The aim here is to describe how to complete the constructive logicist program, in the author’s book Anti-Realism and Logic, of deriving all the Peano-Dedekind postulates for arithmetic within a theory of natural numbers that also accounts for their applicability in counting finite collections of objects. The axioms still to be derived are those for addition and multiplication. Frege did not derive them in a fully explicit, conceptually illuminating way. Nor has any neo-Fregean done so.

We examine the sense in which logic is a priori, and explain how mathematical theories can be dichotomized non-trivially into analytic and synthetic portions. We argue that Core Logic contains exactly the a-priori-because-analytically-valid deductive principles. We introduce the reader to Core Logic by explaining its relationship to other logical systems, and stating its rules of inference. Important metatheorems about Core Logic are reported, and its important features noted. Core Logic can serve as the basis for a foundational program that could (...) be called Natural Logicism, an exposition of which will build on the (meta)logical ideas explained here. (shrink)

Sections “Introduction: Hume’s Principle, Basic Law V and Cardinal Arithmetic” and “The Julius Caesar Problem in Grundlagen—A Brief Characterization” are peparatory. In Section “Analyticity”, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle, bearing in mind that with its analytic or non-analytic status the intended logical foundation of cardinal arithmetic stands or falls. Section “Thought Identity and Hume’s Principle” is concerned with the two criteria of thought identity that Frege states in 1906 and (...) their application to Hume’s Principle. In Section “The Nature ofion: A Critical Assessment of Grundlagen, §64”, I scrutinize Frege’s characterization of abstraction in Grundlagen, §64 and criticize in this context the currently widespread use of the terms “recarving” and “reconceptualization”. Section “Frege’s Proof of Hume’s Principle” is devoted to the formal details of Frege’s proof of Hume’s Principle. I begin by considering his proof sketch in Grundlagen and subsequently reconstruct in modern notation essential parts of the formal proof in Grundgesetze. In Section “Equinumerosity and Coextensiveness: Hume’s Principle and Basic Law V Again”, I discuss the criteria of identity embodied in Hume’s Principle and in Basic Law V, equinumerosity and coextensiveness. In Section “Julius Caesar and Cardinal Numbers—A Brief Comparison Between Grundlagen and Grundgesetze ”, I comment on the Julius Caesar problem arising from Hume’s Principle in Grundlagen and analyze the reasons for its absence in this form in Grundgesetze. I conclude with reflections on the introduction of the cardinals and the reals by abstraction in the context of Frege’s logicism. (shrink)

The idea that mathematics is reducible to logic has a long history, but it was Frege who gave logicism an articulation and defense that transformed it into a distinctive philosophical thesis with a profound influence on the development of philosophy in the twentieth century. This volume of classic, revised and newly written essays by William Demopoulos examines logicism's principal legacy for philosophy: its elaboration of notions of analysis and reconstruction. The essays reflect on the deployment of these ideas (...) by the principal figures in the history of the subject - Frege, Russell, Ramsey and Carnap - and in doing so illuminate current concerns about the nature of mathematical and theoretical knowledge. Issues addressed include the nature of arithmetical knowledge in the light of Frege's theorem; the status of realism about the theoretical entities of physics; and the proper interpretation of empirical theories that postulate abstract structural constraints. (shrink)

Simple-type theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simple-type theory. Even structures that violate simple-types are isomorphic to structures that can be studied in simple-type theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simple-type theory fails to assure an infinity of natural numbers as objects. This paper argues that the problem of infinity (...) is based on a metaphysical prejudice in favor of numbers as objects — a prejudice that mathematics can get along without. (shrink)

What is at stake philosophically for Russell in espousing logicism? I argue that Russell's aims are chiefly epistemological and mathematical in nature. Russell develops logicism in order to give an account of the nature of mathematics and of mathematical knowledge that is compatible with what he takes to be the uncontroversial status of this science as true, certain and exact. I argue for this view against the view of Peter Hylton, according to which Russell uses logicism to (...) defend the unconditional truth of mathematics against various Idealist positions that treat mathematics as true only partially or only relative to a particular point of view. (shrink)

