Can you be sure the number 3 isn't Julius Caesar?

0:00 A hotbed of reading logicists. And it's not really quite correct. I mean, it's rather in one of the main logicists in Glasgow rather than in St. Andrews. And also, there are other members of the St. Andrews School who are not logicists. I'd rather like to characterise all this as after truth, and some of us are logicists and some of us don't. A century ago, Russell was profoundly optimistic about the process of fascinating mathematics and logic, and sharing this model like one of them. An example of an optimism. He thought, when you discovered how to derive that number theory, It comes to you with all the overwhelming force of revelation, like a palace emerging from the autumn mist as the traveller ascends an Italian hillside. The stately story of mathematical edifice appear in every order and proportioning the new perfection in every part. And he thought that if in fact it could be demonstrated that mathematics could be a whole new era in philosophy will open up. If this could be achieved, successfully accomplished, he said, there is every reason to hope the near future will be as great an epoch in pure philosophy as the immediate past has been in the principle of mathematics. Great triumphs inspire great hopes, and pure thought may get achieved within our generation. Such results will place our times, in this respect, on a level of the greatest age of priests. Well that's what Russell wrote for mathematics and metaphysicians in terms of what can we say now about logicism as a human to make a sense of it. To answer that question I'm going to go back and just briefly think about what Frager did in his Foundations of Arithmetic. In the foundation of arithmetic, Fringer sought to explain how we can perhaps finite cardinal numbers by deriving the general laws governing those numbers from logic and a general definition of the term cardinal number. And he thought that any such adequate definition of cardinal number would be of identity conditions of cardinal numbers. And therefore he made his first proposal. His first proposal was that we derive arithmetic from Q's principle, a principle that says the number of F's equals the number of G's, if and only if the F's and the G's are in one-to-one correspondence.

2:30 However, soon he became dissatisfying with it because whilst it tells you whether or not the number of X is identical to the number of G's, the number of Wounds of Mars is identical to the number of G's or Jupiter or not, it feels to tell you anything about whether or not the number of X is Caesar. simply doesn't tell you, it tells you the necessary and sufficient conditions that one number is identical to another number, but not whether our context is described differently in any particular ways, or identical or distinct from anything so described. A figure therefore abandoned this principle as a foundational principle, consult thought to derive it from a more fundamental principle governing extensions. The famous axiom 5 tells us that two extensions are identical, their relevant concepts are extensive. Of course that project was fated to the catastrophe, as it turned out the fundamental principle of governing extensions was inconsistent. When Frager realised this, he struggled away for a few years, but it was eventually led the view that logic could not cast any The neagogicist says that Flinger's big mistake was to think that the problems that confronted this principle, Hume's principle, were indeed insurable. In fact, they're not. They can be solved, and there's no need to dig down to any more fundamental principle liable to contradiction, such as accident time. So to avoid the contradiction, the neologesis sticks with Hume's Principle. That's the fundamental principle of a number theory. And that's adequate identity conditions to cardinal numbers. What I'm going to do today is to assess whether or not the neologesis is correct in that claim, whether indeed we can use Hume's Principle as a fundamental principle governing a technical object in the world. Now, I think it's important to assess whether Hume's Principle can serve as a foundational principle in that sense, that we look at some of the background assumptions from which the Neo4jian argues, which are the principles which we have on the first page

5:00 in our handout. The background assumptions reverse a more familiar step of the theorising, According to which language and reality lead a radically distinct life, with a naturalistic perspective, which is familiar with us from people like Armstrong, Lewis and Villa, who say that really what's in the world is there independently of language, and you can speak a certain way, and there's no guarantee that there's anything in the world corresponding to your existence. That's it. The mock methodology of the Neo-Phredean, our neologicist, runs precisely counter to that. Rather, the neologicist thing is that insofar as you speak truly, there must be objects and concepts in the world answering to the way you speak. The world can't fail to mirror the contents of our language in that sense. I don't get with more substance that's a rather imprecise mirroring thought. Really I think the Neo-Frame thesis can be divided up into two components. What's usually called the syntactic priority thesis and what I'll call the method of abstraction. Two combined provide the underlying methodology of the Neo-Rochism. Syntactic priority thesis is what guarantees the mirroring effect. First we put into syntactic decisiveness. This says that if an expression exhibits the characteristic syntactic features of a single letter, then it is indeed a single letter. It is an expression with the semantic function of referring. Referential minimalism then says that if you have a referring expression in a true sentence, there must be an object corresponding to the referential programs. Finally, linguistic priority tells you that what is referred to as an object, because linguistic categories are prior to ontological categories. To be an object is just to be referred to by a singular term, and constantly is guaranteed that it will be referred to as an object. Given that conception of language and reality, it follows those two main components to establishing that indeed there are numbers and that we can know about them. The first thing to establish is that numerical terms are singular, and the second is that they occur in contexts that we know to be true. It then follows, given the C1 to C3, that we know that there are objects corresponding to that way of speaking. And the way that E.F. again goes about establishing that conclusion is to undertake a logical reconstruction of our arithmetical practice. The way the reconstruction functions is by laying down rules for a normal language game

7:30 in which new syntactic expressions are introduced by Hume's principle. It's established that in this language game, the expressions in it are singular and the context in which they occur are true. It's then showing that this new language game exhibits a pattern of use which is the same pattern of use as that which can be associated with ordinary arithmetic. Finally, the fourth thesis is appealed to at this point to show us that we really do know about numbers as a result of this. And this is the doctrine that says meaning supervenes on use. I'll express this thesis as one word reference, although it's also primarily one word true. And it says that two expressions exhibit the same pattern of use, then one refers to an item, then the other with the same use also refers to that item. So what we've done is we've invented a language game containing singular terms filled with true sentences. We know that there are the corresponding objects. We then established that those singular terms have the same pattern of use as ordinary arithmetic expressions, and then by this original, meaning that means on use, we then know that ordinary arithmetic refers to these objects which we find out about to undertake technological reconstruction. That's the fundamental methodology. The actual application of the strategy depends on the method of abstraction, which is the device used to ensure that the expressions that you introduce in your novel language game are indeed singular. The method of abstraction takes expressions of sentences in the form AP and uses them to introduce normal expressions. The basic idea here is that what's characteristic of singular terms is the figure in identity context as the key syntactic feature. So what we're trying to do is to lay down stipulations that guarantee that there are indeed identity contexts in which the expression curve The way we guarantee a singularity is to produce a normal operator, sigma, on expressions we already understand, alpha's here, and then laid down the truth conditions for the whole context in which they occur in terms of truth conditions we already understand

