Introduction (contd.)

0:00 The problem I'm interested in, and I think anybody is interested in, is the problem of how you might describe a world in which there is fundamentally only one kind of stuff. You might think that the world consists only of elementary particles, and yet you want to describe it in terms of more sophisticated things. Relations or opportunities to do possessions of minds or in the mind. There's nothing in physics that makes it easy to do things in the mind or in the mind. It's the linguistic term. It's natural to try and explain the endeavour to think in these non-monistic terms in terms of... to explain a way of understanding in terms of the linguistic term. To try and do that in terms of languages. What I've answered is a particular project I've been interested in. It's a project of understanding how one might believe, say, that there are minds, or that there's, you know, there's, it might be a good example, how the world can contain minds, and how those, you know, those, and to understand this in terms of translating a language describing a world that contains mentalistic. Properties, mentalistic properties. Translating such a language into a language that only contains the language of physics. Now, interpretations between translations between one language or another are, if you do your interpretations properly, these are mathematical objects. Ever since the last hundred years or so, theories have been mathematical objects, languages are mathematical objects, and translations between them are mathematical objects. So the endeavor to understand this philosophical process of reduction or emergence in terms of translation can be seen legitimately as a mathematical act.

2:30 I'm not the only magician in this. I have a professor, and he has written about this. I don't think I have a professor who is a philosopher. He writes in the category of philosophical significance. His work is not flagged as much as it should be. The particular example that I'm interested in, that I've studied in some detail, is really what one might consider a toy example. I'm interested in very, very simple things. I'm starting off with a language of sets, a language which just has variables involved in the preference of sets, and interpreting into that a language that describes numbers as well. If you don't like symbols, you might like it. Now is the moment to go to sleep. Is this visible? Can you see it from the back? Yes? This is the language of epsilon. Epsilon is the membership relation between sets. And I'm interested in the endeavour of interpreting into this the language of cardinal arithmetic. Now there are two very, very different ways in which one can approach this. There are two words that I would like to introduce you to. An interpretation is simply a map from this language to that language, but also, specifically, special kinds of interpretations are called implementations. In this language, we're talking not only about sets, but also other kinds of objects, in particular cardinals. And what do we know about cardinals? Two sets have the same cardinal, if and only if there's a bijection between them.

5:00 Now, you may or may not, in doing a translation like this, wish to implement cardinals as sets. You may wish to say, for example, that the cardinal number three is a collection of all sets of a couple of numbers, and cardinal number one is a set of all sets with one number. You may or you may not. If you choose to do that, then we say you have implemented cardinals. So you have a property in here, a building of cardinal numbers, such that two sets of the same cardinal numbers can only be valued at equal. But actually you don't need to do this. And I think it is quite important to remember that it is possible to make sense of cardinal numbers by making sense of talk about cardinal numbers without believing that such things actually exist. This language here is going to have the same variables as we have here, variables that range over sets. It will also have variables that range over cardinal numbers, which I'll write as Greek letters. It will have operations on those things, plus, times, and it will have a cardinality operator. The cardinal number of x equals alpha. It can say things like that. Now we want to know how to interpret this language into that language. And we do this as follows. Variables that range, variables of the old style simply get interpreted as themselves. The translation from this language into this language will send set variables, these things like x, to themselves. It will send variables over cardinal numbers. I've taken these variables in Greek letters, simply send them to the corresponding lowercase running, And then I will have to prepare things like alpha equals beta. How will I translate the equality symbol between cardinals? That will have to go to A is equiponed to B. But this relation is the relation that says A, there is a bijection between the sets A.

7:30 And if I want to say something like alpha times beta equals gamma, that will go over to A. Cross, where this cross here is Cartesian product, is equivalent to, I think, is the lowercase Roman letter corresponding to Gamma. Now, in this kind of interpretation, what have we got? We have here a language that talks about cardinal numbers, and we have interpreted it into a language that only talks about sets. This account of cardinals has no ontological commitment, whatever, beyond the ontological commitments in this language here. And the reason for that is, if I have a standard Tarski-style semantics for this stuff, and I interpret my binary relations as sets of ordered pairs, and I have assignment functions that take variables to elements in the domain, I can simply compose that semantics, which I think of as a function or family of functions, I compose that semantics with this interpretation on the right, and this gives me a semantics. So ontologically, this is very, very important, but the cost arises in a different way, because although what we can say about cardinal numbers, we can say certain things about cardinal numbers here, in an ontologically cost-free way, which doesn't commit us to the existence of cardinal numbers. But we're very restricted in what we can say. And here, I've interpreted this equality sign between two variables of type cardinal. I know that that goes over to this equipollent symbol between the corresponding variables. But it doesn't enable me to make sense of alpha equals x, where x is an old-style variable.

