundings). Our last great breakthrough in physics occurred in 1973. After that. It's just about properties of materials, physics of biology, dynamics of vast clusters of stars. We didn't have new fundamental discovery like the Dirac Equation or unification of electric and magnetic field. The Large Hadron Collider can spell the difference but they decrease it from 14Tev to mere 7Tev from the SCC plan of 40Tev in the 1980s. What if new phenomenon occurs at 8Tev, Then we miss it.

 

[nismaratwork]

Where did you hear about replacing particles with resonances or solitons? Any references? Superstrings is the only game in town. If it was false, everything flows down the drain and billions of dollars lost. Our last great breakthrough in physics occurred in 1973. After that. It's just about properties of materials, physics of biology, dynamics of vast clusters of stars. We didn't have new fundamental discovery like the Dirac Equation or unification of electric and magnetic field. The Large Hadron Collider can spell the difference but they decrease it from 14Tev to mere 7Tev from the SCC plan of 40Tev in the 1980s. What if new phenomenon occurs at 8Tev, Then we miss it.

You're so wrong in so many ways it makes my head spin. The LHC is mostly probing for another particle to fit the standard model... only indirectly in ANY way MIGHT it support string theory... and that's unlikely. Keep reading.

 

[apeiron]

Where did you hear about replacing particles with resonances or solitons? Any references?

I thought you were reading Smolin's Trouble?

Even he mentions Laughlin, Volovik, Xiao-Gang Wen, and Olaf Dreyer - see p247.

Superstrings is the only game in town. If it was false, everything flows down the drain and billions of dollars lost. Our last great breakthrough in physics occurred in 1973. After that. It's just about properties of materials, physics of biology, dynamics of vast clusters of stars. We didn't have new fundamental discovery like the Dirac Equation or unification of electric and magnetic field. The Large Hadron Collider can spell the difference but they decrease it from 14Tev to mere 7Tev from the SCC plan of 40Tev in the 1980s. What if new phenomenon occurs at 8Tev, Then we miss it.

This is sounding hysterical now. I mean superstrings might believe it is the only game in town but...

I agree that money could have been much better spent supporting systems science of course. Yet the LHC is still going to probe the EW breaking scale and find something important. That seems like 90 percent certain. If it is not the Higgs, not Susy, it still has to be something going on at that energy scale.

You have to admit that you don't actually know much about the physics you want to criticise. That is plain in that you are quoting from popularisations and have made some very basic mistatements.

That is not a problem if you are just interested in learning. But you have to earn the right to be as critical as you are now being.

Smolin may sound critical, but he of course has earned the right - and his criticisms are not exactly what you seem to think.

 

[Ken G]

Certainly it is hoped that the LHC might find new phenomena not currently predicted, which would reinvigorate the need to discover fundamentally new theories. The fact that it would probably not interface with string theory is what many view to be the primary problem with string theory-- it's just too far from what we can observationally constrain.

 

[rogerl]

I thought you were reading Smolin's Trouble?

Even he mentions Laughlin, Volovik, Xiao-Gang Wen, and Olaf Dreyer - see p247.



This is sounding hysterical now. I mean superstrings might believe it is the only game in town but...

I agree that money could have been much better spent supporting systems science of course. Yet the LHC is still going to probe the EW breaking scale and find something important. That seems like 90 percent certain. If it is not the Higgs, not Susy, it still has to be something going on at that energy scale.

You have to admit that you don't actually know much about the physics you want to criticise. That is plain in that you are quoting from popularisations and have made some very basic mistatements.

That is not a problem if you are just interested in learning. But you have to earn the right to be as critical as you are now being.

Smolin may sound critical, but he of course has earned the right - and his criticisms are not exactly what you seem to think.

Oh. When I mentioned billions of dollars down the drain. Of course I didn't mean the Large Hadron Collider. I mean all the Ph.Ds and research programs and research funding all over the world is on superstring theory. Lee Smolin mentions that. This is the context of what I meant billions are down the drain (or more like million of dollars).