10:00 which is what this subtraction is doing. It tells us the identity condition introduced on the left-hand side coincides with the true condition in the sense we already understand a certain equivalence relation between alphas of which are only familiar. What are three theses which are associated with that subtraction? The first is that by undertaking this process of stipulation, you will succeed in introducing singular terms. two expressions on the left-hand side constructed by the application of a novel operator, sigma, to already understood expressions. These would be novel singular terms. The second is that where the intended domain of quantification of which you began, and we were simply talking about alphas in this case, contains insufficient entities to conform with the stipulation, then we can understand these novel expressions introduced as picking out novel objects in an extended domain, who may not have been previously familiar with. Finally, referential realism, the idea of the objects that I refer to, are indeed real objects. They deserve as much of a realist interpretation as those other expressions which were willing to construe in a referential way. Well, let's apply this methodology. Hume's principle is an instance of an abstraction principle. We are to imagine it's an equivalence relation one-to-one correspondence on expressions we already understand in concepts. We introduce novel expressions via the application of novel operator A on these concept words to produce identity conditions which are stipulated to coincide with expressions we already understand. The idea is that by doing that we're now in a position to prove that the numerical object exists, and here's the most straightforward proof that numbers exist, but numbers either exist. We know as a second odological truth that the concept of non-self-identity is actually worse itself. After all, nothing falls under the concept of one's self-identity, and unfortunately it must be. One to one corresponds with a self. So we have a logical truth. That gives us an instance of the right-hand side of Hume's principle. So given the stipulation of Hume's principle that says context of this form coincide with context of that form, we're now entitled to derive the conclusion

12:30 is identical to the number of non-self-identical things. Well, let's think back to the basic underlying methodology. If there are singular terms occurring in a true sentence, we know there are objects corresponding to them. The method of abstraction has introduced singular terms of this form. Here's one, and it's occurring in a true sentence. We know it's true because it's equivalent in true conditions to a sentence of second So there must be an object corresponding to this way of talking. In other words, it's legitimate for us to quantify into this context and say there is a y that is the number of x's, the number of non-superidentical things. Which we might, but otherwise, by saying that zero exists, since zero may be defined as the number of non-superidentical things. So that's the basic truth that usually gets the existence of zero at some fundamental assumptions about her language and being able to relate. More generally, Frego's theorem establishes that the system that may be derived in two-principal and second-order logic exhibits the same pattern of use as ordinary arithmetic. The idea here is that what Frege's theory shows us is that Frege arithmetic, the system that follows from secondary logic and Hehm's principle, is such that piano axioms can be interpreted and then interpretation improved within the system of Frege arithmetic. And that's why he's to establish the same issue. Since we can know Hehm's principle by mere stipulation in combination with secondary logic, it follows that we can know a priori and Frege arithmetic. Then using the Virch principle that meaning supervenes on use, it follows that we can know piano arithmetic, a priori as well. Well that's the story. But what about the fundamental objection that made Frege Stalin's proposal, the Caesar problem? He slightly transposes where he's speaking from section 62 of the written icon, he says, naturally no one is going to confuse Caesar with a number, but that's no thanks to our definition. about directions in this context, which doesn't really make any difference. Well, what exactly is this problem that makes Trader a score and move on to the fatal axiom file? Well, there are a whole bundle of Caesar problems, and I don't think those are usually sufficiently distinguished. Here, I think I'm going to distinguish three, too, which I'm going to hand out. The first is that you might just think that this is an epistemological

15:00 that Caesar is his number. Hume's principle doesn't tell us anything about Caesar's number. Therefore, Hume's principle fails to provide an adequate foundation for an unloved room. How can this be a logical reconstruction of arithmetic if it fails to tell us that one of the most basic things we know about arithmetic Maybe that his objects are destined to order your material to take those like Caesar. Well, if that's the Caesar problem, I don't think it presents an insuperable objection to imply Judas' principle to give us a theory of knowledge of the sort of pure applied arithmetic, which you can describe using Cameron's axioms. Together, while I think of that, I'd like you to ask the question, what weight do we usually suppose that Julius Caesar is not a number? Is the knowledge that Caesar was not a number part of what we grasp when we comprehend our truth? Is that part of what you learn when you're at school that Caesar was not a number? Would you take away someone's GCSE because they failed to say whether or not Caesar was a number, even though they were very good at performing arithmetical competitions, even at a very high speed. Maybe, in fact, the knowledge you have, I mean granted, it's really folk metaphysics. In fact, it's really good folk metaphysics and there are good arguments for thinking why it is that Caesarism is a number. But nonetheless, why should we grant that it's mathematical knowledge, that your mathematical knowledge is corrupted if you don't know whether or not Caesarism is a number? Well I don't see any reason to think why it should be. Of course if we go further here, in the full version of this paper, I do question whether or not we can claim to know the season of the number. Supposedly much influenced by this quotation from Russell from the relation of sensitive physics, he talks about the notion of thing, an object, was invented by the prehistoric metaphosicians to whom common sense is due. So you may say it's just common sense, but common sense can be questioned. Putting that issue to one side, let me just grant that we know that Caesar isn't a number. Does it follow from the fact that someone failed to know that, that they would fail to know the truths of the algorithm for not having that piece of knowledge? I want to say not entirely just because of their intuitions, but because it seems to me we can have a notion of knowledge which is sufficiently fine-grained to allow that you know the truth without knowing whether or not Caesar or not.

17:30 What we need is something like a distinction between a knowledge of thoughts, where what you know is sensitive to the mode of presentation under which you know those things, and knowledge of propositions, or singular propositions, which is knowledge of certain objects irrespective of the mode of presentation we use to describe them. So the idea I have here is that you might have perfectly good knowledge of numbers equating Hume's principle in terms of the introduced cardinality operator, but nevertheless not know the propositions which those sentences express. Even the thought expressed, you didn't know the proposition in the sense that you didn't know whether or not the objects which are picked out by that thought are identical or distinct to other objects or the same objects described in different ways. Here's an analogy. Why shouldn't it be the case that I know perfectly well that water is wet without knowing whether or not water is H2O or X1Z. You can't have perfectly good knowledge about water being wet without knowing whether water is wet or water is wet. I don't then think that the duly used problem is best understood as the abysmological problem. I think it's better understood as a semantic problem. It's not that we know we definitely see the difference in number and whose principle must cover that knowledge. principle fails to confer a content on the expressions it purports to introduce. Now, the truth of zero was important because there we have this quantitivative. In the transition from there are a number of non-self-identical things is identical to itself to there is something, which is a number of non-self-identical things, to perform that transition we presuppose the logical complexity. That is, that what we have here is a singular term, the number of non-self-identical things, or the object, referred to by the singular term, satisfying or failing to satisfy the open sentence, is identical to the number of non-self-identical things. So, in order to have logical complexity, in order to justify the existential introduction, we have to have some prior justification for thinking of the statements introduced by

