The origin and status of our concept of cardinal numbers

0:00 Thank you. abstraction principle, of which Hume's principle and their concept of number are primarily 10. For Christman, the philosophical significance of Trayvon's theorem is that it shows the definite infinity of the natural numbers to follow from an explanation of the concept of number in terms of an abstraction principle. and this is familiar to most of you, I'm sure. So an extractive principle has this form. It's the universal closure of an expression of this form where R's equivalence relation and the variables X and Y may be of any order and the function signal may be of mixed type. In the case of Hume's principle, the equivalence relation is the relation on concepts of a one-one correspondence, and the cardinality function is a map from Freudian concepts to objects. As we will see in greater detail, it's important for this program that concept formation via an abstraction principle be distinguished from the ordinary practice of axiomatization and indeed that the case we made that not all cases of constant formation via an extraction principle are to be regarded in the heart. In the case of Hume's principle, the basic idea is to be how Hume's principle is a stipulation, one which gives two class of statements of numerical identities, namely those who perform the number of S identical with the number of G. Although the specification of two conditions is partial, that is, it

2:30 concerns only a restricted class of statements of numerical identity with very appropriate cognition judgment. The resulting explanation of the concept of number is healthy and complete insofar as it suffices for the second-order derivation of the basic laws of arithmetic. Since concept formation via Hume's principle, like concept formation in accordance with any other abstraction principle, involves the introduction of a new concept, it would be a mistake to do the principle of the analytic of the concept of number in the way in which this notion, the notion of analytic problem, has traditionally been understood. Rather, Nietzsche's principle has held to be a trivial consequence of the nature of the concept of number because it's a stipulation governing the introduction of that concept rather than the analysis of a pre-existing concept. The stipulative character of the principle is important Since it is this feature which allows it to fulfill, albeit in a restricted and modified sense, logicism is promised to deliver arithmetic from logic plus some species of stipulation. Thus, originally, Crispin knew numbered theoretic logicism because it derives the basic laws of arithmetic from a stipulation governing the concept of number, without, however, providing an explicit definition of the individual numbers or immediately precedes on the basis of a geological vocabulary. Although numbered theoretic logicism falls short, of the goal of explicitly defining the vocabulary for arithmetic in truly logical terms, its explanation of numerical identity by Hing's principle is achieved in terms of one-one correspondence, and this is the concept of geologic. There can be no question of the importance of Helen Wright's contributions and of the vitality that have imparted to a subject long regarded as having, at most, historical interest.

5:00 Crispin's rediscovery of Frege's theorem has been a major impetus to the re-evaluation of Frege's philosophy of arithmetic. And it is in hell's defense of a modified form of Frege's logicism has elicited a critical reaction, which has essentially clarified the issues that need to be addressed. Nevertheless, I think that their final view I will begin by reviewing some of the critical reaction that has been directed at, especially at Crispin's formulation of the NLJDN program, with the end of the thoroughbred where I believe the central difficulty with the position lies. I will then present what I did to be the correct account of the status of Hume's principles and with the significance of Craig's theory. For reasons that will soon become evident, the first-class objections that have been urged against this program are known as bad company objections. Crispin addresses two such bad company objections to his position. First, this is not what Crispin says. This is what follows are the objections in Summers. First, concept introduction via an abstraction principle can fail, and can fail spectacularly, as the case of the abstraction principle expressed by, from Gasset's basic law, Bob, and Shogh. But the objection goes, how can we accept a view which invites us to rely upon methodology, which is known to be seriously flawed? And secondly, second, bad company objections, given our freedom to stipulate, or another abstraction principle that is necessary to supplement the account with a criteria capable of governing the choice of one abstraction over another. Those are the objections raised in one form or another in this very brief explicitness by right, on starting time, by George Colossum. Apropos of the first objection, I propose a first objection to one-time inconsistency. Crispin makes a convincing case

7:30 that the fact that a methodology sometimes leads to flawed conclusions does not mean that it is itself. It is immediately flawed. And he reminds us that Hume's principle is consistent relative to analysis. Why should it be necessary to show Now, one difficulty with this response is that it tends to assimilate concept introduction via an abstraction to the case of ordinary axiomatization, and this makes it difficult to appreciate the insistent on the part of both Bob and Crispin, but what they are proposing is sharply different from mere axiomatic stipulation. However, the consistency of Hume's principle fails to settle the second bad company of judgment, which effectively raises the question whether, and in what sense, the principle is true. For suppose we grant that Heame's principle stipulates or lays down the two conditions for certain statements of numerical identity. Is the truth of Heame's principle merely a matter of stipulation? But this is how Crispin intends to be understood, suggested by many passages. Here is one from Richard Hex's collection in honor of Donna. the state of affairs is initially given to us as the obtaining of a certain equivalence relation, but we have the option by stipulating that the abstraction is to hold of so reconceiving such states of affairs that they come to constitute a new kind of sentence. The Christian use abstraction principles of stipulations is reinforced by a contrast he draws between the basis of our knowledge of the truth of the human principle and her knowledge of the existence of the known group. Abstract objects are not creations of the mind, brought into being by a kind of situation. What is formed, created by such an abstraction, is rather a concept. The effect is really fixed. The truth conditions of identity statements concerning a new kind of thing, and it is quite another question whether those truth conditions are ever realized.

