I've been working for several years on a project, on a course I call ``The Limits of Mathematics.'' And this course in the past three years has gone through a number of different versions. It has a lot of software attached to it. And I've been changing this software. I've been trying to explain this software to people as the course. In fact a few weeks from now I'll be giving this course in Rovaniemi, Finland, on the arctic circle, for two weeks.
The way this course is now is that it's on this Web site that I gave you guys the URL for in the abstract. It's an HTML document and it has a lot of LISP in it, and just to make life worse this is not normal LISP. It's pretty close to pure LISP, you know, the heart of LISP that everybody learns. But I've had to add a few features to LISP. So this stuff is all available and you're welcome to play with it. The course is there.
And as general background I suggest my lecture on ``Randomness in Arithmetic and the Decline and Fall of Reductionism in Pure Mathematics'' which was published, among other places, by Oxford University Press in Nature's Imagination, that's the name of the volume. That's a talk that I gave here.
One new thing I add to the stew is that an idea that was missing which is significant is the idea of program-size complexity, or how big a program is, how many bits of information there are. Not the time! In that sense it's just like the old recursive function theory and Turing machines. I don't care about time. But I do care about the size of a program in bits. Okay? So that's one new idea.
The other new idea I've added just in the past three years is that I want to really be able to run the programs on interesting examples on a computer. In other words, it's not enough to wave your hands and talk about an algorithm. I not only want a universal Turing machine that I can prove theorems with, but one that is fun to actually program and that you can run interesting examples on with a computer.
John McCarthy invented LISP not just for use in AI [artificial intelligence], not just for use as a practical language. If you look at his 1960 paper on LISP in the Communications of the ACM, he said, ``This is a better universal Turing machine. Let's do recursive function theory that way!'' [Of course these aren't McCarthy's words, it's my paraphrase.] And the funny thing is that nobody except me has really I think taken that seriously. And the reason of course is that theoreticians didn't really much care about programming, about really playing with the computer. So the thing I've added is I think nowadays you have no excuse! If a theory has to do with the size of computer programs, you want to damn well see the programs, you want to be able to run them, it ought to be a programming language that is easy to use.
So I've done that using LISP because LISP is simple enough, LISP is in the intersection between theoretical and practical programming. Lambda calculus is even simpler and more elegant than LISP, but it's unusable. Pure lambda calculus with combinators S and K, it's beautifully elegant, but you can't really run programs that way, they're too slow. So I would say that I believe that LISP is a powerful language that can actually be used on a computer to run stuff in a finite amount of time, but on the other hand it's simple enough and elegant enough that you can prove theorems about it. So that's the kind of game that I've been playing. Okay, so you'll find this on my Web site.
The best complexity measure says, you have a LISP S-expression, which is a computer program, and then you have binary data on the side:
lisp binary S-exp dataThe problem with measuring the size of a LISP program is that due to LISP syntax the LISP S-expression is redundant. So you really want to add to the LISP S-expression, which is a powerful way to express an algorithm, some raw binary data. That's like having data on a tape in addition to your program. So in the data the bits can be 0 and 1 independently, and there's some way that the program can get access to the data, and then you look at the size of the whole thing.
So basically the universal Turing machine I'm using as my yardstick to talk about program size, to use to measure program size, has programs which are big binary programs:
BINARY PROGRAM lisp binary S-exp dataThat sounds bad, right? We have this awful machine language in binary. You're not going to want to write in binary. Well, it's not so bad. The beginning of a program is a LISP S-expression converted to binary, 8 bits per character. And you know where the LISP S-expression ends, because by convention there is always a special ASCII character at the end.
BINARY PROGRAM lisp binary S-exp data 8 bits/char
So the universal Turing machine starts reading the binary tape bit by bit. It doesn't sort of gobble it all in and start running. It reads each bit of the program as it needs to—that's very important. And it starts off by reading a LISP S-expression, and then every time the LISP S-expression says ``Read another bit,'' it gets it from what's left on the tape, that is, in the binary program. And if the S-expression attempts to read beyond the end of the binary data, the program is a failure and is aborted. So that's how it goes. And then you just add the total number of bits, which is 8 times the number of characters in the S-expression plus the number of bits in the data. That gives you the total number of bits in the program, okay?
So that's done because just using the size in characters of a LISP S-expression (or multiplying it by some factor to have it in bits) is no good. LISP syntax is pretty simple, but even so there's redundancy there. So you've got to add this extra binary data on the side.
Define U of p, U is the function, p is the program, as follows:
define (U p) cadr try no-time-limit 'eval read-exp pNow this is it, this is my universal Turing machine that I described in words. Try is a primitive function in this LISP sort of like eval in normal LISP. Try has three arguments, here they are:
no-time-limit 'eval read-exp pSo this, argument 2,
'eval read-expis a LISP expression that we're going to evaluate with a time limit. In fact there's no time limit here:
no-time-limitYou also have associated binary data. Try is the way that you give a LISP expression binary data. So this
pis the binary data, it's the program for the universal Turing machine. It's a list of 0's and 1's. This is try's third argument.
Oh by the way, in normal LISP you have to put lots of parentheses. In my LISP I leave out a lot of them because in my LISP every built-in function has a fixed number of arguments, so I don't bother to write all the parentheses.
Back to the LISP code for my UTM.