20:00 Hume's principle of logically complex, decomposable, and desamble terms of predicates. The Caesar problem can be understood as a challenge to the idea that really these are logically complex, and the challenge turns on the thing of what we know about the predicates, which these expressions need to be decomposed into. The worry is that in order to understand this predicate, you have to do something general. you have to understand the contribution it makes, independent of the other expressions with which it's combined. In other words, you've got to be able to understand this contribution to other sentential contexts, not just this one. In particular, if you really understand that predicate as it occurs here, you ought also to understand... The idea is that in order to understand its occurrences as it occurs in the pure identification, a certain entity between different sorts of numbers, you ought also to understand the predicate's contribution to those contexts. But look, human's principle tells us nothing about the contribution of that predicate to these particular contexts. It just does nothing, there's no corresponding claim about one-to-one correspondence between concepts available here to give any substance, semantic substance, to the occurrence of this expression in this context. Consequently, we don't know what this means here. And if we don't know what it means here, we don't know what it means in the pure mathematical case either. Because if we knew what it meant in that context, we would have any kind of grasp of the meaning of that predicate and we could say what it meant here. And we don't, therefore we don't. Hume-Smith's fault fails to introduce a logically complex sentence in the first place, and therefore a quantification into it is logistical. I just briefly like to distinguish that from a slightly different objection, which is often confused with it by North American writers. The North American writers conclude this problem along with the problem which has been asked by our phrase and what numbers could not be. It's usually speaking with the question given that we have these numerical expressions and given that we've got all these sets, which sets are we talking about when we use these numerals? This problem isn't which objects are we talking about, it's the problem that we fail to confer content on the expressions introduced in such a way as to suppose they're talking about any objects whatsoever.

22:30 It's not which objects they refer to, do they refer to any objects? That's the primary worry here. Now the Neo-Frugean does have a response to it, a response to the semantic problem. What I'm going to do now is to say that I'm not convinced by the solution, the Neo-Frugean or Neologesis, or Morgan Christen, have given to this problem. What I'm then going to go out and argue is that given the underlying methodology, the Neofrugean would do better to actually just dismiss the problem as spurious in the first place. Okay, well, the Neofrugean diagnosis of the semantic problem introduces some extra multiple machinery that we get to talk about. Their diagnosis of the problem, if I understand right, is that Hume's principle aims to introduce a new sort of object, namely numbers. And to achieve the introduction of a new sort of object, what you've got to do is introduce a new sort of concept, concept number, cardinal number. An understanding of a sort of concept consists in two capacities, the capacity of having a criterion of application and the capacity of having a criteria of identity. What a criteria of application does is it tells you whether or not a concept that you have written is going to discriminate between whether a concept applies to something or it doesn't. The criteria of identity doesn't be different, it tells you that given things fall under these concepts, whether or not the things falling under it are identical or distinct. So something like cow is going to be sort of consistent in the criterion application because when you understand that you know whether or not any interoperatory individual is a cow or not, and you're also quite good at telling whether or not this cow is the same as the one in the other field and so forth, whether this region is identical to this island's cow or whatever it might be. The problem here is that Hume's principle doesn't give us that two-fold capacity. What it does is it gives the criterion of identity. It's very good at telling us whether or not the numbers that show up on the left-hand side of Hume's principle are indeed identical or distinct, depending on whether or not our corresponding concepts succeed or fail to figure out a relation of one-to-one correspondence. Criterion of identity is fine, but by contrast there is no criterion of application here. We're not able to tell whether or not an

25:00 is such that the concept of number applies to it or doesn't. It simply doesn't take a square of the series for a number. Consequently, Hume's principle fails to introduce a sort of concept of number, and consequently fails to introduce objects and numbers. Well, the neo-fregaining solution to this problem is to try and sort of extract from, squeeze out of a criterion of identity, which Hume's principle, as a good Greek provides, squeeze out a criterion of application, and then indeed you will have a genuine social concept introduced. So that here is that QS provides the ingredients for introducing a social concept when it's combined with what's called a social inclusion principle, which The basic idea behind the sort of inclusion principle says that in general one sort is included within another sort just if the content of a range of identity claims involving those different sorts coincide in content. So suppose we have a sort of F and a sort of G. If it indeed is the case that some F is a G, then some statement about identity concerning an F must be the very same claim about identity as concerns some G. These two claims will coincide in content if indeed G1 is F1. Sorry, is there some way you could arrange to use the left hand white board? Correct. Because the light on that board seems to have broken or something. Sorry, but it's very dull on that side, but quite bright on that. It shouldn't take us too long to move it if you want. Right. Okay, so... I'll go on. Sorry. Can we move the stack? Oh, we can all move it. Okay, so I'm just going to speak to the sort of inclusion principle.

27:30 Sort of objects act as included in a sort of object's G, meaning the content of a suitable range of identity statements where G's is the same as that asserting satisfaction of the criterion of entity for the cross-trips. What does that mean when we apply that thought to numbers in the human's principle? Well, the concept number, numbers will fall under concept person just if there are some statements governing identity numbers that coincide in content with statements of identity governing persons. What that means is that there will be some statements governing persons and numbers that assert satisfaction of the very same criteria of identity. The other thought is that that's never going to happen, because the criteria of identity associated with numbers involves one-to-one correspondence between concepts. By contrast, the criteria of identity chronically associated with persons will involve something like bodily continuity or psychological continuity, or whatever story it is you want to tell about such two little bits of persons. So the criteria of identity associated with identity claims and persons in numbers are almost different. Plus when their content doesn't coincide, that can never be the case for the persons in numbers. The two sorts just can't overlap. And since we know they can't overlap, we have a criteria of applications, or a high-negative criteria of application that tells us that the numbers just can never apply to things which have a different sorts of identity criteria associated with them. The sort of criterion of application gets so extracted from the characteristic nature of the criteria of identity associated with numbers, which is so different from the criteria of identity associated with objects. Well, I don't feel satisfied with that sort of explanation. And there are only two reasons for this. One is that, as it stands, it's presently incomplete. And the other is that I don't really see how it can be completed. Okay, so first the sense in which it's incomplete. What we've got, basically the way the green strategy is working here, is we're getting truth conditions for identity context