10:00 According to Crispin, the representation of Hume's principle as a kind of stipulation should not prejudice the question whether the existence of the numbers is a matter of stipulation. In fact, Crispin maintains that the existence of the numbers is something discovered rather than stipulated by holding that our knowledge of their existence is ultimately derived from a principle whose truth is a matter of situation. Now, although this might be credible as a view of conventional explicit definition, I don't think it can be maintained in the case of a contextual definition like Hume's principle. It is precisely because ordinary definitions are not creative that we can say that anything which is established or discovered on their basis does not really depend on them, and in consequence does not share the conventional character of their ophthalmic basis. But even if Hume's principle can be regarded as a stipulation, it is certainly not a stipulation of this character, since it is creative over the theory to which it is ahead. Indeed, this is precisely the point where the analogy by an abstraction principle and the methodology of concept introduction by where an explicit definition can break down with the result that concept introduction by abstraction becomes difficult to distinguish from axiomatic stipulation. In a recent paper written with Bob Hale, Brisbane again in English I don't believe there's out here generously what's the title of what's the title of I forgot implicit definition the one in the the one in the the one in the the one in the just out Because of ridiculous modern methods of bibliographic referencing,

12:30 you just don't remember important things like the titles of the articles which you're referring. You just remember the names of the editors of the volumes in which they appear. So it's in that place. So in a recent paper written with Bob Gale, Crispin again emphasizes the difference between treating Hume's principle as a stipulation, something which he and God both will be unproblematic, and treating the existence of the numbers as a matter of stipulation, which they agree is problematic. Their point is that Hume's principle lays down partial satisfaction conditions for the relation of numerical identity, thereby establishing the existence of the relation in analogy with the use of axioms as definitions of the terms they introduce. Helen Wright insists that it is not part of their position that the stipulation of such satisfaction conditions should secure the existence of the objects related by the relation of numerical identities rather their existence is secured by proof the proof that there are objects related by the relation thus introduced the discovery of this proof gives the sense I think the existence of the numbers might be set to be discovered. The methodology here described is only a variation on a familiar and clearly unexceptionable sub-theoretic practice. In standard axiomatic sub-theory, we freely lay down the defining condition and then proceed to verify, to prove on the basis of our axioms of set existence, that our definition is not vacuous and that we have not simply redefined the empty set. In the absence of an account of our basis for believing the set theoretic axioms, which justify the non-acuity of our definition, this methodology does not give and does not pretend to give an account of belief about sub-existence. If the sub-theoretic axioms are represented as stipulations, this

15:00 is naturally assumed to transfer to the basis where our belief in the sub-existence and non-vacuousness we are able to prove on their basis. But except for the fact that we are here working in credo arithmetic rather than ZF sub-theory, this is exactly the methodological by Bob and Crispin, why should their claim to have given us an account of the basis for our belief in the numbers be viewed any differently? In particular, if Hume's principle is represented as a stipulation, then the basis for our belief in the existence of the numbers must be regarded as a trust in that stipulation, just as in the case of Scepter. Perhaps the stipulative character of Hume's principle On a model of reference-fixing stipulation after the matter of Saul Kripke's example of the introduction of a term like meter, this is an idea that was suggested to me by a paper of Paul Legosian's reprinted in Bob and Crispin's Methodistics and Philosophy Language textbook. again well anyways the idea would be that we stipulate that at time t bar b is a meter long from which it follows that at t the length of b falls under the concept meter although we have laid down a stipulation a reference facing stipulation in doing so we have also succeeded in making a factual assertion namely that B has a particular length at T, something that obtained before the convention was laid down, and something that will continue to obtain even if the convention is withdrawn. I found something similar, oh, for concept introduction by an abstraction principle, so that it, too, may be seen to consist in laying down stipulation while at the same time having a factual content. Couldn't right inhale argue, but the opposition, fact versus convention, has been overdrawn, obscuring the fact that the stipulative character of Hume's principle, on which they have correctly insisted, does not preclude it from having a factual character as well.

17:30 Now, the difficulty with this response, I think, is that while Turkey's model of reference fixing might suffice to show that it is not in general true that stipulations are incapable of a factual content which is independent of the stipulations themselves, it is of no help in clarifying how the stipulative character of concept introduction by abstraction the factual content of a principle like Hume's. This is because the factual content of the assertion that R.D. is a meter long is carried by the possibility of displaying or ascending, independently of the reference-fixing stipulation, the length to which the concept of the meter is to be applied. But this possibility is precisely what is lacking in the account of the number-theoretic case in terms of concept introduction by extravagant. The idea that the truth of Hume's principle is a matter of stipulation that is unconstrained by any antecedently determined truth allows the position to sidestep the difficulties which confront showing that it is true captures our pre-analytic notion of numerical identity. However, this comes at a cost. The problem is that there are many abstractions, all of them satisfiable, but relative to certain assumptions, not necessarily mutually satisfiable. For example, in the standard of equality for numbers, Ullos introduced several sentences with very different model-theoretic properties from one another, but with an equal length being treated as abstraction principles that introduce a class of abstract singular terms. In the philosophical significance of Traeus' theorem, Rudd considers a modification of one of Houlos' examples in order to motivate a constraint on admissible abstraction principles. The example involves a type-lowering function which takes two concepts to the same object