Anyway. I read Smolin book a few years back and now just rereading it and at the same time listening to Brian Greene new book in mp3. Also I mostly forgot what I read after a few months.

I just want to know how math is related to reality. In the initial message of this thread. I wonder how our reality can work with the dirac equation enough to predict the existence of the positron. This is not answered and my title was supposed to be "why does dirac equation work?" but changed it last minute. I mean. In Newtonian mechanics, trajectory can be modeled by math, but dirac equation, it's very abstract and yet reality can conform to it. Why. This is what I simply want to know.

I own about 50 pop-sci books like Lisa Randall Warped Passage so I know stuff like the Hierarchy Problem but I still don't know how math correponds to reality, which is not discussed in most of the books.

About the LHC. I know Higgs, Supersymmetry is its main target. But it also can detect missing energy enough to discover hidden dimensions.. which could say a thing or two about whether Superstrings are on the right track or not.

 

[Pythagorean]

I asked about the current state of string theory in the "beyond.." forum above if you would like to hear the replies from our forum advisers:

https://www.physicsforums.com/showthread.php?t=482610

 

[rogerl]

You're so wrong in so many ways it makes my head spin. The LHC is mostly probing for another particle to fit the standard model... only indirectly in ANY way MIGHT it support string theory... and that's unlikely. Keep reading.

Oh. When I mentioned billions of dollars down the drain. Of course I didn't mean the Large Hadron Collider. I mean all the Ph.Ds and research programs and research funding all over the world is on superstring theory. Lee Smolin mentions that. This is the context of what I meant billions are down the drain (or more like million of dollars).

 

[rogerl]

Admit that millions of dollars given in institution funding can go down the drain if string theory is wrong. Smolin said that "The aggressive promotion of string theory has led to its becoming the primary avenue for exploring the big questions in physics. Nearly every particle theorist with a permanent position at the Prestigious Institute for Advanced Study, including the director, is a string theorist... "" The same is true of the Kavli Institute of Theoretical Physics. Eight of the nine MacArthur Fellowships awarded to particle physicists since the beginning of the program in 1981 have also gone to string theories. And in the country's stop physics departments (Berkeley, Caltech, Harvard, MIT, Princeton, and Stanford), twenty out of the twenty-two tenured prefessors in particle physics who received PhDs after 1981 made their reputation in string theory or related approaches".

Sorry for quoting it (would quote no more). I pay big taxes and don't want it down the drain. Had they contribute the money instead on the SCC funding in the 1980s. We could already have answers. What they do instead is kill thousands of civilians in Iraq with the money! So what do you say we rally in front of the Senate.. lol.. just kidding...

If the LHC misses on the higgs and supersymmetry and the new phenomenon occurs at the 15Tev. Then we may miss it in our lifetime. And for some of us.. it is unacceptable. TOE can benefit our lives greatly. I want it to happen in my lifetime. Don't you?

 

[Maui]

My only beef with Smolin is that he has not really understood the systems view (he just seems busy trying to reinvent it eventually).

Perhaps he has understood what others have failed in his field - that probing the limits of reality is actually asking "What is existence?"

Having read the book, i'd say that Smolin does understand that the primitive view of an ever-expanding, existential solid piece of matter(we happen to call Universe) existing into the uncreated and non-existent void(we naively call "non-existence") is deeply wrong.

The problems with QM are problems of existence(what it is and how it takes place - it turns out existence is much weirder), not ones of reality.

We don't have to separate reality from existence and I don't think that we can continue the pursuit to understand reality and leave existence to philosophy. Because they are actually one and the same. There is no difference between reality and existence, is there?

 

[JDStupi]

[...]but I still don't know how math correponds to reality[...]

Welcome to the club! Nobody KNOWS for sure, but it certainly is not something simple. You set out to learn science because you think you will get all of "The Answers", as though the world were a collection of answers, but this is never what happens. I don't know why this is a popular way to think, is it slightly fed by the way education is presented? As a static body of facts? I don't know. Point of the random rant is, you must seek to view things from a new perspective and thing about things in new ways and understand the complex relationships between things rather than look for The Answers.

