The Emergence of Math
post by AnotherIdiot · 2012-11-02T01:08:20.318Z · LW · GW · Legacy · 44 commentsContents
Abstractions Axiomatic Systems (for example, Peano Arithmetic) I Can Prove Things About Numbers Using Apples Apples vs Peano Arithmetic To Conclude, Predictions are "Proofs" None 44 comments
In his latest post, Logical Pinpointing, Eliezer talks about the nature of math. I have my own views on the subject, but when writing about them, it became too long for a comment, so I'm posting it here in discussion.
I think it's important to clarify that I am not posting this under the guise that I am correct. I'm just posting my current view of things, which may or may not be correct, and which has not been assessed by others yet. So though I have pushed myself to write in a confident (and clear) tone, I am not actually confident in my views ... well, okay, I'm lying. I am confident in my views, but I know I shouldn't be, so I'm trying to pretend to myself that I'm posting this to have my mind changed, when in reality I'm posting it with the stupid expectation that you'll all magically be convinced. Thankfully, despite all that, I don't have a tendency to cling to beliefs which I have seen proven wrong, so I will change my mind when someone kicks my belief's ass. I won't go down without a fight, but once I'm dead, I'm dead.
Before I can share my views, I need to clarify a few concepts, and then I'll go about showing what I believe and why.
Abstractions
Abstract models are models in which some information is ignored. Take, for example, the abstract concept of the ball. I don't care if the ball is made of rubber or steel, nor do I care if it has a radius of 6.7cm or a metre, a ball is a ball. I couldn't care less about the underlying configuration of quarks in the ball, as long as it's a sphere.
Numbers are also abstractions. If I've got some apples, when I abstract this into a number, I don't care that they are apples. All I care about is how many there are. I can conveniently forget about the fact that it's apples, and about the concept of time, and the hole in my bag through which the apples might fall.
Axiomatic Systems (for example, Peano Arithmetic)
Edit: clarified this paragraph a bit.
I don't have much to say about these for now, except that they can all be reduced to physics. Given that mathematicians store axiomatic systems in their minds, and use them to prove things, they cannot help but be reducible to physical things, unless the mind itself is not physical. I think most LessWrongers, being reductionists, believe this, so I won't go into much more detail. I'll just say that this will be important later on.
I Can Prove Things About Numbers Using Apples
Let's say I have 2 apples. Then someone gives me 2 more. I now have 4. I have just shown that 2+2=4, assuming that apples behave like natural numbers (which they do).
But let's continue this hypothetical. As I put my 2 delicious new apples into my bag, one falls out through a hole in my bag. So if I count how many I have, I see 3. I had 2 to begin with, and 2 more were given to me. It seems I have just shown 2+2=3, if I can prove things about numbers using apples.
The problem lies in the information that I abstracted away during the conversion from apples to numbers. Because my conversion from apples to numbers failed to include information about the hole, my abstract model gave incorrect results. Like my predictions about balls might end up being incorrect if I don't take into account every quark that composes it. Upon observing that the apple had fallen through the hole, I would realize that an event which rendered my model erroneous had occurred, so I would abstract this new event into -1, which would fix the error: 2+2-1=3.
To summarize this section, I can "prove" things about numbers using apples, but because apples are not simple numbers (they have many properties which numbers don't), when I fail to take into account certain apple properties which will affect the number of apples I have, I will get incorrect results about the numbers.
Apples vs Peano Arithmetic
We know that Peano Arithmetic describes numbers very well. Numbers emerge in PA; we designed PA to do so. If PA described unicorns instead, it wouldn't be very useful. And if PA emerges from the laws of physics (we can see PA emerge in mathematicians' minds, and even on pieces of paper in the form of writing), then the numbers which emerge from PA emerge from the laws of physics. So there is nothing magical about PA. It's just a system of "rules" (physical processes) from which numbers emerge, like apples (I patched up the hole in my bag ;) ).
Of course, PA is much more convenient for proving things about numbers than apples. But they are inherently just physical processes from which I have decided to ignore most details, to focus only on the numbers. In my bag of 3 apples, if I ignore that it's apples there, I get the number 3. In SSS0, if I forget about the whole physical process giving emergence to PA, I am just left with 3.
So I can go from 3 apples to the number 3 by removing details, and from the number 3 to PA by adding in a couple of details. I can likewise go from PA to numbers, and then to apples.
To Conclude, Predictions are "Proofs"
From all this, I conclude that numbers are simply an abstraction which emerges in many places thanks to our uniform laws of physics, much like the abstract concept "ball". I also conclude that what we classify as a "prediction" is in fact a "proof". It's simply using the rules to find other truths about the object. If I predict the trajectory of a ball, I am using the rules behind balls to get more information about the ball. If I use PA or apples to prove something about numbers, I am using the rules behind PA (or apples) to prove something about the numbers which emerge from PA (or apples). Of course, the proof with PA (or apples) is much more general than the "proof" about the ball's trajectory, because numbers are much more abstract than balls, and so they emerge in more places.
So my response to this part of Eliezer's post:
Never mind the question of why the laws of physics are stable - why is logic stable?
