 Hello my philosophic friends and welcome to episode number 16 of Patterson in Pursuit. You guys are probably tired of hearing me say this, but I am so excited about this interview. This is another one of my favorites ever. Today I'm asking the foundational question, what is logic? And to help me answer this question I spoke with Dr. Timothy Williamson, who is the White Ham professor of logic at Oxford University. We spend a lot of time talking about the nature of logic, the constraints of logic, paradoxes and contradictions in logic, and we also get in some really interesting discussions about the metaphysics of concepts and mathematics. If you have an appreciation for logic, I can bet the farm that you are going to love this interview. Dr. Williamson is the author of several books, most recently a book called Tetralog. I'm right, you're wrong, which is a fictional conversation that takes place on a train between four people. I'll have a link to that book at the show notes page this week, which is steve-patterson.com slash 16. And shortly I will also put up a link to my upcoming book Square One, The Foundations of Knowledge, which is specifically on this topic as well. So I hope you enjoy my conversation with Dr. Timothy Williamson of Oxford University. So first of all, thank you very much for sitting down and speaking with me today. You're welcome. I have several questions for you because as somebody who's interested in philosophy, I think that logic is a really big deal. And I think this is kind of universally understood in pretty much every area of thought. People treat logic with a great deal of respect and they say things like, oh, you're not being logical and that's supposed to be no, you've made a big error. But when you try to ask the question, well, what is logic? You get a lot of different answers. Some people say, well, logic is the rules for reasoning or it's the rules of thought or it's the rules of existence. There's lots of different interpretations and explanations. Try to identify what logic is. In your own worldview, what do you think logic is? Well, logic is actually several things because it's studied in different areas for different purposes. It's studied in philosophy. It's studied in mathematics. It's studied in computer science. And of course, all those people have different aspects of it which they find most interesting. But from my point of view, as a philosopher, the most fundamental part of logic is really just concerned with very, very broad structural generalizations about how things are. So it is just as much concerned with reality as any other kind of investigation. But just at this very abstract structural level, so we're interested in principles like everything either is the case or is not the case. So it's basically the law of excluded metal and that sort of typical law of logic. So when you say you're concerned with kind of, would you say your logic has to do with principles of existence in the most abstract form? Yeah, you could say existence just in the sense of what there is and how it is. I think in some ways, mathematics is a bit like that, but logic is concerned with even more fundamental kinds of generalizations about how things are. I mean, people talk about it as the laws of thought, but I think it's not so different from a subject like physics. I mean, physics isn't really concerned with the laws of thought. It's concerned with physical reality. But then of course, if you know something about physical reality, that tells you how you ought to think about physical reality. And in the case of logic, I think it's similar that it's not primarily concerned with giving us instructions about how to think, but if you do know something about the sort of broadest structural features of reality, that should constrain the way that you think in the same way that any other knowledge should. So there's a lot of questions that are evoked by that response. One of them you say at the beginning, you said it's studied by a lot of different areas of thought. I think that is certainly the case. But does that imply that it's just one thing, that there's just one logic out there and there's different approaches pointing at the same thing? Or are there different logics out there? Well, there are different logics in various senses. I mean, there are, of course, just as in physics, there are different physical theories. I mean, in logic, there are different logical theories. Some of those clash with each other. Some of them are rival theories of the same subject matter. Others are not really rivals at all. They're just concerned with answering different questions. So I mean, there's a lot of variety within logic itself. It would be true to say that there's different logics, kind of from an academic standpoint, that there's different competing claims to what logic is. Or different theories about logic, if you like, about logical reality. But the question perhaps not necessarily about the fact that there are different competing claims about logic, but this idea that there are, in reality, multiple logics. So you have like a pluralistic, logical framework that you have your logic, somebody else has their other logic, another culture has another logic. What do you think about that idea? I'm not a pluralist about logic. It seems to me that the questions, the most fundamental questions that we're asking have right and wrong answers. And if two people are giving inconsistent answers to a question, then at least one of them is wrong. Isn't that an application of, let's say, classical logic? So what you're saying is something like, if two propositions contradict one another, one's got to be, at least one of them is wrong, is fair to say. But isn't that only true within your framework of classical logic? Well, that is a claim that classical logic make. It's not a claim that absolutely everybody accepts. But the fact that some people disagree with it doesn't make it false or doesn't even make it anything other than absolutely true. So then what do you think? I completely agree, by the way, with everything you just said. I'm just playing devil's advocate. I had an interview at Columbia University where I was talking to a gentleman about this topic, and he was a dilatist. Where the grand priest version of there are some true contradictions that you just have to accept things. It is not the case that everything is true or false. There are some exceptions. So you would say, you think there's an error in that way of thinking. Yeah, I'm not a dilatist. I think that when somebody contradicts themselves, something is going wrong. I think Dirty Theism, it's an interesting view because it's not just a completely irrational