Math and Reality. What is the deep connection?

Ken G

Did I say that out loud?

Pythagorean

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

http://www.dartmouth.edu/~matc/MathD...ng/Wigner.html

apeiron

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

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.

apeiron

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

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

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?

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

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.
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

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

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.

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).

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.

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/photomorphos...ments/qpdf.pdf

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

rogerl

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

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

apeiron

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.

http://www.physicsforums.com/showpos...43&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.