10:00 So the semantics, the ontologically free semantics for this language... Which we inherit from this one restricts us in what we're allowed to say. So let's look at the things that we are allowed to say and contrast them with the things that we're not allowed to say. We are allowed to say that two cardinals are equal. We're allowed to say that one cardinal times another equals another. I can save myself a bit more space on this board, make sense of things like the art of lessening and depleting. What do I mean by what does that get translated into? That simply becomes there is a function from a into b, which is the variable alpha has to become the variable a, and the variable beta has to become the a. So in this context, there are various things I can say about cardinals. I can persuade myself that in some sense they exist, but the things I cannot say, for example, are equal, like this thing here, equality between a cardinal and a discerner. That simply doesn't make sense. It's not to say that we cannot make sense of it, but merely that this semantics, the ontologically cost-free semantics, doesn't give us an account of the meaning of things. And given the theme of identity in this conference, and another point worth emphasising, I wrote a formula like this. The output in the interpretation, the variable output becomes the variable a, and the variable beta becomes the variable b. And cardinals are just sets, but they are sets with different identity criteria, maybe a physical rule, or a new rule, a law, or a particular rule, a law, okay.

12:30 On this account, there are cardinals, but we are restricted in what we can say about them. And, and this is the important point, the identity criteria for cardinals are not the same identity criteria for sets. If I want to think in here that everything is being a set, while making the variables that support the range of other cardinals, I have the feeling that everything is a set, and when I want to talk about cardinals, I use this different style of variable for them, but my identity criteria for these objects, when I think of them as cardinals, is different from my identity criteria for them as sets, but I'm thinking of two objects as cardinals. Then I think they're identical, if and only if, as sets, they are equi-posit. And let's think a little bit more about what... Can I remove this picture now? Think a little bit more about what we're not allowed to say on this account.

15:00 I'm not allowed to say that something like this doesn't make sense. The semantics doesn't give us an account of the meaning of expressions like this. The function of Euler's, Euler's torsion function, which is so important in the RSA, I think, Euler's torsion of n is the number of n, the number of natural numbers n, so that n is less than n and 5n is, consider this set, the set of natural numbers that are less than n but are prime to n, take that set and consider its cardinality. Now, it's standard that if n is prime, then 5n is just equal to 5. But consider this. This is apparently an uncontroversial piece of arithmetic of natural moments. What I want to warn you about is that on the account I've just given, this actually doesn't make sense. Because the cardinals that emerge where this big L was, these are cardinals of sets. Then this thing here isn't just an innocent set. It's a set of cardinals. And in my original theory, I just had sets of sets. There were no cardinals in the original. So this thing doesn't really make sense. I've got to equip the theory, I've got to enhance the theory, by adding to it gadgetry for thinking about sets of cardinals. Now, here I come across a trivial, a disastrous problem, namely that there aren't enough fonts around. What I want is a new, I want a new font so I can have a variable as it might, I mean a Greek A or a Gothic A or something.

17:30 A new style of variable for arranging over sets of coordinates. So, variables arranging over sets. Let's use capital Roman letters. Very annoying that capital Roman letters tend to be so often the same as capital Greek letters. Extremely irritating. But let's suppose I've got an expression A and B, where A and B, these are variables that range over sets of cardinals. How am I going to make sense of variables ranging over sets of cardinals? What is the identity criteria for sets thought of as sets of cardinals? And if you think about it for a while, it becomes clear that the correct notion for this is that A is equal to B. A is equal to B as sets of cardinals if for every member of A, thought of as an ordinary set, there is a B in B such that A and B are the same size, and for every B in B there is an A in A such that A is the same size as B. In other words, any size of a member of A is a size of a member of B, so the cardinals that are manifested that appear as sizes of members of A also appear as sizes of members of A. It turns out that that gives us a smooth running count of equality between sets of parts.