(read-exp)is a primitive function with no arguments. So we're trying to evaluate this
('(eval(read-exp)))normally with a time limit, but in this case with no time limit. And read-exp reads a LISP expression from the beginning of the binary data p. And then we evaluate it, in other words, we run it. And while you're running it, if within the LISP expression that you read you ask for more binary data, you get it from what's left of the binary data.
It works! Try does exactly what it should, I added try to normal LISP just for this. And try is the real difference between normal LISP and my LISP. Okay, so this
(define (U p) (cadr (try no-time-limit ('(eval(read-exp))) p)))is my universal Turing machine, which I think is not a very big, complicated program.
And the nice thing about this is that you see lots of examples in my course. What I do is I define LISP functions then I run them on examples. And there are lots of comments saying why I'm doing it and what it proves.
Actually there are two versions of each of these LISP runs, one of them with no comments and running it just on the case that counts, and another one with all the comments in the world and with lots of test cases for all the auxiliary functions. I do this because without comments a program is incomprehensible, but when you put all the comments in it becomes incomprehensible for another reason—it's so big that you can't find anything. So I present it both ways.
Okay, so one of the things you can do is you can actually get a program and run it on this machine. So how do you do it? Well, you take a LISP expression, and then you want to convert it to binary, so I have a primitive function for doing that, and then you concatenate it to the binary data that it needs, and then you feed this into the function U of p that we defined. And in my Web site I have runs of the universal Turing machine that show how it works.
H(x, y) ≤ H(x) + H(y) + c.
What this means is this. H(x) is the size in bits of the smallest program for the universal machine U of p that calculates x; x is an S-expression. And H(y) is the size of the one that calculates y. And putting them together and adding a fixed number of bits you get a program that calculates the pair (x, y), which won't have a comma by the way, since we're doing LISP, which is without commas.
So how does this work? Well, let me show you what the program looks like. What you do is you take the minimum size program for x, the minimum size program for y, and I'll call that x* and y*, and you concatenate in front a prefix φc, a magic prefix that's going to be c bits long, and I'll show you this prefix. This gives you
φc x* y*.Then when you give it to the universal machine U of p it's going to produce the pair (x, y).
Now what is this prefix φc that you're concatenating in front of these two bit strings x* and y*? Actually you can put any program for x and any program for y here in φc x* y*. I don't know the minimum ones. You'll see that this prefix works because if you give it any program x* to calculate x and any program y* to calculate y, this prefix φc when you feed it to my universal machine U of p is going to end up being a program that calculates the pair (x, y).
So what is this prefix φc? Well, the prefix is this. Start with
(cons (eval (read-exp)) (cons (eval (read-exp)) nil))Here I'm putting in all the parentheses. Then convert this to a bit string φc. So you write this expression and use a primitive function to convert it to a bit string and then you use append to concatenate these lists of bits and get this long bit string:
φc x* y*.And then you feed it into U and it produces (x, y).
How does this
(cons (eval (read-exp)) (cons (eval (read-exp)) nil))work? It says, ``Read from the binary data.'' The binary data when φc is running is going to be x* y*, the rest of the program. So it's going to read and run the program x* to calculate x, and that's going to give it x. And then it's going to read off of the rest of the binary data and run the program y* to calculate y, and that's going to give it y. And then it's going to cons these two things x and y up into a pair. Okay?
So it's the number of bits in this S-expression φc
(cons (eval (read-exp)) (cons (eval (read-exp)) nil))which is the constant c in
H(x, y) ≤ H(x) + H(y) + c.So count the number of characters here in φc, multiply by 8, and that's the constant c in this theorem. Okay? So this is pretty straight-forward.
Okay, so we're going to define a function that counts how many programs halt, let's call it count-halt, which has two arguments, one is a prefix, and the other is bits-left.
define (count-halt prefix bits-left)What's the idea? I'm going to count how many programs for my universal Turing machine halt... Oh, I should put in time... Let me put in time, there's another argument here:
define (count-halt time prefix bits-left)So I'm going to count how many programs halt on my universal Turing machine that have this prefix with bits-left bits added on afterwards, within this amount of time. So this count-halt is the auxiliary function that I use. Okay?
So the first thing to show you is how you get from this the halting probability. So before I define the auxiliary function, let me show you how to define a function omega, the main function, that uses this auxiliary function. It's only one main function with one auxiliary function. Let's call it omega of n:
define (omega n)This is the nth approximation, the nth lower bound on the halting probability. As n goes to infinity, this will give you the halting probability in the limit from below. The problem is it converges very, very, very, very slowly! [Laughter] Noncomputably slowly! Thanks for the laughs.
Let's define this function. Well, what I want to do is I want to cons up a rational number. And the way I do it is, I count how many programs halt within time n that have as prefix nil, the empty bit string, and that add n bits to it.
define (omega n) cons (count-halt n nil n)And I cons that with division.
define (omega n) cons (count-halt n nil n) cons /And cons that with, I use ^ for power, 2 raised to the power n, and nil.
(define (omega n) (cons (count-halt n nil n) (cons / (cons (^ 2 n) nil))))
This depends on the representation you pick for rational numbers; I just write out a fraction. And so (omega n) is a triple, I'm creating a triple. So it's (the number of programs n bits in size that halt within time n) divided by 2n. Okay?
So that's the main function. Now let me define the auxiliary function. The auxiliary function goes like this. You start off with the empty prefix nil and recursively you start adding bits until you get a full n-bit program and then you run it for time n and you see whether it halts. Now how do you program that out in LISP? Well, recursively it's very easy, you go like this.