Logic is stable for the same reasons the laws of physics are stable. Logic emerges from the laws of physics, and the laws of physics themselves are stable (or so it seems). In this way, I dissolve the question and mix it with the question why the laws of physics are stable -- a question which I don't know enough to attempt to answer.
Edit: I'm going to retract this and try to write a clearer post. I still have not seen arguments which have fully convinced me I am wrong, though I still have a bit to digest.
44 comments
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comment by Larks · 2012-11-02T01:20:03.022Z · LW(p) · GW(p)
Axiomatic Systems ... can all be reduced to physics. I think most LessWrongers, being reductionists, believe this.
I would be suprised if this were true. In fact, I'm not even sure what you mean by it.
However, I think you're confusing the finitude of our proofs with some sort of property of the models. I mean, I can easily specify models much bigger than the physical universe.
At the very least this warrants further explication.
Replies from: AnotherIdiot↑ comment by AnotherIdiot · 2012-11-02T01:26:06.326Z · LW(p) · GW(p)
Axiomatic Systems ... can all be reduced to physics. I think most LessWrongers, being reductionists, believe this.
I would be suprised if this were true. In fact, I'm not even sure what you mean by it.
Well, given that mathematicians store axiomatic systems in their minds, and use them to prove things, they cannot help but be reducible to physical things, unless the mind itself is not physical.
However, I think you're confusing the finitude of our proofs with some sort of property of the models. I mean, I can easily specify models much bigger than the physical universe.
You can specify models much bigger than the physical universe. But that's just extrapolating the rules by assuming they would keep on working. We do have good reason to believe that they would keep on working, though, because if they stop, then contradictions take place, and from contradictions anything is true, so we would be living in one very strange universe.
Edit:
It has also occurred to me that it doesn't even matter if a number is larger than the total amount of atoms in this universe. Because as I've said in my post, a number is what you get when you abstract away all the little details which aren't shared by all the places where numbers emerge, like the fact that it's atoms you're counting.
So a system for representing a number larger than the total number of atoms represents a number anyway, as long as it follows the rules of numbers. And a model of something much bigger than the universe works simply because the details of how large the universe actually is are ignored in the model.
Replies from: tim, mwengler↑ comment by tim · 2012-11-02T06:43:09.497Z · LW(p) · GW(p)
Is your claim that because the mind is itself physical, any idea stored in a mind is necessarily reducible to something physical?
Because this seems like a map-territory confusion.
ETA: minds can contain gods, magic and any number of wonders that are fundamentally irreconcilable with physical reality.
Replies from: mwengler, mwengler, AnotherIdiot↑ comment by mwengler · 2012-11-04T14:21:45.201Z · LW(p) · GW(p)
Is your claim that because the mind is itself physical, any idea stored in a mind is necessarily reducible to something physical?
Indeed, the mind seems empirically to be something we experience due to the workings of visibly finite machine.
My mind contains a concept of 4 which is pretty dang useful, I can visualize 4 in so many ways and see it so often in reality without even trying to look for it. My mind's concept of 1000 doesn't suck, but is starting to look more like fluid measure (continuously variable) than discrete. With neocortical help I can improve my concept of 1000 around the edges, learning what 1000 dollars looks like, or a thousand grains of sand, or 1000 people in a stadium, but the amount of the conception of 1000 that my mind actually misses, when held side by side with my conception of 4 is, I think, very obvious.
I can think of 10 billion, which is more than the neurons in my brain, 100 billion which is more than the cells in my body (including the bacteria), 10 trillion (which is more than the neuronal interconnections in my brain) but all of these concepts are so fuzzy as to be honored primarily in the logarithm base 10. That is, virtually the only thing I know about 10 billion is that it is 10x as much as 1 billion, and similar such indirect conceptions.
My point in this is that if we think about how we actually think about numbers, the physicalism seems clear, including the limits in clarity we might expect as we exceed the number of parts of the physical system we are using for doing such thinking.
So how, then, do we come up with theorems about different kinds of infinities if our conceptions of these things are so finite? I believe it to be true (please correct me) that 1) the number of things we know about infinities is quite finite (maybe 1000 things at most?) 2) the proofs we use to know these finite number of things about infinities are also quite finite (comprising perhaps 10,000 pages, perhaps 100,000 to prove everything humans know about math, including everything we know about infinities)
PUNCHLINE: Therefore it is very suggestive to think the physical substrate is a gigantic part of the story, and I am at a loss to see an opening for any serious contribution from a non-physical source.
↑ comment by mwengler · 2012-11-04T14:09:48.880Z · LW(p) · GW(p)
ETA: minds can contain gods, magic and any number of wonders that are fundamentally irreconcilable with physical reality
Are not the non-existence of gods and magic empirical truths? I can imagine someone with a map tailored to the universe-as-is would say if they started seeing magic and gods that there was a deeper sense in which the magic was really technology we don't understand and the gods were life that exceeded our abilities in dimensions where the individual from our world had never seen such strong evidence of any life even tying humans in ability.
But I should think in a universe with beings who exceeded our intelligence by factors of a trillion that magic and gods would be a lot better map of that territory than the map we use in our universe without such beings. Is a dog who believes "slavery is wrong" actually better in some real way in a world like ours (where dogs are not very smart and are domesticated by humans)?