30:00 things like Caesar equals number three, by playing off the identity criteria associated with the singular terms in those sentences, to see that these two sorts of things can never coincide, because the criteria and the entity are so very different. That's it. If that works, it works for explaining why sentences like Caesar equals 3 have indeed got a truth value in false. But what it doesn't do is explain other sorts of identity contexts which Hume's principle fails to provide an account of, which are where we in fact have a quantified claim, the line where we in fact get to say that zero exists. But this is a claim that Hume's principle is directly silent about, because we can't transpose a claim of this form into a claim of by one-to-one correspondence between concepts, anymore than we can transpose Caesar equals three. But But we can't apply a strategy I've just outlined appealing to criteria of identity to ring out about a criterion of application because what that strategy presupposes is that there's two singular terms with two opposing criteria of identity to play off against each other. But here we've only got one singular term and only one criteria of identity is mentioned. So this is, though, to solve the Caesar problem fully, not just for mixed identity claims but also for quantified statements, we've got to extend the strategy. And to extend the strategy we've got to understand what this predicate means. Because what you understand when you understand a fortified claim is the idea that that open sentence is satisfied by some arbitrary term. That is a term independently of how it's named. That's precisely what we're not told how to do when the strategy was in play, since it only tells us how to assess the dissatisfaction on all of that predicate

32:30 relative to how everything which we're attributing it to is named. So this means, as we're here, where our quantification of language is generally objectual, We must be able to grasp how an object may satisfy or fail to satisfy a predicate, independently of whether or how it's named, and this particular strategy won't explain that. So that's the sense in which I think the solution to the Neo3-gain problem is incomplete. I'd like to go a little bit further and issue a challenge to the Neo3-gain solution, which turns on asking what exactly does it mean when we talk about the content of identity claims coinciding with one another. The content of claims about Caesar just doesn't coincide with claims about three being identical. But what exactly does content mean? What I have in mind here is something of a dilemma. Either we can put the notion of content on the side of sense or we can put the notion of side of reference. And either way, I want to play in the near-to-game so I have no trouble. If the notion of content means sense and mode of presentation, then the social inclusion principle doesn't seem to be plausible. Because it must be plausible for two socials to overlap even though they differ in sense. For example, man and animal are two socials that overlap even though they differ in sense. So that can be what content But no one can it mean reference, but was it certainly true that two socials can only overlap if identity claims involving one involves reference to the very same individuals as identity claims involving the other? It doesn't follow from any of that that numbers count the objects. It's because it doesn't follow from the fact that identity claims are settled in different ways that they're not claims concerning the same objects. So I don't really see any of you being able to see this strategy. So what I'm going to close with is the suggestion that they should in fact try starting a more radical metaphor, which is in fact to deny there's really a problem in the first place. And this is a suggestion that goes back to Kardin. Here's a quotation in the elimination of metaphysics in the Therese. Explicitly consider, as an example, when in sentence two, Caesar is a prime number.

35:00 He says of it, true is meaningless. Prime number is a predicate of numbers that can neither be affirmed nor denied of a person. The fact that the rules of grammatical syntax are not violated easily, seduces one at first glance into the erroneous opinion that one still has to do with a statement, albeit a false one. The sentence A is a prime number is false, given only if A is divisible by a natural number different from A and from 1. Evidently, it is illicit to prove Caesar here for A. This is an example that has been chosen that the nonsense is easily detected. So the idea that I was suggesting that is that indeed, rather than Caesar equals 3 having a truth value or a truth condition which Hume's principle fails to supply, rather C2 equals 3 is a nonsense statement and there's no flaw in Hume's principle that's given to supply a truth condition for that context. Put the point more generally, the idea is that the semantic problem demands more than is generally required for the introduction of expressions of content. The idea is that if Hume's principle does give enough content to our numerical singling terms and sentences to allow us to treat them as open to quantification, to do the sorts of things that we want to do with figure arithmetic, there's no requirement that we be able to say anything about the occurrence of these expressions in these strange identity cases. Well, of course, Carnot wasn't himself explicitly interested in this sort of issue, at least not in that context where I've pointed from him from. And in fact, there are several different ways of dealing, of sort of developing this thought about the not really being a solution required to the semantic thought, or that the semantic threshold for infusing expression can take too high. I'm going to distinguish two types of development of that thought. One is to say that really Hume's answer succeeded in introducing expressions with content that he understood to refer, but sometimes the expressions introduced are indeterminate. It's not really clear in some sense what they're referring or what sort of thing they're referring to, and that's why they'd be able to settle whether or not CISO is the number 2 or 3. There's no flaw there because indeterminacy is a phenomenon

37:30 that we're quite generally familiar with in natural language and we just need to apply the strategies which we normally use to understand indeterminacy in language generally to understand the indeterminacy which is exhibited when we start messaging whether or not CISO is 3. The old Times book which I'm going to finish on, says that these contexts aren't indeterminate, they're just unsemantical, there's really no meaning at all. And that I will claim is constant for the underlying methodology which informs the theory of the theory. Let me begin by just quickly running through why I don't think indeterminate-safe solutions are the right way to go here. Let me just divide up the range of solutions here into three. Epistemicism is the idea, familiar with Timothy Williamson, that basically there is a fact of the matter about whether Caesar is the number three or not. It's that the two's principle gives us no mechanism for ever finding out the truth or falsehood of that sentence. Just as was moving. Williamson thinks there's a fact of the matter about one of the primary bodies. There is a fact of the matter between red and orange, it's just that we're never going to find it. Another solution would be to appeal to vague objects, or the like Parsons does. The idea here would be that really what Hume's principle is doing is letting you catch on to the nature of a fundamentally vague object, a number. But it's perfectly precise in its relations to other numbers, it's entirely vague with respect to its relationship to other objects. So here I'm thinking not that there's indeterminacy in the expressions numbers, they definitely refer to some certain things, but what they refer to are indeterminate in rare. And that's why we're failing to settle these mixed-identity contexts. And finding this suggestion from the field where you explicitly consider the Caesar problem, I suggest you can solve it by doing some sort of super-evaluations technology. The idea here roughly would be that there are lots of different arbitrary ways of stipulating what the reference to the New Morse could be, but there is no unique way, and so we should just apply our supervaluations to say that the resulting sentences are neither true nor false. And that's what Hume's principle is telling us. Well, I don't think any of these strategies will do for some different reasons. Let me give you a flavour and sort of objection which I think applies to all of them. So the point is particularly clear with respect to epistemicism and vagueness idea. What both those thoughts do is they suggest that really you have succeeded in referring to something.