Let's first see if bits-left is not equal to 0. If bits-left is greater than 0, then add a 0 bit to the prefix and subtract one from the number of bits left, see how many of these programs halt within time n, and add that to the same thing when you append a 1 to the prefix instead of a 0.
So we've got this:
define (count-halt time prefix bits-left) if > bits-left 0 + (count-halt time (append prefix '(0)) (- bits-left 1)) (count-halt time (append prefix '(1)) (- bits-left 1))
I'll put parentheses in occasionally and other times not. [Laughter] Well, actually in my LISP you don't put them in for primitive functions and the interpreter adds them, and writes it out for you just so that you can check that it's what you meant.
So this is the recursion. This is going to make count-halt look at all n-bit programs. And what happens when it finally has no bits left to add? It started off with n bits to append to the prefix, and with the empty prefix. So now you've finally got an n-bit string, you get all possible n-bit strings. So then you take the n-bit string, and you try for time n eval read-exp applied to the n-bit data string which is the prefix. If this try is a success then count-halt is a 1 otherwise it's a 0.
So now we've got this:
define (count-halt time prefix bits-left) if > bits-left 0 + (count-halt time (append prefix '(0)) (- bits-left 1)) (count-halt time (append prefix '(1)) (- bits-left 1)) if = success car (try time 'eval read-exp prefix) 1 0
This recursion is going out and seeing whether all possible n-bit programs halt or not. When you've got the whole n-bit program, how do you see whether it halts or not? You try running it for the time that you were given, which is going to be n for omega of n. And this part
(try time 'eval read-exp prefix)does that. It's taking the prefix as the S-expression and the binary data of a program for my universal Turing machine. It's reading an S-expression off of the beginning of the prefix and it's running the S-expression with the rest of the prefix as binary data. Okay?
I don't really have the time to explain this properly. If you were giving this to students in a class you'd probably take a whole class to explain this. And then the students would go and they would play with it and they would use it. And if you look at my Web site you'll find that I'm running count-halt on examples to show that it works. I also have a different definition of omega of n which is more traditional, which builds up the list of all n-bit strings, and then sees which halt, and then counts how many halt. But I think that what I have here is more elegant, it's really all you need.
Let me say it in words again.
What you see here in this version of LISP is the nth lower bound on Ω. With increasing positive integers n, omega of n gives you better and better lower bounds on the halting probability. The problem is that you never know how far out to go to get the halting probability with a given degree of accuracy. To get n bits of the halting probability right, you have to go out Busy Beaver function of n, that is basically the idea.
You look at all n-bit programs, as prefix you start off with the empty list of bits, and the bits left to add is n initially, and you're going to count how many of them halt within time n. That's what
(count-halt n nil n)does.
You really understand an algorithm when you've programmed it and tried it on examples and debugged it yourself. So I think that to use this in a class effectively you need exercises. You need to ask the students to do variations on what you've presented, so they make it their own. I don't do this here. This is a very concise version of the course.
If someone really wanted to use this as a course, they could start with this, which it is at least feasible that someone could understand. Before it wasn't. I could understand it. A few very committed people could sort of try to think they understood it, but I think there's more of a chance with this, it looks more like normal LISP. So that's from the pedagogic point of view.
Now let's look at this philosophically. The point is this. This program is Ω's definition. This is really it. So if this number is going to be very mind-boggling, you'd like to pin down very concretely how it's defined. And we have.
You'd like to pin down very concretely how Ω's defined before you start being astonished by it. Here Ω is really very concrete and you can actually run (omega 0), (omega 1), (omega 2), ... Of course the time grows exponentially as n goes up, but maybe you can get to one interesting example. You can try variations of this program to convince yourself that you understand it. And you can try debugging count-halt separately by running examples to show that it works. That's how to convince yourself that this works: run lots of examples. I haven't tried giving a formal proof that this program is correct. I wonder if that would be an interesting project?
Okay, now I once tried explaining this to an astrophysicist who amazingly enough was briefly interested [laughter], a very bright guy. He was so bright that he could understand the previous version of the program for Ω, the one in which the names of built-in functions are a single character, and where you had to program out arithmetic, you didn't have positive integers like we do here. He took the definition of (omega n) and count-halt, printed them on a small sheet of paper, folded it up and put it in his wallet. He said that it was like a mantra, ``I have Ω in my pocket!'' But it was Ω for him only because he was very bright and he took the trouble to go through this course with me. I think that this new version is much easier to understand than the one that he put in his pocket.
Well, the next thing I do in the course is I point out that (omega 0), (omega 1), (omega 2), ... is a monotone increasing sequence of rational numbers. So, this isn't constructive, but we all know that there is a real number which is the least upper bound of this sequence. So (omega n) defines in a nonconstructive way a real number Ω. It's almost constructive. It's pretty close to being constructive. Really constructive would be if you could calculate each digit of Ω, right? I'm getting better and better lower bounds, the only thing I don't have is I don't have a way to calculate how far out to go to get a given degree of accuracy.
Okay, so this defines a real number that I call Ω. Now let's imagine this number being written in base-two binary, so there's going to be a 0, a binary point, and then there're going to be some bits, right?
Ω = 0.011101...And let's think of this bit string as a LISP S-expression, it's going to be a list of 0's and 1's separated by blanks. And let's define ΩN to be the first N bits of this, the LISP S-expression for the first N bits of Ω. It's a list with N elements.
