Mathematics as a lossy compression algorithm gone wild

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.

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.

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.

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.

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) all aspects of the universe. We suspect that there is a lossless compression 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.

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.

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.

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

While numbers might not be useful in some universes, they would nevertheless be correct. 1 + 1 would still equal 2 because they are defined that way. Whether the universe contains an entity capable of expressing that fact doesn't matter.

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

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

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 account for the inconsistencies between classical mechanics and Maxwell's equations using primarily the observation that the speed of light is absolute regardless of the motion of the light source and the postulate that the laws of physics are the same in all inertial reference frames. He just worked with the numbers under the rules of mathematics to (independently) develop the Lorentz transformations which lead to his Special Relativity.

The key feature here is that Einstein did not perform experiments. He knew form the null-result of the Michelson-Morley experiment that the speed of light is constant, but besides that, the vast amount of the work done was what Richard Hamming called "Scholastic" in its approach. I've even heard it said that as far back as Newton, the idea of non-locality was considered preposterous and that in itself gives the idea of a universal speed limit which might have been enough, along with Galileo's Principle of relativity to get very close to Special Relativity using only the tools of mathematics he had available to him and the current theories of motion and electromagnetism.

Now obviously a lot of work went into that, but so many strange predictions fell out of it that it really is amazing, some might say "unreasonable". For example, time dilation, mass/energy equivalence, length contraction, and eventually (after GR) black holes, relativistic cosmology, gravitational lensing, and the existence of dark matter.

That we can derive knowledge about how the universe works because of inconsistencies between simpler mathematical descriptions of various phenomena really does seem to suggest that the universe "runs on math". Now I don't mean to suggest that the equations are written somewhere in the sky and that some entity "breathes fire" into them; Just that the structure of the universe is isomorphic to the ideal math that we could use to explain and predict it. I would not be at all surprised to find out that somehow, "they are the same thing," whatever that might mean.

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 other dimension, is just about as reasonable as claiming that apples exist in another dimension.

If there's someone who can proove that they've broken the laws of physics while performing mathematics, then they have a valid basis for mathematical realism.

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 version of the actual piece.

Also, I don't think it makes sense to say math is part of nature (except in the sense everything is part of nature), though it may be that math is a psychological analogy to some feature of nature--like vision is an analogy to part of the EM spectrum. It would be a strange coincidence otherwise, considering how useful math is helping us make certain predictions. At the same time, many features of nature are utterly unpredictable, so either math only images select parts--we can't see x-ray--or we haven't fully understood how to use mathematics yet.

Incidentally, I think it is true that math--all our secular, materialist pretense aside--is still widely felt to have magical properties. In this regard Descartes' god is still with us.

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.

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 way a universe which doesn't fall under that definition? Is there an even more Ultimate Ensemble that we can't conceive of because we're creatures of a Turing universe?

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 on logic. Imagine that foreign language but all the symbols are highly compressed in an area, like one side of a numerator, with all kinds of relations between them, representing different things. I simply don't have the working memory to make any sense of it by the time I've looked up what a particular thing is and it's relation to a few others.

• 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, for them, but it is contingent on choice of axioms. This seems to lead to a vision multiple conflicting mathematical truths, which will seem counterintuitive to some. Formlists can buttress their case by by appealing to two ur -axioms: maths is about minimising contradiction whilst maximising richness of structure,; and speculating that these provide considerable constraints to axioms choice.

• Starting from the premise that maths talks us about the world, you are faced with a vision number of puzzles; whether maths is as real as the world, whether the world is as abstract as maths, why physicists and mathematicians are different people doing different things, why some maths is incorrect physics... Starting from the premise that apple's are oranges, you are likewise faced with puzzles about why they look and taste different... Physics is the science of explaining the world with maths. That maths can be used to explain the world has the samemcontent as saying that physics works. It is not statement about maths per se. Historians succeed in using words to describe events. That isn't a fact about lexicography.

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 doesn't suck. Huckleberry Finn doesn't either. Were both "discovered" out of that infinite space of structures? I guess you could say so, but that seems quite a peculiar way of looking at it.

To say that human beings "invented numbers" - or invented the structure implicit in numbers - seems like claiming that Neil Armstrong hand-crafted the Moon.

Doesn't seem that way to me. The moon is not a conceptual structure.