No, the money will not be "wasted". Wrong doesn't mean "wasted" science needs wrong, it provides perspective on how nature works. Just because it isn't "The Answer" doesn't mean it doesn't help us think about reality in a new way. Doesn't mean it hasn't spawned useful research and developed new mathematics.

I pay big taxes and don't want it down the drain

Oh please! Now your being nitpicky. Of all the things you could worry about for where your tax money goes your worrying about the tiny percentage that goes to string theory? I have no stats, but considering Science funding as a whole is tiny compared to where your money goes, string theory is nothing. Complain about something else, don't use your precious money as a justification for why it is "wasted".

TOE can benefit our lives greatly. I want it to happen in my lifetime. Don't you?

TOE? Popular books seem to peddle the TOE, what makes you think the TOE is even possible? What is a TOE? Some magical machine that if we put numbers in we get some laplacean universe where we know everything? Why do you seek to know everything?

It seems you think science is in some one-to-one correspondance with how nature "actually is" and once we find the right equations we have "figured out" lawful reality. Maybe maths relation to reality seems so confusing to you because you believe they are at opposite ends of a dichotomy. Physics is "concrete" and Math is "abstract". Look closer and you may find that the boundaries between black and white are not quite so defined.

Perhaps Maths is a psychological tool we use to conceptualize the world, and we use experimentation to act in the world and subject phenomena to a greater number of combinations and theorize about them.

Gasp! but then everything I think about what science is telling me could be false?
Maybe, depending on what false is, what your conceptions of truth are and if science is even actually telling you that as opposed to you reading that into science?


Sorry for quick rant, I don't mean to sound harsh if I do that isn't my intent. Please be willing to fundamnetally alter the way in which you view reality. If you are not willing to do that, you don't want "The Answers" all that much.

 

[Ken G]

And another way to make that point is, science is not supposed to be about replacing evidence-free belief systems with evidence-based belief systems, it is supposed to be about a completely different attitude toward the meaning of what "truth" is. It starts with a healthy skepticism that there is any such thing as truth, other than a "current state of understanding." I've never liked "TOE" language because it's really a false lesson in science.

 

[nismaratwork]

And another way to make that point is, science is not supposed to be about replacing evidence-free belief systems with evidence-based belief systems, it is supposed to be about a completely different attitude toward the meaning of what "truth" is. It starts with a healthy skepticism that there is any such thing as truth, other than a "current state of understanding." I've never liked "TOE" language because it's really a false lesson in science.

SSSHhhhhhh!... publishers will hear you...

 

[Ken G]

Did I say that out loud?

 

[Pythagorean]

Since no one responded to this, I figure I'll reduce the workload for you:

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

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

 

[apeiron]

Since no one responded to this, I figure I'll reduce the workload for you:

What's the point you're trying to make? Apart from the fact Wigner represents the mysterian take on the question.

As I have argued, maths is "unreasonably effective" because it separates the constraints from the construction, the universals from the particulars.

For anything to exist, to develop and persist, it must be organised as "a system". It must have the self-organising form of global constraints and local constructions gone to no longer changing equilbrium.

Maths is a way of modelling this truth, without ever acknowledging this truth (it seems).

What maths does is freeze the global constraints (which in the systems view, are actually subject to development and change) and so greatly simplifies the task of modelling. The global constraints become axiomatic - things that just are. Things that themselves don't get explained. Then this sets up a world of all possible combinations of constructive action made permissible by a certain collection of fixed global constraints.

Take the famous case of Euclid's postulates of geometry. There was that axiom about parallel lines. It seemed very sound as an unchangeable constraint on reality. Then relax that constraint - make flatness something which has a developmental history rather than something that is frozen in - and a new more general view of geometry can be seen.