The theorem you want to show is that H(ΩN) is pretty complicated, that there's a lot of information in those first N bits of Ω. In fact it's algorithmically irreducible. You cannot compress the first N bits of Ω into a program substantially less than N bits in size. The exact result is that
H(ΩN) > N − 8000.This turns out to be the case.
Why is there a lot of information in the first N bits of Ω? It's not difficult to see why. If you knew the first N bits of Ω, it solves the halting problem. It would enable you, slowly, but it would enable you to solve the halting problem for all programs up to N bits in size. So once you do that, you run all the programs up to N bits in size that halt, you see what they calculate, and then you just put what they calculate together into a list. That gives you something which cannot be done by a program less than or equal to N bits in size.
And it takes 8000 bits to explain how to do this. There's a prefix φ8000 which is 8000 bits long that if you put it in front of a program that calculates the first N bits of Ω, this gives you a program to calculate something whose complexity is greater than N. This shows rather concretely that this inequality
H(ΩN) > N − 8000has to follow. Okay?
So I'm giving my universal Turing machine U a program which starts with an 8000-bit prefix φ8000. That sounds like a lot, but in fact it's only a thousand characters of LISP. And then next to it I concatenate any program ΩN* for calculating the first N bits of Ω, if somehow I had one. I don't know where to get one! So this is what we've got:
U( φ8000 ΩN* ).And then you run this and what the machine U does because of φ8000 is this. First it reads in and runs ΩN* to calculate the first N bits of Ω. Then it uses (omega n) to get better and better lower bounds on Ω until ΩN is correct, until it gets the first N bits of Ω right. God, or an oracle, is giving us the program for the first N bits of Ω!
Once U has calculated (omega n) for n high enough that the first N bits of Ω are correct, at that point it's seen all N-bit programs that are ever going to halt. In fact, if they ever halt, they halt within time n.
And then U sees what these programs produce, and it forms the list of the output of all programs up to N bits in size that halt. And to finish up everything, U outputs this list and halts. In other words, this list is the value of
U( φ8000 ΩN* ).It's precisely the list of all LISP S-expressions x with program-size complexity H(x) ≤ N.
This list cannot itself have program-size complexity ≤ N, because it can't be contained within itself. So it can't be the output of a program less than or equal to N bits in size, it's got to have program-size complexity greater than N. Therefore this program φ8000 ΩN* for producing the list
U( φ8000 ΩN* )must be greater than N bits in size. In other words, (the size of any program that calculates the first N bits of Ω) plus 8000 has got to be greater than N. So you get this inequality
H(ΩN) > N − 8000.Okay?
And in my Web site I actually show you this program. I program out in LISP the algorithm that I just described in words. You program it out, it's not a big deal, φ8000 is about one page of LISP, of my new LISP. But if you put in a lot of comments and run lots of examples then it's more than one page. Okay?
Martin Gardner had an explanation of this algorithm in an article in Scientific American on Ω that you'll find in one of the Martin Gardner collections. In my Web site this algorithm is actually written out in LISP and you can run it on some examples. You can certainly run the auxiliary functions and other pieces of the algorithm and convince yourself that it works. But when you put it all together, you're not usually going to be able to run it on the real data ΩN, because the whole point of this is to show that the bits of Ω are hard to know, they're algorithmically irreducible.
By the way, it's easy to see that
H(ΩN) > N − 8000implies that Ω is violently non-computable. Let's compare the two real numbers Ω and π. π has the property that the string of its first N bits has very small program-size complexity. Given N, you can calculate the first N bits of π. So the first N bits of π only have about log2 N bits of complexity. But the first N bits of Ω are irreducible, can be reduced at most 7999 bits, maybe down to N − 7999, we've proved that.
By the way, program-size irreducibility implies statistical randomness. So Ω's irreducibility implies that Ω is a normal real number. In any base, its digits all have the same limiting relative frequency.
H(ΩN) > N − 8000which states that Ω is algorithmically irreducible. By the way, I don't show in the course that this implies statistical randomness. That's a detour. I want to go straight to the incompleteness result.
(But I think you can sort of wave your hands and try to convince people that irreducibility implies statistical randomness. Or you can actually go and develop algorithmic information theory in excruciating detail! But in my course my goal is to get to the major incompleteness results as quickly as possible, illustrating them with interesting LISP programs, not to develop the entire theory. Now you could also take the proof of every theorem in my Cambridge University Press book Algorithmic Information Theory and write down a LISP program for the algorithm in it. I once started doing that but I quickly gave it up. It was too much work, and you don't really want to see every algorithm in the theory in such detail. I'm concentrating on programming in LISP what I think are the fun algorithms in the theory, especially the ones connected with the basic incompleteness results, the ones I want to use to present what I think are the fundamental concepts in my theory.)
So the last program in this course is the one that yields an incompleteness result for Ω.
Let's start with this question: How do I represent a formal axiomatic system in LISP? Well, this is the way I do it. Think of it as a black box which every now and then throws out a theorem. So it's a program that you start running, and it may or may not halt, normally it doesn't halt, and every now and then it prints out a theorem.
So why is this a formal axiomatic system? Well, the idea is that I don't care about the details of what's going on inside, what the axioms are, or what the logic used is like. What I care about is the criterion, which I think was enunciated rather clearly by Hilbert, that states that the essence of a formal axiomatic system is that there should be a proof-checking algorithm. So if that's the case, you can run through all possible proofs in size order, see which ones are correct, and print out all the theorems. Of course this is impractical. The time it takes to do it grows exponentially, and I wouldn't really want to run such a program.
