Mathematics as a lossy compression algorithm gone wild

post by shminux · 2014-06-06T23:53:46.887Z · LW · GW · Legacy · 80 comments

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.

 

80 comments

Comments sorted by top scores.

comment by gwern · 2014-06-07T01:14:21.585Z · LW(p) · GW(p)

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.

comment by maxy · 2014-08-15T17:27:09.800Z · LW(p) · GW(p)

Those "very short programs" are useful even if the universe has no bias towards them. It's just Occam's razor. I think it has more to do with the process of knowledge gathering than with the universe itself.

comment by gwern · 2014-08-17T00:34:11.590Z · LW(p) · GW(p)

Those "very short programs" are useful even if the universe has no bias towards them.

They are? How? If they have no privileged status and phenomena are due to long programs as likely as short programs (leaving aside the issue of how they works given that there are so many more long programs than short ones), then they don't predict well. That doesn't sound useful.

It's just Occam's razor.

And what justifies Occam's razor if the universe has no bias towards short programs?

comment by Lumifer · 2014-06-07T01:20:03.626Z · LW(p) · GW(p)

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.

comment by The_Duck · 2014-06-07T16:51:17.862Z · LW(p) · GW(p)

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.

comment by shminux · 2014-06-07T18:56:08.868Z · LW(p) · GW(p)

We suspect that there is a lossless compression algorithm, i.e., a theory of everything.

Yeah, I don't see this as likely at all. As I repeatedly said here, it's models all the way down.

comment by The_Duck · 2014-06-07T20:46:39.277Z · LW(p) · GW(p)

Fair enough. I can see the appeal of your view if you don't think there's a theory of everything. But given the success of fundamental physics so far, I find it hard to believe that there isn't such a theory!

comment by shminux · 2014-06-07T21:23:31.160Z · LW(p) · GW(p)

Given that every time we discover something new we find that there are more questions than answers, I find it hard to believe that the process should converge some day.

comment by The_Duck · 2014-06-07T22:23:05.927Z · LW(p) · GW(p)

every time we discover something new we find that there are more questions than answers

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 ask about 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!

comment by DanielLC · 2014-06-07T19:28:38.867Z · LW(p) · GW(p)

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.

comment by CronoDAS · 2014-06-07T06:54:01.956Z · LW(p) · GW(p)

Possibly related:

The Reasonable Ineffectiveness Of Mathematics (warning: PDF)

comment by shminux · 2014-06-07T07:21:47.166Z · LW(p) · GW(p)

Neat. While this article does not go far enough for my tastes, I am quite happy that it confirms my "strong non-Platonism" intuition (I called it anti-Platonic.)

comment by torekp · 2014-06-07T20:59:35.723Z · LW(p) · GW(p)

Not having read the whole thing, it seems to make much of the failure of classical models to describe and predict electronic circuits. But if the correct model is quantum, this isn't necessarily surprising.

It seems possible that a set of physical laws will be discovered that allows lossless compression. Maybe QM as now understood will be included in that set. If so, it seems that those laws, and the mathematics involved therein, describe something real.

But that's not Platonism by the Wikipedia definition.

comment by JeremyHahn · 2014-06-09T22:06:22.875Z · LW(p) · GW(p)

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?

comment by shminux · 2014-06-09T23:06:53.486Z · LW(p) · GW(p)

I agree I was too brief there. The original motivation for math was to help figure out the physical world. At some point (multiple points, really, starting with Euclid), perfecting the tools for their own sake became just as much of a motivation. This is not a judgement but an observation. Yes, sometimes "pure math" yields unexpected benefits, but this is more of a coincidence then the reason people do it (despite what the grant applications might say).

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 heuristics. Is that not what pure math is?

The main reason is pure curiosity without any care for eventual applicability to understanding the physical world. Pretending otherwise would be disingenuous.

Theoretical physics is not very different in that regard. To quote Feynman

Physics is like sex: sure, it may give some practical results, but that's not why we do it.

comment by JeremyHahn · 2014-06-10T20:56:38.707Z · LW(p) · GW(p)

I guess I always took the phrase "unreasonable effectiveness" to refer to the "coincidence" you mention in your reply. I'm not really sure you've gone far toward explaining this coincidence in your article. Just what is it that you think mathematicians have "pure curiousity" about? What does it mean to "perfect a tool for its own sake" and why do those perfections sometimes wind up having practical further use. As a pure mathematician, I never think about applying a tool to the real world, but I do think I'm working towards a very compressed understanding of tool making.

comment by shminux · 2014-06-10T23:57:28.045Z · LW(p) · GW(p)

I guess I always took the phrase "unreasonable effectiveness" to refer to the "coincidence" you mention in your reply.