↑ comment by AnotherIdiot · 2012-11-02T07:35:42.714Z · LW(p) · GW(p)
Is your claim that because the mind is itself physical, any idea stored in a mind is necessarily reducible to something physical?
...
ETA: minds can contain gods, ...
No, I'm claiming that the idea of god exists physically.
In our universe, the map is part of the territory. So the concept of god which a human stores in his mind is something physical. God himself might not exist, but the idea of god, and the rules this idea follows, exist, despite being inconsistent. And these rules which the idea of god follows can be represented in many ways, all of them physical.
For example, in the human mind, in computers, in mathematical logic (despite inconsistencies), etc. All these ways of representing god are done using completely different configurations of molecules. What is the common ground between them? Certainly not that the idea of god and it's rules are some special thing with special properties. So what do the hard drive and the human mind have in common when representing the idea of god?
By my theory, what they have in common is abstraction. Ignore all the specific details about how hard drives and human minds work, and just look at the specific abstract rules which we define as "god". These are complex, so we can't easily visualize this removal of details. It's much easier when talking about apples and numbers. You can see that when you have 2 apples, you can get the idea of 2 by ignoring the fact that it's apples, and that they're in a bag, and that gravity is affecting them. It's also easy to see when talking about balls. You get the idea of a ball by taking a sphere of matter, forgetting what it's composed of and forgetting it's radius. This abstract idea of a "ball" fits many things, because it's just ignoring details which vary from ball to ball.
So my claim is that the idea of axiomatic systems exists in the physical universe. In fact, all the ideas we ever have, and there rules, exist in the physical universe. But if we take PA as an example, the idea of PA exists in a mathematician's mind, and numbers emerge inside this idea of PA, because numbers do emerge inside PA. So by removing the details of how PA is stored in the mathematician's mind, we obtain numbers, which is just like getting numbers by removing the details about apples.
This still leaves the question of why numbers emerge in so many places. My best guess is that they do because the universe is built upon simple and universal laws of physics, so it's only natural that the same patterns would be appear everywhere.
↑ comment by mwengler · 2012-11-04T14:02:03.449Z · LW(p) · GW(p)
So a system for representing a number larger than the total number of atoms represents a number anyway,
There seems to be no problem representing numbers using machines that have many fewer pieces than the number represented. With only 10 bits I can represent more than 1000 numbers as a trivial example. WIth only 10 billion neurons I can represent infinity (in the human mind) although it might be difficult to prove my representation was perfectly accurate.
comment by Decius · 2012-11-02T03:08:10.590Z · LW(p) · GW(p)
Numbers are not relatable to quantities.
2 apples added to 2 apples yields 4 apples, because that is a physical fact.
2+2=4 because arithmetic was designed to be useful.
If the physical facts regarding apples were to change, 2+2=4 would still be true, but it would not be useful to use the symbols 2 and 4 in the description of quantities of apples.
Replies from: AnotherIdiot↑ comment by AnotherIdiot · 2012-11-02T03:14:14.306Z · LW(p) · GW(p)
If the physical facts of apples were to change such that 2 apples added to 2 more apples did not give you 4 apples, then removing the detail that it's an apple would not yield numbers. In such a case, you would not be able to abstract apples into numbers. They would abstract away into something else.
Likewise, if you changed the mental processes which makes Peano Arithmetic, you would not change numbers; you would merely have changed what Peano Arithmetic can be abstracted into.
The thing to get from my post is that numbers are an abstraction: they are apples, when you forget that it's apples that you're talking about. They are Peano Arithmetic, when you forget that it's mental processes you're talking about. They are bits, when you forget it's bits you're talking about. But this does not make them special, just like the concept "sphere" is not special. We're just lucky that numbers emerge all over the place in the universe, probably because the laws of physics are the same everywhere so everything is built upon the same base.
Replies from: Decius↑ comment by Decius · 2012-11-02T03:46:50.785Z · LW(p) · GW(p)
You can't get a number out of a quantity- it's simply that just about all of the rules that apply to numbers also apply to quantities.
That's because numbers were created to be useful in manipulating quantities. Mathematics interacts very well with physics from a combination of factors: Physics appears to obey similar underlying laws as mathematics, and mathematics is created by people who want to explain physical phenomena.
When you forget that it's apples and miles that you are talking about, 2 apples and 2 miles never yields 4 anything, even though 2 apples plus 2 apples is 4 apples, and 2 miles plus 2 miles is 4 miles.
There's even a term for what you describe in the academic world- Magic Units. It is used to provide partial credit for showing mathematical calculations on numbers rather than showing mathematical calculations on quantities.
Replies from: AnotherIdiot↑ comment by AnotherIdiot · 2012-11-02T04:14:12.353Z · LW(p) · GW(p)
Very nice and convincing argument. There were some moments when thinking about it when I though your argument defeated my view. Sadly, we're not quite there yet.
