Mathematics as a lossy compression algorithm gone wild

by shminux 5y6th Jun 20144 min read80 comments

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This is yet another half-baked post from my old draft collection, but feel free to Crocker away.

 

There is an old adage from Eugene Wigner known as the "Unreasonable Effectiveness of Mathematics". Wikipedia:

the mathematical structure of a physical theory often points the way to further advances in that theory and even to empirical predictions.

The way I interpret is that it is possible to find an algorithm to compress a set of data points in a way that is also good at predicting other data points, not yet observed. In yet other words, a good approximation is, for some reason, sometimes also a good extrapolation. The rest of this post elaborates on this anti-Platonic point of view.

Now, this point of view is not exactly how most people see math. They imagine it as some near-magical thing that transcends science and reality and, when discovered, learned and used properly, gives one limited powers of clairvoyance. While only the select few wizard have the power to discover new spells (they are known as scientists), the rank and file can still use some of the incantations to make otherwise impossible things to happen (they are known as engineers). 

This metaphysical view is colorfully expressed by Stephen Hawking:

What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?

Should one interpret this as if he presumes here that math, in the form of "the equations" comes first and only then there is a physical universe for math to describe, for some values of "first" and "then", anyway? Platonism seems to reach roughly the same conclusions:

Wikipedia defines platonism as

the philosophy that affirms the existence of abstract objects, which are asserted to "exist" in a "third realm distinct both from the sensible external world and from the internal world of consciousness, and is the opposite of nominalism

In other words, math would have "existed" even if there were no humans around to discover it. In this sense, it is "real", as opposed to "imagined by humans". Wikipedia on mathematical realism:

mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one sort of mathematics that can be discovered: triangles, for example, are real entities, not the creations of the human mind.

Of course, the debate on whether mathematics is "invented" or "discovered" is very old. Eliezer-2008 chimes in in http://lesswrong.com/lw/mq/beautiful_math/:

To say that human beings "invented numbers" - or invented the structure implicit in numbers - seems like claiming that Neil Armstrong hand-crafted the Moon.  The universe existed before there were any sentient beings to observe it, which implies that physics preceded physicists. 

and later:

The amazing thing is that math is a game without a designer, and yet it is eminently playable.

In the above, I assume that what Eliezer means by physics is not the science of physics (a human endeavor), but the laws according to which our universe came into existence and evolved. These laws are not the universe itself (which would make the statement "physics preceded physicists" simply "the universe preceded physicists", a vacuous tautology), but some separate laws governing it, out there to be discovered. If only we knew them all, we could create a copy of the universe from scratch, if not "for real", then at least as a faithful model. This universe-making recipe is then what physics (the laws, not science) is.

And these laws apparently require mathematics to be properly expressed, so mathematics must "exist" in order for the laws of physics to exist.

Is this the only way to think of math? I don't think so. Let us suppose that the physical universe is the only "real" thing, none of those Platonic abstract objects. Let is further suppose that this universe is (somewhat) predictable. Now, what does it mean for the universe to be predictable to begin with? Predictable by whom or by what? Here is one approach to predictability, based on agency: a small part of the universe (you, the agent) can construct/contain a model of some larger part of the universe (say, the earth-sun system, including you) and optimize its own actions (to, say, wake up the next morning just as the sun rises). 

Does waking up on time count as doing math? Certainly not by the conventional definition of math. Do migratory birds do math when they migrate thousands of miles twice a year, successfully predicting that there would be food sources and warm weather once they get to their destination? Certainly not by the conventional definition of math. Now, suppose a ship captain lays a course to follow the birds, using maps and tables and calculations? Does this count as doing math? Why, certainly the captain would say so, even if the math in question is relatively simple. Sometimes the inputs both the birds and the humans are using are the same: sun and star positions at various times of the day and night, the magnetic field direction, the shape of the terrain.

What is the difference between what the birds are doing and what humans are doing? Certainly both make predictions about the universe and act on them. Only birds do this instinctively and humans consciously, by "applying math". But this is a statement about the differences in cognition, not about some Platonic mathematical objects. One can even say that birds perform the relevant math instinctively. But this is a rather slippery slope. By this definition amoebas solve the diffusion equation when they move along the sugar gradient toward a food source. While this view has merits, the mathematicians analyzing certain aspects of the Navier-Stokes equation might not take kindly being compared to a protozoa. 

So, like JPEG is a lossy image compression algorithm of the part of the universe which creates an image on our retina when we look at a picture, the collection of the Newton's laws is a lossy compression algorithm which describes how a thrown rock falls to the ground, or how planets go around the Sun. in both cases we, a tiny part of the universe, are able to model and predict a much larger part, albeit with some loss of accuracy.

What would it mean then for a Universe to not "run on math"? In this approach it means that in such a universe no subsystem can contain a model, no matter how coarse, of a larger system. In other words, such a universe is completely unpredictable from the inside. Such a universe cannot contain agents, intelligence or even the simplest life forms. 

Now, to the "gone wild" part of the title. This is where the traditional applied math, like counting sheep, or calculating how many cannons you can arm a ship with before it sinks, or how to predict/cause/exploit the stock market fluctuations, becomes "pure math", or math for math's sake, be it proving the Pythagorean theorem or solving a Millennium Prize problem. At this point the mathematician is no longer interested in modeling a larger part of the universe (except insofar as she predicts that it would be a fun thing to do for her, which is probably not very mathematical).

Now, there is at least one serious objection to this "math is jpg" epistemology. It goes as follows: "in any universe, no matter how convoluted, 1+1=2, so clearly mathematics transcends the specific structure of a single universe". I am skeptical of this logic, since to me 1,+,= and 2 are semi-intuitive models running in our minds, which evolved to model the universe we live in. I can certainly imagine a universe where none of these concepts would be useful in predicting anything, and so they would never evolve in the "mind" of whatever entity inhabits it. To me mathematical concepts are no more universal than moral concepts: sometimes they crystallize into useful models, and sometimes they do not. Like the human concept of honor would not be useful to spiders, the concept of numbers (which probably is useful to spiders) would not be useful in a universe where size is not a well-defined concept (like something based on a Conformal Field Theory).

So the "Unreasonable Effectiveness of Mathematics" is not at all unreasonable: it reflects the predictability of our universe. Nothing "breathes fire into the equations and makes a universe for them to describe", the equations are but one way a small part of the universe predicts the salient features of a larger part of it. Rather, an interesting question is what features of a predictable universe enable agents to appear in it, and how complex and powerful can these agents get.

 

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