Alan Saunders: Hi, this is The Philosopher's Zone, I'm Alan Saunders and let's take a walk with Albert Einstein. In the 1930s, Einstein bought a house in Princeton and from there he would walk every day to and from his office at the Institute of Advanced Studies, along with the great logician Kurt Gödel. Nobody knows what they talked about but Einstein later said that he came to the Institute merely to have the privilege of walking home with Gödel.

So who was this man with whom Einstein was so keen to chew the fat? He was a logician who proved a number of surprising results, most significantly his celebrated incompleteness theorem which tells us that there are parts of mathematics that traditional methods of proof cannot access. To tell us about that, here's Mark Colyvan, Professor of Philosophy at the University of Sydney and Director of the Sydney Centre for the Foundations of Science.

Mark Colyvan: I think he's one of the great logicians; I think he belongs right up there with Aristotle, Descartes, but just in general philosophy, in mathematics, I think he ranks in any one of those areas; he was someone who did bridge across both mathematics and logic and philosophy. I think he stands as one of the greats in any of those three areas.

Alan Saunders: Let's start at the end. He died in 1978. How did he die?

Mark Colyvan: He basically died of malnutrition. He degenerated physically and no doubt mentally as well.

Alan Saunders: He was in Princeton at the time, and it wasn't as though there was a famine in Princeton in the United States, so this was a willed death in some way, wasn't it?

Mark Colyvan: Yes, that's right. It's hard to know exactly what went on, but he certainly wasn't in full possession of his cognitive faculties at the time, and he just let himself go, as it were. I think the accepted view is that he did basically an act of slow suicide.

Alan Saunders: And actually if we're talking about fundamental issues like that, we should say that he was a believer in God, wasn't he? I don't know that he was a believer in orthodox religions, but he certainly believed in God and worked on proofs of the existence of God.

Mark Colyvan: Yes, that's right. He's famous for the couple of results, the completeness theorems but he is so amazing a polymath, you know, he played around with proofs of existence of God, he made major contributions within mainstream mathematics.

Alan Saunders: But his proof of the existence of God was a version of the famous ontological proof from the Middle Ages of St Anselm which basically says God is a being greater than which nothing can be conceived. Therefore God must exist, because if you conceive God is not existing, then you're conceiving something that is less than the greatest being that could be conceived, therefore it is necessary that God exists. And he actually developed a highly mathematical version of that, didn't he?

Mark Colyvan: That's right. It was a very strange proof for someone who worked in set theory of all places, because you don't get existence that cheaply in set theory, you can't just define the set of all sets for instance.

Alan Saunders: Let's just pause for a moment, because we're sort of on the cusp, as it were, of mathematics and logic here, and let's just pause to ask what logic is for. Logic doesn't actually tell you the truth, that's not what it's for, is it?

Mark Colyvan: No. No, logic is about inference if you like, it's about consequence. So when someone gives you an argument, whether it be an editorial in a newspaper, or a philosophy class or a mathematics class, or whatever, when someone says, 'Such-and-such plus such-and-such, therefore ... something else', they're making an inference, they're claiming that the conclusion follows from what went before, let's call them the premises. And logic is about that relationship. It doesn't tell you that the premises are true, because for instance all pigs can fly, Percy is a pig, therefore Percy can fly, that's a valid inference, that's for a logician, as good as it gets, but of course one of the premises is false, so it doesn't tell you that the premises are true, and therefore that conclusion is true, it's just about the relations between the premises and the conclusion.

Alan Saunders: And mathematical logic is essentially the same thing but carried to a higher or more complex level.

Mark Colyvan: That's right. So mathematical logic is just fleshing out that basic idea but using quite a bit of mathematical machinery to do it. And the other side of it is using that mathematical machinery to understand the logic within mathematics itself. Because mathematicians do a great deal of this kind of work, drawing conclusions from premises.

Alan Saunders: When Gödel was a young man in the early 20th century, the foundations of mathematics were in crisis. What was the nature of that crisis and what did it have to do with logic?

Mark Colyvan: A couple of sources for the crisis I think. Perhaps most serious was that (in) a very famous letter to the mathematician and logician, Frege, Bertrand Russell wrote that he'd found a contradiction in his formalisation of this mathematics of set theory. And it turns out it very generalisable, although Russell was particularly interested in a contradiction he'd found in this formalisation that Frege had run with, one of the foundation mathematical sciences of set theory. It turns out it's perfectly general, so Russell had shown that there was a liar-like paradox.