Trying to add 2 miles to 2 apples does not make sense. There is no physical representation of such an operation. So you can't try to abstract that into numbers. Here's an example, to clarify:
Let's say I've got a bag of 2 apples, I add 2, and one falls through the hole in my bag. The number of apples in my bag is 2+2-1=3. The first 2 is an abstraction of the original number of apples in my bag, the second 2 is the number of apples I added, and -1 is the number of apples that fell out through the hole, and 3 is the total. Everything in this equation can be mapped back to reality by adding in details. But in 2 apples + 2 miles, the + cannot be mapped back to reality. And so it is not representative of apples.
I can think of an easy attack on this, by saying that 2 apples + 2 oranges does make sense in our world. But this is a disguised incorrectness. The oranges do not fall into the criterion of "apple", so it does not actually make sense to add them using the '+apples' operator. Of course, we could generalize the '+' operation to mean '+fruit', but then it becomes 2 fruit + 2 fruit, which does make sense, and is true whether you reject my current views or not.
I'm looking forward to your next argument :).
Replies from: Decius↑ comment by Decius · 2012-11-02T04:58:27.833Z · LW(p) · GW(p)
I put two apples in my one bag, then walk two miles and add two more apples to the bag. There is one hole in the bag, and one of the apples falls out.
2 (apples) + 2 (miles) + 1 (bag) - 1 (hole) - 1 (apple) + 2 (apples) = 5
I end up five miles from where I started, because I dropped units from my quantities and did operations on the numbers.
Or, I could apply a constant force of 5 newtons to an object massing 25 kg for a duration of 8 seconds. I change it's velocity by 5N8S(1MKG/(NSEC^2))/25KG 1.6 M/S. By conserving units on all quantities, I convert force-time against a mass into acceleration.
Those units can be preserved through all mathematical operations, including exponentiation and definite integration.
Replies from: mwengler, AnotherIdiot↑ comment by mwengler · 2012-11-04T13:57:20.130Z · LW(p) · GW(p)
I end up five miles from where I started, because I dropped units from my quantities and did operations on the numbers.
Don't we have the same problem with complex numbers?
2 + 3j = 5
I end up with 5 because I ignored imaginary numbers?
I'm not sure what my point is, this is after all a question. I am wondering, does the fact the same error occurs from dropping physical units as from dropping the very non-physical, seemingly quite mathematical concept of the sqrt(-1)==j?
Replies from: Decius↑ comment by Decius · 2012-11-04T18:54:08.936Z · LW(p) · GW(p)
It's a different error, because quantities aren't inherently algebraic, even though they very often behave as though they were.
For example, arranging apples in a grid three wide and two apples deep requires 6 apples, not 6 apples^2, even though the area of a grid two inches wide and two inches deep is 6 in^2.
↑ comment by AnotherIdiot · 2012-11-02T05:54:24.632Z · LW(p) · GW(p)
Or, I could apply a constant force of 5 newtons to an object massing 25 kg for a duration of 8 seconds. I change it's velocity by 5N8S(1MKG/(NSEC^2))/25KG 1.6 M/S. By conserving units on all quantities, I convert force-time against a mass into acceleration.
Those units can be preserved through all mathematical operations, including exponentiation and definite integration.
Hmm... Another good argument. This one is harder. But that's just making this more fun, and getting me closer to giving in.
If I abstract a whole bunch of details about apples away, except the number and the fact that it's an apple, I get math with units. So if I have a bag of 3 apples, I can abstract this to 3apples. I can add 1apple to this, and get 4apples. Why? Some ancient mathematicians have shown that when you retain this extra bit of information, the abstraction "math with units" is formed, which is pretty much exactly like normal math, except that numbers are multiplied by their units. "math with units" and plain old math are very similar because they are both numbers, but "math with units" happens to retain a few extra details about what the numbers are talking about.
1apple 1apple=1apple^2 is true, but it doesn't make sense. You can't add in details to turn this into apples because of physical limitations, so 1apple 1apple does not emerge from apples. But it is consistent with the rules of mathematics, which you obtain when you remove the complex details of what an apple is, so this is considered true.
1m * 1m=1m^2, however, does make sense, and we can add in details to transform this into an area. m/s can be transformed into the speed of an object by adding in details. m/s^2 can be transformed into an actual acceleration. kgm/s^2 can be transformed into an actual force by adding in details. I think this is true of all useful units, but if you can think of one for which this is not true, please share, because that will be a big blow to my theory. If this theory survives all the other arguments, I will still need to prove that this is true, and until I prove it, my theory is weak. Thank you for showing me this.
Anyway, a summary of this comment: math with units is an abstraction, one level below plain math, and it mostly follows the rules of math because it's just math+(some simple other thing). Units which don't make sense cannot be un-abstracted into things in the world, but because math is consistent, we can still manipulate them like anything else because we have abstracted the details which makes it not make sense away.
So I needed to extend my theory for this, but thankfully not by making it more complex at the base. This is all just emergent stuff in the universe. So if my theory holds up, I expect Occam's Razor will be in favor of it.