Unreasonable effectiveness tends to refer to the observation that the same mathematical tools, like, say, mathematical analysis, end up useful for modeling very disparate phenomena. In a more basic form it is "why do mathematical ideas help us understand the world so well?". The answer suggested in the OP is that the question is a tautology: math is a meta-model build by human minds, not a collection of some abstract objects which humans discover in their pursuit of better models of the world. The JPEG analogy is that asking why math is unreasonably effective in constructing disparate (lossy) models is like asking why the JPEG algorithm is unreasonably effective in lossy compression of disparate images.

The "coincidence" part referred to something else: that pursuing math research for its own sake may occasionally work out ot be useful for modeling the physical world, number theory and encryption being the standard example.

As a pure mathematician, I never think about applying a tool to the real world, but I do think I'm working towards a very compressed understanding of tool making.

When I talk to someone who works in pure math, they usually describe the motivation for what they do in almost artistic terms, not caring whether what they do can be useful for anything, so "tool making" does not seem like the right term.

comment by William_Quixote · 2014-06-07T00:43:40.397Z · LW(p) · GW(p)

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.

comment by shminux · 2014-06-07T01:10:25.484Z · LW(p) · GW(p)

What's your prediction?

comment by DanArmak · 2014-06-07T13:42:22.673Z · LW(p) · GW(p)

He shouldn't tell it to you, but in secret to a third party. Otherwise the experiment would be ruined.

comment by Luke_A_Somers · 2014-06-07T17:52:03.706Z · LW(p) · GW(p)

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

comment by shminux · 2014-06-07T19:56:23.658Z · LW(p) · GW(p)

Never underestimate the race for status.

comment by DanArmak · 2014-06-07T16:07:04.513Z · LW(p) · GW(p)

He has sent me his prediction in a private message.

comment by shminux · 2014-06-07T19:54:17.073Z · LW(p) · GW(p)

Well?

comment by DanArmak · 2014-06-07T20:13:02.199Z · LW(p) · GW(p)

The prediction was:

My prediction is that the writer has a strong background in calculus including multi variate but has not taken more theoretical proof courses like set theory / logic 2 / or number theory.

A graduate physics degree no doubt requires calculus. What about the rest?

comment by shminux · 2014-06-07T21:21:56.127Z · LW(p) · GW(p)

That's about right. I did learn elementary set theory and logic, and had to prove theorems in the course of my research, but I was not interested in, say, computability, advanced logic, formal systems or the number theory. I guess Platonism is more pervasive among pure mathematicians.

comment by solipsist · 2014-06-08T13:59:42.059Z · LW(p) · GW(p)

You've studied abstract algebra too, no? At minimum, you've dealt with groups.

comment by shminux · 2014-06-08T17:48:54.807Z · LW(p) · GW(p)

Oh, for sure, had to learn some, like finite groups, Lie groups/algebras.

comment by shminux · 2014-06-07T16:18:26.660Z · LW(p) · GW(p)

How would it be ruined? My "mathematical background" does not depend on his prediction and I cannot change it after learning what the prediction is.

comment by DanArmak · 2014-06-07T16:20:26.015Z · LW(p) · GW(p)

But his prediction could change subtly after learning what your background is.

"Ruined" is too strong; I should have said "influenced".

comment by shminux · 2014-06-07T16:23:48.527Z · LW(p) · GW(p)

Right, that's why I asked what it is before revealing the answer.

comment by DanArmak · 2014-06-07T16:27:43.836Z · LW(p) · GW(p)

Your way of formulating the answer may have been influenced by his theory?

I'm quibbling. I recommended the "full procedure" out of habit. It's wasn't really necessary in this particular case.

comment by shminux · 2014-06-07T17:10:41.114Z · LW(p) · GW(p)

Anyway, I have about 3 years of college math (geared toward physics, not math students), plus some elements of grad-level math required for my grad degree, such as differential geometry and algebraic topology.

comment by TylerJay · 2014-06-07T18:34:29.510Z · LW(p) · GW(p)

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.

comment by anonym · 2014-06-16T16:08:50.488Z · LW(p) · GW(p)

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.

