So you would prefer that, instead of having one all-purpose number system that we can embed just about any kind of number we like into (not to mention do all kinds of other things with), we had a collection of distinct number systems, each used for some different ad-hoc purpose? How would this be an improvement?
You might consider the fact that, once upon a time, people actually started with the natural numbers—and then, over the ages, gradually felt the need to expand the system of numbers further and further until they ended up with the standard objects of modern mathematics, such as the real number system.
This was not a historical accident. Each new kind of number corresponds to a new kind of operation people wanted to do, that they couldn’t do with existing numbers. If you want to do subtraction, you need negative numbers; if you want to do division, you need rationals; and if you want to take limits of Cauchy sequences, then you need real numbers.
I don’t understand why this should cause computer-programming types any anxiety. A real number is not some kind of mysterious magical entity; it is just the result of applying an operation, call it “lim”, to an object called a “sequence” (a_n).
Real numbers are used because people want to be able to take limits (the usefulness of doing which was established decisively starting in the 17th century). So long as you allow the taking of limits, you are going to be working with the real numbers, or something equivalent. Yes, you could try to examine every particular limit that anyone has ever taken, and put them all into a special class (or several special classes), but that would be ad-hoc, ugly, and completely unnecessary.
I think you’re being a bit uncharitable here. You’ve just moved the infinitude/”mysterious magicalness” from talking about real numbers to talking about sequences of rational numbers, and it is in fact possible to classify sequences as definable vs. undefinable, as well as computable vs. uncomputable. (Though definability personally seems a bit ad-hoc to me, seeing as it depends on the ambient theory.) I don’t think it’s really extraordinary to claim that an undefinable or uncomputable sequence is a bit mysterious and possibly somehow unreal.
EDIT Apr 30: Oops! Obviously definability depends only on the ambient language, not the actual ambient theory… that makes it rather less ad-hoc than I suggested.
EDIT: I should probably add, though, that this whole argument seems mostly pointless. Aside from where cousin_it and Jonicholas got to talking about how to represent numbers in computers—and that seems to have hardly anything to do with actual real numbers seeing as those can’t be represented in computers—it seems to be basically just Jonicholas saying “I don’t like the reals for common constructivist reasons” and other people saying “Regardless, they’re valid objects of study”. Is there more to it than that?
I think you’re being a bit uncharitable here. You’ve just moved the infinitude/”mysterious magicalness” from talking about real numbers to talking about sequences of rational numbers
That was deliberate. (How was it uncharitable?)
I don’t think it’s really extraordinary to claim that an undefinable or uncomputable sequence is a bit mysterious and possibly somehow unreal.
It may not be extraordinary, but it’s still a confusion. A confusion that was resolved a century ago, when set theory was axiomatized, and the formalist view emerged. The Cantor/Kronecker debate is over: Cantor was right, Kronecker was wrong.
The source of this confusion seems to be a belief that correspondences between mathematical structures and the physical world are properties of the mathematical structures in question, rather than properties of the physical world. This is a kind of map/territory confusion.
My understanding is this thread was started, and to some extent kept rolling, by an unrelated thread, where I was behaving extremely hostile to EY, and several people went through all my back posts, looking for things to downvote. Patrick found something.
To put myself in the clear, I came across this old comment because I was looking through Doug S’s old posts (because I was idly curious). I replied to your comment because I’m ridiculously pedantic, a virtue in mathematics. I haven’t downvoted any of your comments and I harbor no feelings of antipathy towards you. Eliezer’s a big boy and he can take care of himself.
Now, back to the math debate! I don’t think it’s legitimate to conflate countable sets with sets with finite information content. Here are two counter examples. 1. The set of busy beaver numbers (a subset of the naturals). 2. The digits of Chaitin’s Omega [make an ordered pair of (digit, position) to represent these] (see http://en.wikipedia.org/wiki/Chaitin%27s_constant). It’s been proved that these sets can’t be constructed with any algorithm.
“The set of busy beaver numbers” communicates, points to, refers to, picks out, a concept in natural language that we can both talk about. It has finite information content (a finite sequence of ascii characters suffices to communicate it). An analogous sentence in a sufficiently formal language would still have finite information content.
Note that a description, even if it is finite, is not necessarily in the form that you might desire. Transforming the description “the sequence of the first ten natural numbers” into the format “{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}” is easy, but the analogous transformation of “the first 10 busy beaver numbers” is very difficult if not impossible.
As nshepperd points out, an element of a countable set necessarily has finite information content (you can pick it out by providing an integer—“the fifth one” or whatever), while generic elements of uncountable sets cannot be picked out with finite information.
I think the finite information content comes from being an element of a countable set. Like every other real number, the digits of Chaitin’s constant themselves form a countable set (a sequence), while that set is a member of the uncountable R. Similarly, the busy beaver set is a subset of N, and drawn from the uncountable set 2^N.
Countable sets are useful (or rather, uncountable ones are inconvenient) because you can set up a normalized probability distribution over their contents. But… the set {Chaitin’s Constant} is countable (it has one element) but I still can’t get Omega’s digits. So there still seems to be a bit of mystery here.