Replies from: Decius↑ comment by Decius · 2012-11-02T14:30:06.372Z · LW(p) · GW(p)
Quantities can be converted to and from numbers:
(32 ft lbf / (lbm sec^2))=1
(64 ft lbf / (lbm sec^2))=2
It is true, but not useful, to say that the area of a circle with radius R is equal to
or equivalently,
Taking the sec^2lbf root does not have an analogue in reality, but the units output from taking the time^2*Force root of a distance raised to the power of scalar*distance*mass is area in this specific case.
comment by Kawoomba · 2012-11-02T09:39:29.722Z · LW(p) · GW(p)
Abstract models are models in which some information is ignored. Take, for example, the abstract concept of the ball.
Taking your definition of an abstract model (so we don't squabble over mere definitions), I don't think that just by removing information you'll go from an actual baseball to the 'abstract concept' of a sphere. You'll also be adding information. For example, for your model you can provide the formula that will yield the exact volume of the sphere - you can't do that as precisely for your baseball. Will your abstract models typically be more compact / contain less information than your baseball, sure. However, the information may be partially different, not just a subset, which it would be if you were just ignoring information.
So to explain the properties of infinity, we simply defined it and some rules it follows, and from there proved other truths about infinity. Infinity may not exist in the real world, but it does exist as our definition, which is just physical stuff. If infinity exists in the real world and I don't know about it, we have probably observed it and created a model of it which follows the same rules as it.
I'm told that to the best of our knowledge the actual universe (as opposed to just our Hubble volume, or the observable universe) is infinitely large. Let's not get started with infinities of higher aleph cardinalities ...
Replies from: AnotherIdiot↑ comment by AnotherIdiot · 2012-11-02T10:27:15.181Z · LW(p) · GW(p)
Taking your definition of an abstract model (so we don't squabble over mere definitions), I don't think that just by removing information you'll go from an actual baseball to the 'abstract concept' of a sphere. You'll also be adding information. For example, for your model you can provide the formula that will yield the exact volume of the sphere - you can't do that as precisely for your baseball. Will your abstract models typically be more compact / contain less information than your baseball, sure. However, the information may be partially different, not just a subset, which it would be if you were just ignoring information.
That's true. Balls are very complex, so there isn't actually much you can ignore about them without invalidating your results. But you can ignore a lot of things and get approximately correct results, which is usually good enough when talking about balls.
Numbers, however, tend to be a little more convenient. If there's a hole in the bag of apples which you don't take into account, you'll get bad results, because that's a detail which impacts the numeric aspect of the apples. But we don't really care that it's a hole when talking about the number of apples. All we need to keep in mind is that the number decreased. If 1 apple fell through the hole, you can abstract that to a simple -1.
Anyway, this post has gotten out of hand, mostly because I was unclear, so I'll retract it and use these comments to write a hopefully clearer version. Thanks for the feedback.
comment by TimS · 2012-11-02T04:15:27.691Z · LW(p) · GW(p)
I might take this a bit more seriously if you could explain the relative sizes of infinite sets using apples. (And bananas, if you must).
Replies from: AnotherIdiot, mwengler↑ comment by AnotherIdiot · 2012-11-02T04:47:30.023Z · LW(p) · GW(p)
Bananas are constrained by the laws of physics, so when you reach the maximum number of bananas possible in our universe, the '+' operation becomes impossible to apply to it. So using physical bananas, it is impossible to talk about infinity.
But even if bananas aren't suited for talking about infinity, where does infinity come from?
Given that we reason about infinity, I infer that infinity can be represented using physical things (unless the mind is not physical). Also, given what I know about mathematics, I expect that infinity is thought about using rules/axioms/(your word of choice).
So to explain the properties of infinity, we simply defined it and some rules it follows, and from there proved other truths about infinity. Infinity may not exist in the real world, but it does exist as our definition, which is just physical stuff. If infinity exists in the real world and I don't know about it, we have probably observed it and created a model of it which follows the same rules as it. And then, by abstracting our model to ignore the fact that it's just a model, and abstracting the real infininty to ignore whatever it's composed of, we get the same thing, and that's what infinity is.
So while bananas are constrained by the amount of matter in our universe and can therefore not represent any number (some of the details about the bananas just can't be safely ignored), PA (or whichever set of axioms we use to think about infinity) can. And from this, we can find rules about infinity, using a purely physical process.
This is sort of guesswork on my part. I am not certain that this is how we have reasoned about infinity, though I would be surprised if it wasn't. So if it isn't, just say so, and I'll probably be convinced that I'm incorrect, and retract my post.
Replies from: TimS↑ comment by TimS · 2012-11-02T07:36:24.615Z · LW(p) · GW(p)
Ok, if you want a serious response instead of a snarky one, here goes:
You may have learned about Euclidean geometry in school. Two points define a line. A line and a point not along the line define a plane. As Euclid defined the geometry, parallel lines never intersect.
However, we don't live in a Euclidean geometry. To a first order of approximation, we live on a sphere. If Line A is perpendicular to Line X and Line B is also perpendicular to Line X, they are parallel in Euclidean geometry. Nonetheless, on a sphere, Line A and Line B will eventually intersect.
So we've got all this neat mathematics deriving interesting results from Euclidean axioms, but nothing in the real world is Euclidean. If we take your thesis (that math can be reduced to physics) seriously, that means that Euclidean geometry is not simply invalid - it is incoherent (i.e. wronger than wrong). You might be willing to bite the bullet and throw Euclidean geometry in the trash, but no one who takes math seriously is willing to do so.
