M: You've been making some provocative statements regarding artificial intelligence, such as that the computer program Mathematica could be considered a sort of AI. Some people would define a true AI as having an inner life; as feeling what it is to be itself. Under this definition, is Mathematica an AI?
C: If that is the definition, then Mathematica is not an AI, because it knows mathematics, but it does not have a human personality at all. If it did it would be a lot more interesting, and more scary.
M: So is there a fundamental difference between something conscious in the human sense and something like Mathematica? Once a program like Mathematica gets smart enough it will be as conscious as a human?
C: Well, I don't know. I think consciousness is a big mystery, and it doesn't fit into our current science very well. A few years ago you couldn't talk about it, it was an unmentionable subject. If you mentioned consciousness, it meant that you weren't a scientist, your career was wrecked. Now people are talking about consciousness again, which I think is good.
M: You made another provocative comment recently. You said that you didn't believe in the real numbers. [Numbers like 3.1415926... that are measured with infinite precision.] What did you mean?
C: Well, being provocative is good; it kicks people out of ruts. I do and I don't believe in real numbers. And I do and I don't believe in positive integers, [The positive integers are 1, 2, 3, 4, 5, ...] even.
Let me tell you why real numbers are very unreal. This example goes back to Emile Borel. A friend of mine, Vladimir Tasic, found this in an essay by Borel. Vladimir Tasic has a book coming out, called "Mathematics and the Roots of Postmodern Thought". I got this example from reading Tasic's manuscript.
Emile Borel was a well-known French mathematician early in the 20th century. He said to think of the following: Take the French alphabet, including blanks, digits, punctuation marks, uppercase and lowercase, and letters with accents, everything! Then start making a list. You start off with all possible one-character sequences, in alphabetical order, then all possible two-character sequences, then all possible three-character sequences, and none of these are very interesting but you keep listing them, and the sequences get longer and longer. So you'll eventually get all possible successions of characters of any given length, in alphabetical order. Of course most of this is going to be garbage, but you're going to find here every conceivable question in French, it's somewhere in the list --- in fact, everything you can write in French is in the list.
Next, said Borel, you number the sequences in this list you've created. And then you imagine a real number 0.d1 d2 d3... whose Nth digit is 1 if the Nth element of the list is a valid yes/no question in French whose answer is yes, and whose Nth digit is 2 if the Nth element is a valid yes/no question whose answer is no, and it's 0 in the case that the Nth element of the list is garbage, not valid French, or if it is valid French, but it's not a yes/no question.
So Borel has one real number that gives the answer to every yes/no question you can ask in French; about history, about psychology, about religion, about math, about physics --- and it's all in one number! So in a way this shows how unreal a real number is, because it has an infinite amount of information. No physicist can measure a number with infinite precision. I used to have a physicist friend at my laboratory, Rolf Landauer. He passed away, unfortunately. He would always remind me that no number measured in physics has more than about twenty digits of accuracy.
M: So what makes a number real? If we have it in our mind? If we measure it somewhere?
C: Well, real numbers are in our mind. Are they anywhere else, that's the question. [See Lee Smolin, Three Roads to Quantum Gravity, for some new fundamental theories that suggest that the physical universe may actually be discrete, not continuous.] Maybe they're only in our mind. 1, 2, 3, 4, 5, ... we tend to think that it goes on forever. You think of the positive integers as an infinite totality, as an infinite set. And in the world of mathematical fantasy, that works, the ideas fit together nicely, but in reality have we ever seen anything infinite? Do we see infinity anywhere?
So what I'm saying is that in a way these mathematical notions are fantasies, they are ideas that we invent. In a way you could compare them to religious fantasies or myths. But they are ideas that do fit together in beautiful ways. They have patterns that are useful, besides being fun for mathematicians to play with, because you find similar patterns in the real world. Knowledge of these patterns can be applied to computer science or engineering or other fields, and this tends to make us believe that the patterns themselves are real.
When I was your age I had no doubts. I loved mathematics and I was an extremist. Mathematics "über alles", it was math above all else. But as one gets older one starts to have philosophical doubts. And my own work on the limits of mathematics, which builds on the work of Gödel and Turing, makes me doubt even more. And I begin to think that in a way mathematics is just a lovely mental game. So Monday, Wednesday and Friday I have doubts about mathematics, and Tuesday, Thursday and Saturday I'm doing mathematics!
M: Did you once believe that things in mathematics always have some correspondence to the physical world?