sort of reaction. It is a way of attempting to deal with certain kinds of paradoxes, which, as a result, where apparently very plausible principles lead to contradictions. And the dilatist's idea is that those contradictions are really telling us something and that that's where in reality there are simply if you like black holes of contradiction. And I mean, their view is that if you're willing to accept a few contradictions, you can actually get a nicer theory overall than if you avoid contradictions. But in all these cases, it is, in fact, possible to avoid contradiction. That's what I was going to ask. So you would have to have some kind of resolution into something like the Liar's Paradox. What do you have a personal, I mean, your conclusion wouldn't be it's true and it's false. So what do you think is the resolution? So there are certain principles about how we apply the concepts of truth and falsity, which in the Liar paradox, they're actually being applied to sentences. And so they're really to do with the way that we handle language. And I mean, if you're a classical logician, the way you deal with the Liar paradox is you say that we've got to be more careful in the way that we handle language, which means that we can't always make the moves that it's quite natural to make most of the time. For example, in talking about which sentences are true. In ordinary situations, we'd think that saying the sentence, snow is white is true is just the same as saying snow is white. And so we can make those transitions between talking about truth and just talking about the world in a very automatic way. But what the Liar Paradox shows is that those transitions are not as straightforward as we think and that, in fact, in certain circumstances, they've got to be restricted when we're dealing with sentences that may in some subtle way fail to express any real proposition about the world, even though they feel meaningful to us. And if you're willing to make those restrictions, then you can handle the Liar Paradoxes without having to accept any contradictions at all. And what I would argue is that in the end, that gives us a better theory than the Dylatheists do because what the Dylatheists do is rather than making their revisions at the level of how we handle language, they revise the very basic logic, which is a logic that is used in all theorizing whatsoever. So, for example, all over the natural sciences and in mathematics, people are using logical principles. I mean, they're reasoning with logic because when they come to an inconsistency, then they do something to get out of it. And so what the Dylatheists are doing is forcing us to complicate fundamentally all the reasoning that we do. Because they say that absolutely basic laws of logic have to be revised. Whereas if you go the classical way, you can just keep the standard laws of logic. They don't need to be revised at all. And so we don't need to mess with normal mathematics and normal physics and so on. All we need to mess with are the principles that we use in reasoning about language, about the way that we apply the concept of truth and falsity to sentences. And so we actually get a nicer overall view than you do by accepting contradictions. Would you say something like this, because this is my position, that if you were to allow logical contradictions into your theory, not only is a demonstration that you've made an error, but it also kind of abandons any type of conceptual coherence. So for example, a Dylatheist position that is somehow entertaining the idea of a true contradiction, it's not even really a coherent notion to entertain because you can't have a coherent theory, which is an incoherent theory, which is a theory which includes logical contradictions. Would you say that's too extreme? Well, it's too extreme to suggest that Dylatheism involves total intellectual anarchy where just anything goes. I mean, Dylatheists, they do have some principles of logic that they adhere to. And if you talk- That they don't contradict? It's not that they have avoid contradictions, but there are various other principles that they adhere to. So there is some discipline to the way that somebody like Graham Priest talks. There is discipline, I think, certainly that is the case in the sense that they sit down and write books and formally try to develop a theory. But what I'm trying to get at is, can it be that a theory is internally coherent if it's not internally consistent? Can you say, yes, I fully understand this subject matter even though within my understanding there's a contradiction? Well, of course, I think it's wrong to say that because there are no true contradictions on my view. But if by incoherent you mean something like a theory that just totally collapses, then that is not the case for Dylatheist theories. But the situation is that in classical logic, if you have a contradiction, then from the contradiction, you can derive any conclusion that you like. And so as it were, all hell breaks loose when you have a contradiction within Dylatheist logic, you can't actually derive very much just from an individual contradiction. And so having a contradiction in your theory does not mean that the theory totally collapses, that it just that anything goes whatsoever. Well, but it's not any it's not any innocent contradiction, though, because if it's the case that you permit any logical actual logical contradiction in your theory, that that implies in principle that logic has exceptions. Well, because it has at least one. That's an exception to classical logic. It's not an exception to Dylatheist logic. So they I mean that they're not having to qualify what they regard as the true logic. Right, which is that they just think that that classical logic has been erroneously set up as a true logic and that actually a different logic is the true one. Well, this is the one that this is this is great because it's so related to the conversations I've had. Because what I was trying to ask this professor at Columbia is, is there a way to fully make sense of a sentence that contains a logical contradiction? Something just straightforward like, you know, the cat is on the chair and it's not the case that the cat is on the chair. And it seemed like the claim was in Dylatheism and yes, you could make some sort of sense to it. Actually, I give there's an even better example because I remember exactly the response. I said, is there any way to make sense of a proposition like there's a square circle? Is that something that is coherent? My position, it's not even coherent if you understand the concepts. And he said sort of because we can say that square circles are square. Yeah, well, even