This thesis is an examination of Frege's logicism, and of a number of objections which are widely viewed as refutations of the logicist thesis. In the view offered here, logicism is designed to provide answers to two questions: that of the nature of arithmetical truth, and that of the source of arithmetical knowledge. ;The first objection dealt with here is the view that logicism is not an epistemologically significant thesis, due to the fact that the epistemological status (...) of logic itself is not well understood. I argue to the contrary that on Frege's conception of logic, logicism is of clear epistemological importance. ;The second objection examined is the claim that Godel's first incompleteness theorem falsifies logicism. I argue that the incompleteness theorem has no impact on logicism unless the logicist is compelled to hold that logic is recursively enumerable. I argue, further, that there is no reason to impose this requirement on logicism. ;The third objection concerns Russell's paradox. I argue that the paradox is devastating to Frege's conception of numbers, but not to his logicist project. I suggest that the appropriate course for a post-Fregean logicist to follow is one which divorces itself from Frege's platonism. ;The conclusion of this thesis is that logicism has of late been too easily dismissed. Though several critical aspects of Frege's logicism must be altered in light of recent results, the central Fregean thesis is still an important and promising view about the nature of arithmetic and arithmetical knowledge. (shrink)

Frege's logicism consists of two theses: the truths of arithmetic are truths of logic; the natural numbers are objects. In this paper I pose the question: what conception of logic is required to defend these theses? I hold that there exists an appropriate and natural conception of logic in virtue of which Hume's principle is a logical truth. Hume's principle, which states that the number of Fs is the number of Gs iff the concepts F and G (...) are equinumerous is the central plank in the neo-logicist argument for and. I defend this position against two objections Hume's principle canot be both a logical truth as required by and also have the ontological import required by ; and the use of Hume's principle by the logicist is in effect an ontological proof of a kind which is not valid. (shrink)

This paper aims to shed light on the broader significance of Frege’s logicism against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices explicit. The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency : it lies in the basic idea that in arithmetic one can, and should, express everything that is implicitly presupposed (...) so that nothing is left unsaid. This, in fact, is the target of Wittgenstein’s critique. Rather than the Tractatus, with its claim that logicism attempts to say something that can only be shown, it is the Philosophical Investigations, with its argument by regress against the thesis that every rule which one can follow must be of an explicit nature, that is of real significance here. (shrink)

The paper starts with an examination and critique of Tarski’s wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L∞,∞. Also characterized similarly is a natural generalization of Tarski’s thesis, due to Sher, in terms of bijections between domains. My (...) main objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation over functional type structures (with domains M0 of individuals and {T, F} at their base) is introduced to accomplish the latter. The main result is that an operation is definable from.. (shrink)

The paper argues that Frege’s primary foundational purpose concerning arithmetic was neither that of making natural numbers logical objects, nor that of making arithmetic a part of logic, but rather that of assigning to it an appropriate place in the architectonics of mathematics and knowledge, by immersing it in a theory of numbers of concepts and making truths about natural numbers, and/or knowledge of them transparent to reason without the medium of senses and intuition.

A perspective in the philosophy of mathematics is developed from a consideration of the strengths and limitations of both logicism and platonism, with an early focus on Frege’s work. Importantly, although many set-theoretic structures may be developed each of which offers limited isomorphism with the system of natural numbers, no one of them may be identified with it. Furthermore, the timeless, ever present nature of mathematical concepts and results itself offers direct access, in the face of a platonist (...) account which generates a supposed problem of access. Crucially too, pure mathematics has its own distinctive method of confirming or validating results - mathematical proof - which supplies a higher level of confidence and objectivity than that available elsewhere. The dichotomy of invention and discovery is too jejune a framework for analysing creative mathematical activity. The Gödelian platonist perspective is evaluated and queried through scrutiny of the part played by mathematical resources and constraints in relation to human activity. It appears that there can be non-causal mathematical explanations and mathematical constraint on purely natural processes. Valuable implications of Quine’s naturalism are explored, but one must be cautious of his thesis of confirmational holism. The distinction between algebraic and non-algebraic mathematical theories usefully contributes to our understanding of the internally differentiated nature of the subject. (shrink)

A detailed argument is provided for the thesis that Dedekind was a logicist about arithmetic. The rules of inference employed in Dedekind's construction of arithmetic are, by his lights, all purely logical in character, and the definitions are all explicit; even the definition of the natural numbers as the abstract type of simply infinite systems can be seen to be explicit. The primitive concepts of the construction are logical in their being intrinsically tied to the functioning of the understanding.