C: No, I believed that the ideas themselves were real. I believed that infinities were real in some sense, maybe not in the physical world, maybe in some Platonic mathematical universe. But I didn't ask myself where they were. If you start asking yourself where they are, either you have to believe they are in the physical world, or else you have to say they are in some other world, and that begins to sound a bit weird. Where is the universe of mathematical ideas? Is the answer that it's in your mind? Is it real then?
Some people think that that's more real than the physical world! If you read G.H. Hardy's "A Mathematician's Apology", he says that 2 + 2 = 4 is absolutely true, but anything in the real world is not as definite, so he thinks that the world of mathematics is more real than our world. This is a view that goes back to Plato. Plato believed that the world of perfect concepts and ideas is more real than our ephemeral, troubled world down here.
M: The problem is that this world is inaccessible and unsatisfactory to those who want to be able to observe in some way that which they consider real.
C: When I was a child I had originally wanted to be a physicist, so I read a lot of physics and I have a lot of sympathy for physicists. I read a lot of Einstein's essays, and Einstein's view is that mathematics is something that you invent. We invent it because it helps us to understand the physical world. So the set 1, 2, 3, 4, 5, ... is an invention just as much as Einstein's field equations are. The only justification is that they work.
Einstein goes on to say that the idea of the positive integers was invented so long ago that we begin to think that they have some a priori validity, that they are a direct inspiration from God, or eternal truths, or to say it the way that Kant put it, that they are necessary tools of thought. But Einstein disagreed; he insists that we invented them, it's just that free inventions begin to look like intellectual necessities after a while. But they are really just our inventions and we should feel free to replace them with other things if that helps.
M: You've said before that the axioms of mathematics are not self-evident, that Euclid was wrong to define an axiom as a self-evident truth.
C: The normal thing that mathematicians believe is that mathematics is inescapable, logically necessary, for the rational mind, and therefore the axioms that mathematicians use are inescapable, they've got to be self-evident.
When you prove something you keep trying to break the proof into smaller and smaller pieces until you get back to principles that can't be doubted. Those are the axioms. They're things that don't need any justification because they're self-evident. Otherwise you have an infinite regress, and how do you ever finish a proof? You've got to start from something, because if you question every step in the proof, and you need another proof to justify it, then you never finish!
So the Greeks said you stop with something that's self-evident and those are what they called the axioms. But my problem is that I read a lot of physics. And physicists don't think that a principle of physics is self-evident. They think the justification is that it works. Maxwell's equations for electromagnetic waves aren't self-evident, but they work. The Schrödinger equation for the hydrogen atom isn't self-evident, but it gives rise to quantum mechanics. Einstein's field equations in general relativity, which talk about curved space-time, I don't think anyone would consider them self-evident!
So physicists have a different way of thinking than mathematicians do, and my problem is that my own work, plus the fact that I read a lot of physics when I was young, makes me tend to see similarities between pure mathematics and physics. Most mathematicians think that there's no connection at all, that pure math has no empirical content and is purely in the world of ideas, and that empirical sciences like physics are completely different. I agree that mathematics and physics are not the same subject. But I think it's a matter of degree, really. Math is here, physics is there, and you can have many intermediate positions between them.
M: So physics is different, and physical reality is different, in that we don't take anything to be self-evident?
C: The problem with philosophy is that you think about it, and all of a sudden reality fades away, breaks into pieces, and you go mad. Descartes tried to start a philosophy by saying that the one thing I can be sure of is I think, therefore I am. He attempted to derive all of philosophy from that. He starts off well. "I think therefore I am" sounds reasonable. But how he gets all the rest of his philosophy from that doesn't seem too watertight to me.
Bertrand Russell discusses something related. He says that we all start with naive realism, which is the doctrine that things are what they seem, and that we perceive them directly. Then science tells us that a table is actually made up of atoms with electrons whirling around. It tells us that when we see something, we don't see it directly, instead we see light waves, which are actually particles called photons colliding into our retina, so that the connection between the observer and the observed is really rather indirect.
So Russell has an essay that says that naive realism, the view that things are what they seem, actually leads to modern science, and modern science shows that things aren't what they seem. As Arthur Stanley Eddington points out, as science progresses an ordinary household table gets weirder and weirder. The quantum-mechanical explanation for why matter doesn't collapse is very strange.
So naive realism says that things are what they seem and this point of view leads to modern science, which says that things aren't at all what they seem. Therefore naive realism is wrong, it leads to a contradiction. The path that led us to modern science was mistaken!