20:00 in case their symmetric difference is finite. Crispin shows that the resulting abstraction principle, which you call the nuisance principle, holds infinite domains, but fails if the domain of individuals is infinite, and the range of the concept variables is the full power cell of this domain. To see the difficulty that such a bad company example poses, Were we to adopt the stipulation embodied in the nuisance principle, we would be restricted to models having only finitely many options. But since Hume's principle holds only in infinite domains, our adoption of the nuisance principle would preclude us from stipulating that numerical singular terms should be used in accordance with Hume's principle. But if we take seriously the idea that abstraction principles are stipulations, merely stipulations governing the use of the singular terms they introduce, that they are conventions which we freely lay down, you might easily deduce the truth of Hume's principle by an incorrect initial choice, one satisfiable only in finite domains like the usage principle. This is Goulas' argument. This would occur if, for example, we had first chosen the stipulation embodied in this nuisance principle. But if abstraction principles are stipulated, and thus nuisance principle just one among many, I can make sense of the idea that there is a right initial choice. If the truth of an abstraction is a matter of stipulation, then the existence of a domain is sufficiently large to contain the numbers which seem to depend on which abstraction happens to be laid down first. So that whether the domain of objects contains a subdomain capable of modeling the basic laws of arithmetic would come to rest on an arbitrary decision of ours. This is clearly a conclusion that neither Wright nor Carroll wished to endorse and they recognized that what I would call the quasi-conventionalist features of the approach make it incumbent on them to articulate a principal division among abstractions.

22:30 One proposal that has been advanced, there are many other conditions that have been investigated as part of the general theory of good abstractions, but I'm only going to concentrate on one, is that an abstraction principle should be directed as acceptable only if it satisfies a conservativeness requirement according to which an abstraction principle is acceptable if it does not constrain the cardinality of concepts with whose introduction it is itself not explicitly concerned. But to know that Hume's principle constrains the cardinality of the numbers, we must know the cardinality function, not only that it associates concepts with objects, so that in particular, not that the numerous concepts are associated with distinct objects, we must also know that the objects thus associated are numbers. Otherwise, how are we to maintain in accordance with the notion of an acceptable abstraction that it constrains only the numbers. It might therefore be argued that we have at least reduced the problem of demarcating Hume's principle from bad abstraction principles to another that is demarcating the numbers from other objects on the basis of principles in terms of numbers the so-called previous user problems. However, this conclusion is predicated on the assumption that an abstraction principle as rich as Hume's is adequately represented as a stipulation. Frayga's idea was that numerical singular terms are referential because with Hume's principle we are in possession of a clear criterion by which we same number has been given to us in two different ways. It's the number of one or another concept. That is, given the truth of the suitable criteria of identity, we are entitled by the context principle, only in the context of the sentence does the word have reference, to infer the numerical singular terms refer. Our access to the numbers is mediated by our recognition of the truth of Hume's principle. And the same principle also serves as the only substantive

25:00 premise, in the proof of their infinity. Trage's account of our access or reference to the numbers thus relies fundamentally on our recognition of the true confused principles. What we require, therefore, is an account of the sense in which this principle is true and an account of how it is known to be true, since this is what is needed to infer by means of the context principle that numerical singular terms refer. The position of Wright and Hale, according to which the principle is really laid down as a convention, only raises a more basic difficulty, namely that of understanding how it is possible to secure the reality of the domain of objects by mere speculation. Criteria of identity are cheap, corresponding to the plethora of equivalence relations, that are the commensurate abundance of objects. Regarded as mere stipulations, the criteria of identity abstraction principles expressed fail to constitute a significant constraint, both on the preintroduction of concepts by abstraction and on the objects which fall under them. To my mind, number-theoretic logicism in this form has not adequately addressed the consequences of treating this principle as a stipulation freely laid down. Instead, it has chosen to concentrate on the question whether the surface grammar of the left-hand side of the principle where it doesn't be used in numerical singular terms. In other words, the apparent deployment of numerical singular terms as singular terms is justified. as Wright puts it one has to read the left hand sides of the appropriate abstraction principles not merely a sensational variance of the right hand sides but in a way which is constrained by their surface syntax once this is established

27:30 no further question can arise over whether such terms genuinely refer and there is no good sense which their reference could be stigmatized as semantically idle. But the difficulty, as I see it, is with the transition from the characterization of the truth of Hume's principle as a simple stipulation to its deployment as a truth having significant existential presupposition, since it is the latter characterization of the principle that that's used in conjunction with the complex principle he lies in, this transition is going to be farther. Nevertheless, I find it needed perplexing, and this constitutes a chief stumbling block to accept any number of theoretical systems to claim to have secured the reference of numerical singular terms. Although, as we have seen, neologicism is adamant about constraining the methodology of concept introduction by an abstraction principle with mere axiomatic stipulation, the characterization of a fundamental principle like Hume's as a stipulation is reminiscent, it seems to be, of Hilbert's account of his axioms as implicit definition. in his correspondence with Hilbert and in the first of his series of essays on the foundations of geometry Frigg argued that by treating axioms as definitions Hilbert precluded taking them to express truth in the intended and generally accepted sense but if Hilbert's axioms are genuine axioms and therefore for Frege not only true but known to be true the reference of their constituent expressions if Hilbert's axioms are genuine axioms the reference of their constituent expressions Frege argued must already be settled Hilbert's response to this insisted in resisting the notion that his axioms express true arguing that they need only be susceptible of an interpretation might say, they come out of truth. But Sperger's own suggestion for approaching the question