Tegmark's Mathematical universe hypothesis is one answer to what that might mean.

comment by Gavin · 2014-06-07T12:11:31.184Z · LW(p) · GW(p)

Can I nominate for promotion to Main/Front Page?

comment by shminux · 2014-06-07T16:22:40.547Z · LW(p) · GW(p)

Some time ago I suggested that (non-link, non-meta) Discussion posts should be automatically promoted to Main if they are upvoted above 20-30 karma. This post is currently well below.

comment by shminux · 2014-06-09T00:47:39.159Z · LW(p) · GW(p)

OK, talked to a few regulars off-line, the opinions are decidedly mixed, so not going to move this post to Main.

comment by TheAncientGeek · 2014-06-18T14:13:35.379Z · LW(p) · GW(p)

The post us currently on the35, can I vote for the move?

comment by shminux · 2014-06-19T20:58:18.560Z · LW(p) · GW(p)

Thanks. If any of the admins find it worth it, I don't mind if they move it.

comment by [deleted] · 2014-06-14T17:17:35.849Z · LW(p) · GW(p)

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.

comment by TheAncientGeek · 2014-06-18T13:57:10.599Z · LW(p) · GW(p)

Would a different universe have different maths?

comment by [deleted] · 2014-06-18T19:18:50.882Z · LW(p) · GW(p)

What d'ya mean by “different” and by “maths”? Sure they would use different notation but it'd denote mathematical structures isomorphic to our own.

comment by TheAncientGeek · 2014-06-18T19:33:53.718Z · LW(p) · GW(p)

I was trying to follow through 395's premise. If maths is implied by the universe, then other universes would imply other maths, n'est ce pas.

comment by Lumifer · 2014-06-14T20:08:40.020Z · LW(p) · GW(p)

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

What does "objective observations of the universe" mean?

comment by [deleted] · 2014-06-15T01:25:12.008Z · LW(p) · GW(p)

Having knowledge that would not be limited by being an observer, a small part of the universe.

What is time if you're not a creature existing in time?

comment by TheAncientGeek · 2014-06-18T13:59:20.649Z · LW(p) · GW(p)

That is not the usual objective/subjective distinction.

comment by Lumifer · 2014-06-15T02:34:54.979Z · LW(p) · GW(p)

Having knowledge that would not be limited by being an observer

Who/what is the subject of "having knowledge"?

comment by [deleted] · 2014-06-15T10:15:57.846Z · LW(p) · GW(p)

In my opinion that is irrelevant as long as the information is not limited by the nature of the observer. However I don't intend to say that "having knowledge" could have a meaning outside some sort of a subjective thing having it. So I'd like to separate the idea of some kind of truth from a subjective experience of having it. If there is any truth like that. If there was, how could we know, if we don't currently? Does it make sense to contemplate on the possibility of there being knowledge we can't have? We are limited and we can't really think outside the box. Knowing that we can't think outside the box, does not provide the capacity to suggest everything that could be outside the box.

comment by TheAncientGeek · 2014-06-18T14:02:38.509Z · LW(p) · GW(p)

Depends on the value of can't. We can't have the knowledge in the library of Alexandria..but counterfactually we could have had.

comment by Lumifer · 2014-06-16T14:52:15.906Z · LW(p) · GW(p)

So I'd like to separate the idea of some kind of truth from a subjective experience of having it. If there is any truth like that. If there was, how could we know, if we don't currently?

That's a pretty well-trodden philosophy topic :-)

comment by peterward · 2014-06-09T01:13:09.988Z · LW(p) · GW(p)

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.

comment by pianoforte611 · 2014-06-08T03:41:05.499Z · LW(p) · GW(p)

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.

comment by shminux · 2014-06-08T04:51:30.793Z · LW(p) · GW(p)

Shouldn't this be physics as a lossy compression algorithm?

Think of math as the JPEG compression algorithm, and physics as cat pictures (not "real cats"). And some other science as, say, pictures of the sunset. JPEG works on both kinds of pictures, even if there is not much in common between cats and sunsets.

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.

That's the Platonic view. I am suggesting that in a universe without cats and pictures no one would invent JPEG, just like in some other (or even this) universe no one would invent human morality, even though one can state that "morality exists" as an abstract concept.

comment by pianoforte611 · 2014-06-08T14:50:15.675Z · LW(p) · GW(p)

I'm not sure that this is a useful disagreement. But anyway: the JPEG comparison is interesting because JPEG is an algorithm, it doesn't "exist" but it isn't contingent on anything physical. Its just an algorithm. Does Hamlet "exist" in an alternate universe where Shakespeare was never born. The question doesn't really mean anything.

I'm not a Platonist, if anything I would lean to towards formalism but I am very wary of taking a firm position on a philosophical issue that I am not intimately familiar with. And I think that Formalists and Platonists would answer yes to the question of whether 1+ 1 = 2 in an alternate chaotic universe (The logicists and intuitionists are rare). You can read about this here.