I don't know! One of the problems you have is that when you start arguing about these philosophical questions you find that everyone has a different viewpoint, and you can't convince anyone of anything. So if you get a bunch of philosophers in a room and you ask a bunch of questions, you'll have every conceivable opinion on every conceivable subject. You know, if there were eight yes/no questions and 256 philosophers, you would probably get all 256 possible combinations of answers.
M: Science and math are a little neater.
C: Yes... I'm just giving my views, but one can discuss these matters endlessly, which is what makes them so much fun. Philosophy is never exhausted. Every generation asks the same questions and then answers them differently, all over again!
M: You said before that the minimum time required to work on an interesting problem, or even to create something interesting in art, is a lifetime. What did you mean?
C: Well, if you pick a problem that is too easy and you solve it, then what do you do with the rest of your life? I picked a problem that was very difficult, to understand the limits of mathematics, and I've been at it since I was fifteen, so that's going to be forty years soon. I consider myself lucky that the area I've worked on had enough there that I could devote a lifetime to it.
M: Does it seem to you that work on an interesting problem takes a sort of monk-like devotion, at the exclusion of other things in life?
C: Well, I'm certainly not a monk. You have to be passionate about something. It's very important to be passionate. To work on a question for a lifetime it has to really excite your imagination, you need to think it's very important. Otherwise you might as well watch television or rock-climb, or throw your life away in some other way!
M: You say you believe that mathematics should be pursued in the spirit of experimental science.
C: When I was a kid I dipped into Euler's collected works, his "Opera Omnia", in the Columbia University math library. Euler explains every step. He does a lot of calculations, looks for different patterns, then he makes a conjecture, then there's a proof with a hole in it, and then a few more papers down the road he finds a way to fill in the hole, and later he polishes up the proof. So he shows us his whole train of thought, and he does a lot of experimentation.
I think Gauss was the same, but Gauss hid all the steps. Gauss said you have to remove the scaffolding when you finish a building. But when you read Euler he explains every step in his train of thought.
So I do indeed think that there is an empirical component in math: it's computation. You do calculations and you see patterns and you make conjectures. And mathematicians when they discover new mathematics very often behave a little bit like physicists would, they use heuristic reasoning. But when you publish, normally you hide all of that and you present it like a direct divine revelation. You never give your train of thought. In fact, mathematics papers tend to be horrible, because they're written in a very compressed style and they're incomprehensible.
The math community really shouldn't do that. Physics papers tend to be more understandable, I think. In physics it's okay to say why you guessed that something was the case, and to give some explanation. In math it's sort of bad if you explain anything. The reader should be smart enough to decode the cryptic way you present it! But I don't think that's fair.
So I guess what I'm saying is that I agree that mathematics isn't physics; mathematics deals with the world of mathematical ideas, not with the physical world. But mathematics in the process of discovery is a little bit like physics. The way you discover something new in mathematics is "quasi-empirical". A Hungarian philosopher Imre Lakatos at Cambridge coined that word; I didn't invent "quasi-empirical", I used to always say it the way you did in your question.
Also there's a book by Polya, who was at Stanford, though I think he started in Eastern Europe and had to flee, called "How to Solve It", which is a high-school level book written by a good mathematician. He has a higher-level version of that book, a two-volume set called "Mathematics and Plausible Reasoning". In the first volume he gives a lot of case histories, telling how people really discover mathematics. And Polya's message is that you have to learn the art of discovery, and that's heuristic reasoning, it's inspired guesswork.
My own work has something to do with this, because I have information-theoretic results on the limits of reasoning, and that leads me to think that to prove more you have to assume more, and this is a little bit more like the way that physicists work. Mathematicians think that you can start with a few self-evident principles and get to all of mathematical truth. Physicists don't think that. Physicists know that when you go to a new kind of phenomenon you need new physical laws to understand it. My own work says that mathematical truth has an infinite amount of information and any finite set of axioms only has a finite amount of information, therefore you have to add new axioms. Well, where are you going to get them? You have to work intuitively in a quasi-empirical way, it seems to me. In a pragmatic, empirical way, like a scientist does. At least that's my feeling.
M: How will the computer be involved in the mathematics of the 21st century?
C: I think that computers are changing the way we do science completely, and mathematics too. The computer can provide an enormous amplification of our own mental abilities, and it's really changing the way everything is done. George Johnson just had an essay on simulation in the New York Times in the Week in Review where he points out that now it doesn't matter what field of science you work in, the computer is fundamental in the work you do. [NYT 3/25/01, "In Silica Fertilization; All Science Is Computer Science"]
I find the computer fascinating. When I was a kid the computer was just beginning to exist, and I read some of the first things on the computer that were published, like "Giant Brains" by Edmund C. Berkeley. I managed to program computers at a time when it wasn't easy for a high school student to program computers, and I loved it. And I loved it as a plaything, as a game. I don't play video games. The computer is my video game, because you can make it do things. I view it as an artistic medium, like clay or oil paints. It's a very malleable medium, the computer, and you can create things with it that actually do something.