40:00 Number two is just that certain fundamental things about it just can never be found out. In one case, because we're just epistemically enshrined from the finding out of the true nature of the number two, that's the epistemicism, or alternatively, because there is simply no fact in the matter of the way to be discovered. That's the big object's thought. And neither of those solutions will tell us anything about the semantic problem that confuses with neurocogranism, because what's being assumed there is that really the terms introduced by Hume-Strains will have content, either in the particular object space that you're really referring to an object, it's just that you can't really, there are certain things that cannot be done in various ways. But that presupposes that maybe we've got content. And what the semantic problem was was that we've got to worry about whether there's any content in the first place because we've failed to impose sufficient multiple structure on the sorts of expressions which he's principal introduces. So any of these strategies are just going to beg the question, do these terms refer, not do they refer to things with such and such a characteristic. Reference is being called into question. And simply saying, oh well, it's some sort of vague reference won't do, because we're worried about whether there's any reference at all. Supervaluation treatments are a little different. Now here, as I said, is something you can just stipulate in lots of different frames and divide out the content by a class and model theory. But I don't think that's going to succeed either, because in effect what you're modelling in your model theory is the idea that there's a whole range of objects available, and we can stipulate for any element of a series of numerals just which object that numeral will refer to. It's just there are lots of different things. The problem with that way of thinking is presupposing that you have access to an infinite totality of objects whereby you can explicitly stipulate that any given numeral stands for any one of those objects, even though there are lots of different ways of doing that. That's precisely what we're not allowed to do. We're trying to get to knowledge of an infinite totality, not just simply assume that we already have access to it, and then by improving model theory, generate content for certain sort of expressions which otherwise seem to be less meaningless.

42:30 So I think all of these approaches face a general problem, which is in effect we're assuming that we can use numerical expressions introduced by Hume's Principle in contexts which Hume's Principle says nothing about. That we can project from the stipulation their occurrence in Hume's Principle to their occurrence in other sorts of contexts and then apply certain types of semantic technology to explain what they mean there. But what's precisely in question is the legitimacy of projecting from the use of these expressions in Hume's principle to other contexts. The problem roughly is that what seems to be going on here is we're assuming the expressions introduced by Hume's principle obey all the usual compositional principles of expressions in ordinary language. But expressions in ordinary language are to introduce contextually in this way, or at least not only so. And so they've got no right just to assume that these expressions should be understood in the usual compositional way. Their use and understanding could be projected from one context, from the contextual context in which they're introduced, to another. Because of this, I want to suggest that the unsemantical approach is the better one. Now I think this particular approach has been mentioned in the literature. You'll find it in Benassarath, and I guess Richard Hecht's caper in the UAC's problem is something in his territory, although he doesn't explicitly say lots of others to you. You'll find this as a suggestion in Van Asser asked what numbers could not be. He says that the notion of object is relatively relative. He says, I propose to deny that all identities are meaningful, in particular to discard all questions as senseless or unsemantical. Identity statements only make sense in the context where there exist possible individuating conditions. That's his proposal. I want to offer a variant on that, which is to say that really the contextual definitions succeeded in introducing a few expressions. It's just that they're not expressions which function in the way that we usually associate with expressions that stand for objects in the usual traditional sense. In fact, I'm trying to follow out Ramsey's thought, there may not be a objects and properties, or between objects and other sorts of entities, or between names and prejudice. There may be intermediate sorts of expressions that can be introduced. And Hume's

45:00 principle was that it was designed to introduce a singular term in a form of others, and in fact succeeded in introducing another sort of expression. A term which has syntactic affinities with singular terms understood in the ordinary way, which figure in true sentences which we do know to have a realist-truth value and which therefore legitimate us in thinking there is some entity or other in the world corresponding to those ways of speaking. The problem with that way of thinking is it seems to come into conflict with the idea of realism associated with Freire. Craig asked the question, how then are numbers to be given to us if we can't have any ideas of intuition. You'll see the quotation from the top of your handout from section 62. The punchline is that if we were to use the symbol A to signify an object, we must have a criterion for deciding in other cases whether B is the same as A, even this is not always in our power to apply this criterion. He links the notion of talking about an object to being able to verify, in some sense, a recognition statement concerning those objects. A notion of object is something that you can see from different points of view, recognized in different sorts of ways. To be a realist about object is to be able to see that object from different perspectives. But what I would suggest is that that notion of realism is perfectly compatible with the idea that what Hume's principle is introduced is not a single term at all, but some other sort of expression. An expression which we can still understand as picking out something in the world use the idea of recognition statement with respect to that entity. Because the fact of the matter is that we can identify whether or not the number of x, non-self-identical things, is identical or distinct to other numbers. Hume's principle does indeed tell us that, and we know that the sentences which make those claims, by virtue of their equivalence to claiming the right-hand side of Hume's principle, do indeed have a greater strength value. And in virtue of the syntactic similarities to ordinary singular terms, the fact that we can form those recognition That gives us perfectly good reason to think that really we are talking about objective entities out there in the world. I want to close with the thought that this is indeed where the near three games should go, and is entirely constant with the underlying methodology.

47:30 The underlying methodology was that language and reality exist in a certain sort of harmony. We're not in a position to discover that the scales fall from our eyes in reality be quite other than the sentences that we use to correctly describe it. Not correctly, I mean, describe it correctly at the level of truth values. Reality just couldn't fail to be that way. If you really do think that language and reality do have that intimate harmony, then there really shouldn't be any objection to thinking that there are entities corresponding to a way of speech, Suppose that we had introduced a way to speak anything, which involved many analogies to what we usually talk about, syntactic analogies to what we usually think of talking about objects, and the sentences containing those terms were indeed true, would affect the speech and reject the syntactic priority pieces in the first place. That's why I don't think it's true. So on the one hand, I don't think the Caesar problem is insurable because I think given the underlying methodology it should simply be rejected, and what that really shows is that what's crucial, though near again, isn't so much the Caesar problem as the underlying methodology itself. What's really required is a defense of the conception of language and reality which underlines the whole conception. Thank you. we have until three questions yeah we might want to have a two minute break between the papers if it's convenient apart from that okay questions these this thin conception of objects like you are thinking of these numbers of being do you take these thin objects to the range of quantum powers It might go on the right-hand side of it. Yes, that's a good question. In a way, you can really start thinking about these things between the set of law, you can use a very unique particular character, which is very small. Well, I'll say yes. The other thing has to say yes. Right, because if you didn't say yes, then the proof of break is going to be true. but then if you do say yes does that make them inevitably make them fool?

50:00 Well they're not fooled in the sense of well I suppose what I'm questioning here declining a link between constitution and identity is not received in a certain way in a certain sort of maximal way So if you think objecting, if you think I want to quantify it as something that has to be by entity conditions, so that for any other object you're able to tell a level of identity or something to it, and indeed if I were to, if I were to allow an implication I would have to allow entity conditions by having thought those efforts have to have the same objects in the first place, that would need to be correct. What I'm suggesting is that really there's no reason to accept that link between, let me accept there is a link between quantification and identity, but why should quantification be linked to the notion of complete identity conditions? when one has one of these category gaps where identity feels different. Well, one thought might be if you have any, if you by a total notion that to understand the quantification on the right-hand side, then you have to have in some sense a prior grasp of the name of the quantification. Then you might think that that was intentional with your suggestion, just because your suggestion gives you too free if you can create thin objects like this then presumably you're going to be happy with lots of other ways of creating singular term-ish sorts of things like things of whereabouts and all these other things that we don't normally put in our ontology but we can never let's use fairly singular term-ish in their behaviour and wouldn't that just make it seem less and less plausible that you did have the prior grasp of the domain I suppose I have the same problem with that sort of objection. The most basic objection to this whole way of doing things is that it's just not plausible about the stipulations of simple principle or anything that you think.