from the point of view of a classical logician, the sentence there are square circles, it has to be meaningful because it's negation is true. It's true to say that it's not the case that there are square circles. But if there are square circles was just nonsense, then negating it would also be nonsense and so it couldn't be true. So it has to have a meaning. But the meaning is just such on a classical view that it can't be true. Well, okay, I guess maybe it depends on what we mean by nonsense. Because what I would say is if you unpack the concept of square and you unpack the concept of circle, you find these two things are mutually exclusive. So it's like claiming two mutually exclusive things can be put together. Well, that doesn't understand what being mutually exclusive is. Well, I think dialysis would agree that in a way nothing can be both square and circular but they also think that some things are square and circular and that's a contradiction. But it's a problem though, isn't it? Well, it's certainly from a classical point of view it's a problem. But if by making sense you mean that it's something that can be defended in a way that doesn't involve as we're complete madness, then they manage to defend it. And I mean, that way the- Depends on what you mean by complete madness. Well, I mean- Depending, believing in a true contradiction, I would say is a pretty good qualifier for complete madness even if you are very composed in having that belief. Yeah, but as, you know, in fact, if you meet a dialysiast like Graham Priest, he's considerably saner than for example, you know, Holocaust deniers or people who think that the world is being run by a conspiracy of lizards or whatever. Okay, so let me ask you this because the attack if you will on classical logic doesn't just come from the dialysiast. It also comes from other angles. Do you have, I know this is, I'm just kind of springing this on you, but do you have then a response to the claims of people who come from, let's say they appeal to quantum physics and they say quantum physics shows that reality is logically contradictory because reality is in two mutually exclusive states at the same time. Do you think that even if they think that's what the evidence shows actually isn't and there's alternative theories which don't include logical contradictions that should be believed? Yeah, I think that the most serious form of non-classical quantum logic is not one that includes contradictions, but it's one which does revise certain classical laws of logic. What it actually revises is the law that if x is the case and either y or z is the case, which is, let's say, one of the distributive laws. And in the 1970s, particularly people like Hilary Putnam at Harvard who's actually just died, put forward quantum logic as a non-classical logic and claimed that we could better understand the logic of the law. As far as I can see, that program of quantum logic has fundamentally failed and Putnam, in fact, withdrew his support for quantum logic. And it's very clear that quantum logic is a non-classical logic. It's a non-classical logic. It's a non-classical logic. It's a non-classical logic. And it's failed, not as it were, just because it's non-classical, but it's simply turned out that even if you adopted this non-classical logic, it didn't really help you understand the physics any better. See, what quantum logicians had been claiming was that it was somehow our prejudices in favor of classical logic that were preventing us from taking quantum reality seriously and that once we could get rid of these logical prejudices, we'd be able to understand what was going on. But it turned out that just abandoning these classical laws didn't make any more sense of the experiments than it had before. So as we're for reasons internal to the physics, it just turned out that going non-classical in logic was not a helpful move in understanding what on Earth is going on with quantum reality. Well, so then I have to re-ask the question, though. I mean, does that mean that you're saying the claims coming from physicists or even philosophers saying that quantum physics demonstrates the limitations or even the... demonstrably demonstrates that classical logic isn't as rigorous as we think it is. Do you think that they're mistaken? Because there are obviously alternative interpretations of quantum mechanics that doesn't include in terms of the Copenhagen interpretation things being here and not here at the same time. There are several different interpretations. Would you say that those are automatically superior just by virtue of the fact that they aren't logically contradictory? Well, I don't think any of the plausible interpretations of what's going on in quantum reality from the point of view of physics are ones that involve postulating contradictions. I think they involve these much subtler and milder changes in logic. But, of course, I prefer classical logic on purely logical grounds, but even if you're willing to go non-classical in an attempt to understand the puzzles of quantum mechanics, it actually doesn't help you as far as physics is concerned. So there's no real advantage from a theoretical point of view. Even if you start off without any special prejudice in favour of classical logic in doing anything non-classical from a purely logical point of view in quantum mechanics. And so I think it's just a mistake to think that quantum mechanics provides any support for non-classical logic. I mean, of course, it's a deeply puzzling area. I don't think anybody denies that, and it's very unclear what the correct account of it is. But there's no reason to think that the correct account involves any meddling with logic itself. Now, let me ask you this. If it were true that you had physicists claiming, maybe in the future, or let's just say that there was a fundamental theoretical disagreement that the physicists were saying, look, we have evidence of non-classical logic, and it was pitted against classical logic. Do you think that there is a, is it hubris to say from a logical standpoint you've only demonstrated that physicist you've made an error? So in other words, a lot of people have the intuition that physics has the final say, that physics is kind of the bedrock justification for our other beliefs. Other people say that no logic and even philosophy is kind of the bedrock through which we interpret physics. Do you have a response to that? Do you think one is more fundamental than the other? Well, one thing that's very clear about physics is that it depends on mathematics. Physicists are using mathematics all the time, and mathematical reasoning itself depends on logic. Logic is the sort of