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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
In my imagination, I have a chalkboard, but no other ability to communicate. So, lots of drawing circles (with emphasis on diameters and circumferences).
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?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Sure, but I can’t name an accessible but deep reference about e or pi of the top of my head.
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.
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.
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.
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.
In my imagination, I have a chalkboard, but no other ability to communicate. So, lots of drawing circles (with emphasis on diameters and circumferences).
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?
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.
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.
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.
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.