But you can cheat and you can just have a LISP S-expression which every now and then outputs a theorem... It won't be the final value... Every now and then it uses a primitive function, I call it display, but you could also say ``output intermediate result.'' So I think of a formal axiomatic system as a LISP S-expression that every now and then calls a particular new LISP primitive function which outputs an intermediate result. And that way it puts out a lot of LISP S-expressions, which are the theorems.
And you can cheat and instead of putting a real formal axiomatic system in there with a proof-checking algorithm and running through all possible proofs, which is terribly slow (and never halts), you can cheat and put in a little example of something that just display's a few sample theorems and then halts. That way you can run a little example and debug algorithms which work with formal axiomatic systems.
How can algorithms work with these toy formal axiomatic systems? Let's say that you have a formal axiomatic system. We've agreed that it's a LISP program that is going to put out intermediate results using a new primitive function called display. Actually all normal LISP's have a function for outputting intermediate results; you use it for debugging, right? But it becomes more important in this framework. It plays an official role in my computerized version of algorithmic information theory, because try captures all the intermediate results. That's very important.
Try is a time-bounded execution/evaluation of a LISP expression. You use try to run a program with a time limit, giving the program raw binary data on the side, and capturing all the intermediate output as well as the final value (if any). That's very important. You see, the new primitive function try that I added to LISP is the way that you can run a formal axiomatic system for a while and see what are the theorems that it produces. That's put in as a primitive function in this LISP. The value returned by try is a 3-element list (α β γ). α is success or failure. β is the value of the expression being tried if the try is a success, and is out-of-time or out-of-data if the try is a failure. And γ is the list of captured display's, the list of theorems. The theorems don't get displayed, they end up in this list instead. Okay?
Let me say by the way that this LISP started off as three-hundred lines of Mathematica. I invented this LISP using Mathematica as my programming tool; I wrote the LISP interpreter in Mathematica. That way I could play with my LISP and try it out as the design evolved. Mathematica is the most powerful programming language that I know. But it's slow. So I rewrote the interpreter for this LISP in C. And it's a thousand lines of C, it's a hundred times faster than in Mathematica, but the program is incomprehensible of course.
I think that the Mathematica code is pretty nice. The C code is incomprehensible! Like all good C programs, it's too clever. You can deal with it while you're writing it, and immediately afterwards it's already incomprehensible, even for the programmer!
The Mathematica code I have to say I really like. It's 300 lines of Mathematica. I think it's pretty understandable, I think it's pretty clean, and I think that if I were presenting this as a course I would go through the LISP interpreter with the students. I think it's important that they should see the code. It's only 300 lines of Mathematica, and they could make changes to the interpreter as a way of showing that they understand it. So it's there, the Mathematica code is there on my Web site. (So is the C code. Use it, but don't try to understand it!)
Okay, so what's the next (and the last) program? The next program works with a formal axiomatic system, which is just a mechanism for producing an infinite set of theorems. It's given to us as a binary program for U, FAS*, which is a LISP expression plus binary data, and we're going to measure its complexity in bits the same way that we did before. I'll prove that if the formal axiomatic system has program-size complexity N, then it can enable you to determine, to prove what is the value of at most N + 15328 bits of the halting probability Ω.
If a FAS has program-size complexity N, then it can enable you to determine at most N + 15328 bits of Ω. [Perhaps one should refer to this as the logical (or deductive) irreducibility of Ω, to distinguish it from the computational (or algorithmic) irreducibility of Ω, namely the fact that H(ΩN) > N − 8000.]
So how do I show this? Well, I have a Berry paradox kind of proof. The idea is that if you could prove a lot of bits of Ω, then Ω wouldn't be this irreducible
H(ΩN) > N − 8000.That would give you a way to compress Ω into the axioms of the FAS. It would give you too concise a way to calculate Ω. If you could prove what the bits of Ω are, you'd do it systematically by searching through all possible proofs, and that would give you a way to calculate the bits of Ω. That's all we're saying, that it would contradict this
H(ΩN) > N − 8000.
So deduction and computation are very close, as Turing already noticed in his famous 1936 paper ``On Computable Numbers...'' where he proves an incompleteness result using the unsolvability of the halting problem. My argument is at the same level, it's analogous.
The new business here is roughly like the paradox of the first uninteresting positive integer. You know, that's sort of an interesting number. So if you could prove that a number's uninteresting, that would be an interesting fact about it, and ``uninteresting'' is the notion of algorithmically incompressible. So it turns out that you can't prove that an N-bit string is algorithmically incompressible if N is larger than the complexity of your axioms. You can't prove that an N-bit string is algorithmically irreducible if it has more bits than the axioms you're using for the proof. And similarly it turns out that with N bits of axioms you get at most N + 15328 bits of Ω.
That's the general idea. Here are the details.
I write out a 7328-bit program φ7328, it's about one page of LISP code. Why 7328 bits? Because you have to add that to the constant in the inequality for H(ΩN) to get the constant in our incompleteness result. So the difference between these two constants 8000 and 15328 is the size of the next program in this course, the last program. Divided by 8, you get the size of a LISP expression in characters.
And what this LISP program φ7328 does is this. Using try it starts running the formal axiomatic system that you're assumed to be given that's N bits of code. So we're looking at this:
U( φ7328 FAS* ... ).The prefix φ7328, the program that outputs the theorems of the FAS, and some extra stuff that I'll explain later are concatenated and fed to U.