basic framework within which mathematical reasoning takes place. So it's not as though the physicists are not relying on logic in their own work. They're simply taking logic for granted, and the logic that they're normally taking for granted is classical logic. Now, that doesn't mean that we can't conceive of challenges to classical logic coming out of physics. That's something like the law of non-contradiction. Well, I think, as I say, I think that's actually the least plausible candidate to be challenged by concerns about quantum mechanics. But let's just say... In some scenario, they'll say, look, this is demonstration of a true contradiction coming from physics, just in that hypothetical world. You say you don't buy it. All my instincts make me utterly skeptical of any such claims, but at the same time, I don't want to suggest that classical logic has some kind of transcendental justification so that, as a word, just by its kind of inner glow of rationality or whatever, that it will automatically trump considerations from anywhere else. I think that it's not something that's beyond question. If we are dealing with very, very recalcitrant problems in natural science and somebody who knows what they're talking about suggests that the best way of understanding some weird phenomenon is by becoming skeptical of some classical law. I mean, we can look at what arguments they're providing and see whether they're really offering a better theory. And we don't have to go into that taking for granted from the beginning that there's no way we're ever, ever going to allow modifications of classical logic. But my own view is that when we take these proposals to modify logic seriously, we find that, in fact, they don't help. But as we come and look at them in an open-minded way, I've done that in a number of areas, including, for example, the logic of vagueness and these non-classical logics, they end up not really having the advantages that are claimed for them. But the best way of arguing that is not by saying that, as when nobody's allowed to question classical logic, is just by looking at what these alternatives are and seeing how they work, and then seeing what the problems are with them. So in evaluating a claim that implies a contradiction, by what tools would you justify, or by what tools are you saying, this is a plausible evaluation if you give up the law of non-contradiction or the law of identity, for example. How would you even try to evaluate a proposition as being reasonably true? Without those? Well, what we're doing is we're comparing alternative theories. The theories include, as it were, logics. I mean, that's one component of them. They may also include various claims about the physical world if we're thinking of challenges to logic from physics. And we compare them against each other by normal scientific standards. So on the one hand, we're looking to see which theories fit best with the evidence. We're looking at which theories are simple and elegant, but also which are strong, which have a lot of explanatory power. Doesn't all of that presuppose the law of identity, for example? No. The law of identity is a particular logical law that everything is identical with itself. That law is not being used in many logical deductions. You can entertain it. Isn't the law of identity something that is inescapably presupposed whenever you're coming up with a proposition about anything? The reason for physical phenomena X is such and such. I'm saying the reason for physical phenomena X is such and such, and I'm not saying the reason for physical phenomena X is not such and such. Isn't it something that's inescapable? So any proposition you make, you're specifically not claiming the opposite of your proposition. You're claiming my proposition is as it is. I think if you try to show that the law of identity is being presupposed by things people say, very often you would actually have to invoke the law of identity in trying to argue that it was implied by what they were saying. How could you have an argument that didn't invoke it? Because all arguments are arguments. There are actually a number of different things that people mean by the law of identity. All arguments are arguments. One thing is that everything that has a property has a property, and another is that everything is identical with itself. Those turn out not to be exactly the same principle. But for example, even if you say all arguments are arguments, that's something that would be challenged by some logics of vagueness. If it's a bit vague which things are arguments, then there may be borderline cases for arguments. And there are non-classical logics of vagueness, and they're not ones that I accept, but they're ones on which you can't actually assert all arguments are arguments. So for borderline cases or vague cases, we could say whatever things are, they are exactly as they are. So if they're a borderline case, then they're still over what they are. I'll give you an example of like in metaphysics. It's funny because somebody actually was trying to demonstrate the limitations of classical logic when they gave this example. And they said, is George Costanza bald? George Costanza is in Seinfeld. He's this half-bald character. Well, is he bald or is he not bald? And supposedly this is supposed to say vagueness. Well, this is a vagueness of language, but it's not a vagueness of logic and it's not a vagueness of existence. So what we could say is, if we want to be precise, George Costanza has exactly as many hairs on his head as he does. He certainly has no more and he certainly has no less. Now, where the vagueness comes in is just whether or not we use a word, bald, which is just a subjective evaluation of somebody's, the amount of hair on their head. Then, yes, I'm fine with that being a spectrum. Some person could say he is bald, some person could say he isn't. No big deal. It doesn't change the idea that he is exactly how he is. There's no vagueness there. Well, even to talk about exactly how many hairs he has on his head may involve vagueness because if you pull a hair off his head there will be a point as it's sort of beginning to rip off where it's a borderline case of being a hair on his head. But I don't think so because the position of the hair is exactly where it is and it isn't where it isn't. So I don't think reality is infinitely divisible. I think that reality is fundamentally finite and therefore at that point where there's no contact, there's no contact and therefore the hair is removed. Well, whether reality is infinitely divisible or not is a matter for physics. There's absolutely no contradiction even in classical logic with assuming, for