The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this paper, it is shown that the graph of the indefinite Denjoy integral f↦∫xaf is a coanalytic non-Borel relation on the product space M[a,b]×C[a,b], where M[a,b] is the Polish space of real-valued measurable functions on [a,b] and where C[a,b] is the Polish space of real-valued continuous functions on [a,b]. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, (...) called ACG∗[a,b], is a coanalytic but not Borel subclass of the space C[a,b], thus answering a question posed by Dougherty and Kechris. Some basic model theory of the associated spaces of integrable functions is also studied. Here the main result is that, when viewed as an ℝ[X]-module with the indeterminate X being interpreted as the indefinite integral, the space of continuous functions on the interval [a,b] is elementarily equivalent to the Lebesgue-integrable and Denjoy-integrable functions on this interval, and each is stable but not superstable, and that they all have a common decidable theory when viewed as ℚ[X]-modules. (shrink)

I argue that the best spacetime setting for Newtonian gravitation (NG) is the curved spacetime setting associated with geometrized Newtonian gravitation (GNG). Appreciation of the ‘Newtonian equivalence principle’ leads us to conclude that the gravitational field in NG itself is a gauge quantity, and that the freely falling frames are naturally identified with inertial frames. In this context, the spacetime structure of NG is represented not by the flat neo-Newtonian connection usually made explicit in formulations, but by the sum of (...) the flat connection and the gravitational field. 1 Introduction2 Newtonian Gravity: The Orthodox Approach3 Newtonian Gravity: Additional Symmetries4 Cosmological Considerations5 A Newtonian Equivalence Principle: Inertial Frames in Newtonian Gravitation6 Theory Equivalence?7 Conclusion. (shrink)

This paper looks at the relationship between spacetime functionalism and Harvey Brown’s dynamical relativity. One popular way of reading and extending Brown’s programme in the literature rests on viewing his position as a version of relationism. But a kind of spacetime functionalism extends the project in a different way, by focussing on the account Brown gives of the role of spacetime in relativistic theories. It is then possible to see this as giving a functional account of the concept of spacetime (...) which may be applied to theories that go beyond relativity. This paper explores the way in which both the relationist project and the functionalist project relate to Brown’s work, despite being incompatible. Ultimately, these should not be seen as two conflicting readings of Brown, but two different directions in which to take his project. (shrink)

I argue that the need to understand spacetime structure as emergent in quantum gravity is less radical and surprising it might appear. A clear understanding of the link between general relativity's geometrical structures and empirical geometry reveals that this empirical geometry is exactly the kind of thing that could be an effective and emergent matter. Furthermore, any theory with torsion will involve an effective geometry, even though these theories look, at first glance, like theories with straightforward spacetime geometry. As it's (...) highly likely that there will be a role for torsion in quantum gravity, it's also highly likely that any theory of quantum gravity will require us to get to grips with emergent spacetime structure. (shrink)

Several modern accounts of explanation acknowledge the importance of abstraction and idealization for our explanatory practice. However, once we allow a role for abstraction, questions remain. I ask whether the relation between explanations at different theoretical levels should be thought of wholly in terms of abstraction, and argue that changes of the quantities in terms of which we describe a system can lead to novel explanations that are not merely abstractions of some more detailed picture. I use the example of (...) phase transitions as described by statistical mechanics and thermodynamics to illustrate this, and to demonstrate some details of the relationship between abstraction, idealization, and novel explanation. (shrink)

This paper aims to understand how recent discussion of novel and robust behaviour in physics might be applied in biology and other special sciences. In particular, it looks at the prospects for extending an account of novel explanation to biological examples. Despite the differences in the disciplines, the prospects look good, at least when we look at a biological example in which a certain kind of reduction is possible.