For further reading, you might consider following the links from this post by Wei_Dai. In short, the issue here - how to talk about the "truth" of mathematics - is a basic problem with the correspondence theory of truth. Eliezer is making an attempt to bridge the gap in the post you highlighted, but he is deliberately avoiding the philosophical choice you made - I suspect because he is unwilling to throw out non-physical mathematics, which I've argued above is a requirement for your theory of mathematical truth.
Replies from: AnotherIdiot, mwengler, mwengler↑ comment by AnotherIdiot · 2012-11-02T08:10:36.527Z · LW(p) · GW(p)
Take a look at my response to tim. Replace god with Euclidean Geometry, and forget the fluff about god being inconsistent, and you can see that Euclidean Geometry is still coherent, because our minds can represent it with consistent rules, so these rules exist as an abstraction in the universe. So my view doesn't make Euclidean Geometry incoherent. I'm not sure what exactly you mean by validity, but the only thing that my view says is "invalid" about Euclidean Geometry is that it is not the same as the geometry of our universe.
Now it gets a bit difficult to write about clearly, I'm sorry if it's not clear enough to be understandable. Things we figure out about numbers using Euclidean Geometry can still be valid, simply because when we abstract the details about Euclidean Geometry to be left with only numbers, we get the same thing as when we abstract apples to numbers, and the same thing is true about our mental representation of PA. So proofs from one can be "transferred" over to another. But "transfer" doesn't really describe it well. What's really happening is that from the abstract numbers, you can un-abstract them by filling them in with some details. So you can remember that the apples were in a bag, and that gravity was acting on them. If, when you add in the details, the abstract number behavior still holds, then the object follows the rules of numbers. So if the added details about apples don't affect the conclusions you make using PA, by abstracting PA into numbers, and then filling in the details about apples, you have shown that things that are true about PA are true about apples too. And all this is done using physical processes.
So my view doesn't entail anything about accepting or rejecting mathematical statements. What it says is that mathematical concepts are abstract concepts, which we obtain by ignoring details in things in this world, and thanks to our awesome simple and universal laws of physics, the same abstract concepts emerge again and again.
Replies from: TimS↑ comment by TimS · 2012-11-02T08:33:06.741Z · LW(p) · GW(p)
Sorry, unintended inferential distance. In a previous post, Eliezer distinguishes between "true" and "valid" because only empirical things can be true, and he doesn't think mathematics is empirical. Thus, propositions that follow from proposed axioms are "valid" - what a mathematician would call true - to avoid confusing vocabulary.
You avoid the confusion by asserting that mathematical assertions really do correspond to some physical state (i.e. are empirical). Under the correspondence theory of truth, that allows some mathematical statements to be true, not simply valid. Nonetheless, I assume you don't think all mathematical statements are true (2 + 2 != 3, etc).
The problem with asserting that mathematical statements are empirical is that there are certain mathematical assertions that are valid but do not have any physical basis. Consider the proposition, "The Pythagorean theorem follows from Euclid's axioms." The statement is valid, but cannot meaningfully be called true because there is no physical fact that corresponds to the assertion by virtue of the fact that the physical universe is not a Euclidean space. But the statement is not false because there is no physical fact that corresponds to "The Pythagorean theorem is not deducible from Euclidean axioms."
In other words, your theory of mathematics has no room for "validity", only "truth." The Pythagorean theorem is interesting to mathematicians, but adopting your philosophy of mathematics would hold that generations of mathematicians have been interested in a theorem that we now know can never be true or false. That's just too weird for most people to accept.
Replies from: AnotherIdiot↑ comment by AnotherIdiot · 2012-11-02T10:16:22.557Z · LW(p) · GW(p)
You seem to misinterpret what I mean, but that's my fault for explaining poorly. This post has been getting out of hand with all the clarifications, so I will retract it and post a hopefully clearer version later on. Maybe as I write it, I'll notice a problem with my view which I hadn't seen before, and I'll never actually post it.
Replies from: mwengler, TimS↑ comment by mwengler · 2012-11-04T13:36:11.592Z · LW(p) · GW(p)
Or you can leave it, take your karma lumps which seem to be somewhat finite, and expect that to the extent there is something useful here, people stumbling across it will be influenced.
Missing the nuance is the right thing to do with most nuances, IF you are interested in making technical progress. And to the extent that there is a bias towards things that would help you build an AI, that is a valid purpose of this board.
↑ comment by TimS · 2012-11-02T13:45:00.352Z · LW(p) · GW(p)
Don't worry about it - I may be missing some nuance.
I would recommend reading some more advanced math before trying to make a philosophy of math. Integers make intuitive sense in the physical world in a way that more advanced math tends not to. Godel, Escher, Bach gets high marks around these parts, and rightfully so.
Replies from: DaFranker↑ comment by DaFranker · 2012-11-02T14:14:33.108Z · LW(p) · GW(p)
Don't even need to go that far. Just take e from logarithms and compound interests and you're already in NOT-INTUITIVE-TO-HUMANS-land.