In lieu of an abstract, here is a brief excerpt of the content:82 Reviews c:\users\arlene\documents\rj issues\type3501\rj 3501 061 red.docx 2015-07-10 4:07 PM LOGICISM BEYOND PRINCIPIA MATHEMATICA Chris Pincock Philosophy / Ohio State U. Columbus, oh 43210–1365, usa [email protected] William Demopoulos. Logicism and Its Philosophical Legacy. Cambridge: Cambridge U. P., 2013. Pp. xii, 272. isbn: 9781107029804.£60.00; us$104.99 (hb). his book brings together eight previously published essays along with three new essays and a brief introduction. In one way or another, (...) each essay pursues either logicism or some broader implication of logicism that a major figure like Russell or Carnap explored. In a narrow sense, logicism is just the position that grounds the concepts and claims of arithmetic (or all of mathematics) in logic. However, in Demopoulos’ hands, logicism becomes a project of much greater significance. A note added to Chapter 5, the classic 1985 article “Bertrand Russell’s The Analysis of Matter” (co-authored with Michael Friedman), indicates this broader scope and Demopoulos’ own take on logicism. Originally Demopoulos and Friedman had concluded their article with both the observation that there are serious “intellectual tensions produced by logicism’s attempt to account for both pure mathematics and applied mathematics (mathematical physics)” and the pessimistic conclusion that “it appears that we can account for the distinctive character of the one only at the expense of the other” (p. 107). Now Demopoulos indicates that even though “something close to [this] is true of Carnap’s Ramsey-sentence q= Reviews 83 c:\users\arlene\documents\rj issues\type3501\rj 3501 061 red.docx 2015-07-10 4:07 PM reconstruction of the language of science”, “the four previous chapters are an extended argument against” the earlier conjecture. For Demopoulos, the main problem is to make sense of both pure and applied mathematics in something like the original logicist form found in Frege, Russell and Carnap.The solution to this problem arises only when we have pinpointed exactly where these logicist projects failed and how their failures can be overcome. Chapter 1 focuses on Frege and his logicism for arithmetic. Demopoulos’ main contention is that Frege aims to reduce arithmetic to logic in order to demonstrate the autonomy of arithmetic from geometric intuition and empirical experience. Frege’s worry about intuition is not motivated by scepticism or by a concern about the cogency of our knowledge of arithmetic (p. 11).The point of logicism is to free arithmetic from the obscurity associated with intuition and to get a clearer grasp of the generality that is characteristic of arithmetic. (Chapters 8 and 9 develop this point in more detail and use it to motivate a brand of logicism for arithmetic that is taken to improve on current neo-Fregean approaches.) For this reason Hume’s Principle takes a central role in Frege’s reconstruction of arithmetic for it not only establishes the autonomy of arithmetic, but also secures the core use of arithmetic in application, i.e. in counting (p. 19). Unlike Dummett and other commentators Demopoulos does not think that this aspect of Frege’s logicism is flawed. Instead, flaws arise only with the accounts of applied mathematics offered by later figures like Russell and Carnap.The root of these problems is that they extend a particular logicist strategy to other areas of mathematics, such as geometry, and to our scientific knowledge more generally. Chapter 2 considers “Carnap’sThesis”, which Demopoulos summarizes as “the assertion that certain applied mathematical theories are not factual” (p. 28). Although the thesis is clearly central to Carnap throughout his career, Demopoulos ultimately argues for a non-Carnapian basis for the factual/nonfactual distinction.The root of the difference is the kind of criteria of identity that are appropriate for the objects at the heart of the theory: … the factuality of an applied mathematical theory will be shown to turn on the recognition that the criteria of identity appropriate to its objects are empirically constrained in a way that the criteria of identity appropriate to the objects of a nonfactual theory are not. (P. 28) For arithmetic, the natural numbers associated with two sortal concepts are identical just in case the objects that... (shrink)

A detailed argument is provided for the thesis that Dedekind was a logicist about arithmetic. The rules of inference employed in Dedekind's construction of arithmetic are, by his lights, all purely logical in character, and the definitions are all explicit; even the definition of the natural numbers as the abstract type of simply infinite systems can be seen to be explicit. The primitive concepts of the construction are logical in their being intrinsically tied to the functioning of the understanding.