I look at the ‘flavour-oscillation clocks’ proposed by D. V. Ahluwalia and two of his arguments suggesting that such clocks might behave in a way that threatens the geometricity of general relativity (GR). The first argument states that the behaviour of these clocks in the vicinity of a rotating gravitational source implies a non-geometrical element of gravity. I argue that the phenomenon is best seen as an instance of violation of the ‘clock hypothesis’ and therefore does not threaten the geometrical (...) nature of gravitation. Ahluwalia’s second argument, for the ‘incompleteness’ of general relativity, involves the idea that flavour-oscillation clocks can detect constant gravitational potentials. I argue that the purported ‘incompleteness-establishing’ result is in fact one that applies to all clocks. It is entirely derivable from general relativity, does not result in the observability of the potential, and is not at odds with any of general relativity’s foundations. (shrink)

Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a (...) model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension. As an application, we resolve an analogue of the joint consistency problem in the predicative setting. (shrink)

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms (...) of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions. (shrink)

The Routledge Companion to Philosophy of Physics is a comprehensive and authoritative guide to the state of the art in the philosophy of physics. It contains 54 self-contained chapters written by leading philosophers of physics at both senior and junior levels, making it the most thorough and detailed volume of its type on the market – nearly every major perspective in the field is represented. The Companion’s 54 chapters are organized into 12 sections. The first seven sections cover all of (...) the major physical theories investigated by philosophers of physics today, and the last five explore key themes that unite the study of these theories. I. Classical Mechanics II. Special Relativity III. General Relativity IV. Non-Relativistic Quantum Mechanics V. Quantum Field Theory VI. Quantum Gravity VII. Statistical Mechanics and Thermodynamics VIII. Explanation IX. Intertheoretic Relations X. Symmetries XI. Metaphysics XII. Cosmology. (shrink)

This paper presents new constructions of models of Hume's Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results of this paper (...) are: (i) there is a consistent extension of the hyperarithmetic fragment of Basic Law V which interprets the hyperarithmetic fragment of second-order Peano arithmetic, and (ii) the hyperarithmetic fragment of Hume's Principle does not interpret the hyperarithmetic fragment of second-order Peano arithmetic, so that in this specific sense there is no predicative version of Frege's Theorem. (shrink)

This book critically examines Leo Strauss s claim that the philosophers of antiquity, especially Plato, wrote esoterically, hiding the highest truths exclusively between the lines.

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...) that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work. (shrink)

Two puzzles with regard to the Kritik der reinen Vernunft are incongruent counterparts and causality. In De mundi sensibilis atque intelligibilis forma et principiis, Kant indicates that the experience of things like left and right hands, so-called incongruent counterparts, involve certain pure intuitions, and hence constitute one line of evidence for the claim that the concept of space itself is a pure intuition. In KrV, Kant again argues that the concept of space itself is a pure intuition, but does not (...) cite the experience of incongruent counterparts as evidence for this claim. Since there is ostensibly nothing in KrV which tells against the existence of the experiences of incongruent counterparts, the natural question is: “Why, in KrV, does Kant not cite the experience of incongruent counterparts as evidence for the claim that the concept of space is a pure intuition?” The problem with causality is as follows. One of the most primary and basic claims of KrV is that empirical experience is structured by non-empirical concepts, such as substantiality and causality. In a portion of KrV entitled the Transcendental Deduction, Kant gives section 20 the heading “All sensible intuitions stand under the categories, as conditions under which alone their the manifold can come together in one consciousness”. Since categories are those concepts which structure empirical experience, section 20 has demonstrated that all sensible intuitions are subject to substantiality and causality. However, there is a later portion of KrV entitled the Second Analogy with the heading “Principle of temporal sequence according to the law of causality: All alterations occur in accordance with the law of the connection of cause and effect”. Thus, the natural question is: “What does the Second Analogy demonstrate about causality which the Transcendental Deduction did not already demonstrate about causality?”. (shrink)

It is well-known that Newton’s theory of gravity, commonly held to describe a gravitational force, can be recast in a geometrical form: Newton- Cartan theory. It is less well-known that general relativity, an apparently geometrical theory, can be reformulated in such a way that it resembles a force theory; teleparallel gravity does just this. This raises questions. One of these concerns theoretical underdetermination. I argue that these theories do not, in fact, represent cases of worrying underdetermination. On close examination, the (...) alternative formulations are best interpreted as postulating the same spacetime ontology. In accepting this, we see that the ontological commitments of these theories cannot be directly deduced from their mathematical form. The geometrical nature of a gravitational theory is not a straightforward consequence of anything internal to that theory as a theory of gravity. Rather, it essentially relies on the rest of nature (the nongravitational interactions) conspiring to choose the appropriate set of inertial frames. (shrink)

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...) from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence (...) relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic. (shrink)

International Studies in the Philosophy of Science, Volume 25, Issue 2, Page 199-202, June 2011.