30:00 of reference to abstract object in terms of the context principle and the use of one or another contextual definition seems vulnerable to the same circularity with which he charged Hilbert over the matter of being as implicit definition. There is even a sense in which Frege's application of the context principle to contextual definitions requiring, as it does, that the contextual definitions are true simplicity, is a more appropriate target of these criticisms than Hilbert's use of the doctored of implicit definition. and indeed the role of Hume's principle in Traeger's analysis of our reference to the numbers appears no less ambiguous than the role of Hilbert assigned to axioms Traeger had argued against Hilbert that treating axioms as definitions puts us in the position of having a single equation which do I know but we might equally well ask regarding a comparable circulatory to Traeger's own how it is possible to know of Hume's principle that it is true if we have not first specified the reference of this constituent expression. It would seem, therefore, that Hume's principle is being asked to fulfill roles which are typically in tension with one another. It must, first of all, be a substantive truth, one which implies the basic laws of arithmetic and forms the philosophical basis of our knowledge of them. A point which may be put by saying, that it is the principle's account of our application of the numbers in counting which underlies our knowledge of their infinity. Secondly, it's supposed to give the sense of the cardinality operator a function usually reserved for definitions, or in any case, for sentences which are stipulated or otherwise laid down. Now, I believe there is a way of securing the truth of Hume's principle which addresses both the ambiguity stipulation versus substantive claim and the termed circularity in this methodology without, however, having to accept the conventionalist features of neo-logicism account or focus account for that.

32:30 In order to motivate this suggestion, let's reflect for a moment on a feature of our ordinary model-theoretic analysis of logical truth, according to which a logical truth is one that is true in all models. Even though this analysis appeals to truth in a model and therefore assumes the notion of interpretation or reference, in the case of the logical truth, this dependence turns out to be, in a certain sense, inessential, since the holding of a logical truth in some particular model does structures, from any other structure. By contrast, an ordinary truth, such as the commutative law of group theory, holds precisely in both structures which are commutative groups, not in all structures. Hence, the truth in a model of a mathematical law as opposed to a logical law depends essentially on the reference of some of its constituent expressions. Thus, even though the use of the model-theoretic framework to explain truth in a model, the truth in a model of a statement requires the notion of reference, it allows that the situation is importantly different according to whether we are considering logical or non-logical truths. Briefly put, the logical truths make no special demands on reference. This is one way of spelling out the topic neutrality of logical truth in a model-theoretic setting. To put the point another way, the referential demands of the logical truth are reflected in a model-theoretic framework as a whole rather than any part of it. It is in this sense that the topic neutrality of the logical truth shows not to have a special status vis-a-vis the existential commitments of the model theoretical framework. The account of the logical truth requires only the minimal existential commitment that any model would meet. My suggestion is that this topic neutrality of the truth of logic can be naturally extended to an account of the truth of King's Principle. As will become clear, the plausibility of this extension does not require us to take a stand

35:00 on whether or not Hume's principle is regarded as a logical truth. Let me begin by addressing the question of the truth of Hume's principle. Although there are, of course, models in which Hume's principle fails, it nevertheless seems plausible to expect of any model which purports to constitute a representation of the structure of our concepts of divided relevance sort of concept, that it must make a provision for the unrestricted application to such concepts of the magnality operator, thereby generating the skeleton of concepts deployed in Frege's proof of his theorem. Such a provision is constitutive of any representation of our concept of divided revelry, and of our concept of numerical identity, so that its satisfaction is a feature of any model that might plausibly represent our notion of truth. Hence, if Hume's principle yields the correct analysis of numerical identity, it will be true in any model which is a candidate for representing our notion of truth, simplicity. And any such model will therefore contain both the parallel numbers, in quotation marks, and the concepts associated with them in the development of Trader's proof. Even if we never achieve a complete specification of the model which the world comprises, it would suffice, so far as this account of our knowledge of arithmetic is concerned, that we should know of any such model that it is constrained to satisfy Hume's principle. For if any such model is constrained to satisfy Hume's principle, then we are justified in supposing it true, not just consistent, not just true in some model or other, but true in that model, truth in which coincides with our absolute notion of truth. The problem of securing the generality of Hume's principle consists in establishing the correctness of Brady's analysis of numerical identity

37:30 in terms of the cardinality operator and the relation of one-one correspondence so that it can be seen to be a feature of every family of concepts of divided regiments which belongs to a possible model of our conceptual structure. where Hume's principle would then enjoy a status in the class of models containing such families of concepts, which is entirely analogous to the status of the truth of logic in the class of all models. Just as logical truth makes no demand on truth in reference, which is not already implicit in every other truth, The referential demands of the truth of Hume's principle would be held in common by every truth, and in analogy with the truth of the laws of logic and the model theoretic framework, its truth would be reflected in our framework, our conceptual framework, as a whole. On this conception, Hume's principle is therefore not merely true in that model which represents our concept of truth, simplicity. It is true in any model to which such families of concepts of divided reference as we have or as we might have and the concepts of numerical identity belong. The principle derives its truth from the fact that the notion of the numerical identity of concepts of divided reference of which it is analytic belongs to any model that can be expanded to a representation of our notion of truth implicit or absolute truth. It derives its generality from the fact that it is a truth that holds in every model capable of representing the structure of our concepts of divided reference and our notion of numerical identity. The Hume's principle is true and we are justified by the context principle in holding to a limited realism concerning the practical numbers, limited as the author of the book word, the realism to which we are entitled fails to be a complete vindication by Trayvon's program because the basic laws of arithmetic derivable for Hume's principle allow us to capture the natural numbers only up to isomorphism. In particular, using only the resources of the context principle and the contextual definition