φ7328 starts running the formal axiomatic system using larger and larger time bounds, capturing the intermediate output, which are the theorems. And it looks at the theorems to see how many bits of Ω it got. And I allow partial determinations where you get some of the bits but you leave holes with unknown bits between some of the bits that you do know.
And what φ7328 does is it looks for a small set of axioms that enable you to prove substantially more bits of Ω than there are in those axioms—at least 15329 bits more. And then it just fills in the holes, the missing bits, which costs just one bit per bit, one bit for each missing bit. So φ7328's final output and the value of
U( φ7328 FAS* missing bits of Ω )will be one of the ΩN's. For some N, it'll be the list of the first N bits of Ω. By the way, φ7328 may not need all the bits of FAS*, because it only keeps a finite part of the potentially infinite computation for the formal axiomatic system.
So if you could use N bits of axioms to get essentially more than N bits of Ω—more than N + 15328 bits, in fact—then you could fill in the missing bits at a cost of one bit each, and you get into trouble with this inequality
H(ΩN) > N − 8000.That's the point.
So that's how it goes, the basic idea is straight-forward. And I actually have this 7328-bit LISP program φ7328 and you can run it. Well, you can certainly test the auxiliary functions. You can certainly run all the auxiliary functions on examples to convince yourself that they work. But to test the whole algorithm, its main function, you have to cheat, you don't give it a real formal axiomatic system, because if you really try to run through all possible proofs in size order it would take too long. So you cheat, you give the algorithm a LISP expression that just throws out a few test theorems that will put the algorithm φ7328 that's running the formal axiomatic system through its paces.
That's how you convince yourself that this all works. And you may have to ``tweak'' things a little bit, you may have to slightly change the LISP code so that it works with your simple test cases. So you can convince yourself by running little examples that this would all work if you really gave it a genuine formal axiomatic system, say Zermelo-Fraenkel set theory.
I would like to have somebody program out Zermelo-Fraenkel set theory in my version of LISP, which is pretty close to a normal LISP as far as this task is concerned, just to see how many bits of complexity mathematicians normally assume. You see, the whole effort here has been to get these constants 8000 and 15328. And if you programmed ZF, you'd get a really sharp incompleteness result. It wouldn't say that you can get at most H(ZF) + 15328 bits of Ω, it would say, perhaps, at most 96000 bits! We'd have a much more definite incompleteness theorem. I hope that somebody will program ZF in LISP. I stop at this point [laughter], you know, programming fatigue! And you want to leave something for the students to do, right? So this is a little job for them. And if they're really clever maybe it'll put out the theorems in Zermelo-Fraenkel set theory reasonably quickly, not by running through all possible proofs in size order, which would take too long, but by doing some kind of tree search so that you can actually have something interesting happen in a reasonable amount of time.
This looks like a kind of programming madness, right? Up to now, it looks like what I've been telling you is, ``Oh how much fun it was to program something that nobody's ever programmed before!'' Or maybe just that this is neater than it was in the previous versions of my course. Yes, programming can be an obsession, it can ruin your life. You can sit at your terminal watching your life fall apart as you stay there hacking away until 4 in the morning! But is there any other justification for this except as a kind of drug? Well, I think so! I think this has some philosophical significance.
What's the point about this incompleteness theorem, about this technical result?
If a FAS has program-size complexity N, then it can enable you to determine at most N + 15328 bits of Ω.
Well, what it's really telling you is that you might get some bits of Ω, the first ones, say. You might be able to calculate the first few bits of Ω. In fact, in a different version of this theory, with a different UTM, (this was with my one-character LISP, my one character per atom LISP), I did get the first 7 bits of Ω, and they were all 1 bits. You see, once your lower bound on Ω is 127/128ths, you know that these bits can't change. The first 7 bits were 1's. And now I have a much friendlier LISP, but I lost this, I can no longer determine the first 7 bits of Ω this way.
But anyway, you might be able to get some bits of Ω without contradicting Ω's logical and computational irreducibility, without contradicting this incompleteness result and this inequality:
But in spite of this, the basic point is that Ω really shows that some areas of mathematics have no structure, have no pattern at all.
Let me put it this way. Normally you think that if something is true, it's true for a reason, right? In mathematics, the reason is called a proof, and the job of a mathematician is to find the reason that something is true, to find a proof. But the bits of Ω are mathematical facts that are true for no reason, they're accidental!
Now this is a very specific Ω, I've programmed it out. Imagine it being written in binary. And you ask, an individual bit, say the 33rd bit, is it a 0 or a 1? Let's say you're trying to prove which it is. And the answer is, you can't! The reason you can't is because, whether that particular bit is a 0 or a 1 is true for no reason, it's true by accident. It's so delicately balanced whether it's going to be a 0 or a 1, that we will never know!
It's like independent tosses of a fair coin. Independent tosses of a fair coin has got to come out heads or tails in each case, but there's no reason that it comes out one or the other, right? So it's exactly the same story with these mathematical facts, with the bits of Ω. There is no pattern or structure in the sequence of bits of Ω.
I don't know why anybody would want to try to determine bits of Ω. Although people have played with the Busy Beaver function, which in a way is like trying to calculate the bits of Ω. I don't know why you'd try to determine the bits of Ω. But if you were to try to do this, what this incompleteness result shows you is that you're in big, big trouble! Because essentially the only way to prove what a bit of Ω is, is to add the theorem that you want to prove as a new axiom. It's irreducible mathematical information. Now you can prove anything by adding it as a new axiom. The point here is that for Ω that's essentially the only way to do it. No compression is possible.