In this paper I examine the prospects for a successful neo–logicist reconstruction of the real numbers, focusing on Bob Hale's use of a cut-abstraction principle. There is a serious problem plaguing Hale's project. Natural generalizations of this principle imply that there are far more objects than one would expect from a position that stresses its epistemological conservativeness. In other words, the sort of abstraction needed to obtain a theory of the reals is rampantly inflationary. I also indicate briefly why (...) this problem is likely to reappear in any neo–logicist reconstruction of real analysis. (shrink)

Bertrand Russell's contributions to last century's philosophy and, in particular, to the philosophy of mathematics cannot be overestimated.Russell, besides being, with Frege and G.E. Moore, one of the founding fathers of analytical philosophy, played a major rôle in the development of logicism, one of the oldest and most resilient1 programmes in the foundations of mathematics.Among his many achievements, we need to mention the discovery of the paradox that bears his name and the identification of its logical nature; the generalization (...) to the whole of mathematics of Frege's idea that it is not possible to draw a demarcation line between logic and arithmetic; the programme, carried out with Whitehead, of derivation of mathematics from the logical system of Principia Mathematica ; and the ramified theory of types, devised by Russell to protect the system of PM from the known paradoxes.Although there is an ample literature on these topics, it is quite important to reconsider Russell's contributions to the foundations of mathematics at a time when, as a consequence of the crisis of the classical programmes in the foundations of mathematics, new trends are beginning to develop within the philosophy of mathematics. These are trends which move in a very different direction from that of logicism, intuitionism, and Hilbert's programme.To see this we need to consider that, in spite of profound disagreements on the nature of mathematical activity, on the relationship existing between logic and mathematics, on the causes of and therapies for the paradoxes, etc., logicism, intuitionism, and Hilbert's programme share an important metaphor: the idea that mathematics is an edifice built on unshakable foundations,2 an edifice which makes possible only a cumulative growth of mathematical knowledge.Such a metaphor—which, together with more specific theses belonging to these schools of thought, remained unsupported …. (shrink)

Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternative consistent principle governing value-ranges. Given some natural (...) assumptions regarding what an acceptable neo-logicistic theory of value-ranges might look like, successfully implementing this alternative approach is impossible. (shrink)

A central part of Frege's logicism is his reconstruction of the natural numbers as equivalence classes of equinumerous concepts or classes. In this paper, I examine the relationship of this reconstruction both to earlier views, from Mill all the way back to Plato, and to later formalist and structuralist views; I thus situate Frege within what may be called the “rise of pure mathematics” in the nineteenth century. Doing so allows us to acknowledge continuities between Frege's and other (...) approaches, but also to understand better the motivation and the significance of his innovations, as well as their limits. (shrink)

Chapter 3 surveys objectivity in the natural sciences. Thomas Kuhn problematized the logicist understanding of the objectivity or rationality of scientific change, providing a very different picture than that of the cumulative or step-wise progress of theoretical science. Theories often compete, and when consensus builds around one competitor it may be for a variety of reasons other than just the direct logical implications of experimental successes and failures. Kuhn pitted the study of the actual history of science against what (...) Hans Reichenbach referred to as the “logical substitutes” logicists used to reconstruct the rationality of theory change. Such substitutes were promoted to show that theory confirmation is a purely epistemic affair, where purely epistemic implies purely inferential. But where the positivists saw logical compulsion as the reason for theory change, Kuhn saw psychology in values as relevant to how theories are actually confirmed or accepted by the scientific community. (shrink)

There are two ways of thinking about the natural numbers: as ordinal numbers or as cardinal numbers. It is, moreover, well-known that the cardinal numbers can be defined in terms of the ordinal numbers. Some philosophies of mathematics have taken this as a reason to hold the ordinal numbers as (metaphysically) fundamental. By discussing structuralism and neo-logicism we argue that one can empirically distinguish between accounts that endorse this fundamentality claim and those that do not. In particular, we (...) argue that if the ordinal numbers are metaphysically fundamental then it follows that one cannot acquire cardinal number concepts without appeal to ordinal notions. On the other hand, without this fundamentality thesis that would be possible. This allows for an empirical test to see which account best describes our actual mathematical practices. We then, finally, discuss some empirical data that suggests that we can acquire cardinal number concepts without using ordinal notions. However, there are some important gaps left open by this data that we point to as areas for future empirical research. (shrink)

Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on whether it (...) applies to definitions of both natural and real numbers. Our aim is to trace the origins of$FC$and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish$AC$from$FC$(§2). We discuss six rationales which may motivate the adoption of different instances of$AC$and$FC$(§3). We turn to the possible interpretations of$FC$(§4), and advance a Semantic$FC$(§4.1), arguing that while it suits Frege’s definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege’s conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of$FC$is indeed faithful to Frege’s definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of$FC$to Frege and appreciating the role of the Architectonic$FC$can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5). (shrink)