example, that time is infinitely divisible between any two moments is another moment. Well, that's a whole other thing that we'll have to talk about because I struggle with the conceptions of infinity. This is one of the things I'm talking about as I'm travelling around. I'm talking to people about infinity because I think there might be some errors in the way that there's been a mainstream conceptualization of infinity since Canter. I think there might be some errors there which is a presumptuous claim but that's what I think. But I do want to keep talking about vagueness if we can because I think it's directly important to claims about if you can have a counter argument to classical logic that isn't somehow a counter argument to classical logic. In other words, a counter argument that doesn't presuppose the law of identity. So it's something like, we said arguments, we're not exactly sure what arguments are. Well, what that means is there's some circumstance in which a proposition is not necessarily clear whether or not we categorize it as an argument or not. Well, regardless, it is precisely what it is and it isn't how it isn't. But there's like a taxonomic categorization to say this is an argument or it's not an argument, whether or not there's blurriness there really doesn't make any difference in terms of the laws of identity and non-contradiction. Let me give you one more example and I'm sorry, this is a monologue but I'll let you respond. An example I like to talk about is somebody standing halfway in a doorway. Is it the case that they're halfway in the doorway? Yes. Is it the case that they're not halfway in the doorway? Yes. Because you could see, if you were being imprecise somebody could say, oh look he's halfway in and it's not the case that he's halfway in and halfway out. It's vague. Well, we can very clearly clear that up just by precise communication. So there's no contradiction there. If it's true that they are halfway in the doorway then they are halfway in the doorway though they may have another half outside of the doorway. I think this is the same case in all cases of vagueness just precision clears it up. I agree with you because I'm coming from classical logic but I do think it's important to understand that there are alternative logics in the sense of alternative theories on which there can be vagueness in reality itself and in which it is not the case that one's always assuming that everything is what it is. What do you mean the vagueness in reality itself? That things could somehow be ways they aren't? It doesn't have to be a matter of contradiction. It can just be that... That they aren't some way. I mean the view is that it's indeterminate how they are. Yes, but isn't that an evaluation of our knowledge of something? So there may be some finite amount of particles that exist and we may not know what it is but that doesn't mean reality is vague. That is an issue about our knowledge and ignorance is that's my view of vagueness but it's not the view that some other people have and although I think they're wrong I don't think that their view just collapses into complete nonsense or anything like that. It's a theory that you can work with. As I say I think it's a mistaken theory but I don't think that one should underestimate one's opponents. Okay, so this is great. This is awesome. I'm really enjoying this because I have yet to have a conversation that has talked specifically about this. I'm writing a book right now on this topic and I want to run an idea about you in real time. So my claim is that existence by what we mean by the concept of existence implies perfect precision. It implies identity, that you cannot have existence without identity. So to the extent that we have existence it is certainly the case that we have existence and therefore we have identity. Existence is existence, therefore identity. So any claims that you could have existence without identity doesn't make any sense. What do you think about that idea? Well, again, I think it depends what you mean by doesn't make any sense. It is internally inconsistent. So existence implies identity. What do you think about that? Well, my own view is that it's true that whatever exists is identical with itself. And that's a standard classical view. But you could in fact have a consistent view. I think it would be a wrong headed one but it would be consistent in the sense that you couldn't just end up implying absolutely everything which said that some things are not self identical. And in fact, even within classical logic you can prove that such a view is consistent in the sense that you could develop such a view without ever just ending up just saying absolutely everything. It sounds like your criteria for determining incoherence is that you could say absolutely everything. My criteria for incoherence is some kind of internal contradiction. But when you gave the example of some things might not be themselves what I would say is what do you mean by things? Because what I think you mean by things or what anybody means by things is existent things. And if that's true then it implies they are exactly as they are. Well, what you're relying on there is a certain principle of classical logic. It's a principle that I accept. I think it is a principle of the best logic that we have. But it's important to understand that it is possible to formulate alternative theories. I mean, it's important to know what you're up against. I agree. And when we're talking about whether a view makes sense if you just say well that you're going to regard any view that implies a contradiction as not making sense then people will are likely to reply well alright you can use the phrase doesn't make sense in that way if you like but then maybe we should take seriously some theories that as you would put it don't make sense but actually it is possible to work with such theories because they don't end up saying absolutely everything. But it's possible to work with them in the sense that you can work with the assumption that 1 plus 1 equals 3. You can work with that in the sense that you can make all kinds of theories technically speaking you write all kinds of books but that doesn't mean that it makes sense. It makes sense at the very least in that it's meaningful and even 1 plus 1 equals 3 is meaningful. It's not just complete gobbledygook because its negation is true. It's true to say that 1 plus 1 is not equal to 3 but that can only be true if it's negating something which itself is meaningful and so that means that 1 plus 1 equals 3 has to be meaningful and so you can have these alternative theories which are meaningful even though they may be from our point of view