20:00 What I have to do now, I have to have a notion of a bijection between two of these things. That's what it is for them to be identical. What would it be for them to be bijected as sets of cardinals? It would have to be, have a relation between A and B. If it relates, something relates to something in B, it must relate to something of the same size as B. If it relates this guy to him, and this guy is the same size as this guy, then it must relate those two as well. And if these two, and if, and if, and in fact that must be a diphenomia. If I have somebody here that's related to this, and these two guys are not the same size, then I can't have this activity, but I should rethink about it. And once I've done that, I have a notion of bijection between sets thought of as sets of cardinals. That means that I can make this abstractive move again. Cardinals and cardinals of sets of cards. And I can do this as often as I want. And I say the chief difficulty in writing this act is the lack of fonts. There just aren't enough alphabets around to be able to do this smoothly. So what is the picture that emerges from it? We have a... all of this is ontologically costly. This gives us a way of thinking about cardinals of sets, cardinals of sets of cardinals, and so on. Because no, we don't commit ourselves to the existence of these entities.

22:30 We've achieved a state of effects in which we can say there are no cardinal numbers, there are only sets. But there are facts about cardinal numbers. Facts about cardinal numbers are simply facts about sets. Now, I can imagine that many of you will have an appetite for this kind of mathematical analysis. So what are the morals of this philosophy in general? What other situations are there in philosophy where one might want to use an analysis of which this is a toy version? Put back historically, there was a time when people thought that living matter would have been sort of stuck in non-living matter. So you had a radically different vocabulary for describing living matter than non-living matter. And moreover you thought there was no possible interpretation of the discovery that... Living matter was actually just non-living matter behaving in a special way. It was a huge liberation. And nobody nowadays thinks that an analysis of this kind is not in principle possible. So we nowadays have no difficulty that the world really only consists of non-living matter, basically. That we can interpret all the things that we want to say about matter that is alive in terms of physical laws. So vitalism has gone away, isn't it? We expect to have an analysis of this kind that explains mental phenomena in terms of physical ones aren't at all complement to the prospects of equipment and analysis of this kind.

25:00 Appendices, I think, is, of course, you cannot deduce how important it is. Now, I thought for some time that the plausibility of this training field arises from intuitions by the interpolation level. If you have a language that just contains ises and doesn't contain any oughts, then clearly there are an application of completeness to it, and you cannot derive any moral judgment from it, whether that's just a simple fact about the nature of logic and language. So clearly you can't do so much. But the interesting question is, can you perhaps interpret oughts as ises in such a way that you can reason sensibly about oughts on the basis of what you know about the physical world? I said in my abstract that I wanted to connect this with the ideas of the Vienna School about categorizing things. And I think in this case I want to return to a point I was making earlier. The maps that you get from the later languages, from the earlier ones, that you compose with the maps that you use as semantics for this, do not touch certain kinds of problems. Because this thing here is a cardinal of a set of cardinals, because this thing here, you can't sense a certain equality between them. Here again, this is the ball example, and the equality between the variable ranging over cardinals and the variable ranging over sets is not given any sign.

27:30 Now, if you look at this with a straight line, but the entities that you describe in this way, Widgets, Womats, whatever it is, or objects of some kind, which arise from an intelligence. They have a kind of restricted kind of existence because there are certain things that you're not allowed to say about them because the semantics isn't there. The semantics doesn't make sense. And so, just to use Carnap's famous example, the stone is thinking about the enemy. The stone, Carnap says this is a cataclysm. You can't ask, you can't sensibly ask whether or not this stone is thinking about you. I'm sure he said it in German. Carnap's point was that this assertion isn't false, it's that it's meaningless. Now, on the analysis of this stuff, we have a very good explanation of why things like this appear non-standard. They appear non-standard not for any metaphysical reason, but for logical reasons. There is simply no way in which this account of their meaning provides any semantic support. I think I might leave it there. The point I want to make is not that this particular example of interpreting cardinal arithmetic inside set theory is an important one, but that the technique that it appeals to is something that can give a way of explaining to us How they can be sensible enough to talk about entities of a certain kind, even though those entities don't exist, and moreover explain how, if you want to think that those entities exist, they do have this rather strange metaphysical nature in that there are certain things you can say about them and certain things that you really can't.