40:00 We are unable to characterize the natural number. There is, therefore, a sense in which the realism concerning the numbers which these considerations vindicate is attenuated relative to that which attaches to realism regarding the physical world. You do not think that the furniture of the world is specifiable, only up to isomorphism. Often, though, on other occasions, and I've said that I compliment you, but I actually do think that, but I do not think of the furniture of the physical world as specifiable when it comes back to more of this. But regardless of something we will have captured uniquely, should we have succeeded in characterizing the model which the physical world comprises. Draguer seems also to have partially conceded this qualification when he saw, in account of the numbers, that would single out one natural sequence of numbers. one from all the rest by characterizing the numbers as extensions, which comprehends all their applications. But if the numbers are captured only structurally, there is a certain conventionalism which attaches to the assertion of their existence. As far as the basic laws of arithmetic derived from Hume's principle are concerned. Any omega sequence will serve as the sequence of canceling. But this is a feature about parallel and our consensus of the constituents of the physical world. A complete vindication of the idealism would require being able to distinguish the natural language, something that memory of resources is going beyond the contextual definition. Nevertheless, we have achieved something more than the conclusion that the contextual definition is merely true in its structure. Instead of this rather weak result, we have explained how it might be seen as true. As such, it would have a distinguished status among mathematical principles, as would the referential commitments of this constituent expression. Leading to one side, the historical question of the view Trader himself came to favor. I am claiming that a novel and plausible philosophy of arithmetic is, at least suggested by Grunbach,

42:30 one which, moreover, is free of the conventionalism, largely free of the conventionalism that his character actually develops. On this view, the explanation of the significant insistence principle and fairness theorem is this. the contextual definition is a constraint on the classes of models that are capable of adequately representing the structure of our concepts of divided reference and our notion of truth simplicity a constraint which bears a direct analogy to that imposed by the logical truth on the class of all structures it is not however a constraint which is arbitrarily laid down like a stipulation of any means adoption of any concept. Rather, the sense in which the contextual definition is definitional is that it advances an analysis of a concept in use, namely our pre-analytic notion of numerical identity. The argument for the correctness of this analysis depends on several important and non-trivial intermediate conclusions throughout the course of Grundlag. The most significant of these are, first, the fundamental thought according to which a statement of number involves the predication of something of a concept. Secondly, the assumption that numbers are arguments to concepts of first level and that the cardinality operator is therefore properly interpreted by a type-lowering function. and thirdly, the contention that as many as and therefore same as if number are to be understood in terms of one-on-one correspondence. By Frege's theorem, in order for any such structure to satisfy being's principle, it must contain the numbers. The existence of the numbers, again, the existence of the numbers, the limited realism of this view, is justified to the degree that we are justified in assuming the correctness of this analysis and to the degree that we are justified in the supposition that the world is modeled by a structure in which it is reflected, in which this analysis is reflected. At this point, it might be appropriate to address a concern dumb as expressed,

45:00 if only to further clarify what is and is not achieved by the view presented here. According to Dometh, Frege's approach to the numbers is fundamentally misguided. Russell's discovery of the contradiction in Frege's account of classes serves only to bring the difficulty into chakra focus. But Frege's basic approach would have been problematic even if no inconsistency had been discovered, since there is an unacceptable circularity in Frege's procedure. The abstraction principle, which introduces the numbers, contains an implicit first order quantifier. So the numbers introduced on the left occur within the range of the variables bounded on the right, and this is made explicit by the explicit definition of 1-1 correspondence. problems. Moreover, this feature is essential if Jung's principle is to support the proof of the infinity of a number sequence, and if that proof turns on the possibility of forming for each n a concept under which the numbers up to and including n fall. But this is possible only if the numbers fall within the range of first order variables in order to allow for the formation of concepts in order to of the formation of the concepts under the success of the argument depends. I'm standing down that stone. It is possible to avoid these difficulties by explicitly assuming at the outset that the numbers are given to us independently of the abstraction principle or contextual definitions which introduces them. But the promise of the Neophradians' abstractionist methodology was that it would replace of the numbers as something which we might say are extended in intuition was an altogether different and non-methorical account of our reference to them. Domingue concludes that not only Drago but also Neofragians who have succeeded him are unable to solve the two problems to which Locacism originally directed its attention. For Domingue's of the objects of number theory.

47:30 And secondly, to show us how to attain the conception of a contemplative infinite domain, to show what our conception of a contemplative infinite domain rests upon, without in either case relying on an appeal to intuition or the facts of experience. Does our account bear any value? Is our account able to address these objections? In order to see in what sense it is able to address these objections, it is necessary to distinguish two rather different ways of viewing the significance of Dragan's theorem. if we think of the proof as purported to show how the numbers might be constructed as Mark Wilson suggested in the recent The Love Memorial Fund at Notre Dame if we think of the proof as purported to show how the numbers might be constructed there is it seems to me an obvious sense in which it legitimately presupposes what it says out to prove how are we view the proof that ambitiously and see it as a verification that Hume's principle implies the delicate infinity of the numbers so that we are therefore justified in claiming of it, both that it captures a central feature of our notion of number and that it reveals the assumptions on which our conception of their infinity may be based then it seems to me there is no In fact, we may take the practice of recovering a central feature of a concept in use by revealing the assumptions on which our use of the concept depends as a characterization of what traditionally passes as a conceptual analysis. Thus understood, Traeger's theorem confirms that its analysis of numerical identity in terms of Thiem's principle is a compelling solution to the second of the two problems Janet poses, that of explaining how we attain the concept of a countably infinite domain. Of course, the debigant infinity of the numbers can be obtained directly from debigant and Keanu's Dedekind's and Piano's well-known astigmatization.