So what does that tell us about "unreasonable effectiveness"? It tells me that we start off assuming that the state of the world we appear to observe (populated by solid objects, limited to three flat dimensions, etc) is fixed that way. The global constraints just are.

Yet when we formulate that as a set of axioms, we then make it very clear to ourselves that they are assumptions. Which can be relaxed. And then in the history of maths, it became clear that relaxing the constraints - seeking the less constrained story that is more general - was a productive route for developing maths. The game became, let's throw overboard any fixed assumptions we can, because what we remove can always be added back in the form of a particular constraint on our imagined system.

So for example, you can shift from geometry (with its definite distances) to topology (with its indefinite distances). Distance becomes a constraint that can be added back into the more generalised description as need be. Just as curvature (or its lack) became an additive ingredient in the shift from Euclidean to non-Euclidean geometry.

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.

You presume the global constraints as axioms. And if anyone queries this, you claim this is a free choice that commits you to no ontology. It is the mathematician's prerogative to state any axiom and explore its consequences just because it is interesting or beautiful (Wigner's argument). Maths claims this fundamental disconnect from reality, from experience. Even if, on closer examination, we discover the axioms being justified by their "natural logic" - and so derived in fact from experience.

Yet because maths has also taken a systematic approach to relaxing the constraints - becoming ever more general - it has paved the way for physics to do the same.

We live in a highly constrained reality (a universe with many very particular features). An unconstrained state is symmetric. A constrained state is symmetry broken. So to see our reality in terms of more general laws, we need to unwind the symmetry breakings. We must describe reality in less constrained terms.

Which is why maths is unreasonably effective. It is a method of successively relaxing constraints - while at the same time, keeping them always frozen, always something that can be added back in at will. This is a very tractable approach - it allows for calculation. All the dynamics gets reduced to the play of numbers - local, atomistic, additive, constructive action, or effective causality.

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.

 

[Pythagorean]

What's the point you're trying to make? Apart from the fact Wigner represents the mysterian take on the question

I don't see it that way. That seems like a view you'd hold if you'd only read the introduction that I quoted. I see Wigner's paper as an exposure of the problem and a foundation for the question. To me, it's an excellent place to start from.

By the length of your reply, I think at least your unconscious brain agrees. And you have sought to answer the question. We should be aware of the language barrier between us now. But I will try to work through your reply, anyway.

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

So my point, in response to the OP's "look, math doesn't work" is that "we'll fin d a way". As you have demonstrated yourself:

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.

Ah, "semiotics" is the term then. I tried looking for some introductory "organic logic" formalism (it's the neighborhood, no?) but all I could find was sensational walls of text. I'll have to look into Peircean.

 

[apeiron]

I don't see it that way. That seems like a view you'd hold if you'd only read the introduction that I quoted. I see Wigner's paper as an exposure of the problem and a foundation for the question. To me, it's an excellent place to start from.

By the length of your reply, I think at least your unconscious brain agrees. And you have sought to answer the question. We should be aware of the language barrier between us now. But I will try to work through your reply, anyway.

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

So my point, in response to the OP's "look, math doesn't work" is that "we'll fin d a way". As you have demonstrated yourself:

Ah, "semiotics" is the term then. I tried looking for some introductory "organic logic" formalism (it's the neighborhood, no?) but all I could find was sensational walls of text. I'll have to look into Peircean.

Surprise, surprise. I have read Wigner's paper in full. And I referenced it much earlier in the thread. Although not by name as it is so well known.

And as usual, you are making a non-reply. No points I raised have been addressed. Instead you say my unconscious somehow secretly agrees with you. If so, it must be horribly confused as well. .

 

[Pythagorean]

As I said, I still have to work through your reply.

Be patient, sheesh

 

[apeiron]

As I said, I still have to work through your reply.

Be patient, sheesh

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.

 

[nismaratwork]

I do so adore when lovers spat...

 

[Ken G]

At least you have signalled your conclusion. Now you are just working on the argument that gets you there.

That could well be the best comeback I ever saw, remind me never to debate you.

 

[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.