30:00 So I think this is an interesting opening of our debates, André, and afterwards... And then, the question is, actually, and then that leads of course to some friction on the, what do you mean by that, but still, this notion of competition, that's actually the question, how much, I don't see that there might be more than one rate of competition, right? And of course, much, very much depends on what exactly you require from the other competition. Yes, I mean, point one, the composition is not problematic. You're right, there are various ways in which you can do it. I'm particularly interested, I mean, at the beginning of my talk, I wrote down two words here. They're probably gone now. They're implementations and interpretations. I'm particularly interested in the kind of implementations where you can do that. The question is, do you interpret the equality in this language between the new entities as equality here or not? That's a key question. Because if you interpret equality between the new entities as equality, then what you've got is an implementation. You've decided that the cardinal numbers are particular kinds of sets, that there is a cardinal number 3 and it is a particular set. And in fact, that is the way in which it's usually done, but I think it's very important to bear in mind that you don't have to do it that way, that you can make sense of talk about these entities without thinking that they are entities of the old kind. So yes, you're right, there are many ways.

32:30 Can I write something down in order to illustrate my question? It's a very simple point, you know, just that suppose I'm posing as a pure logician, I'm not a philosopher, I would say that, look, I have a language with sets here and I have a theory here, say set FC, I have this theory here too, perhaps with some accommodation, and then I must have some axioms for my cardinals, right? Yes, because I have a cardinal theory, let's say, okay. The real key here is that what I can do is I can change the formation rules for this language slightly so as to allow this into my set of well-formed formulas. And then I add an axiom to k saying that for all x. I'm adding this axiom. So you have two sorts of language. So what I've done, I have not changed the lexicon of the language. I haven't changed any of... I haven't added... I have only allowed this as a well-formed formula and added this as an axiom. Now, I suspect that the theory I am obtaining in this fashion will be a conservative extension of what I had. And if I then dig my heels into the ground and say, no, this is my theory, then your objection will... Well, it's another website or something. Well, is it the same thing? Yeah, that's really the question. Because now I do have a semantics for this, but all the models will probably be the same.

35:00 But I do have a semantics for this, and this is now allowed. So I'm not allowed to say it's a category theory. You now turn it into a two-sorting theory, or an axiom that says that this is a two-sorting theory. But it was already a two-sorter theory, because you already had sets and cardinals. I haven't changed that. The question of whether or not these quotient sets are sets is incredibly sensitive. So your choices are set here and there. Because if you start off with ZFC, then the quotient sets aren't sets. They are set locally, but globally they are quotients. But then of course, the whole point of this analysis is that you can make sense of talk about cardinals without taking cardinals to be particulars. It is certainly true that most of the ways, most of the ways, every way, that the cardinal number 2 is a particular. Each card number is a particular sector of knowledge. The point of this analysis is that you don't have to do it, that you can do it without taking it, you can do it in a virtual way, but that there is a cost, there's a cost attached to it, and that cost is the category distinction.

37:30 Intuitively, the first one who studies a conditional probability thinks that it is not a probability of the whole universe. And the one who studies a cardinal thinks that it is not a set, that it is a state. The second state is to say that a conditional probability is the probability that is determined by the actions of the module. And a cardinal is a set of quotients. The difficulty I have with this is not that you're speaking in French, the difficulty I have with this is that I know that I don't understand probability. There may be useful parallels between the two cases, but I'm so terrified of what you're going to say. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. And this is the first approach. So that's the implementation and interpretation of structures not appear to be defined, you see, by a general approach to the problem, then it should be, I would say, as best as it could work, the fact that I went to a plant because it was a little bit of a joke. So I'll say, that is something that I know I don't understand. There are many things I don't understand. I liked your talk, and I think I followed it, but I got most interested in you getting to the end, when you wrote Hume, and you started thinking about opening up in a different direction.

40:00 And, to be honest, if my first proposition is a misunderstanding, then I'll just drop the whole question. But I think you said... That even if there weren't any cardinals, we could say interesting things about them in terms of physics. Okay. I think you then went on to say something like, even if there weren't any orcs, we could say interesting things about them in terms of isids. But at the very beginning of your talk, you said something about minds and the material universe, and I wondered if another step would be, there aren't any minds, that we can say interesting things about them in terms of material. But I think you're beginning to get into trouble there, and I think people like Churchman and so on do. I think we could actually change our vocabulary and talk about, you know, material properties as expressing what we mean by mathematics, but I don't think that ever really worked, and I just wonder if there would be something that... I mean, the semantics is fascinating, but... If I'm talking about cardinals in terms of sets, do I have to sort of keep thinking that there aren't any cardinals? And if I talk about odds in terms of indices, do I have to keep thinking that there aren't any odds? And then if I talk about minus in terms of material, do I have to keep thinking there aren't really any minus? Because I think the way we normally talk about these things, we assume that the things we're talking about in fact do have a kinder existence. I think you allowed for that, a kinder existence. Could you say more about that? I don't want to, I mean, I suppose I like to think that I'm a physicalist about philosophy of mine, but, and this gives an indication of the way in which a physicalist analysis might work, but if it doesn't work, that's cool too, I mean, but my feeling is that an approach like this is, you know, you... If you think that analyses of this kind are possible, then you've got to take the possibility seriously and you've got to attack it very hard. I think this is the way to do it. This is, as it were, a paradigm. Even if it doesn't work, I think we might find the exercise helpful. Well, I think pursuing the analysis would be really interesting. I think it would beautifully set together the level. That is a particularly simple case. The interesting thing about this is that it's quite clear that people who do mathematics are categorists of computer scientists.

