from complexity, universality

October 24, 2010 7:11 AM

A brief tour of the mysteriously universal laws of mathematics and nature.

BONUS
- The Bootstrap [1,2]
- The Physics of Wet Dogs

posted by kliuless (33 comments total)

Oh, I love this. Will have to spend more time reading this later.
posted by empath at 7:55 AM on October 24, 2010

You might have mentioned it was Terry Tao, the title on first glance sounded a bit timecube ranty and I almost skipped over. Mr Tao is the rare mathematician that can write for non-experts.
posted by sammyo at 8:31 AM on October 24, 2010

Here is a thing from MIT's Max Tegmark positing that the universe is math and everything that can be described mathematically exists literally in a multiverse.
posted by East Manitoba Regional Junior Kabaddi Champion '94 at 8:33 AM on October 24, 2010 [1 favorite]

It's terrific. We need much more of this out there in the world, and people who can communicate like this about science on the national media scene.

Thanks for posting this. Exactly what I love most about Metafilter, this kind of discovery.
posted by fourcheesemac at 8:33 AM on October 24, 2010

I've just been made aware that I spent all of April unaware that it was Mathematics Awareness Month.
posted by Obscure Reference at 8:43 AM on October 24, 2010 [1 favorite]

Two things:

1. Benford's law is one of the weirdest, most counterintuitive things in all of mathematics for me. It bothers me a lot more than, for example, Banach-Tarski.

2. I believe I can win that game of dots and boxes in Fig 14, if it's my turn.
posted by Wolfdog at 8:53 AM on October 24, 2010 [1 favorite]

Do you play that you are required to take a box if 3 sides are already taken?
posted by Obscure Reference at 9:18 AM on October 24, 2010

Ordinarily yes, but that rule is apparently not in force in the game in progress.
posted by Wolfdog at 9:22 AM on October 24, 2010

But where do these universal laws come from? How did they arise in the first place, and how do they have hold over reality the way they do?

Going through the wiki on the Philosophy of Mathematics is an awesome way to spend a rainy Sunday afternoon... or a career.
posted by Slap*Happy at 10:00 AM on October 24, 2010

I never thought I'd see a mention of Sierra's mildly popular MMORPG, The Realm, anywhere, let a lone in a paper about math and nature!
posted by cman at 10:24 AM on October 24, 2010

Modeling a White Hole in your sink
posted by rosswald at 10:34 AM on October 24, 2010

Here is a thing from MIT's Max Tegmark positing that the universe is math and everything that can be described mathematically exists literally in a multiverse.

In the Well World novels by Jack L. Chalker, the universe is a mathematical construct maintained by a planetary supercomputer. Much of the story in the later novels revolves around what would happen if the program ever needed to be "reset".
posted by JaredSeth at 10:47 AM on October 24, 2010

1. Benford's law is one of the weirdest, most counterintuitive things in all of mathematics for me. It bothers me a lot more than, for example, Banach-Tarski.

Benfords law makes more sense if you realize that fluctuations in numbers happen in amounts proportional to their size. Lets suppose most bank accounts remain within 30% of an average value. My account with $1500 in it on average is always going to have a "1" in the first digit, whereas your account averaging $5000 will often have different leading numbers.

Put another way, numbers that start with "1" are more "stable" to relative perturbations, whereas numbers with larger first digits are less "stable". Going from 8 to 9 is a 12.5% change, whereas going from from 1 to 2 is a 100% change.
posted by jpdoane at 11:01 AM on October 24, 2010 [12 favorites]

The mathematical substance of the piece is terrific, and it's an enjoyable read. The ornamental epigraphs are really irritating, though — self-congratulatory posturing, seemingly meant to produce some superficial feeling of philosophical/literary depth, but totally unconnected to the actual subject.
posted by RogerB at 11:16 AM on October 24, 2010

Here is a thing from MIT's Max Tegmark positing that the universe is math and everything that can be described mathematically exists literally in a multiverse.

I call shenanigans on that paper. Tegmark repeatedly conflates "can be completely modeled mathematically" with "is (ontologically) purely mathematical" but doesn't justify the conflation. He's confusing the map for the territory. Look:

... all physics theories that I have been taught have two components: mathematical equations, and words that explain how the equations are connected to what we observe and intuitively understand. When we derive the consequences of a theory, we introduce new concepts -- protons, molecules, stars -- because they are convenient. It is important to remember, however, that it is we humans who create these concepts; in principle, everything could be calculated without this baggage. For example, a sufficiently powerful supercomputer could calculate how the state of the universe evolves over time without interpreting what is happening in human terms.

All of this raises the question: is it possible to find a description of external reality that involves no baggage? If so, such a description of objects in this external reality and the relations between them would have to be completely abstract, forcing any words or symbols to be mere labels with no preconceived meanings whatsoever. Instead, the only properties of these entities would be those embodied by the relations between them.

But this is to construct an abstract mathematical model of the universe, the elements of which are partly derived from observations of the empirical world, and then pretend that the abstractions relate only to one another and not to the empirical world ("mere labels with no preconceived meanings whatsoever") -- at which point you are no longer talking about reality, but rather about the mathematical model you made of it. It's like drawing a really detailed and accurate map, and then saying the territory is lines of ink on paper rather than actual rivers and roads. None of the analogies and tests in Tegmark's article overcome this simple objection.
posted by twirlip at 1:58 PM on October 24, 2010 [2 favorites]

a sufficiently detailed model is indistinguishable from reality.
posted by empath at 2:37 PM on October 24, 2010

The older I get, the more it seems to me that the chain (or tree structure) of cause and effect (which has many formal resemblances to mathematical proof) is actually more fundamental and more powerful than any kind of mathematical analysis, and is necessary in order to rescue mathematical explanations of physical phenomena from the otherwise vitiating effects of the many unavoidable self referential paradoxes of mathematical logic.
posted by jamjam at 2:46 PM on October 24, 2010

a sufficiently detailed model is indistinguishable from reality.

