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.

This ties in well with the intelligence-as-compression paradigm: much of mathematics can be interpreted as a collection of very short programs, and so in a predictable universe with a bias towards short programs, it's unsurprising if a lot of them turn out to be useful somewhere or other.

Well, yes, and the reason isn't mysterious.

In order to compress a stream of data you need to discover some structure in it. If there is no structure -- e.g. if the stream is truly random -- then no compression is possible. And if the structure you found is "really there" and not an artifact of your structure-searching techniques, then it just as useful for extrapolation and prediction.

I think when we say that the universe "runs on math," part of what we mean is that we can use simple mathematical laws to predict (in principle)

allaspects of the universe. We suspect that there is alosslesscompression algorithm, i.e., a theory of everything. This is a much stronger statement than just claiming that the universe contains some predictable regularities, and is part of what makes the Platonic ideas you are arguing against seem appealing.We could imagine a universe in which physics found lots of approximate patterns that held most of the time and then got stuck, with no hint of any underlying order and simplicity. In such a universe we would probably not be so impressed with the idea of the universe "running on math" and these Platonic ideas might be less appealing.

I don't think that's really true though. The advances in physics that have been worth celebrating--Newtonian mechanics, Maxwellian electromagnetism, Einsteinian relativity, the electroweak theory, QCD, etc.--have been those that answer lots and lots of questions at once and raise only a few new questions like "why this theory?" and "what about higher energies?". Now we're at the point where the Standard Model and GR together answer

almost any question you can askabout how the world works, and there are relatively few questions remaining, like the problem of quantum gravity. Think how much more narrow and neatly-posed this problem is compared to the pre-Newtonian problem of explaining all of Nature!Mathematics is generally thought of more of a method of predicting things. Since prediction and compression are equivalent (a good compression algorithm is precisely one that has shorter statements for more likely predictions), it's equivalent to say that math is a compression algorithm.

Possibly related:

The Reasonable Ineffectiveness Of Mathematics (warning: PDF)

So what does "gone wild" mean? Your paragraph about this is not very charitable to the pure mathematician.

Say that mathematics is about generating compressed models of the world. How do we generate these models? Surely we will want to study (compress) our most powerful compression heurestics. Is that not what pure math is?

I think this is interesting even if I don't fully but into it's argument. May I ask what your mathematical background is? I have a mental prediction based on the post that I'd like to test.

I think we can trust shminux not to go get additional degrees in response to WQ's answer.

Never underestimate the race for status.

Shouldn't this be physics as a lossy compression algorithm? A lot of math has nothing to do with anything in the real world. I guess I agree, the mathematical nature of physical laws is simply an expression of the predictability of the universe.

While numbers might not be useful in some universes, ... (read more)

Great article, though I've always been a bit more of a mathematical realist myself.

The part that still fascinates me is how taking a couple of different mathematical descriptions of certain phenomena and working solely with the numbers under the "laws" of mathematics can lead to mathematical theories and predictions of seemingly unrelated phenomena.

For example, Einstein developed Special Relativity to accou... (read more)

Can I nominate for promotion to Main/Front Page?

What is a universe without humans?

We are limited to subjective observations, and can not confirm what objective observations of the universe would be.

I'd like to argue in this comment that mathematics is an implied property of the universe. We might "mistake" to think that mathematics are governing the universe, but rather the way the universe works can be described from our subjective perspective with the seemingly abstract entity of mathematics. The universe contains mathematics in the way it exists.

Claiming that mathematics exist in some ot... (read more)

All analogies are suspect, but if I had to choose one I'd say physics' theories--at best--are if anything like code that returns the Fibonacci sequence through a specified range. The theories give us a formula we can use to make certain predictions, in some cases with arbitrary precision. Video, losslessly- or lossy-compressed, is still video. Whereas

fib n = take n fiblist where fiblist = 0:1:(zipWith (+) fiblist (tail fiblist))

is not a bag holing the entire Fibonacci sequence, waiting for us to compress it so we can look at a slightly more pixelated v... (read more)

Can you actually imagine or describe one? I intellectually can accept that they might exist, but I don't know that my mind is capable of imagining a universe which could not be simulated on a Turing Machine.

The way that I define Tegmark's Ultimate Ensemble is as the set of all worlds that can be simulated by a Turing Machine. Is it possible to imagine in any concrete... (read more)

For observers to exist some parts of the universe must follow patterns, but not necessarily all of it. Could the "Unreasonable Effectiveness of Mathematics" relate to why all of the universe can probably be modeled with math?

While I love your analogy and agree that maths is simplifying

It's also, not. I wish Wikipedia editors and jurors were more sympathetic to the idea that the vast majority of Wikipedia's audience doesn't have have capacity to overcome the cognitive complexity of mathematical formalisms in many technical articles and would appreciate an english in instead as well. Simple english Wikipedia doesn't have its share of technical articles if that's where you'd rather the english went.

Mathsy people, to help you put yourself in our shoes, consider this Wiki article o... (read more)

if you could figure out physical laws from apriori maths, that really would be unreasonably effective. As it is, physicists have to carefully select maths that works physically from the much larger amount that doesn't. That this is possible is about as surprising as finding a true history in the library of Babel. The unreasonable effectiveness of maths is the reasonable effectiveness of physics.

For formalists (anti Platonists), 1+1=2 is true in all universes, providing you adopt axioms from which it can be derived. It isn't a truth about the universe, f

I believe I'm in basic agreement. Definitely in the nominalist camp.

Math is an evolved conceptual structure. Why does the math we use work? About the same reason the hammers we use work. Things that work, get used. We make changes, see which ones work better, and use those.

How is it that math can work? Well, how is it that the conceptual structures we use work? We try to use the ones that do, and move on from those that don't. Nothing to see here folks, move along.

There's an infinite space of conceptual structures. Most of them suck. Some don't. Math doe... (read more)