Jan 12, 2008

40 comments

Yesterday I talked about the cubes {1, 8, 27, 64, 125, ...} and how their first differences {7, 19, 37, 61, ...} might at first seem to lack an obvious pattern, but taking the second differences {12, 18, 24, ...} takes you down to the simply related level. Taking the third differences {6, 6, ...} brings us to the perfectly stable level, where chaos dissolves into order.

But this (as I noted) is a handpicked example. Perhaps the "messy real world" lacks the beauty of these abstract mathematical objects? Perhaps it would be more appropriate to talk about neuroscience or gene expression networks?

Abstract math, being constructed solely in imagination, arises from simple foundations - a small set of initial axioms - and is a closed system; conditions that might seem *unnaturally* conducive to neatness.

Which is to say: In pure math, you don't have to worry about a tiger leaping out of the bushes and eating Pascal's Triangle.

So is the real world uglier than mathematics?

Strange that people ask this. I mean, the question might have been sensible two and a half millennia ago...

Back when the Greek philosophers were debating what this "real world" thingy might be made of, there were many positions. Heraclitus said, "All is fire." Thales said, "All is water." Pythagoras said, "All is number."

Score: Heraclitus 0 Thales 0 Pythagoras 1

Beneath the complex forms and shapes of the surface world, there is a simple level, an exact and stable level, whose laws we name "physics". This discovery, the Great Surprise, has already taken place at our point in human history - but it does not do to forget that it was surprising. Once upon a time, people went in search of underlying beauty, with no guarantee of finding it; and once upon a time, they found it; and now it is a known thing, and taken for granted.

Then why can't we predict the location of every tiger in the bushes as easily as we predict the sixth cube?

I count three sources of uncertainty even within worlds of pure math - two obvious sources, and one not so obvious.

The first source of uncertainty is that even a creature of pure math, living embedded in a world of pure math, may not know the math. Humans walked the Earth long before Galileo/Newton/Einstein discovered the law of gravity that prevents us from being flung off into space. You can be governed by stable fundamental rules without knowing them. There is no law of physics which says that laws of physics must be explicitly represented, as knowledge, in brains that run under them.

We do not yet have the Theory of Everything. Our best current theories are things of math, but they are not perfectly integrated with each other. The most probable explanation is that - as has previously proved to be the case - we are seeing surface manifestations of deeper math. So by far the best guess is that reality is made of math; but we do not fully know which math, yet.

But physicists have to construct huge particle accelerators to distinguish between theories - to manifest their remaining uncertainty in any visible fashion. That physicists must go to such lengths to be unsure, suggests that this is not the source of our uncertainty about stock prices.

The second obvious source of uncertainty is that even when you know all the relevant laws of physics, you may not have enough computing power to extrapolate them. We know every fundamental physical law that is relevant to a chain of amino acids folding itself into a protein. But we still can't predict the shape of the protein from the amino acids. Some tiny little 5-nanometer molecule that folds in a microsecond is *too much information* for current computers to handle (never mind tigers and stock prices). Our frontier efforts in protein folding use clever approximations, rather than the underlying SchrÃ¶dinger equation. When it comes to describing a 5-nanometer object using *really* basic physics, over quarks - well, you don't even bother trying.

We have to use instruments like X-ray crystallography and NMR to discover the shapes of proteins that are fully determined by physics we know and a DNA sequence we know. We are not logically omniscient; we cannot see all the implications of our thoughts; we do not know what we believe.

The third source of uncertainty is the most difficult to understand, and Nick Bostrom has written a book about it. Suppose that the sequence {1, 8, 27, 64, 125, ...} exists; suppose that this is a fact. And suppose that atop each cube is a little person - one person per cube - and suppose that this is also a fact.

If you stand on the outside and take a global perspective - looking down from above at the sequence of cubes and the little people perched on top - then these two facts say everything there is to know about the sequence and the people.

But if you are one of the little people perched atop a cube, and you know these two facts, there is still a third piece of information you need to make predictions: "Which cube am *I* standing on?"

You expect to find yourself standing on a cube; you do not expect to find yourself standing on the number 7. Your anticipations are definitely constrained by your knowledge of the basic physics; your beliefs are falsifiable. But you still have to look down to find out whether you're standing on 1728 or 5177717. If you can do fast mental arithmetic, then seeing that the first two digits of a four-digit cube are 17__ will be sufficient to guess that the last digits are 2 and 8. Otherwise you may have to look to discover the 2 and 8 as well.

To figure out what the night sky should look like, it's not enough to know the laws of physics. It's not even enough to have logical omniscience over their consequences. You have to know *where* you are in the universe. You have to know that you're looking up at the night sky *from Earth.* The information required is not just the information to locate Earth in the *visible* universe, but in the entire universe, including all the parts that our telescopes can't see because they are too distant, and different inflationary universes, and alternate Everett branches.

It's a good bet that "uncertainty about initial conditions at the boundary" is really indexical uncertainty. But if not, it's empirical uncertainty, uncertainty about how the universe *is* from a global perspective, which puts it in the same class as uncertainty about fundamental laws.

Wherever our best guess is that the "real world" has an *irretrievably* messy component, it is because of the second and third sources of uncertainty - logical uncertainty and indexical uncertainty.

Ignorance of fundamental laws does not tell you that a messy-looking pattern really is messy. It might just be that you haven't figured out the order yet.

But when it comes to messy gene expression networks, we've *already found* the hidden beauty - the stable level of underlying physics. *Because* we've already found the master order, we can guess that we won't find any *additional *secret patterns that will make biology as easy as a sequence of cubes. Knowing the rules of the game, we know that the game is hard. We don't have enough computing power to do protein chemistry from physics (the second source of uncertainty) and evolutionary pathways may have gone different ways on different planets (the third source of uncertainty). New discoveries in basic physics won't help us here.

If you were an ancient Greek staring at the raw data from a biology experiment, you would be much wiser to look for some hidden structure of Pythagorean elegance, all the proteins lining up in a perfect icosahedron. But in biology we already know where the Pythagorean elegance is, and we know it's too far down to help us overcome our indexical and logical uncertainty.

Similarly, we can be confident that no one will ever be able to predict the results of certain quantum experiments, only because our fundamental theory tells us quite definitely that different versions of us will see different results. If your knowledge of fundamental laws tells you that there's a sequence of cubes, and that there's one little person standing on top of each cube, and that the little people are all alike except for being on different cubes, and that you are one of these little people, then you *know* that you have no way of deducing which cube you're on except by looking.

The best current knowledge says that the "real world" is a perfectly regular, deterministic, and *very large*
mathematical object which is highly expensive to simulate. So "real life" is less like predicting the next cube in a sequence of cubes, and more like knowing that lots of little people are standing on top of cubes, but not knowing who *you personally *are, and also not being very good at mental arithmetic. Our knowledge of the rules does constrain our anticipations, quite a bit, but not perfectly.

There, now doesn't *that* sound like real life?

But uncertainty exists in the map, not in the territory. If we are ignorant of a phenomenon, that is a fact about our state of mind, not a fact about the phenomenon itself. Empirical uncertainty, logical uncertainty, and indexical uncertainty are just names for our own bewilderment. The best current guess is that the world is math and the math is perfectly regular. The messiness is only in the eye of the beholder.

Even the huge morass of the blogosphere is embedded in this perfect physics, which is ultimately as orderly as {1, 8, 27, 64, 125, ...}.

So the Internet is not a big muck... it's a series of cubes.