50:00 What is the advantage of Frege's development of erythritic from Hume's principle? The answer, I think, is that by contrast with Dedekind and Piano, Frege derives the number theoretic or pure properties of the numbers from an analysis of their application in the practice of counting. It thus shows how a mathematical analysis, such as Deuteron's and Keanu's, arises out of the most common everyday applications we make of the numbers. But what is that as first problem? Have we given up on securing the existence of the numbers? If Hume's principle is regarded as a conceptual analysis of our arithmetical knowledge, rather than vindicates the existential commitments of what we take ourselves to know. Such an analysis can clarify the nature and extent of those commitments, and it can clarify the assumptions from which they derive, but it may still fall short of persuading someone who, for whatever reason, denies the existence of the numbers. We have simply not addressed the question whether it is coherent which, on the analysis on author, the analysis on the presupposed, our account does not yield what Dummich has called a swath of argument for the existence of the numbers. It bested the fourth explanation of how it is possible to arrive at the subject of a number sequence, an explanation that is based on our use of the numbers in counting. but although such an explanation falls short of the kind of justification Dumbach is demanding it may be said in its defense that it is not at all clear how many things stronger might be achieved so I'm just about done the foregoing proposal has some affinity with earlier views Thus, from Frege and Brussel, I've taken the idea that logic and arithmetic deal with truth of great generality,

52:30 while from Carnap, I have appropriated the idea that the truths of arithmetic are partly constitutive of our conceptual framework. I differ from Carnap over the matter of the basis of the truth of constitutive principles like Hume, for Carnap as I think with Bach and Kristen his two threats have found the decision of ours Carnap, more plausible I think than they, takes this to undermine the notion that a constitutive principle can result of actual content I also differ from Carnap by preserving the notion that a constitutive truth can incorporate significant existential presuppositions it can't have a concept. Karnak would dismiss this contention as a conclusion of internal and that's kind of a question. Where I diverge from Drake and Russellism, in my conception of the generality that institutive principles enjoy, human principles, it is not going to be a general truth where I am, but a principle which holds any possible problem of what the size of this book. By way of conclusion, I would like to remark on the role of count of the truth of the although I appeal to the notion analytic of arguing that he's principle is analytic of the notion of numerical identity I have said nothing of his status as analytic truth. Indeed I conceded that he's principles need not be regarded as a truth of logic and is thereby renounced one of securing its analyticity I've also argued that it is not a mere convention, because it also presents the triviality of the situation, something which the traditional use of the doctrine of analyticity, especially by Carnac and followers of him, was supposed to support. Therefore, on the present account, Hume's principle is a general truth, not just in the sense that it holds universally of everything, in the relative domain and its classification, but in the stronger sense that it is true in every world, that it's in every model which is capable of representing our conceptual structure.

55:00 while it is possible to introduce a notion of analyticity according to which a principle like Hume's would come out analytic insofar as it satisfies this notion of generality. It is unclear what would be gained from doing so. Certainly there is nothing in such a notion of analyticity which would justify the usual conclusions that have been followed from such a characterization. For example, it would not follow that the principle is trivial, and it would not follow that it is without existential commitment or factual content. The plausibility of the idea that Jung's principle is analytic of the concept of numerical identity depends on plausibility of a particular conceptual analysis. But the truth of the principle, this analysis depends on the presuppositions of the framework of which the analysis is an analysis. And these presuppositions are rather strong. How, then, are we to answer the question as human principle analytic? We should answer yes if our goal is to emphasize that the principle expresses the results of conceptual analysis. The significance of such a positive answer is that it indicates an important methodology. The method of analysis is exhibited in Brunelach in the first great work of the analytic tradition. If, however, the point of the question is to suggest that the principle is distinguished in some manner other than what is implied by its generality, that it is a convention of language or without significant existential chief opposition, our answer must be no. Thank you. We have certainly time to send questions. Would it help to compare radio development or radio development age or senior with Now, this may not be historically accurate, but as I'm picturing it, what Vedekin does is he first characterizes the natural number structure.

57:30 And then he has a section towards the end about applications, and he ends up proving something like influence. Right? But it ends up through an explicit definition. Now, is he doing the same thing, in your sense, that is describing, I'm trying to tell you how you put it, but sort of what any model of our number? Yeah. Yeah. Yes and no. No, I mean, I take the Dedekind axiom to be Dedekind's proposed analysis. Right. So what counts as the analysis is the starting point, which is supposed to capture the notion that he sets out to capture. Yes. My contention is that the starting points of Dedekind and Traga differ, and that the significance of this difference is not merely a difference in axiomatic, but that one is proposing a different kind of analysis than the other. One is capturing, and Devin is capturing, in my view, as the fundamental feature of number. It's purely mathematical character, which makes no reference whatsoever to its applicability. whereas Frege, by starting from this different point of view, is emphasizing what might be called a philosophical analysis by making this connection with everyday applications. Well, Dedican makes the connection, too. It just comes afterwards. So, let me see, is it this simple? The Dedican gets the structure first, and then the applications are tacked on, right? Whereas Frege starts with the applications and then gets the structure out of those. But do you see a difference between those, given the way you're understanding the philosophical significance of the two projects? Yes, I do. I mean, I see the starting point as being what is to be called the analysis of the notion of number, or the notion of numerical identity, or whatever. And what is, therefore, constitutive of the notion, or what is the fundamental future of it? And the peculiarity of, I mean, the peculiarity of Frank's approach is that, and that makes it rather different, is that it should tie our concept of the number to this feature of how we're in it.