ridiculous. The question is whether there's any reason for taking them seriously and in the case of logical paradoxes they do provide a bit of a reason for taking some of these very strange alternative views seriously because the logical paradoxes are really hard to solve and whatever is the correct view of them it's going to be pretty strange because we've tried all the non-strange things and none of them work so we have to be willing to take seriously some pretty weird theories just in order to do justice to the depth of these problems and some of these theories may involve saying things that are mad. In the case of logic I don't in fact think that we need to go all the way to contradictions but the case of infinity that you mentioned before that's a case where we've developed theories or cantor theories which have become part of absolutely standard mathematics which do contradict things that people regarded as self-evident before I mean for example that in some sense a part of a thing can be the same size as the whole thing and so we've had to revise our ideas about what's really obvious. Why wouldn't you conclude then that a cantor was wrong? Well I think the success of modern mathematics is a pretty clear indication that a cantor was not on the wrong track. What do you mean by success? Well it's both its role in providing the mathematical framework for the rest of science and cantor's set theory when once it's rigorously developed does not involve any contradictions I mean suddenly nobody's found any contradictions. Even something like a part being the same size as the whole you would say is not a contradiction. There's no contradiction there because it's not of the form something is the case and it's not the case. You cannot derive anything. But it's conceptually contradictory so what we mean by the term part and what we mean by the term whole implies unequalness. Well unequalness in the sense of it's not their identity but it doesn't imply a difference in size and what cantor did was to take apart a number of different things that were lumped together in kind of vague ordinary discussion of parts and holes and bigger and smaller and so on. He separated them very carefully and showed how we can reason in a completely consistent way about as well the different components of them and we now have the theory of finite and infinite sets and we've also got logics of part and hole and when we put all these sorts of things together we don't get any contradictions and we can maintain that there are things which are the same size as one of their parts that's less than the whole and I mean there are simple examples of that so that if you take the natural numbers 012345 and then you take the even numbers 02468 I mean they can be put in one to one correspondence with each other the same number of each even though the even numbers are just a proper part of the natural numbers So let me ask you, I do want to go, I have another question I want to ask you about certainty but this is mathematics so relatable to this If it were the case that there could be a logic which did not allow for or which could demonstrate the errors in Cantor's set theory and specifically his diagonal proof do you think that would be superior in terms of logical palability to the conclusions that we find in set theory so for example, this is my honest belief that I think the way that Cantor went about doing that does not understand the nature of numbers so Cantor was a Platonist so the way that he's formulating his questions take all the natural numbers I don't think you can do that, I don't think numbers work that way numbers are something that are conceptual so you take exactly as many numbers as you conceived of you can have certain properties and there's no infinite here so the diagonal proof doesn't work because in a sense all sets are necessarily finite because they are conceptual, they aren't existent out there separate of our conception of them and we can't conceive of an actual infinite so therefore as he's working through his diagonal proof and every one of the lines it has dot dot dot and I have an issue with the dot dot dot because it implies something that there's an actually existent infinite amount of numbers out there and I don't think that's the case I think numbers are conceptually created and by conceptually created you think that they're created by our minds or something yeah, they're concepts one is not something that's out there one is a concept I think there are ideas and there are what the ideas are of and I mean there's the number one and there's a concept of the number one but there's a distinct I don't think they are I think that is the presupposition of Platonism that one is out there separate of our conception of it and I don't think that makes sense I think it's the same thing the analogy I use just where in England is Harry Potter does what is the ontological status of Harry Potter well if nobody conceived of him he would have no existence he has no existence out there and to the extent he exists it's only as a product of people thinking but J.K. Rowling's in and out and writing thinking about him he exists in the conceptual world we can say things like Harry Potter has circular glasses yeah that's a true statement but we have to bracket it and say in terms of the mental world if nobody can ever conceive of him that wouldn't even make sense the same is true I think of numbers numbers have the exact same type of existence as fictional characters they do not exist when we're not thinking about them yeah but there's a difference because it would be ridiculous to try to use the fictional character of Harry Potter to explain what happened a million years ago before J.K. Rowling ever wrote the books but in the case of mathematics we use numbers in doing physics and we use them to explain events that happened millions of years ago yes but these aren't arbitrary concepts it's not fiction maybe I'm glad you brought that up maybe I won't use that example because that implies something I don't mean to imply that they're just capricious that it's just some fanciful notion that we come up with that's useful I would say they're conceptual but their grounding is in logic one is it's a logical concept amount quantity is something that is a logical concept that can have an actual direct reference to existent phenomena in reality so if I say there's one microphone here that means something very very concrete and we can abstract from it so we can say one-ness is nice to think about in the abstract you can apply it to all sorts of different areas even a million years in the past but it doesn't exist when it's not being conceived of it's a