The Platonist view is that 1 + 1 = 2 expresses some truth about the entities "1", "+", "2" and "="

The formalist view is that 1 + 1 = 2 follows from the axioms, its isn't a truth but rather the predicate of a system of rules. If "mathematical axioms are granted" then "1 + 1 = 2"

But pretty much no philosopher thinks that mathematical statements are contingent on something. And I'm not sure if you're saying that. Mathematical statements would not be generated in some universes yes, but that's not relevant to whether the statements themselves are true. The statements are not contingent on the nature of the universe.

If you think that they are contingent, how far does this go? Is the law of non-contradiction contingent because in a sufficiently weird universe, no one would discover it?What about the statement "If X then X"?

comment by shminux · 2014-06-08T17:53:16.278Z · LW(p) · GW(p)

Mathematical statements would not be generated in some universes yes, but that's not relevant to whether the statements themselves are true.

I guess I failed to present my view clearly enough. See if this works better:

Is the law of non-contradiction contingent because in a sufficiently weird universe, no one would discover it?

Is the virtue of mercy contingent because in a sufficiently weird universe, no one would discover it?

comment by pianoforte611 · 2014-06-08T18:49:15.195Z · LW(p) · GW(p)

Is the virtue of mercy contingent because in a sufficiently weird universe, no one would discover it?

I don't know if I'm a moral realist. But assuming I were for a second: mercy essentially means "be nice to people even if they are bad/ do bad things". Its not a statement about the world. Its not a truth claim. Its not a definition. Its not an algorithm. Its an imperative, an instruction. As such it is not subject to being contingent or necessary. Is the statement "Eat your vegetables" true? Well no it doesn't have a truth value. Likewise with ethical statements.

Which isn't remotely similar to mathematical statements which most philosophers either think express truths (Platonists) or formalisms - sequences of symbols that are generable from simple rules. But no philosophers that I know of think that mathematical statements require referents in the real world. There are some very strange mathematical fields that have nothing to do with this universe. My take on this is that it doesn't really matter all that matters is that mathematics is useful.

comment by Gavin · 2014-06-07T12:08:41.348Z · LW(p) · GW(p)

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?

comment by DanArmak · 2014-06-07T13:34:56.775Z · LW(p) · GW(p)

The set of all Turing Machines is merely countable. Instead, imagine an ensemble of universes corresponding to the real numbers, some of which aren't even computable.

For example, universes running on classical mechanics, where values can be measured with infinite precision, and which also have continuous space and time to represent those infinitely precise values. In other words, a set of universes where two universes can be different in the position of a particle, position being defined as a real number.

This seems easy to imagine - it's a pretty standard Newtonian world.

comment by Luke_A_Somers · 2014-06-07T17:53:52.139Z · LW(p) · GW(p)

Even in a Newton world, you can make predictions. You just won't be able to make them to infinite precision.

I, too, am having trouble imagining a perfectly unpredictable universe with agents in it.

comment by DanArmak · 2014-06-07T19:12:10.897Z · LW(p) · GW(p)

I can't either. But I can imagine a partially predictable world, which cannot be perfectly simulated by any Turing machine (that's what Gavin's question was about), and yet is completely deterministic and has a simple mathematical description. For instance, a continuous, deterministic world where a fundamental physical constant is an uncomputable number.

comment by Luke_A_Somers · 2014-06-10T02:53:52.818Z · LW(p) · GW(p)

I see - you're going after a much weaker criterion than the one described in the OP.

I'd note that my 'finite precision' comment covers uncomputable constants. Unless no computation can even converge on them...

comment by DanArmak · 2014-06-10T18:29:19.744Z · LW(p) · GW(p)

I see - you're going after a much weaker criterion than the one described in the OP.

I was responding to Gavin's comment and its criterion was "a universe which could not be simulated on a Turing Machine".

I'd note that my 'finite precision' comment covers uncomputable constants. Unless no computation can even converge on them...

The set of all constants on which some computation converges is still only countable (each Turing machine's output converges on at most one constant).

A universe with uncomputable constants in the laws of nature might still be predictable in practice. We don't know if the constants in our own universe are computable or not, because we can't even measure them beyond a few hundreds bits each. And as long as we keep measuring them instead of deriving them from some equation, our knowledge will presumably remain finite, so computability does not matter for us in practice.

On the other hand, full simulation does depend on the precise values. (If it didn't, the constants wouldn't have those precise values in any meaningful sense.)

comment by Luke_A_Somers · 2014-06-10T20:34:09.154Z · LW(p) · GW(p)

The set of all constants on which some computation converges is still only countable (each Turing machine's output converges on at most one constant).