So you would prefer that, instead of having one all-purpose number system that we can embed just about any kind of number we like into (not to mention do all kinds of other things with), we had a collection of distinct number systems, each used for some different ad-hoc purpose? How would this be an improvement?
You might consider the fact that, once upon a time, people actually started with the natural numbers—and then, over the ages, gradually felt the need to expand the system of numbers further and further until they ended up with the standard objects of modern mathematics, such as the real number system.
This was not a historical accident. Each new kind of number corresponds to a new kind of operation people wanted to do, that they couldn’t do with existing numbers. If you want to do subtraction, you need negative numbers; if you want to do division, you need rationals; and if you want to take limits of Cauchy sequences, then you need real numbers.
I don’t understand why this should cause computer-programming types any anxiety. A real number is not some kind of mysterious magical entity; it is just the result of applying an operation, call it “lim”, to an object called a “sequence” (a_n).
Real numbers are used because people want to be able to take limits (the usefulness of doing which was established decisively starting in the 17th century). So long as you allow the taking of limits, you are going to be working with the real numbers, or something equivalent. Yes, you could try to examine every particular limit that anyone has ever taken, and put them all into a special class (or several special classes), but that would be ad-hoc, ugly, and completely unnecessary.
I think you’re being a bit uncharitable here. You’ve just moved the infinitude/”mysterious magicalness” from talking about real numbers to talking about sequences of rational numbers, and it is in fact possible to classify sequences as definable vs. undefinable, as well as computable vs. uncomputable. (Though definability personally seems a bit ad-hoc to me, seeing as it depends on the ambient theory.) I don’t think it’s really extraordinary to claim that an undefinable or uncomputable sequence is a bit mysterious and possibly somehow unreal.
EDIT Apr 30: Oops! Obviously definability depends only on the ambient language, not the actual ambient theory… that makes it rather less ad-hoc than I suggested.
EDIT: I should probably add, though, that this whole argument seems mostly pointless. Aside from where cousin_it and Jonicholas got to talking about how to represent numbers in computers—and that seems to have hardly anything to do with actual real numbers seeing as those can’t be represented in computers—it seems to be basically just Jonicholas saying “I don’t like the reals for common constructivist reasons” and other people saying “Regardless, they’re valid objects of study”. Is there more to it than that?
That was deliberate. (How was it uncharitable?)
It may not be extraordinary, but it’s still a confusion. A confusion that was resolved a century ago, when set theory was axiomatized, and the formalist view emerged. The Cantor/Kronecker debate is over: Cantor was right, Kronecker was wrong.
The source of this confusion seems to be a belief that correspondences between mathematical structures and the physical world are properties of the mathematical structures in question, rather than properties of the physical world. This is a kind of map/territory confusion.
A good point.
Sorry, uncharitable was the wrong word there. I meant you didn’t address the actual apparent problem. Your new comment does.
No, there’s nothing substantive beyond that.
My understanding is this thread was started, and to some extent kept rolling, by an unrelated thread, where I was behaving extremely hostile to EY, and several people went through all my back posts, looking for things to downvote. Patrick found something.
To put myself in the clear, I came across this old comment because I was looking through Doug S’s old posts (because I was idly curious). I replied to your comment because I’m ridiculously pedantic, a virtue in mathematics. I haven’t downvoted any of your comments and I harbor no feelings of antipathy towards you. Eliezer’s a big boy and he can take care of himself.
Now, back to the math debate! I don’t think it’s legitimate to conflate countable sets with sets with finite information content. Here are two counter examples. 1. The set of busy beaver numbers (a subset of the naturals). 2. The digits of Chaitin’s Omega [make an ordered pair of (digit, position) to represent these] (see http://en.wikipedia.org/wiki/Chaitin%27s_constant). It’s been proved that these sets can’t be constructed with any algorithm.
“The set of busy beaver numbers” communicates, points to, refers to, picks out, a concept in natural language that we can both talk about. It has finite information content (a finite sequence of ascii characters suffices to communicate it). An analogous sentence in a sufficiently formal language would still have finite information content.
Note that a description, even if it is finite, is not necessarily in the form that you might desire. Transforming the description “the sequence of the first ten natural numbers” into the format “{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}” is easy, but the analogous transformation of “the first 10 busy beaver numbers” is very difficult if not impossible.
As nshepperd points out, an element of a countable set necessarily has finite information content (you can pick it out by providing an integer—“the fifth one” or whatever), while generic elements of uncountable sets cannot be picked out with finite information.
I think the finite information content comes from being an element of a countable set. Like every other real number, the digits of Chaitin’s constant themselves form a countable set (a sequence), while that set is a member of the uncountable R. Similarly, the busy beaver set is a subset of N, and drawn from the uncountable set 2^N.
Countable sets are useful (or rather, uncountable ones are inconvenient) because you can set up a normalized probability distribution over their contents. But… the set {Chaitin’s Constant} is countable (it has one element) but I still can’t get Omega’s digits. So there still seems to be a bit of mystery here.