Alan Saunders: Let's back up a bit here. And first of all, look at what set theory is. Now a set essentially is just a collection of things isn't it? I mean you can have a set of socks, a set of pens, a set of anything really.

Mark Colyvan: That's right. So it's just a way that you collect things together, not necessarily physically, so in mathematics there will be a set which is my right big toe and the Eiffel tower, that's a set. So you can just collect things together in this formal sense. And the thing that's collected together it's important to note, is a set, it's not the things in the set. So the set of people in this room is not a person, it's a set.

Alan Saunders: OK. Now Frege is working on the foundation of mathematics, and in order to do that, he's actually asking very fundamental questions. What is the number two? for example. And he decides that a number is a set, doesn't he?

Mark Colyvan: That's right. Yes, he's asking these very fundamental questions, so fundamental in fact that many of the mathematicians at the time just didn't get what he was on about. He was worrying about what the number two is, whereas most mathematicians were worried about all sorts of fancy infinities. And Frege thought, well no wonder you can't understand all of these fancy infinitives. We don't even know what two is, so let's get a grip on that. So his proposal was that two is a bunch of sets that have two things in them.

Alan Saunders: So we have a way independent of counting from one to two, we have a way of determining that there are two things in this set?

Mark Colyvan: Yes. So typically if you want to say that two sets have the same size, so if you want to say that you've got seven people in the room and you want to say that the number of things in that set is the same as the number of things in the set of days of the week for instance, you typically would count both, notice that they both had seven, and then say that they're the same. But you can do it without counting. So for instance, in a large theatre, you can note that there are the same number of people as there are chairs by just simply noting that no-one is standing and all the chairs are full, you don't have to actually do the counting, you can pair them up. And so this was an insight that was around at the time, and certainly Frege's idea was to pursue this, that it's about one-to-one correspondence as a mathematician would put it.

Alan Saunders: OK, so there's Frege doing this in Germany, and then he gets this letter from his English colleague, Bertrand Russell. What does Russell have to tell him, and why is it bad news?

Mark Colyvan: He points out that there is a paradox in the set theory, and without going into the details of the actual set theory paradox, the basic idea is very similar to the liar paradox, which was well-known to philosophers, which is simply a sentence that says of itself that it's not true. So if I say to you now 'I am lying right now', that sentence turns out to be paradoxical. If it's true, it's false, if it's false, then it's true.

Alan Saunders: Because if what you say is true, then it must be false, because you've just asserted the falsity of what you're saying. And if it's false, given that you've asserted the falsity of what you're saying, it must be true. So we go back and forth.

Mark Colyvan: So it's paradoxical. What Russell did, he'd been thinking about such things for far too long to be healthy, I think, and he realised that you could construct these things in other mediums, if you like, and so he constructed a set theoretical analogue of this, and again, we don't need to go into the details, but it really is just the same kind of trick, this liar-like paradox to be found in Russell in the Frege system. And although the letter was directed to Frege and to Frege's system, the paradox is perfectly generalisable. You can construct such a paradox in any of the set theories of the time. Well that was the really big crisis. It looked like mathematics itself was inconsistent.

Alan Saunders: Let's just look at the way in which it works in set theory. Russell assumes that you can make sets of anything, and you can make sets of sets. You can bung sets together and have bigger sets. And he assumes that there must be... some sets are members of themselves, and some are not. So the set of all short words it's itself a short word. Short is a short word. The set of all French words, well French is not a French word, so that's not a member of itself. And he says then Well if some sets are members of themselves and some are not, what about the set of all sets, which are not members of themselves. Is that a member of itself, or is it not. And you get exactly the same paradox, don't you, because the qualifications for being a member of the set are that you're not a member of yourself, therefore if it's a member of itself, it can't be a member of itself and if it's not a member of itself, then it is. So it's exactly the same sort of, as philosopher's say, 'self-referential paradox'.

Mark Colyvan: That's right, yes. So that was the basic idea. It came from thinking about the liar and just a very, very clever move, to see that the liar paradox wasn't just about truth, which is what the standard story was, but to see that there's a set theoretic analogue of it, and that was devastating for foundations of mathematics, because it looked like the best mathematics of the day was in fact inconsistent.