1:00:00 Because that's what he starts with, rather than that's what he takes on at the end. but if our goal is to get both to get the applications and the structure well of course I mean relative to certain restrictions on Dean's principle the two are entirely equivalent extensively equivalent but the same could be said of many analyses so for example there are different analyses of the notion of computability which turn out to a class of objects in some sense. But we don't regard them as identical analyses for that reason. And the same is true, it seems to me, in this case, of regular determinants. So that's one analogy that is guiding, I think. I'm trying to get clear about what seems to be the main move where you said that the human principle true in any domain that's represented our understanding of the concept of value reference and the medical identity. I'm not quite sure what you meant by that. Could you perhaps say to what extent would the corresponding thing be true, say, about basic law of high? Would basic law of high be true in any domain which represented our understanding of the notion of equivalence of concepts and of identity? Yes, you could say that, and then the unfortunate feature of Basic Law 5 is that it fails, whereas the suggestion is that our conceptual structure, incorporating as it does teams principle in concepts of divided preference that fulfill it, does not. So my account is splitting, in my account I'm trying to do it as a split, the task of capturing a feature which is analytic of a particular concept,

1:02:30 And being able to take that as a basis for the inference that, therefore, the concept and the assumptions that we make regarding it are true. Right. So would you say it's analytic of our concept of extensions that basically all five holds of them in the sense? so is the idea it's analytic of a concept of extension that turns out to be a limited utility right, because nevertheless it's analytical of the concept, so is it more generally looking at the general formulation you wrote in the platform the completely general extraction principle, so is it whatever equivalence relation R is, would you say that It's analytic of the concept introduced by stipulating what you wrote on the blackboard, the concept of sigma. It's analytic of sigma, but that holds. Whatever you want to do. I don't know if I would say that. But suppose I do. Well, I'm just trying to get a grip on what you're saying maps onto the... Right, right. You must say that. I want to ask the same question as Michael, but you're already getting a direction which isn't making the answer clear as I'd like it to be. Can I have something very better, Kevin? Yeah. Let's make a parallel to it. Yeah. What is it for a model to represent the structure of our concept as a whole? there's certainly an impression that phrase which is why it wouldn't be right to respond it has to be already a concept of a certain sort you were generalizing of a concept that's perfect good question I'd be definitely assumption that we have our medical knowledge I'm taking that as something given So that what is required of an appropriate analysis of the concept of numbers is to recover some kind of feature of that knowledge.

1:05:00 I take that to be an infinity in the case of the numbers. I take that to be the basic thing that we know about numbers if we know anything about them at all. so here's the question do the antecedent concepts the divided references, the social concepts is natural number already one of those in this picture or would it rather choose principles to be faithful to any model of concept of divided reference in general maybe they already need their concept of natural number but is the picture all the concepts, but there's something to which he's principal's face. Yeah. A natural number would be one of them. Even if it's not. It would be one of them even if it's not. Even if some tribe have a range of principal's all the concepts, it has yet no color. Is there something of the structure of their concepts which he's supposed to be? Is that the question? Yeah. I guess I want to say regarding that, but I don't want to make this an anthropological and I don't know I don't know naturally how to do that with a symbol and succinct answer and there's a danger to reality on the dilemma if you make an ambitious plan so the thought is roughly the two instruments will reflect something about the structure of any conceivable system It looks like it might just happen with some system of divided directions. Well, I would say of that five, that its conceptual structure would be incapable of accommodating even the elements of our notion of truth.

1:07:30 Why don't you just say the name of the state? Because the model that they would permit would be to offer. Now, it would be an interesting question to investigate, perhaps, what families of conceptual structures might be, and that sounds possible, but that would be a different project than mine. I'm saying of a tribe like ours, in which we have concepts of divided reference of the sort that we do have. King's principle is true in it. But I I have to say, I haven't some, I haven't easy answers, quick answers to the consideration of those sub-cases. Well, okay, the other one said, it shouldn't be the case that the only reason why our concepts, so far as a valid reference, have a structure congenious instruments, because one of those concepts What is characteristic of the structure of our concept is that not only do we have concepts of divided reference, not only do we have a means by which we can make assignments of cardinality to them, but the results of those assignments can themselves be subsumed under other concepts. That's what I meant by the skeleton concepts of underlies straight as proof. Now, the circularity or triviality that is involved there is neither more nor less than the circularity or triviality that would be involved in any form of proof.