42:30 If you show them this kind of analysis, they immediately recognise it as something that they always know, really. A very striking thing about this kind of analysis is that it's almost complete absence from the literature, but it's ubiquity in the folklore. All categories matter. All magicians matter. But in other words, it's enough. And yet this is the kind of thing that philosophers need if they need to make a serious-minded attack. I have seen one more question. In this presentation, we will discuss the importance of the method of implementation, and the importance of the method of implementation in terms of the linguistic approach to the whole thing, the method of implementation, because the method of implementation was derived from the material, from the sneak in, which sometimes results in questions. You also said that there is both. On the one hand, things are neutral, but you do lose the present power if you choose the implication one. So my question is, is there a reason why you think that people went rather the other way? Are there other reasons why implementation has been taken over? That's a very good question. I think, and I may be quite wrong about this, the impression I have is that humans like to get their hands on things. They like to believe that the things they're studying are real. So if you offer some of those accounts in which they can talk about widgets or loci or whatever they are, while pretending that these things don't exist, then they must be authors. Humans really, I think it's something to do with the architecture of human beings. I think that all there is to it. And I may be quite wrong about it. So it's not even a point of wanting to name their bodies, but how they're viewed as human beings. I don't think so. No, but let me say, I mean, to take the example that you give, is that the way that order pairs in section, order pairs any math that you need to, an order pair is not a set. An order pair is a mathematical logic, so it's not a set. Now, in fact, you can do an awful lot of stuff with it, which is quite, by taking...

45:00 All of these things are structured with facts, and never saying that these facts are the basis for the topology, algebra, or quantum or algebraic algebra. It doesn't matter. And moreover, if you ask any secularist to know what mathematics is, they'll say, of course it does, basically. And yet, you will find that any good concept theory or definition of mathematics is just because people like to have these things known about them. It doesn't actually serve any purpose at all. I think that the good construction of L is the mind of the primitive publishers. They do actually trade on the fact that they're really passive in the process. I think it's just done for the fact that he constructed it. That's the only thing. If you've got another explanation, tell me. It still comes to kind of, I don't know, I think there's one choice between whether to define, say, world affairs, you know, for a lot of the professors, you know, that's really right, and then of course you have to start with mathematics, or somehow change the basic things, you know, we should somehow choose the world affairs, kind of, the world affairs, and then people start to talk to you in different ways, so it's still... I think it came back to this question of losing expressive power. If you take all the pairs as primitive, as a primitive operator, then you do lose expressive power. You can't ask whether or not the ordered pair has three methods or has five methods. You can't ask separate questions about it. But then you never want it. The kind of set, the expressive part that you lose is the expressive part that you don't want to lose.

47:30 What could you say a little bit more about the difference between interpretation and implementation? Because if I recall correctly, Tashki's sort of definition of what an interpretation is, just isn't implementation. I don't know Tashki worked for this. In my book I distinguish between, among interpretations, I distinguish between implementations and things that I call vacants. So an interpretation is simply a map of the life time between the cycles in the century. An implementation is one that takes equality between the new variables and takes it to equality again. And most interpretations are of this kind of mechanism. You can do an awful lot without... Okay, I think we can... this is nicely... first rich and interesting contribution. And it's time for our second speaker. I have a nice announcement to make. Grant Rees arrived much earlier than I expected, and he succeeded in staying more or less in the technical suite of climate, so I hope we can give this presentation for the moment in what's perceived to be the most important. Okay, but I would like now to invite Jean-Yves Béziers to come. Thank you very much. Thank you for your attention.