Alan Saunders: In his autobiography, Bertrand Russell says it seemed really, really silly to be worried by this. This was just a sort of child's game, and I was - I don't know whether we need to believe his autobiography but what he says is 'I spent months and months and months puzzling over this, over what seemed such a silly problem.' Why was it perceived by him to be such a big problem, and by Frege to be such a big problem, and was it really that big a problem?

Mark Colyvan: I think that it was. There is a sense in which it is this little trick, indeed I called it a trick a moment ago, I didn't mean to trivialise it by that, but there is a sense in which this is a kind of trick. Once you've seen how to construct liar paradoxes, you can see them all over the place and you can instruct them. But as you said in the opening, mathematics is supposed to be this bastion of rigour and truth, and here right in the heart of mathematics is a contradiction. So here is one reason why it's so serious, because from the Russell set, I mean the Russell set is a perfectly well-formed set of standard set theory, the so-called naïve set theory, and the fact that that set is both a member of itself and not, from that you can prove just about anything using classical logic. In fact you can prove that one equals two starting from the Russell set. That's strikes me as bad.

Alan Saunders: It's at this stage that David Hilbert enters the story. Who was he and what did he have to say?

Mark Colyvan: Well Hilbert was one of the, if not the leading mathematician of the early part of the 20th century, an absolute major figure.

Alan Saunders: He was German, wasn't he?

Mark Colyvan: German, that's right. And he took this very seriously as well. I think Russell's comments about it not being that serious, I think were later in life when he was worried about other things, but it should be said Russell himself spent decades on something that he later thought was silly, but Hilbert thought it was very serious as well. And he was intent on finding rigorous foundations for mathematics, and set about trying to get a systematic story about what mathematics was about. And part of that story was that he required a finite consistency proof of mathematics. That is, he wanted to prove consistency of mathematics, but it had to be done by finatory means. That actually doesn't really worry us too much for the purpose of getting to Gödel's theorem, but he did require this consistency proof of the certain portion of mathematics.

Alan Saunders: so what we're looking for here is consistency and we're looking for completeness. In other words, we want a system in which everything that is true can be proved, that's your completeness, and nothing can be proved that isn't true. That's your consistency.

Mark Colyvan: Yes. Perhaps a little bit more stringently: consistency is you don't want to be able to prove something and its negation, you don't really want to prove that two plus two equals four, and that it doesn't as well.

Alan Saunders: OK. So we know what we want, and now Gödel comes on the scene with his first theorem, which as you've said, he actually arrives at a very early age. What does he have to say and what does it have to do with the paradoxes that troubled Russell?

Mark Colyvan: Well it connects up with paradoxes in two ways. One of which was Gödel had also spent far too much time thinking about the liar paradox to be healthy, and realised that this trick could be used to actually motivate some positive results as well. So in a way, what Russell did was generate an analogue of the liar paradox in set theory. What Gödel did was to realise that you can use this same kind of trick, this self-reference trick, things that talk about themselves, to show that if you like there are kind of blind spots in mathematics. So the first theorem, first incompleteness theorem of Gödel's says that there will be mathematical sentences that will not be provable within a particular axiomatic system of mathematics, if that system is consistent. So if you've got a consistent system, it will have blind spots, if you like.

Alan Saunders: So in the original liar paradox, I'm saying what I am now telling you is false, and that's paradoxical. With Gödel, what I'm saying is, This cannot be proved, and that's paradoxical.

Mark Colyvan: Yes. So with a very clever technique which is often brushed over because it's mere technicalities, but it's actually quite complex technicalities, but by this little process, he is able to encode these sentences in mathematics. So effectively it's a sentence of mathematics that says of itself that it's not provable, and again, just go through the familiar liar reasoning that you went through earlier, you see that if it's provable then it's not, and if it's not, then it is. You can't prove that sentence within the mathematical system. So there's an example of if you like to think of it as a blind spot.

Alan Saunders: Given what Hilbert was after, he wants a mathematical system that is consistent and complete, this is problematic for Hilbert, but it's not a disaster, is it?

Mark Colyvan: No, that's right. It's a problem with the completeness, it means that Hilbert's hope of being able to get all the true sentences out in a kind of algorithmical sort of way by just deriving them within the system, it looks like that's going to have some limitations. That's what the first theorem effectively tells us. There will be some sentences in the system, moreover recognisably true (we'll get to this perhaps later, the consequences of Gödel's theorem) but the sentence that I am not provable, it's recognisably true from someone outside the system, as it were. So there's a truth of mathematics that's not provable within mathematics. So it looks like Hilbert's hope of getting access to all the truths in mathematics via this formulaic computational method was not going to work.