1:10:00 But the issue of having the possibility of a tribe that has a very limited conceptual repertoire and of which one would say of them that Hume's principle is not analytic of their conceptual structure is interesting. I suppose animals have conceptual structures in some extended sense and probably don't have a conception of the definite infinity of the number. But the conceptual structure of a representative chimp is likely not going to even be rich enough in its overall architecture, of the work alone in the actual contents of the concepts it contains. Just one of the basis of a model of what everything that we have to be true about the world. That's the way I would incline on that. Let me just say one other thing. the quasi-continent. there seems to be a classical difficulty that emerges from any quasi-continent view of the sort of anthropocentrism and it's as it were I think this bears on what we've just been talking about but I'm curious can you or can anyone else tell me his principle. I have a long footnote to that fact, which I, unfortunately, not in this paper. Hume put the principle in terms of number. And by number, he meant what you mean by erythma or plurality. Yeah. And Hume says something to the effect, when two numbers are so combined, by which I think he means compared, as that to every unit of the one there corresponds a unit of the other

1:12:30 if you pronounce them equal where by equal he means perhaps what we would mean by equivalent in size and so for Hume I mean this is why the designation of this principle is Hume's principle drives Michael Dummett well it's a bit unfair to Hume but it's un-Humean as well to think that he generated infinitely many abstract objects, wouldn't he? But, I mean, part of the... I mean, but plurality, this is probably the difficulty, right, because the use of Hume's principle Freyland, Neopreneans, and myself is one that involves the application of numbers of concepts, and Hume's plurality is not really either a concept or even a sub. It's operating with the conception of the number of the set of units that Frank actually criticizes. A bunch of units are both plurality. But of course, I mean, what causes the flap here is the fact that you count empty concepts as having number. If you didn't, there'd be nothing in... An empty... If an empty... I mean, I don't know what you'd get if you'd ask the guy on the street, but if you say, what's the number of unicorns in your garden? Well, there is no number because there aren't any. And if you took that view, then Hume's principle doesn't generate an axiom of infinity. It's just consistent with any, I mean, you can add it to any model, and it's perfectly okay. Because there are only n non-empty subsets of an n-element set. I mean, sorry, n different sizes of subsets of an n-element set. so it's it's a kind of creation ex nihilo isn't it really I mean if you can stick that empty set in there the whole shooting match comes sort of pops into existence I think Hume might not have been happy with that I don't know I get that impression Yeah, I do. I mean, he's very careful to point out that there's only finite many points on a line, for example.

1:15:00 Can I come back to your answer to Crispin earlier? I'm still trying to get clear about what sort of domain this is of the job that the human principle is true in. I mean, if you have a whole bunch of ordinary sort of concepts, and you have a domain which never left, you didn't have any numbers in it, why would that, I mean, that would therefore be a domain which was not true. So you'd have to say that's not a domain which represents art. That's right, yeah. But in what sense does it fail to represent? I mean, it's got sort of concepts in it, so it's representing our understanding about those. And I think perhaps it's, I mean, so put it on a few numbers in it. You know, if the number's up to five, say, then you could easily say it represents an understanding of the notion of numerical identity because you'd understand what numerical identity was applied to the numbers that there were in that. So where does it fail your criterion? Well, as Stray articulates his truth, whenever he assigns a number, he forms a new concept and then asks after its number. and this process of formation of new concepts in this manner about to proceed indefinitely for it to do so there must be there must be in the domain numbers so why doesn't it help your criticality to do the matrix members in one sense in one sense the criterion in one sense I'm not making claims for I'm not making claims for non-ridiality what I in an older tradition one would say Hume's principle is a conceptual truth I'm trying to give you a picture in a person with which you can think of what that might mean to say that it is a conceptual truth

1:17:30 without using that terminology which I find has been overused and is now illuminated. I'm not, it isn't part of my program that it should come out as a significant discovery that there are infinitely many, that there are infinitely many non-polis. Right, and I thought I got the impression it was part of your program that you should distinguish the existential commitment part of it from the idea of being analytic of some concept. So there should be two statements. First, you can see human principles analytic of a concept. Right. And then you can ask the question, yeah, are there any numbers around to be analytic of? but now the domains that you're talking about that are on which depends the first stage of seeing this analytic of something all the domains in question are domains which seem just to have a full existence Yeah, but what I take to the novel is the notion our access to that fact about that domain should proceed via a principle that, like Hume's, relates to the counting of concepts, relates to the counting of things that fall under concepts. I take that to be the illuminating aspect of the analysis, that it should be derivable from that path. capturing that aspect of our nation of number of numerical identity that suffices for the derivation that suffices for a derivation the definite infinity of that domain not that not that I'm taking it to be that I'm taking that to be our regarding the domain, but it does contain intimately high-eating numbers. Right. And how does

1:20:00 this account avoid the projection of circularity, which you were earlier, I think, placing against the more traditional account, to prove the consistency of whose principle you have to appeal to that number, so that circular projection. Well, I didn't actually raise I was embracing that objection. The circularity objection that I was considering is actually not unrelated to what you're now, I think, putting forward, and that is that this is the objection of Delmet, but in order for Craig's argument to work, the numbers have already to be assumed within the range of the first-order variables. And so, as a result, you can't say of Hume's principle or an analysis based on it or an explanation of the concept of a number based on it, that it has in any manner whatsoever secured the existence of the group. Now, with respect to that objection, I've conceded it and said, indeed, it has not. But I don't have any account that does do that. With respect to the other objection, which came essentially to saying, it gives us no access to a concept of a comically intimate domain, which is an epistemic objection of a different sort a just logical objection of a different sort I'm trying to disagree I mean the fact that the domain already contains the numbers can't be an objection to reanalysis providing us with a means of knowing that the numbers are there and so that stops on my view but the significance of the of the analysis of number based on this principle is the way it addresses that it is a logical concern of what our conception of a company and the domain is based upon it's based upon our experience of counting what falls under concepts in that sense.

1:22:30 But that, of course, presupposes that the objects known are indeed already present. It doesn't construct them in any sense. OK, there is a large, but not infinite number of sandwiches. In the common rule in the philosophy department, there is no matter where it moves. And if you would like to come for a million new restaurants.