Frege's main contributions to logic and the philosophy of mathematics are, on the one hand, his introduction of modern relational and quantificational logic and, on the other, his analysis of the concept of number. My focus in this paper will be on the latter, although the two are closely related, of course, in ways that will also play a role. More specifically, I will discuss Frege's logicist reconceptualization of the natural numbers with the goal of clarifying two aspects: the (...) motivations for its core ideas; the step-by-step development of these ideas, from Begriffsschrift through Die Grundlagen der Arithmetik and Grundgesetze der Arithmetik to Frege's very last writings, indeed even beyond those, to a number of recent "neo-Fregean" proposals for how to update them. One main development, or break, in Frege's views occurred after he was informed of Russell's antinomy. His attempt to come to terms with this antinomy has found some attention in the literature already. It has seldom been analyzed in connection with earlier changes in his views, however, partly because those changes themselves have been largely ignored. Nor has it been discussed much in connection with Frege's basic motivations, as formed in reaction to earlier positions. Doing both in this paper will not only shed new light on his response to Russell's antinomy, but also on other aspects of his views. In addition, it will provide us with a framework for comparing recent updates of these views, thus for assessing the remaining attraction of Frege's general approach. I will proceed as follows: In the first part of the paper (§1.1 and §1.2), I will consider the relationship of Frege's conception of the natural numbers to earlier conceptions, in particular to what I will call the "pluralities conception", thus bringing into sharper focus his core ideas and their motivations. In the next part (§2.1 and §2.2), I will trace the order in which these ideas come up in Frege's writings, as well as the ways in which his position gets modified along the way, both before and after Russell's antinomy.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:360 HUME'S NATURALI Z EP PHILOSOPHY In "Epistemology Naturalized," Quine claimed that the failure of reductive-foundationalist attempts in epistemology, after the model of Carnap' s Aufbau, must lead to a redefinition of epistemology's task. Instead of setting out to reconstruct the whole fabric of our knowledge from absolute data through deductive operations, we should investigate how human subjects derive their knowledge of nature from sensory inputs. Thus epistemology is (...) to be conceived as a branch of natural science, cognate with psychology, not as the study of the foundations of science any more. In the present paper, I will first outline Quine's position and his related interpretation of Hume's philosophical enterprise. Then, I will try to assess the relevance and fairness of Quine's account and consider in what measure it allows us to understand the novelty of Hume's conception of philosophy, since Hume also advocated a nonfoundationalist conception of philosophy which bears some striking analogy to naturalized epistemology. Finally, I will suggest which consequences are to be drawn from this unusual view of Hume, especially as regards the self-reference of philosophy. Quine's position in the matter of epistemology is well known. For him, under its traditional form and program, epistemology used to deal with the foundations of science, be it mathematics or the science of nature. This foundationalist approach had two connected aspects, one which Quine calls conceptual; the other he calls doctrinal. As Quine 361 puts it "the conceptual studies are concerned with clarifying concepts by defining them, some in terms of others" (Quine, p. 69). In the same manner, the doctrinal studies attempt to establish laws "by proving them, some on the basis of others" (Quine, p. 69). Derivative concepts will be defined in terms of the clearer ones and complex laws will be proved from primitive and obvious ones. Thus the traditional or classical conception of epistemology involves simultaneously a theory of concepts or meanings and a theory of truth. These two branches (Quine speaks of a duality of structure or bifurcation) converge so as to provide an analysis of the way our knowledge of nature -- and our knowledge in general — stems from and is built upon our sensory experience which validates it. In so characterizing epistemology, Quine simply restates in his own terms the f oundationalist and reductivist program of positivist epistemology, which itself tied in with the logicist program in mathematics. Then Quine suggests that Hume dealt with both sides of this program, the conceptual and the doctrinal. To begin with, Hume developed what might be called a theory of concepts or meanings in terms of our sensory data. Such is the point of his theory of ideas which consists in tracing them back to the original impressions they are derived from. In spite of its limitations, which hinge on the lack of an approach in terms of statements, Hume's phenomenalism does the job of a theory of meaning and concepts. Hume also attempted to give an answer to the question of the truth of our knowledge. In this case again, his phenomenalist stand allows him to account for the truth of our singular statements concerning our perceptions, even though this result is achieved at the cost of identifying objects and sensory data. 362 However, when Hume comes to general statements and singular statements concerning future events, his attempt fails. This is not surprising. According to Quine, if it were possible to improve on Hume's theory of meaning and concepts, especially through the use of contextual definition techniques, by contrast, all the attempts to logically derive the truth of our knowledge from true premises have failed. In matter of induction, we are not one step further than Hume was. "The most modest of generalizations about observable traits will cover more cases than its utterer can have had occasions actually to observe. The hopelessness of grounding natural science upon immediate experience in a firmly logical way was acknowledged" (Quine, p. 74). Or "the Humean predicament is the human predicament." For Quine, foundationalist attempts of the sort Hume engaged in involve two cardinal principles of empiricism which remain unchallenged. One of these principles is to the effect... (shrink)