I.e. How does a "perfect sphere" even remotely make sense in the real world? What the hell does e correspond to in the universe? A ton of trig, logarithm and limit stuff can prove problematic to simpler philosophical analyses of mathematics. And it's not like you can just throw out e and π either, since they yield accurate predictions so obviously there's something "true" or at least valid about them.
Replies from: mwengler, TimS↑ comment by mwengler · 2012-11-04T13:50:52.880Z · LW(p) · GW(p)
There is no PRACTICAL difference between a theory that treats e and pi as "rational but not yet perfectly determined" and "irrational." And by practical I mean as used by practitioners, people who build stuff, people who survey, even people who need to know the parallax to the most distant object in the universe between two relatively closely spaced telescopes on earth.
"infinite" broken down to its roots means "not finite." The practical value of "not finite" is probably "so large that we need to be sure we always get the same answer when we assume it is a million times larger," that is, we verify that we have a PRACTICAL solution that has converged as we make x larger and larger, and whether x is > 1 km (when designing a microcircuit) or x > 1 quadrillion light-years, we never need to know, PRACTICALLY, what happens when x finally reaches infinity.
Engineering principles which are WRONG when quantum and relativistic considerations are taken in to account stand firmly and valuably behind quadrillions of dollars worth of human infrastructure.
Philosophical theories which are limited in validity only to these principles are possibly not useless.
↑ comment by TimS · 2012-11-02T14:37:31.691Z · LW(p) · GW(p)
Sure, but I can't name an accessible but deep reference about e or pi of the top of my head.
What the hell does e correspond to in the universe?
It's a number than happens to have interesting mathematical properties - but it is no harder to explain physically than any other irrational number. Even if one thinks numbers are made of apples, one ought to be able to conceive of numbers of apples that aren't integers or rationals.
In short, I don't think the interesting constants are cleanest examples of the problems with mathematical pure physicalism.
Replies from: DaFranker↑ comment by DaFranker · 2012-11-02T14:57:02.834Z · LW(p) · GW(p)
Hah. The real hidden question was actually "How does one arrive at e specifically by looking at the universe, and why does it work like that?", I think.
I agree that they're not the most clear stuff, but I've listed them as the most accessible wonder-inducing mathematics-related points of interest.
Replies from: TimS↑ comment by TimS · 2012-11-02T15:12:13.005Z · LW(p) · GW(p)
That's an interesting question, and I have no idea about the answer.. If aliens asked me to define e, I'd start talking about exponential functions that were their own derivative. But I have no idea if that's the historical motivation for noticing e.
Pi is obviously much easier, since it is part of the ratios linking circle diameter to circle perimeter and circle area.
Replies from: DaFranker↑ comment by DaFranker · 2012-11-02T15:21:50.202Z · LW(p) · GW(p)
Pi is obviously much easier, since it is part of the ratios linking circle diameter to circle perimeter and circle area.
If I had to explain Pi to real aliens that somehow understood English but not our mathematics, I would start with straight lines of a fixed length (radius) that share one (fixed) endpoint and where the other (movable) endpoints get gradually closer and closer.
Some multiple of pi is the ratio you apparently get as you compare those lengths and extrapolate for infinitely-closer-and-closer lines.
Sounds simple enough, as far as explaining abstract concepts to real aliens goes.
Replies from: TimS↑ comment by mwengler · 2012-11-04T13:32:55.528Z · LW(p) · GW(p)
However, we don't live in a Euclidean geometry.
Millions, perhaps billions of humans have put food on the table and built machines with 100s of times the power or calculational ability of human individuals without ever needing to concern themselves with the breakdown of euclidean geometry. Perhaps there are machine designs that have failed due to this breakdown, but failure has so much entropy that it teaches us infitely (and I do not mean that literally) less than our successes do.
One idea I love in lesswrong is the "how do I code that in to an AI" bias in evaluating efforts. Even if there is some frontier where a deviation from euclidean geometry is necessary to understand in our design of the ultimate AI (or at least the last one built by humans)? Why would anyone be uninterested in the theory behind 99.99...% of the progress we are likely to make?
Replies from: gwern, TimS↑ comment by gwern · 2012-11-04T16:35:40.496Z · LW(p) · GW(p)
Millions, perhaps billions of humans have put food on the table and built machines with 100s of times the power or calculational ability of human individuals without ever needing to concern themselves with the breakdown of euclidean geometry.
Try sailing an ocean, as millions of humans have had to do (even just the ones doing so involuntarily like the Africa->America slave trade) with plain Euclidean geometry and then tell me how practical alternative forms of mapmaking, direction-setting, and locations are.
As it happens, Nick Szabo is slowly blogging how exactly one does that: http://unenumerated.blogspot.com/2012/10/dead-reckoning-and-exploration-explosion.html and http://unenumerated.blogspot.com/2012/10/dead-reckoning-maps-and-errors.html
Just because many millions of people don't need to concern themselves with that doesn't mean there aren't many other millions who don't.
↑ comment by TimS · 2012-11-04T18:42:22.482Z · LW(p) · GW(p)
In addition to the object level mistake that gwern has pointed out, you've made a meta-level mistake.