Alan Saunders: But unfortunately for Hilbert, Gödel had something else up his sleeve, hasn't he, he's got his second theorem.

Mark Colyvan: That's right, so the second incompleteness theorem says if you like informally one of the blind spots is a consistency of the system itself. So you can only prove the consistency of the system if in fact the system's inconsistent. That's one way of putting the second incompleteness theorem. So no formal system, that's rich enough to embed the mathematics that we want is able to prove its own consistency.

Alan Saunders: Let's look further at the significance of this. For something that might not have been too much of a problem in Gödel's time but certainly is in ours: the computer. Now the computer is a system for cranking out axiomatic proofs. How is that affected by what Gödel has to say?

Mark Colyvan: Well there's his very tempting thought. It's been put forward by a number of people: the philosopher Lucas and physicist Roger Penrose, have advanced a thesis along these lines. And the basic idea is very seductive. The thought is that in a sense what Gödel has shown is that these formal systems have blind spots, if you like. Let's keep it in those terms. There are things that these things can't prove, including they can't prove their own consistency. And yet a human can see the truth and recognise the truth of the sentence that says that it's not derivable, not provable. So there's a sense in which the system is blind to the truth of that, but a human is not. It's a very tempting thought then to say 'Aha! Here's something that humans can do that machines cannot do even in principle, that a human can recognise a sense of the so-called Gödel sentence, but a machine cannot do that.'

I actually think that that's misguided, but it's worth sort of pausing on it for a moment, just to reflect on how tempting that thought is. It does seem to be a natural extension of Gödel's incompleteness theorems.

Alan Saunders: Well it does reflect on whether this has consequences for our understanding of the human mind, and I suppose one immediate thing you want to say is that if it's a trade-off between consistency and completeness, well in those circumstances, I don't really care. Almost certainly completeness isn't going to trouble me very much in my daily life. Consistency possibly rather more, but I'll have one or the other, I'll have whatever's going, and that's the way the mind works.

Mark Colyvan: It's one of those fascinating, I think mistakes. I'm inclined to call it a mistake in philosophy that he's been incredibly fruitful. I mean there are some mistakes that are just a straight-out mistake and as soon as someone points out the mistake that's the end of it, and the person who made the mistake has egg on their face. But this is not one of those cases. I think it's led to all sorts of interesting responses.

I think there are a number of ways you can come at this Penrose / Lucas argument. So one response is, as you say, Well where's the surprise? Humans don't worry about completeness in fact, and I don't think they worry about consistency, excepting locally: I don't want to sell you that I'm in Ultimo right now and I'm not in Ultimo. But if one reflects upon your overall system of beliefs of course there'll be inconsistencies in that system of beliefs. And so we really don't care about consistency all that much either, it would seem.

But I think there's actually a more fundamental way in which these arguments fail and that's just simply that what the theorem says is that there'll be a blind spot that's peculiar to that system. It doesn't say that it's a blind spot full-stop. It's a little like the idea that there'll be sort of blind spots or trouble spots if you like in your belief system that won't be problems for me, and vice versa. So for instance, physically you can say this, but it's deeply paradoxical for you to say the following: 'We're in Ultimo, but I don't believe it.' But I can say of you, I can say 'We're in Ultimo, but Alan doesn't believe it.' And you can say it of me, so there's a sort of blind spot for you, but not for someone else.

That doesn't mean that somehow or other my mind works differently from yours, or whatever, it's just each system has these little blind spots. And so in a way, what Gödel's theorem shows is there are these blind spots, and someone who tries to make too much out of the fact that some other system, i.e. a human mind, can see the truth of these blind spots that the mathematics can't, I think is missing the slipperiness if you like of these blind spots. Everybody has them and they slide around.

Alan Saunders: Mark Colyvan, thank you very much indeed for joining us.

Mark Colyvan: Thanks, Alan.

Alan Saunders: Mark Colyvan is Professor of Philosophy at the University of Sydney, and Director of the Sydney Centre for the Foundations of Science.

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I'm Alan Saunders, thanks to producer Kyla Slaven and sound engineer Charlie McCune. Hope you can join me next week. Until then, 'bye for now.