52:30 If you have, if you reject reality or language, reality as its nature affects independently of language, then in a way you shouldn't be surprised that reality should turn out a little multiplicity that corresponds to a range of possible situations that you can make about. The thought gets by it because you think that reality does have an independent nature and can be conceived independently of it, and there are ways of talking about it that it will match onto it and make brand clear the better that it would actually be able to satisfy it. So this whole project works on the assumption that that's not. Let me hide my question in response to your question. The first is, do you think that worry is greater with respect to the issue of thin objects than it is with respect to other forms of distractions that are introduced, introducing other thick objects? There is this more risk to the middle of the proliferation of vast quantities of entities that you can generate many different sources to be non-employed abstractions. So that's where a question of that can be. Is that something that seems to be bad with thin objects here? Yeah, I was trying to suggest that it made that particular objection in standard predictability objection just seem worse because there just seemed there were too many of them. So many thoughts we haven't even thought of yet. you know, I can just produce some more, just introduce a new way of talking, which introduced a new sort of thing which clearly wasn't in any sense in my ontology before and I was using quantifiers before and I'm of the same, my understanding of what I meant then is the same as what I mean now, so I've got some stable I suppose I am arguing about it in a fine way but some stable notion of option that has to carry on through your discourse So, your understanding of objects now has to be the same as your understanding of objects before you introduced these. Well, what I'm questioning there is the idea that we really do need as a word of a stable, strict, consensual object to sustain the notion of quantification. Rather, the thought is here that the notion of object the concept of the family's influence concept, traditional qualifications are sustained by

55:00 lacking similarities between different sorts of expressions that should be introduced to the various situations. I mean, I do agree with these are a problem and there's one that I'm saying the right insight and the right sort of ways together I really don't see why it is Why isn't it the quantification of the client's history? I mean, you say it seems like just too much is out there. Why shouldn't that be all that stuff out there? I mean, if you think of objects as corresponding to the expression that you use, then there's going to be an end of them, as there are possible ways of security. It seems to flow naturally from the view rather than, you know, sort of characterizing a subject in the discussion, and you're just like, you're cute and you're talking like you, rather than having to move with an object. So, giving context, what you quantify are all the sort of things you can count, and don't you, in order to have, for them to do the sort of things you can count, have to think of them as also the sort of things that, you know, like Gregor on Aftler and Arter, that can, exactly the sort of things that count the other aspects, so that you know that it's actually the same thing you didn't have to count in the past. Because your conception of thin objects is open-ended in the sense that the thin objects are precisely the sort of things that you have to allow as an open possibility, but later you might come around the other side of them and discover that they were actually saying something else you've been talking about from another perspective. As long as that's open, I don't tell you can count them. Well, I suppose I asked, well, do we really need to count them, count them in that sense, in order to be able to assure ourselves that we don't treat the terror? I would suggest not, because nowhere in the point of proof of freedom is different. You actually have to look behind your back and check and see whether or not the new rules are Caesar or not.

57:30 because there's no point in that I mean, throughout that proof you only appeal to pure philosophical contexts there's no need to go away and check and see whether the ideas refer to those contexts but of course even in other ways all you're doing is counting numbers yeah, so this is where this is the sort of thought I had where I thought you were distinguishing between knowledge of propositions and knowledge of thoughts by having a fine grade knowledge of propositions of thoughts can you can you make the distinct some of the distinctions maybe neatly in terms of language with ordinary mathematics like your child learns he doesn't have to know about Caesar in order to pass his O levels you're really saying that he is doing his mathematics in a limited language which does not include the word CA if when extending the language then the question might arise and maybe looking at it in terms of the way you're extending the language of the mathematics you're doing might sort of help to clarify some of the distinctions, would it? Well, I partly agree with that. I suppose it was thinking about how children might really react to, actually as an undergraduate, it's about how they react to the question of Cs or a number, whether or not people can spend any philosophy or answer too much philosophy. They tended to respond to the answer. No, there could be not. But that sort of points of limited utility in this context. Because what the New Yorker game is doing is they're not trying to base biology on principles that actually generate our understanding of words. Not seeking to under a couple of principles that are taught at the school, or some kind of work in your medical module ahead. Rather what we're trying to do is set up a practice, invent a new language data and then show its suspicions and rarities between the new model and the old narrative medical practice and the speed of the other state It's not as aware of a story which goes by explaining how we acquire those skills.

1:00:00 It would be in a way of a relative to introducing expression bias to the relation. Could I move on to the discussion answer to you, Chris? I mean, you slipped along with the idea of still objects, with Michael, but was that really what you wanted? Do you want to think of the numbers of still objects? I did get that idea, and it seems to me that Craig was right, that if we're thinking in terms of objects at all, you must make space for the idea of a multiplicity of loads of concentration, you don't do the same thing. And come to that, I think I see a multiplicity of numbers are deficient, in that way. It does count as loads of presentation, which is something to potentially put in many loads of presentations. Yes, indeed. Okay. Some people want to, some people want to operate them. So, what are some of the issues? In what sense are there fewer loads of presentation that we found in the lab of the box office? Well, there are... What I ought to say is something I'm following. I mean it follows from the fact that the number Caesar is thin, but in a sense Julius Caesar is thin too, because you don't know whether or not we can then find him under Merckham's presentation here. It's a matter of investigation just to what extent Caesars himself did or not. If you're some sort of… If that's really what you want though, just to be fraught. We're trying to mark up some kind of obvious by reference to a relatively So I've spoken with progress on the subject because an object there I can understand in a lost equation way, rather than an analogous way It depends on the screen, so let's just compare to the Caesar cases in the rest of your models.