I wasn't arguing for the usefulness of Euclidean or non-Euclidean geometry. I was trying to shorten the inferential distance. Euclidean axiomatic mathematics is some of the first axiomatic mathematics anyone is taught in school in the West. The OP might not have understand what it was that his theory was missing in reference to infinite sets, so I used an example I expected him to be more familiar with, in an effort to make my point clearer to him.
You may not think my particular point is interesting in a practical sense, but pointing that out is quite rude unless you really think that I'm unaware that the difference between Euclidean geometry and real world geometry does not always make a practical difference.
It's like asserting the difference between Newtonian and relativistic physics doesn't make a practical difference. I don't know how true that is, but saying something like that to Einstein or Hawking is just rude.
↑ comment by mwengler · 2012-11-04T13:26:41.605Z · LW(p) · GW(p)
I suspect because he is unwilling to throw out non-physical mathematics, which I've argued above is a requirement for your theory of mathematical truth.
Do you then like Anotheridiot's theory as a theory of physical mathematics? As an engineer, it seems to me that if you restrict yourself to stuff that is actually useful in creating machines (in a very general sense), you find, perhaps, only 100/infinity % of those creations require non-physical math, and for the sake of loosening the bound, lets take the "littlest" infinity of all the choices, whatever that means.
Even machines that think about infinity are finite, witness the astonishing finiteness of the human mind, even the really good ones.
Replies from: TimS↑ comment by TimS · 2012-11-04T18:33:08.661Z · LW(p) · GW(p)
No, you've misunderstood me. The OP's theory of mathematical-truth-as-physical-object is hopelessly flawed.
But you are wrong about infinity. It is hard to built modern technology without calculus, and impossible to have calculus without infinite sums (integrals) or infinite limits (derivatives). If you are trying to make a point about academic / non-physical mathematics, you might have a point (depending on how cutting edge physics turns out to work) - but infinity does not advance the ball.
↑ comment by mwengler · 2012-11-04T13:20:27.412Z · LW(p) · GW(p)
Presumably you are aware of the host of problems introduced to mapping reality when concepts of infinity are included? There is literally no problem on earth in which the concept of infinity is needed for a solution. In the overwhelming majority of problems, a concept of infinity is either not needed or all, or leads to wrong answers when employed. The planet is finite, the mind is finite, the observed universe is finite. Even infinitesimals are make believe, I don't think we have ever observed anything less than 10^-20 m, and if I am wrong make it 10^-50 m to put a wall around it.
Whether you think there are some places where infinity helps or not, would you disagree that there are a vast ream of useful things to be done with numbers where infinity would never be missed?
So why toss out the beginnings of a system that doesn't include something of such limited utility as a detailed property of a set of concepts that aren't even needed to accomplish everything technical in the world that has to date been accomplished?
Replies from: Kindly↑ comment by Kindly · 2012-11-04T15:38:41.736Z · LW(p) · GW(p)
I think the most important reason for keeping infinity around is that the simplest system which describes how numbers work implies the existence of infinitely many numbers. It is much easier to work in this system and then ask "is my answer smaller than 10^50?" than to work in a system using only numbers less than 10^50 and ask "does the result I got actually exist?" at every step.
Similarly, the simplest system that can describe lengths of things is the real numbers. In a way these are make-believe, but it would be ridiculous to try to describe the length of a diagonal of a square without having the concept of the square root of 2.
Comparing the different sizes of infinity is, in its simplest case, a description of how these two systems interact. Don't tell me that's not useful.
comment by Anatoly_Vorobey · 2012-11-02T09:28:36.458Z · LW(p) · GW(p)
There aren't any apples in reality. There are, arguably, atoms, and let's stop there and not go further to quantum states, elementary particles, virtual particles, etc. We can allow for the sake of the argument that there are atoms. But there certainly aren't any apples in reality. There're lots of atoms everywhere, and some of them are bunched together in a somewhat different way than some other ones, and the boundary between them is inherently fuzzy. Apples exist in your mind.
Discreteness on a micro level is arguable, but irrelevant to your everyday perception; discreteness on a macro level only exists in your mind. Out there in reality, there are just atoms.
Replies from: mwengler, AnotherIdiot↑ comment by mwengler · 2012-11-04T13:12:03.739Z · LW(p) · GW(p)
For the sake of argument you are willing to keep atoms irreducibly, even though in the same reality in which there are no apples there are also no atoms, just quarks and neutrinos and a few 100 other "fundamental" particles.
What do we gain by deciding "for the sake of argument" to deny apples but accept atoms?
Replies from: Anatoly_Vorobey↑ comment by Anatoly_Vorobey · 2012-11-04T19:31:03.694Z · LW(p) · GW(p)
We gain simplicity of argument. The "messiness" of atoms is already enough for my argument that apples as discrete entities exist in the mind but not in reality; the reality of quarks, leptons and bosons (perhaps 16-18, rather than a few hundred particles) is even more "messy", which makes my argument stronger, not weaker, but also more complicated to explain.
↑ comment by AnotherIdiot · 2012-11-02T10:02:45.856Z · LW(p) · GW(p)
Out there in reality, there are just atoms.
I know. But it's easier to talk about apples than atoms. And the apples are just another level of abstractions. From atoms emerge apples, and from apples emerge [natural] numbers.