So there is no structure, there is no pattern, Ω has maximum entropy, it mirrors independent tosses of a fair coin. Now to a physicist what I'm saying sounds pretty reasonable, right? Ω has maximum entropy, the bits of Ω are completely uncorrelated. But to a mathematician this all sounds weird!
My new Ω is a particular well-defined real number. It's a real number with a rather simple definition, I can even write the LISP program that defines Ω on one computer screen. So you believe, thinking in Platonic terms, that each bit is either a 0 or a 1, even if I'm never going to know which. But it's black or white, right? And what I'm saying is that I think that it's really better to think that it's grey. It's really better to think that each bit of Ω has probability one-half of being 0 or of being 1—even though it's a particular well-determined bit, because I've written out the program that defines this number. Defines it, not by enabling you to calculate it bit by bit—that would contradict Ω's unknowability. But the program for Ω does enable you to calculate better and better lower bounds on Ω, so Ω is almost a computable real number in the same sense that π is. Almost, but not quite!
So the game is like this. I'm trying very hard to be constructive, as constructive as possible. Writing out programs is a sure sign that you're a constructivist, right? [Laughter] I want to settle all the programming details. But I'm trying to be as constructive as possible about non-constructivity! I want to exhibit something that escapes the power of constructivity, of mathematical reasoning, that you can't calculate, but that's just over the border between the constructible and the non-constructible. And I think that Ω is pretty damn good, it's just on the border.
Now this doesn't mean that all of mathematics falls down in a heap! But the normal notion of mathematics was that there were a small, finite set of axioms and rules of inference that we could all agree on, from which all the infinite mathematical truth would follow. This is the tradition that goes back to Euclid, to Leibniz, to Peano, to Frege, to Russell and Whitehead, to Hilbert. And Gödel showed that there was a problem. And Turing showed that there was a problem, using a different method involving computers. And I think that Ω follows in that tradition and shows that the problem is even bigger. However I don't think this means that you should stop doing mathematics! So what is the significance of Ω and of incompleteness? Should it affect how we actually do mathematics? I'll give you my opinion.
Of course the nature of mathematics has been discussed for a long time! Every generation of mathematicians have their own answer. But let me share with you my own feelings about this, my tentative conclusions.
There's a word that a philosopher coined that's very good. He says that there's an emerging new school, and emerging new quasi-empirical view of the foundations of mathematics. One talks about the formalist school, the logicist school, and the intuitionist school. Well, Thomas Tymoczko has a book, New Directions in the Philosophy of Mathematics, which has a whole bunch of articles, including two of mine, and he thinks that all these articles tend to support a new quasi-empirical view of the foundations of mathematics.
Now what does quasi-empirical mean? I'll tell you what it means to me. Quasi-empirical means that pure math ain't that different from physics! The normal notion of pure math is that mathematicians have some kind of direct pipeline to God's thoughts, to absolute truth! But poor physicists! You know, they tried Newtonian mechanics, it looks good for a while, then Einstein shows it's all wrong. Then—surprise!—quantum mechanics shows that Einstein was wrong! And now there's superstring theory. And is it right? Is it wrong? And mathematicians laugh and say, ``Oh those poor physicists! It's such a messy subject! They always have to backpedal, they don't know what they're doing! It's all so tentative!''
Well, I think that mathematics and physics are not really that different!—Physicists love it when I say this! [Laughter]
Let me try explaining this another way. Euclid said that mathematics is based on self-evident truths. But my impression is that maybe axioms are not self-evident truths. I don't believe in self-evident truths. Maybe it's more like in physics. Maybe mathematics should be done more like physics, where you're willing to add new axioms because they're useful, not because they're self-evident. And then of course you have to be prepared to say ``I goofed!'' and remove an axiom, which mathematicians don't like to have to do.
This may sound completely crazy, but in fact it's not just my opinion. Gödel makes very similar remarks in Volume II of his Collected Works, in his essay on ``Russell's Mathematical Logic.'' This essay was originally published in the Paul Arthur Schilpp volume The Philosophy of Bertrand Russell.
I talked about new axioms to a mathematician once, and he replied, ``Okay, I'm willing to add the Riemann hypothesis as a new axiom if you can prove to me that it doesn't follow from the usual axioms.'' Well, that's hard to do, because if the Riemann hypothesis were false, then there would be a numerical counter-example that one could easily verify that shows that it's false. So if you could show that the Riemann hypothesis is beyond the power of the usual axioms, that would imply that the Riemann hypothesis is true!
So these ideas are very controversial.
But the whole point of algorithmic information theory, the whole point of my information-theoretic approach to incompleteness, is that sometimes to get more information out of a set of axioms, you've just got to put more in. So let's put more axioms in! Physicists have always done that.
I had these ideas a long time ago. I proved my first information-theoretic incompleteness theorem in 1970. Although it's only in the past two or three years that I discovered how to actually program the algorithm in my original proof and run it on examples. And I was recently surprised to discover or rediscover that there are highly relevant quotes from Einstein and from Gödel. Let me throw them into this discussion, let me end with that.
Einstein has a very nice remark that I angered some mathematicians with. But what do they care, he's only a physicist, right? [Laughter] In his essay ``Remarks on Bertrand Russell's Theory of Knowledge'' in the Paul Arthur Schilpp volume The Philosophy of Bertrand Russell, Einstein says that ``the series of integers is obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences.'' So you can see that Einstein's attitude is very empirical.