1:02:30 If you were some sort of air-jack spot that presumably you think that Caesar could be You can present Caesar under a physical mode of presentation and you can present it under a mental one and indeed these things are identical. If that's the right type of body then indeed you can have this one type of project understood under two distinct families of mood of presentations, of physical representation. That's what I'm suggesting isn't the case with the numerical expressions understood in the way in which they're introduced on Thiem's principle. Thiem's principle fails to make the assumption that something like your solution directs to Thiem's principle. Supposing that Thiem's principle doesn't really tell us anything about the mixed-head-head phase, then you take as that as what is showing is that numerical terms, unlike CESA, just don't have these presentations, while number 2 and number 3 only has numerical notes of presentation, but doesn't have some categorically type, theoretically, distinct notes of presentation, where you can see two to one sponsor of both types. That's the sense that we've made available the opportunity for seeing something analogous James, I've got a question about the proof of existence of zero. I was wondering about the status of the first premise, whether it really is a logical premise. I don't know if this is really something pretentious or trivial question but how is I supposed to read that is it supposed to say is it good to read the terms of the left hand side and the right hand side as property terms or something being something that's not identical to itself that's right and we're supposed to understand the term in the middle as meaning corresponds one to one of course so my worry is about really whether independently of our

1:05:00 understanding number when we have the concept of multiple correspondence whether it's is applicable where we don't have a finite plurality so i might say that you know what we understand by that expression is we're entitled to say that the things falling under one kind correspond one things falling under the other kind, but we can kind of take one, take the other, match them up and keep going and not have any left out. Now, I mean, that just, with that understanding of the idea of a wanton correspondent, it just doesn't seem to be determinate whether the empty set is going to be in wanton correspondence with itself. I mean, if there are no things falling under that kind being not identical with itself, then why should there be a... What reason have we got to think as a fact of the matter about whether the set of things falling under that corresponds to itself? Well, there's nothing falling under that kind of set. Okay. I'm not going to pass over to our second condition here. What is the canonical answer to that question? I mean, there are no objects falling under the concept of self-identity, so how is it that those objects can fall in one-to-one correspondence with themselves? Because the definition is true in all of them. For every x, if x falls under one, there's a relation. So it's a condition, like if an object falls under one, then it falls under the other. that is something matching the four to the other. Well, it's quantified. There's a relation such that for every x if x falls under one. The relation in this case is the empty relation. Correct. So if you buy the empty set, you buy the empty relation as well. Yes, of course. Yes, that's right. So if you're presupposing there is something called the empty relation or to get the empty. Without that, it falls apart. Yeah. Okay. The empty relation is just the thing that's false in all pairs of arguments. Okay, thanks. Any relation would do. For every R. Depends on how you formulate the equinumerosity.

1:07:30 Usually there is an R. Yeah, sure. But any R would be. actually I just have a couple of questions for clarifications I didn't catch this thing at the end what's your take then on the syntactic priority thesis is it this that the syntactic priority thesis might deliver only thin objects that is if the conditions are met then we have objects the only thin ones. Is that it? Well, after all we know, given the separatism's depletions, we know that they're thin in the center scale. Oh, we know that they are thin? Yes. Oh, okay. We know they're thin in the center scale. All right, now the other one is, what do you want to make of the following predicate? The objects that were mentioned in your talk. Like, let's suppose that the only objects you happen to mention in this talk are Caesar, Frege, Crispin, Bob 2 and 3 so there's this predicate the number of objects mentioned in your talk sorry, the objects mentioned in your talk it applies to those things what do you want to say about that predicate, that it's not a sorrel? presumably we don't know what its number is no, sorry, there's no fact of the matter about what its number is right? That's correct. I think there are a lot of different ways of responding to that. What I'm going to say is indeed not suitable. Because there's no criteria of identity? Another way of responding, presumably, would be to say that that predicate employs the notion of number in a way which is illegitimate. The notion of number being employed to construct that predicate can't be the one that was introduced by Hughes Principle. Well, you know in Frege's Theorem that we do apply Hume's Principle to predicates that have numbers falling under them, yes? So what you want to say is that you can't have predicates which have numbers plus other things falling under them. Yeah, Hume's Principles. And why not? What Hume's Principles mean to agree with them? So it says for every F and for every G is what it says, right? So what, these aren't within the range of Hume's Principle, and why not? Because, again, because they're not sorrows?

1:10:00 Is there anything to play with restricting quantification then? I mean, does your view have leave in the kind of unrestricted quantified quantification that Frege seems to have accepted? When you say for every x, we need for all x. It certainly doesn't mean that you take constitution for one day. I know, but what are we going to make of that is, you're restricting identity in some way, right? Are you thereby also restricting quantification or no? That is, you can have unrestricted quantification. Are you intended to keep that? That's what it's going to have to do is this is going to be some sort of next sort of quantification. That is the relativity, right, so the relativity of identity, because I noticed, you know, you have this Benassarap quote here, right? identity sort of not all of each but somehow right and we're also going to have some sort of relativity of quantification to okay although you there would be analogies between the different types of quantification to sustain the idea of the liberalized types of quantification. So it's when the world is fragmenting into a different array of objects corresponding to different language, I was wondering whether you might clarify in what sense you hold the realist position if you deny that there must be a matter of fact concerning which objects the numbers are. So in what sense you are a realist if there is no matter of fact about what sort of objects the numbers are? What do you take realism to? Realism might take to be the view that your mathematical statements, realism and not mathematics, they are true of a reality, so they are true of universe topological structures or whatever it is.

1:12:30 So if you deny that there is a matter of fact about numbers, then in what sense can you say that you are a realist about number theory for instance? Well, my question is what does it take for something to be true of an object or true of an entity? The fact that there is, I mean, if you are a realist, that there is a matter of fact for which your statements are true. Well, surely there is a matter of fact, even on this reversal, there is a matter of fact about what the rule of 3 refers to. In a sense, this is something which does not answer my question, because I want to know... You're asking to do something where there's nothing to be known. Why should the idea of there being an objective reality there demand the idea that there's an all-inclusive domain when you can set up the principle that these things are meant to be the same as anything else in the community. What is there to be an object that a man has to be seen as a human being? No, I mean, if you are a realist, the object is at the root of truth and the objectivity of your statements. Now, my question is this. If you don't believe that there is a matter of fact about these objects, namely that there is no factor which allows you to distinguish what you refer to when you are using the number 3, from something else, then why do you call yourself a realist? In what sense are you a realist? Well, I'm a realist in the sense that I can interpret truth values on these sentences in a realist way. There is a fact of the matter of whether or not 2 equals 1, or 3 equals 4, or false, definitely objective. And because I want to look at the context of what I'm having in a syntactic priority thesis, I think the facts about what objects there are are settled by facts about truth. To be a realist about an object is to be a realist about the truth value of the sentences describing those objects.