But I was also fascinated by the ideas. The computer changes the way you think about things. One way to say that is that you only understand something if you can program it. Another way to say it is that the computer is the empirical content of mathematics. The computer is the lab for mathematics, the same way that the physics lab is the empirical content for physics.
Marvin Minsky puts it very provocatively when he says we are a carbon-based life form that is creating a silicon-based life form that is going to replace us! I hope not; I like human beings. I don't want us to be replaced by machines.
But I think the computer is a tremendous philosophical concept. All of my work is based on the computer. I use the computer to define what is meant by randomness, and I use it to define what is meant by a scientific theory. A scientific theory is a computer program that calculates your experimental data, and the more compact or concise the program is, the better the theory. That's my version of Occam's razor. And I say the data is "random" if there is no theory for it, no description more concise than the data itself. So for me the computer is a philosophical concept, though it's true that it also pays for my salary. Yes, computers are useful, but it's the conceptual revolution that I find absolutely fascinating.
You go back to Leibniz, and Leibniz was talking about replacing reasoning and controversy by computation. He had this vision of creating a symbolic logic. There were wars of religion then in Europe, between the Protestants and the Catholics. Leibniz's fantasy goes back to Plato, and it was that instead of killing people, maybe we could reason things out, and ideally reasoning would be just calculation. Instead of having to fight we would sit down at a table as gentlemen and we would reason it out and everyone would agree. That's a beautiful philosophical fantasy, but I don't think it's really possible.
Not all philosophy comes from Plato and the Greek ideal of rational argument. And Greek philosophers believed in rational argument, but the ancient Greeks weren't so rational. They had terrible wars. But I think it's good to try. The results by Gödel and Turing, and my own results, show that pure reasoning isn't black or white; it doesn't answer all questions in an absolutely definitive way, but that doesn't mean that reasoning is useless and that the only way to convince someone is to point a gun at his head. I hope that that isn't true. That would be terrible if it were true.
M: Are there any readings that you would recommend to the humanities student interested in the ideas of 20th century mathematics? The humanities student often leaves out mathematical ideas from their intellectual lives.
C: I think mathematics is like music. It's a music that not all of us can hear, but to me it's definitely an art form. To me the ideas are beautiful, and it's too bad if you make them seem utilitarian.
I can certainly recommend some older books. I don't know much about the newer books. I would strongly recommend "A Mathematician's Apology", by G.H. Hardy. It's a short essay on math as art. G.H. Hardy was a contemporary of Alan Turing, but a little bit older.
Another book I would recommend is by Nagel and Newman, "Gödel's Proof". That's one of the books that got me interested in Gödel. It has to do with the soul-searching in mathematical logic, where mathematics asks what are the limits of mathematics. In many ways that book is obsolete because Turing's work and my own work give a very different perspective, but I still think it's a lovely book.
There's also a lovely book by Tobias Dantzig called "Number, the Language of Science". It's from the 1930's and still in print. It's on the evolution of mathematical ideas and how new fields of math are created and where new concepts come from. [Dantzig's book is anecdotal, but I'd like to see a mathematical theory that explains how mathematics evolves and gets around all the incompleteness results.]
Then there's a collection of very romantic biographies of mathematicians by Eric Temple Bell called "Men of Mathematics". That book is very controversial now. Some say it's a very bad book. It's a very romantic book, and not necessarily accurate. Nevertheless it has inspired a lot of people. You can easily acquire a passion for mathematics by reading that book. So I think it still has good qualities, even though you can argue about the details. Bell had strong opinions and wrote with passion. He was passionate about mathematics, and it shows in every word.
If you write a book that offends no one and make sure everything you write is absolutely, 100% correct, then you end up writing nothing. You end up with very dry books that are infinitely cautious. The problem with those books is that they're safe, but they're not inspiring. A book that has some opinions, you can react in favor of them or you can react against them, but at least there is something that is being said there that you can be for or against!
Then there was a four-volume set that we all carried around with us when I was at the Bronx High School of Science. It's called "The World of Mathematics", and it's edited by James R. Newman. It collects a lot of stuff on mathematics: essays, history, articles, fiction, all kinds of stuff. It's just been reprinted by Dover.