conceptual logical tool so this is where I think the project of Bertrand Russell was justified in trying to underground all of mathematics in logic I think that's true I think all of mathematics follows from logical principles Yeah, but you're on dangerous ground in invoking Bertrand Russell because he said that to do logic properly you have to have a sense of reality and the kinds of things that we're dealing in logic in mathematics have as much reality as we're dealing with in zoology I completely agree, yes, this is funny I was just talking to Dr. Isaacson about this that I think the, this sounds contradictory but I think my own philosophy incorporates logicism and intuitionism which seem like they're mutually exclusive but I think there are good parts in both like the rejection of infinite sets and in terms of the metaphysics of mathematics being something that's conceptual but unfortunately the intuitionists thought that that meant the laws of identity as well as just this conceptual thing as if we come up with it and then it's not referenced to something that's out there in reality I think that's mistaken so in my own view okay well let me ask you this rather than diving into that let me ask you this if it were the case that what I'm saying is accurate about the metaphysical nature of numbers that I think it necessarily implies that there's not an infinite amount of them in the sense that there's an actual infinite it means that you can conceive of a number greater than a number that you've already thought of yes certainly but that doesn't mean that there's an actual infinite out there and I think it does imply that in no sense can you have a hole that is the same size as one of the parts of the hole and it also implies you can't then therefore you can't have different sizes of infinite sets because that idea is not even sensible don't you think all of that all follows from this particular physical conception of mathematics well it's hard to say what follows from it because I think you're oscillating between conceiving of something the something that you're conceiving and then the conceiving of it but those are different things you have to make that distinction for mathematical thought just as for any other kind of thought there's what you're conceiving and this is the conceiving of it so what do you think about the existence of when we're talking about Harry Potter does he exist? there's a fictional character Harry Potter who is not a boy but is a kind of control construct that was created by J.K. Rowling what is it's existence if nobody would have thought of that construction well then it would have been a merely possible fictional character what is the status of a possible fictional character what is the actual status of one is it existent or not depends what you mean by existent but it's self identical yeah but it's not and it wouldn't be anything it would be something it would be a possible fictional character so would you say that all possible fictional characters have existence have some type of existence a thin logical sense of existence really all so the books that are written a million years from now with characters that we know nothing of they have some type of existence to them yes but you have to understand that existence doesn't have to be in space and time it doesn't have to be concrete but where is it well you're saying where but I just said it doesn't have to be in space so you think that future concepts that have never been conceived have a non-spatial existence all future concepts I'm sorry I don't mean to laugh at you that doesn't that strike you as absurd no all possible conceipts so the well everything look everything exists I mean that's I mean Quine made this point that's trivial and if by exist you just mean being something then whatever there all of anything exists because everything is something so a thousand years from now somebody who reads the Harry Potter books and they misread them and they think Harry Potter has square glasses that concept of Harry Potter with square glasses that nobody would ever have thought about prior to that conception of it even if it's 10,000 years and take any misreadings of Harry Potter that particular concept with one little the shoelace is tied differently that still has a real non-spatial existence so a misreading is something that happens and that was a possible misreading all along so that is one theory but wouldn't you find it much more persuasive or comfortable to say it is not the case that all possible concepts have any type of existence that concepts do not exist prior to their conception and they do not exist after their conception entirely dependent on our active conceptualization of them but if you're saying there are some concepts that don't exist is that what you're saying? No, I'm saying at the point of conception that concept exists so it's like there is no concept which exists separate of it being conceived that's what I'm saying by virtue of what we mean by a concept I would say No, I don't think that's that's not right I mean a concept is something that it's a way of conceiving something the concept of London let's say I mean it's a way of thinking of London Yes, but there's no ways of thinking of London without the mind so you wouldn't have the concept of London without thinking of London It was still as were the additional possibility of thinking of London was there all along In a real metaphysical way has some type of real existence This is a I feel like we could talk a whole series on that Okay, one more question though This is just a hard geared shift back to what we were talking about before When you were saying we have to entertain the idea of even potentially giving up the law of non-contradiction it seems implausible we have to enter Does that mean in your world view there's no room for certainty There are plenty of things that we can be confident of that we don't have to worry about to entertain doubts about If that's enough for certainty then certainty is fine but if by certainty you mean that there are some things that in no circumstances would it be reasonable to doubt them then I don't think that there's anything as certain in that sense For any proposition that you like there are circumstances in which a reasonable open-minded person would doubt them Okay, so can I rephrase that would you accept this rephrasing that your position is certainty as understood as 100% confidence in the accuracy of some proposition of any proposition and the belief that there's not even a conceivable possible way of proposition to be false is that what you're saying you reject that idea If what we're talking about is a kind of dogmatic disposition