I think that Einstein's position is that the positive integers are not a priori, they're not God given, we invent them like we invent all of physics. But the positive integers look more a priori than other concepts because they've been around longer, we invented them a long time ago. After all, when an idea has been around for a few thousand years, it's not surprising that people think that it's obvious, that it's just common sense, that it's ``a necessary tool of thought.'' The other extreme is the latest field theory in physics. That looks a lot more tentative. It hasn't been here long, and it'll probably be shot down next week, right? And there are probably thirteen different versions! But in Einstein's view there is no fundamental difference. The positive integers have been around for a long time, but they're still just an invention.
So I like this quote from Einstein, but it doesn't convince mathematicians.
Then there are some very interesting remarks made by Gödel. You can find them in Gödel's Collected Works. Gödel's philosophical position is the exact opposite of Einstein's. Gödel believed in the Platonic universe of mathematical ideas, he believed that the positive integers are just as real as tables and chairs. There are an infinity of positive integers, and they're out there somewhere. They're in the Platonic universe of mathematical ideas, that's where mathematical objects are!
I don't know! I used to laugh at all of this when I was a kid. But think about it seriously. You're young, you're trying to learn mathematics, you're doing elementary number theory, and you don't really start to worry about the fact that elementary number theory presupposes arbitrarily large positive integers, and how do they fit in the universe? Imagine a positive integer that is
10101010digits long. Does it exist? In what sense does it exist? You don't care—right?—you prove theorems about it, you know that it would be commutative, right? a + b is going to be equal to b + a even if neither number fits in the universe! [Laughter] But then later on in life you start to worry about this! [Laughter]
So I'm not sure if the positive integers exist anymore. But Gödel thinks that they do, and that philosophical position is associated with the name of Plato. But it's really just the classical mathematical position that the positive integers really exist, that an infinity of them are really out there. And starting from that, Gödel comes to a very surprising conclusion. Since the positive integers are just as real as tables and chairs, you can do experiments with them by doing calculations, and if you see a pattern, you can just go ahead like a scientist would in dealing with electrons. Since integers are just as real as electrons, why can't we use the same kinds of methods that scientists use?
Here are Gödel's exact words, taken from a previously unpublished manuscript *1951 that is in Volume III of his Collected Works:
``If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics.''
And in his essay ``What is Cantor's Continuum Problem?'' in Volume II of his Collected Works, Gödel says that maybe we'll come up with new versions of set theory, maybe we'll come up with new ideas about sets the same way that physicists come up with new ideas about physical objects. The justification for these new principles would be that they're useful, that they help us to organize our mathematical experience, just as the ultimate justification for physical principles is that they help us to organize our physical experience.
I think it's very funny! Here's Einstein, who's a diehard empiricist, and here's Gödel, who's a diehard Platonist, and they sort of come to the same conclusion! But they were buddies at the Institute for Advanced Study, so it's not surprising that they influenced each other. And the information-theoretic viewpoint that I've explained to you today leads me in the very same direction.
Another funny thing is that I don't believe that all this work on the foundations of mathematics has changed at all the way mathematicians actually work. Gödel's incompleteness theorem was initially very shocking. But then mathematicians noticed that the kind of assertion that Gödel constructed that's true but unprovable is not the kind of assertion that you deal with in your everyday work as a mathematician. And I think it's fair to say that about Ω too. The longer I live with Ω, the more natural it looks to me. But a skeptic would say, ``I don't care about the bits of Ω, so what if there's trouble there!''
But I do think that something else is making a difference in how mathematicians carry on their everyday work. It's the fact that the computer has so vastly expanded mathematical experience, that we just have to deal with it somehow. A lot of physicists do experimental work on the computer now. There's even a journal called Experimental Mathematics.
The computer has so vastly increased mathematical experience, that how do we keep up with all of it? Well, the answer is that sometimes you see that something seems to be the case, and it'd be nice if you could prove it, but for the moment you can't, so you sort of conjecture it. And if you're doing mathematical physics, then you're on the borderline between math and physics, and it's certainly okay to behave in this way. But if you're a mathematician, if you're just over the border, then it's not so clear anymore.
Well, I once had a conversation like this with a mathematician, and the Riemann hypothesis came up, and he said, ``It works fine the way we do things now. You have a paper, and you say the paper is `modulo the Riemann hypothesis.' So why do you need to call it a new axiom?'' And he has a point. But I do think that if a principle is very helpful for a long time, and nobody shoots it down, why not call it a new axiom? But one shouldn't do this too quickly. One has to be careful, the way physicists are, although I sometimes wonder how careful physicists really are! [Laughter]
Okay, that's the idea. And if you go to my Web site you'll find this course there, together with the Mathematica code for my LISP interpreter. So you're welcome to play with it, and you're welcome to send me e-mail about it, and I'd be pleased as Punch if one of you tried giving my course on the limits of mathematics to real students in a normal university setting.
Anyway, you guys have been nice enough to invite me for many years to give talks, and it's been very stimulating to talk with you, and for the moment I think I've exhausted this train of thought. Of course, I've been thinking that since I was fifteen years old, but fortunately every now and then I get a new idea! [Laughter] But for now, that's all I have to say, and maybe someday I'll have more to say, and I'll be lucky enough to come out here again and have a chance to kick it around with you! Thank you very much! [Applause]