What does that mean? Indistinguishable for whom, in what sense, and in which relation to the model or the reality in question? To be honest, it's a bit frustrating to deal with a one-liner conversational bomb like this without having an explanation of why you believe it's a valid counterpoint, or even what the terms mean and how they're situated.
posted by invitapriore at 3:04 PM on October 24, 2010

A sufficiently pseudoprofound stoner aphorism is indistinguishable from philosophical trolling.
posted by RogerB at 3:24 PM on October 24, 2010 [2 favorites]

What does that mean? Indistinguishable for whom, in what sense, and in which relation to the model or the reality in question?

Indistinguishable for someone inside the simulation. I think everyone here should be familiar with living inside a simulation, because we all live inside one created by our own brain. Do you think you have any idea what reality is? All you think you see is just a model your brain makes.

The only access we have to 'reality' is the math we use to describe it.
posted by empath at 4:10 PM on October 24, 2010

All you think you see is just a model your brain makes.

The only access we have to 'reality' is the math we use to describe it.

But you've still prioritized the models backwards. Even given the picture you paint of the situation, the mathematical model has to be based upon a distinct model created by the brain. So the mathematical model can't provide us with the only access to the world.
posted by voltairemodern at 4:56 PM on October 24, 2010

Even given the picture you paint of the situation, the mathematical model has to be based upon a distinct model created by the brain. So the mathematical model can't provide us with the only access to the world.

Observation only gives you access to an infinitesimally small portion of an infinitely large multiverse. Math gives you access to an infinite number of worlds, including all the ones we can't observe.
posted by empath at 6:41 PM on October 24, 2010 [1 favorite]

This (slightly derail-y) argument about whether math is just an ultimately meaningless abstraction or whether mathematical concepts exist independently of humans or what is not new.
posted by jedicus at 6:57 PM on October 24, 2010

a sufficiently detailed model is indistinguishable from reality

All models are wrong; some are useful. The only model of the universe sufficiently detailed is the universe itself.
posted by Mental Wimp at 11:50 PM on October 24, 2010 [2 favorites]

All you think you see is just a model your brain makes.

See, here's the twisty part. Your brain, and the model inside it, is part of the universe.
posted by Mental Wimp at 11:52 PM on October 24, 2010

you win a golden llama award :P

cheers!
posted by kliuless at 3:21 AM on October 25, 2010

Finger pointing at the moon.
posted by kersplunk at 3:24 AM on October 25, 2010 [2 favorites]

oh and speaking of mathematical truths...
posted by kliuless at 3:45 AM on October 25, 2010

oh and speaking of mathematical truths...

The (informal) explanation of Gödel's in your link suggest another interpretation: Gödel just found a new kind of proof. Sort of a meta-proof-by-contradiction; a proof of unprovability, if you will. Very handy for identifying nonsensical mathematical statement. Not a hole so much as an augmentation of the standard theorem-proving armamentarium. I'm pretty sure that interpretation is wrong, but I'm not sure why.
posted by Mental Wimp at 9:24 AM on October 25, 2010

Or, one could say "suggests", thereby matching verb number to subject number. If you want to.
posted by Mental Wimp at 9:25 AM on October 25, 2010

Oh, and that "Gödel's"? That should be "Gödel's theorem", of course.
posted by Mental Wimp at 9:25 AM on October 25, 2010

a sufficiently detailed model is indistinguishable from reality

But the point you seem to have missed here empath is that it is NOT reality. And when it differs from reality, and at some rare and too often critical point it will, we end up with disastrous consequences. This is proven in experiment after experiment showing that what we perceive is sometimes very different from reality (ie. read up on change blindness and other perceptual illusions). These are only the examples we happen to imagine up and test for. The possibility still lurks that there are many more such distortions, but we just haven't figured it out.

ALL maps differ from the territory. There are fundamental limits on what we can know at the end of the day. It's a nice heuristic to believe that everything is knowable, as that pushes us to continually strive for more knowledge. However, the heuristic of fundamental limits is also vitally important as it protects us from the folly of assuming our map is our territory.

I call shenanigans on that paper. Tegmark repeatedly conflates "can be completely modeled mathematically" with "is (ontologically) purely mathematical" but doesn't justify the conflation..

I agree with twirlp here. If empath, you're arguing that since the model is indistinguishable from the reality, then one can safely assume the model is reality, you simply need to review recent and not so recent history to figure how that assumption has turned out.

Getting back to the parent article, this statement seems contradictory:

However, the principle of universality does have definite limitations

Wouldn't universality indicate applicability across all cases? Or is he using universality within a statistical mechanics sense? I'm not sure lay people would necessarily understand the distinction between universality as used within dynamic system modeling and the more popular definition of universality as being applicable in all cases. He doesn't seem to explicitly state the definition of universality, only where universality applies:

Over the decades, many such universal laws have been found, that govern the behaviour of wide classes of complex systems, regardless of what the components of that system are, or even how they interact with each other.

I think this confusion leads people into misapplying the law of large numbers, the central limit theorem, the normal distribution, and others universal laws outside their applicable use cases.
posted by herda05 at 10:26 AM on October 25, 2010

It seems like this is importantizing the quantitative which under certain circumstances is certainly quite important. I'm come from qualitative land though, and in quantifying those qualitative "results," understanding this better will be my task. Unfortunately, I come from graphic designer / editorial land, and math is not my forte, alas. :(
posted by 8175309 at 3:41 PM on November 4, 2010