such that whatever came along it would not tempt you to call something into doubt I think that kind of disposition is not an appropriate one to have if somebody were to say let's say I were to claim Dr. Williamson is dogmatic believes that you can have absolute certainty as the way I just described it you would say this is a false claim would you say you're certain that that's a false claim No, I'm not certain that it's a false claim I mean that particular claim that I'm not dogmatic I think all of us should worry a bit about whether we're being dogmatic So I guess what I'm getting at is for your own internal state your own internal state of mind so you're having the perception of being in a conversation right now if somebody were to claim right now that you are not having the perception of having this conversation right now would you say you're certain that they're wrong I'm certain by ordinary standards that they're wrong but you're not absolutely certain I think what you're probably building into the idea of absolute certainty is an absolutely dead set disposition never to call something into doubt and well in the case of the claim that I'm having this conversation you know because from now I may be inconsiderable doubt about whether I had any such conversation but that's not ten years from now I'm saying right now so you have some quality to your experience you know let's say you ate a strawberry and you're having the experience of eating a strawberry I'm saying is can you be absolutely certain that you're having that experience not necessarily that the strawberry exists or anything like that just that the contents of your perception exist in the present can you have certainty about that well you can know such things and I mean you can be reasonably confident of them you don't need I mean there's no special reason for having any doubts about them but if you mean that you can have a disposition such that under no circumstances will you entertain any doubts about them for example you will not for a moment take seriously some some philosophical view on which consciousness is some kind of myth or whatever then I think that that is not a good sort of disposition to have I think one's one's got to have a disposition to that allows one to question things that one has been completely confident of of course if one just if as soon as anybody just raises a question one throws oneself into terminal doubt then I mean that's obviously a bad disposition to have but the ability to see things from a point of view that is inconsistent with your own is an important one to have what about the middle ground the middle ground being for virtually nothing I am absolutely certain however there is a small limited set of things which I have absolute indubitable certainty one of them is the contents of my present experience couldn't possibly what I would say is it couldn't anybody that were to claim that it is not the case that I'm having some perception right now is necessarily wrong it is certainly the case that perception is a real phenomena that is happening in the universe I'm certain of it and I couldn't be wrong about it because I'm having I have direct knowledge about that and I would also say that that kind of certainty applies to certain mathematical truths and logical truths but outside of that there's not many areas where you get that kind of certainty do you think that's an dogmatic or unhealthy way of thinking about philosophy if you take the case of logical truths there are plenty of logical truths that I'm quite confident of I think I know and I think lots of people know them but at the same time I don't just dismiss out of hand theories like dilithiarism which question or reject those logical laws of course there's no point in being interested in questioning that doesn't give any reasons for questioning because anything whatsoever can be questioned and we'd never get anywhere if we just question things for no particular reason but the case of logic has shown that quite intelligent reasons can be given intelligent well informed reasons can be given for things that we thought were indubitable and progress has been made by a willingness to doubt things that we previously took for granted and so a good intellectual disposition to have is one that is willing to question things when intelligent well informed reasons for questioning are provided. Of course it may be that the things that have been questioned are things that actually are true and on reflection we realise even that we know that they're true but the thing is if you have a disposition to hold certain things above doubt no matter what happens the effect of that is probably a whole lot of false things that you believe that you're going to hold beyond doubt I mean for example in the days of slavery people might have held the proposition that slavery is morally acceptable as something that was beyond doubt. Well I think on that note thank you very much for talking to me this has been an awesome conversation well thank you I've enjoyed it Alright so that was my interview with Dr. Timothy Williamson of Oxford University if you guys are interested in what we're talking about check out the show notes page this week steve-patterson.com slash 16 I'll have a link to Dr. Williamson's books my upcoming book and I'll also have a link to an article that I just wrote the longest article I've written on my site about a topic that we mentioned here which we talked about mathematics and this guy Georg Cantor who revolutionised the world of mathematics by claiming that there were infinite sets and there were in fact different sizes of infinite sets that some infinities are bigger than others my own evaluation is I'm not persuaded by that argument in fact I think his argument is pretty poor and contrary to the very kind and friendly disposition of Dr. Williamson I take a very aggressive stance against Cantor's claims so that sounds interesting to you that will be also at the show notes page my own attempt at refuting Cantor and make sure to tune in next week where I'm going to be doing a breakdown episode on this interview because there's so many nuggets of wisdom that need to be highlighted and expanded upon and if you liked this interview and you want more discussion about rationalism and logic then do me a favour and check out my Patreon page you can support this show by going to patreon.com and pledging one dollar of support whenever I post new content like this it helps make these interviews possible plus you'll also get a free copy of all the books I've written and all the books I will write in the future alright that's it for me I hope you guys enjoyed it and have a fantastic week