I meant to say 'no computation can even constrain them to a useful degree'. Strict convergence is not necessary.

comment by DanArmak · 2014-06-11T08:26:27.978Z · LW(p) · GW(p)

That seems to be the same as 'no computation can output a prefix that is sufficiently long to be useful in predictions'?

This would depend on how long a prefix you need. If there exists a constant K such that K-length prefixes of all universal constants are enough, then certainly all possible bitstrings of length K can be computed.

The question becomes: what does a universe look like where you need to know the exact value of some constants to make useful predictions? (Perhaps you can make some predictions using approximations, but some other possible scenarios remain unpredictable without precise knowledge.)

This requires knowing an infinite amount of information: some kind of physical reification of an uncomputable real number. (I'm only using real numbers as a familiar example; larger infinities could be used too.) Not just to know the constant, but probably also to know the precise initial state of the system whose evolution you want to predict.

One possible reification is the precise position, mass, momentum, etc. of an element in the simulation - assuming these properties can also have any real value, and that you can control them to that degree.

I feel I could still imagine such a universe. But either way that's a fact about me, not about universes. I don't know how to predict the measure of such universes in any multiverse theory.

comment by Gavin · 2014-06-07T16:03:30.808Z · LW(p) · GW(p)

I would think those would all be representable by a Turing Machine, but I could be wrong about that. Certainly, my understanding of the Ultimate Ensemble is that it would include universes that are continuous or include irrational numbers, etc.

comment by DanArmak · 2014-06-07T16:06:07.239Z · LW(p) · GW(p)

Turing Machines have discrete, not continous states. There is a countable infinity of Turing Machines.

comment by shminux · 2014-06-11T17:01:25.720Z · LW(p) · GW(p)

I don't know that my mind is capable of imagining a universe which could not be simulated on a Turing Machine.

I never said it could not be, just that the Turing Machine would not be a concept that is likely to evolve there.

Imagine a universe where there are no discrete entities, so numbers/addition is not a useful model. Whatever inhabits such a universe, if anything, would not develop the abstraction of counting. This universe could still be Turing-simulated (Turing Machine is an abstraction from our universe),

This is the essential point I am trying to make. Mathematics is determined by the structure of the universe and is not an independent abstract entity. I feel like I failed, though.

comment by James_Miller · 2014-06-07T04:00:03.837Z · LW(p) · GW(p)

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?

comment by DanArmak · 2014-06-07T13:40:38.193Z · LW(p) · GW(p)

That seems related to the anthropic questions of why the universe is much more ordered than the minimum required for Boltzmann brains, or much older than the minimum required for life to develop.

comment by [deleted] · 2015-09-03T12:33:07.814Z · LW(p) · GW(p)

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.

comment by TheAncientGeek · 2014-06-18T13:12:20.132Z · LW(p) · GW(p)
  • 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.

comment by buybuydandavis · 2014-06-07T02:42:42.792Z · LW(p) · GW(p)

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.

comment by TheAncientGeek · 2014-06-18T14:04:50.936Z · LW(p) · GW(p)

Work at what? For whom? Mathematicians are happy with maths that is of no use to physicists.

comment by buybuydandavis · 2014-06-20T01:29:30.231Z · LW(p) · GW(p)

And that's fine. In fact, that's great. If people want to enjoy the aesthetics of conceptual structure, I hope they call me over for the fun.

But the "what" in the "work at what" I was speaking, is "predicting other data points, not yet observed", per the OP.

comment by TheAncientGeek · 2014-06-20T15:35:11.880Z · LW(p) · GW(p)

Not actually the job of maths..."this hammer doesn't work, you can't drive in screws with it"

comment by buybuydandavis · 2014-06-20T18:49:02.127Z · LW(p) · GW(p)

Math doesn't have a "job". It's use by people to fulfill their ends. For most people, those ends are prediction.

comment by TheAncientGeek · 2014-06-20T19:05:26.048Z · LW(p) · GW(p)

Grab an arbitrary piece of maths and it won't predict anything. There is a technique and speciality and skill of finding the right pieceof maths to match the territory, and that is called physics.

Meanwhile...no professional mathematician gets sacked for failing to predict or otherwise being empirically correct. It's not their job.

comment by shminux · 2014-06-08T05:46:43.169Z · LW(p) · GW(p)

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

Actually, I am not a nominalist, I only adopted a somewhat-nominalist position for this post to express what I think about math. In actuality I slide all the way down the slippery slope and consider the term "exist" meaningless except in the original sense of "perceived". But that would be a subject for a different post.

comment by TheAncientGeek · 2014-06-21T15:36:05.987Z · LW(p) · GW(p)

Not even "potentially receivable"?