This article is an extended critical study of Kit Fine’s The limits of abstraction, which is a sustained attempt to take the measure of the neo-logicist program in the philosophy and foundations of mathematics, founded on abstraction principles like Hume’s principle. The present article covers the philosophical and technical aspects of Fine’s deep and penetrating study.

In the preface to this interesting book, Achinstein claims to have shifted away from the tradition which deals with explanation in logicist terms. I must confess at the outset that I do not think that he has succeeded in escaping from the main constraint that logicism represents, the attempt to do philosophy without attention to specific features of content. The "content" approach is not even mentioned in this lengthy work. And yet, as I hope to bring out in this (...) review, Achinstein makes one essential step towards a "content" approach, without perhaps quite noticing that he has done so. (shrink)

Wittgenstein gave a clearly erroneous refutation of Russell’s logicist project. The errors were ably pointed out by Mark Steiner. Nevertheless, I was motivated by Wittgenstein and Steiner to consider various ideas about the natural numbers. I ask which notations for natural numbers are ‘buck-stoppers’. For us it is the decimal notation and the corresponding verbal system. Based on the idea that a proper notation should be ‘structurally revelatory’, I draw various conclusions about our own concept of the (...) class='Hi'>natural numbers. (shrink)

This essay aims to retrace the varied influences of Henri Poincaré’s ideas on the thought of Joachim Metallmann (1889-1942). Metallmann was a Polish Jew who was murdered by the Germans during World War II and has largely been neglected in the West. Metallmann’s philosophy of science inspired by figures like A. N. Whitehead and H. Poincaré was contrary to the positivistic and logicistic trends dominant in Poland in his time, veering towards a metaphysical reflection on nature. The influences of Poincaré’s (...) work can be found throughout Metallmann’s major works. My intention is to show how Metallmann’s idea of emergentism is quite probably based on Poincaré’s theory of nonlinear dynamical systems. Moreover, the structuralist tendencies of Metallmann’s thought can be linked to Poincaré’s ideas. This all proves that Poincaré’s impact on the philosophy of science goes beyond his role as a “father of conventionalism.”. (shrink)

ABSTRACT How do non-experts single out numbers for reference? Linnebo has argued that they do so using a criterion of identity based on the ordinal properties of numerals. Neo-logicists, on the other hand, claim that cardinal properties are the basis of individuation, when they invoke Hume’s Principle. I discuss empirical data from cognitive science and linguistics to answer how non-experts individuate numbers better in practice. I use those findings to develop an alternative account that mixes ordinal and cardinal properties to (...) provide a detailed answer to the question: how do we in fact semantically individuate numbers? (shrink)

This essay aims to retrace the varied influences of Henri Poincaré’s ideas on the thought of Joachim Metallmann (1889-1942). Metallmann was a Polish Jew who was murdered by the Germans during World War II and has largely been neglected in the West. Metallmann’s philosophy of science inspired by figures like A. N. Whitehead and H. Poincaré was contrary to the positivistic and logicistic trends dominant in Poland in his time, veering towards a metaphysical reflection on nature. The influences of Poincaré’s (...) work can be found throughout Metallmann’s major works. My intention is to show how Metallmann’s idea of emergentism is quite probably based on Poincaré’s theory of nonlinear dynamical systems. Moreover, the structuralist tendencies of Metallmann’s thought can be linked to Poincaré’s ideas. This all proves that Poincaré’s impact on the philosophy of science goes beyond his role as a “father of conventionalism.”. (shrink)

Frege proposed that his Context Principle—which says that a word has meaning only in the context of a proposition—can be used to explain reference, both in general and to mathematical objects in particular. I develop a version of this proposal and outline answers to some important challenges that the resulting account of reference faces. Then I show how this account can be applied to arithmetic to yield an explanation of our reference to the natural numbers and of their metaphysical (...) status. (shrink)

"Case notes." Ethos: Official Publication of the Law Society of the Australian Capital Territory, (199), pp. 34–35.