1:15:00 might not happen in any way before them to question the realist and truth value of the respect of all sentences in which these terms occur. Sentheses which describe these objects are the realist and the truth value. I see. So do I say about truth, really? Well, to think that realism about truth wouldn't place realism as a very reference for speech would reject this whole way of thinking and think that the context was was first thought in the first place. Objects aren't even independent of their occurrence of expressions. That's what idealism is supposed to come from. The idea is that the thing that was used as you said was truth in reference, built from objects, that's precisely the first part of this campaign for British thinking. Can I just move on to this question? John, can I just paraphrase what you did catch Can I come back to what Stuart was asking about, and how much it recommends things. I suppose, coming back to the relationship between what you're doing and what Nashra was doing and what that could not be, in the quotation you gave on the last page, he seems to be saying something which sounds pretty plausible, if you introduce real numbers and there's a theory of real numbers and there's a contrast in that theory and then completely separately you introduce, say, natural numbers using the P-L-X here to the contrast of the Bayes well, introducing those two different theories doesn't give you any way at all to set the question whether real numbers are the same as natural numbers that's positive but in your case in order for Hume's principle CESAR has to be understood as for the range of confines so if the object is ever to explain it, CESAR in that sense has to be part of the same theory as so you can't use the dog in purely naturalized terms you can't straight forward use that dog and say CESAR is just part of a different theory So, I mean, if you went with what you seem to be saying upon relation to Stuart, we wanted to fragment everything to lots of different theories, or to use your phrase language gains, right, take the language gains here as just theories, yes.

1:17:30 well, you can't do that if Caesar's part of a different language game from the numbers whether he's part of a segment a language game don't you have to answer this to the Caesar question Okay, and I think we will work. We will not be concentrating on the pure cases, which is where the Belgian attempted to employ the intriguing part of this analysis of what a national actor is using. So just to clarify for example, I've got to have been figuring, following under the concepts, which I've gone across, so I need to be able to mention I'm very excited. You know, why is it I need to come to a decision about why is it I need to have a unified theory mention in the move Caesar and numbers fall within the range of a single quantifier. I don't know where to exactly do that. It doesn't fall back when I need to direct to your question. Well, you've got to be able to talk about Caesar within the same linguistic practice or And I take it that part of that game is that I'm not saying the quantifiers that you use to play with that game. If you think of the expansion of the trade 1.12, I suppose to say that the number of geo-cord networks is 5. If you expand that out on the right-hand side, you're going to try to reflect the 1.12 forms between geo-cord networks and pre-incesses of 5 in the greater definitions. so you must have a single domain to study relations of a certain kind between others, from thick things to thin things so you can't get by you can't get through that statement unless you have a single domain you can't see those kinds of numbers

1:20:00 but I don't see why that's a problem for you why don't you just say yeah, that's right, that's a commitment what I'm not committed to is holding the for an arbitrary concept, like the one that Stuart gave mixing up the two kinds of things there has to be a number associated with that concept the NXFX can go partial It's only well-defined, but it has said about your personal life. It seems to me that that thing fails the definition of sortle anyway. I mean, with sortle, you need a criteria of identity. You know, I mean, that's exactly what you don't have for that one. Well, you can have mixed concepts covering different sortles. But how are you going to handle that formula, Chris? How are you going to handle that formula? I mean, how's the logic of the language going to go? So, I mean, part of Craig's problem was to make sure that his sentence has always meant something definite. It's no good saying, well, sometimes they mean something, sometimes they don't. I mean, it might be okay from the point of view of analyzing the facts, but it's not going to be okay from the point of view of getting a formal system going. So how are you going to handle that kind of treatment? How are you going to handle that formal? I mean, we need to watch it at some point. So I don't see why it would... You can't just say, for example, introduce singular descriptive terms and then say, well, sometimes they just don't have a reference and then we don't have a sentence. The reason you can't do that in a formal language is because you've got to know when your sentences are well... That way you have to know facts in order to know whether your sentences actually have truth values. So that's going to be a... The phrase only goes on account of the contrast between synopsis. Yeah. Oh, I agree with you. If we had that, we could then do an informal characterization, which is that very good. You could do that with... Just do a two-sorted language. You could do a two-sorted language. You could have synopsis on one side, a thick one's on the other. I suppose I want to say there are as many synopsis of the same kind as there are synopsis of another. You need relationships with numbers. Uh-huh. That's a problem. So suppose we grab that numerals refer to thin objects, which are from this different realm. I guess it's still the case that Hume's principle doesn't determine which thin objects

1:22:30 numerals referred to. That is, it's compatible that the numeral 1 and 1 to 1, the numeral 2 to 2 and so forth but it's also compatible with it that you permute the numbers and you change reference according to it. North Americans to use your earlier phrase. That includes Mexico. Sorry? Which includes Mexico. We tend to think that unless you've been able to address that worry, then you haven't really solved the Caesar problem. Well, I won't say there are different Caesar problems. What I've been trying to think about ourselves in a dysbological version is also from. And what I suggest to put there is that you might know perfectly well the truths of sentences are pure, I think you took those sentences, I don't have to take it, but then you don't get to the linguational tricks you refer to. You might know the truth of those sentences were pure, you'd probably be familiar with the fine and confident way of applying sentences while still feeling to know the underlying essence of Numbers' work. It would not necessarily fall into question. If you're grasping the truth of wisdom in some sense, it would fall into question. It seems to be a lack of a single knowledge of the individual shapes of the work. So, in particular, that sort of problem about doing those truths of wisdom is an important question by this particular question. version of the Caesar problem. It's not clear to me whether or not exactly... Well, the usual reason why it's thought that the Caesar problem has settled in this context is that the content of identity claims

1:25:00 So I'm not understanding because you can, even thin objects you can describe in a different way. So if you have quantifiers that are... In different numerical places. Sorry? In different numerical places. I'm not sure I understand that. I mean, you can have a quantifier that ranges only over thin objects and still wonder whether, you know, how to understand the predicate x is identical to the number of non-self-identical things, where x is a variable of the thin sort. On the grounds, since Hume's principle doesn't determine which thin objects are the reference of your numerals. Well, the prior issue has to be is there any sense to wondering which of the thin objects these expressions are sent for? I'm wondering whether what they do succeed in doing that. Let's have an example. So, does the immutable two refer to the same thin object as the a pair set up, not a set in a symbol. I'll predict that you know the tombs refer to that thing. Well, Hume's really just assignment about that, it's about Caesar, in a way. There's just nothing, there's no fact about one to one cartoon, or one of the machines to play just the same sort of paper as it does in the over-numerical case. The idea is that the issues about identity really make sense insofar as the stipulations that we've got in a great context which allow all those questions to be genuinely raised. It's not clear to me why it is that for any other sort of abstract object whose principle allows for people to stay questioning whether or not the object's introduced or identity Well, I'll ask you more about it later. I'm going to stop there. Apologies to those people who are still waiting for questions. It remains to thank everyone. Thank you.

1:27:30 Thank you.