I'd also recommend Einstein and Infeld on "The Evolution of Physics", and Feynman on "The Character of Physical Law", even though they're not math books.
M: What about your own books?
C: My books might be too technical. My least technical book is called "The Unknowable". Some of my essays are quite easy to understand. I think that the Scientific American articles on my web site can be read by anyone.
The first chapter of "The Unknowable" is readable, I think, but the other chapters get hard pretty fast. I would start with that book, but if you want something more inspiring, but harder, my book on "The Limits of Mathematics" is good.
My lecture transcripts tend to be more understandable. Look at my web site. And there's also an interview there on the creative life that some people might enjoy. [That's now the second interview in this book.] It talks about mathematics as an art, which it is for a lot of us. It depends on whether you're doing it for the money, you're doing it for fundamental understanding, you're doing it for the beauty of the ideas, you're doing it for a career ---
M: You're doing it because there's nothing else for you to do ---
C: If you mean inner necessity, that's fine, that's what happened to me, I had no choice!
These notes discuss up-to-date, high-tech versions of Borel's number, and the thesis that the computer is in fact Plato's universe of ideas.---G.J.C.
Here's how. In a modern computer, a text in French is just a character string, which is represented internally by 8 or perhaps 16 bits per character, depending on your coding scheme. So just take a text in French, convert it to binary, prefix it with an additional 1 bit to avoid collisions/synonyms (texts that give the same number), convert the bit string into a positive integer, and use it to index the appropriate digit of Borel's number! This can be done easily enough, for example, in Mathematica.
The synonym problem is that if you don't prepend a 1 bit, then the bit strings 011 and 11 both correspond to the integer 3, which is no good.
So this scheme avoids having to generate Borel's enormous list of all possible French texts. Plus we get direct access to the appropriate digit of Borel's number to answer our question.
I will resist the temptation to program this out and exhibit examples! You do it!
For example, use 3 digits for each character in French, and then we'll have Borel's number consist of an infinite list of pairs consisting, in any order, of each question followed by the [best?] answer, with appropriate punctuation for separation. That is, Borel's number is now an infinite list of (question, answer) pairs.
This version of Borel's number answers all questions, not just yes/no questions, but it comes at the price of slower access. Instead of direct access to the appropriate digit to answer a question like we had in Note 2, now we have to run through the digits of Borel's number looking for the question before we can find the answer to it!
And if this version of Borel's idea isn't good enough, you can invent schemes for giving the answers to all possible questions in all possible languages, even in cases where there are multiple answers!
So by now, we've probably made our point and beaten Borel's number to death!
The idea is that computers are really mathematical machines, that they operate in Plato's world of ideas, and that computer engineering is really technology for simulating Platonic worlds, for isolating the programmer from the physical implementation below. And, in fact, the Internet and the World Wide Web may be considered a sort of world of linked Platonic concepts. So by creating computers and the web, humans are in fact raising their level of consciousness, and partly moving themselves from this world into Plato's world of abstract ideas!
What about going all the way? Could there be software life forms that lived entirely in this Platonic world of ideas that we are creating? And would they be conscious like we are? (Computer viruses are a first but much too primitive step in this direction, much too primitive to be conscious.)
Well, in a sense, that's just what WE are! In physics and in biology many times the key step in achieving a higher level of organization is to isolate the higher level of abstraction from its lower level implementation. For example, DNA creates a digital programming interface out of biochemistry, that can be understood to a certain extent as digital software without worrying about the chemical implementation below. And at many levels in hardware and software engineering one creates levels of abstraction, one builds things out of components and one is shielded from having to know how they work inside, one uses them as black boxes, as new levels of abstraction. To a certain extent physics does this, as quarks combine into neutrons and protons, these form atoms, then molecules, etc. And it certainly happens in biology, where the living units at one level become "cells" or components used to build higher-level life forms. But it happens particularly cleanly in computers, where the programmer lives in a fantasy world of perfect software, and this illusion is shattered only if the machine hardware breaks down.
In the interview transcript I have a very materialist view of the world, but here, on the contrary, I'm sustaining the very anti-reductionist thesis that not only biology, psychology, etc. cannot be derived from physics, but, in fact, physics is completely irrelevant, the higher-level phenomena have nothing to do with the lower-level ones, that is precisely what they are trying to achieve, that is how Nature and Life progress...
Perhaps BOTH views, that the world is made of matter, or that the world is made of ideas, both contain some truth. Perhaps these complementary views reinforce rather than contradict each other. Perhaps both are needed for us to be able to make sense of the confusing world that we find ourselves in. Perhaps we are the attempt of matter to create mind!