In this paper, I develop an argument for the thesis that ‘maximality is extrinsic’, on which a whole physical object is not a whole of its kind in virtue of its intrinsic properties. Theodore Sider has a number of arguments that depend on his own simple argument that maximality is extrinsic. However, Peter van Inwagen has an argument in defence of his Duplication Principle that, I will argue, can be extended to show that Sider's simple argument fails. However, van Inwagen's (...) argument fails against a more complex, sophisticated argument that maximality is extrinsic. I use van Inwagen's own commitments to various forms of causation and metaphysical possibility to argue that maximality is indeed extrinsic, although not for the mundane reasons that Sider suggests. I then argue that moral properties are extrinsic properties. Two physically identical things can have different moral properties in a physical world. This argument is a counterexample to a classical ethical supervenience idea (often attributed to G.E. Moore) that if there is identity of physical properties in a physical world, then there is identity in moral properties as well. I argue moral value is ‘border sensitive’ and extrinsic for Kantians, utilitarians, and Aristotelians. (shrink)

In this book, Sean Noah Walsh applies Herbert Marcuse’s observations on counterrevolution to recent developments in education politics. Seemingly disparate issues such as the exercise of state power to reorganize curricula, the derision of intellectuals, the permeation of consumerism into the collegiate experience, and the expansion of online teaching belong to the same strategy in which the faculties of dissent are neutralized before they can develop and dissent is established as the paramount political obscenity.

Many philosophers believe that a moral theory, given all the relevant facts, should be able to determine what is morally right and wrong. It is commonly argued that Aristotle’s ethical theory suffers from a fatal flaw: it places responsibility for determining right and wrong with the virtuous agent who has phronesis rather than with the theory itself. It is also commonly argued that Immanuel Kant’s ethical theory does provide a concept of right that is capable of determining right and wrong (...) in specific cases. I argue, however, that Kant never gives a determinate moral theory of right. Rather, I argue that Kant’s moral theory is similar in many ways to that of Aristotle, in that it still holds that a moral agent with phronesis, rather than the theory, determines what is right. Kant’s practical philosophy was not so much meant to tell us right and wrong as to prevent bad moral theory from corrupting our moral common sense, and it is our moral common sense that determines right and wrong naturally. (shrink)

Aristotle’s best human life is attained through theoretical contemplation, and Confucius’ is attained through practical cultivation of the social self. However, I argue that in the best human life for both Confucius and Aristotle, a form of theoretical contemplation must occur and can only occur with an ethical commitment to community life. Confucius, like Aristotle, sees that the best contemplation comes after later-life, greater-learning and is central to ethical and community life. Aristotle, like Confucius, sees the best contemplation as presupposing (...) full ethical commitment to community life. So, I argue for the theses that: on Aristotle’s view, the best human contemplation requires one be fully morally good; on Confucius’ view, to be fully morally good requires the best human contemplation; being fully morally good for both requires commitment to the good of others and the community. (shrink)

The aim of this article is to address the recently renewed debate pertaining to esotericism, secret messages encoded within writings from antiquity, especially in the writings of Plato. The question of esotericism has assumed a prominent role within debates concerning the history of political thought. Ever since Leo Strauss offered his suspicion that there were secrets ‘buried in the writings of the rhetoricians of antiquity’, the idea that philosophers deliberately concealed their true beliefs in a way that few could detect (...) has been fiercely debated. More recently, the research of J.B. Kennedy has made international headlines for discovering a musical pattern embedded within Platonic writings, a pattern that Kennedy insists is evidence of Plato’s Pythagorean allegiance. The theses proffered by Strauss and Kennedy are empty doctrines of esotericism, or empty esotericisms. These doctrines insinuate the presence of secret or coded writing within Platonic dialogues but reveal no actual secret. These theses of esotericism falsely represent Plato as hyper-cryptic.Without actually providing substantive content, these notions of esotericism compel the reader to merely negate the exoteric writings of Plato, which actually render his already heterodox writings as commonplace and orthodox. (shrink)

Recent discussions of emergence in physics have focussed on the use of limiting relations, and often particularly on singular or asymptotic limits. We discuss a putative example of emergence that does not fit into this narrative: the case of phonons. These quasi-particles have some claim to be emergent, not least because the way in which they relate to the underlying crystal is almost precisely analogous to the way in which quantum particles relate to the underlying quantum field theory. But there (...) is no need to take a limit when moving from a crystal lattice based description to the phonon description. Not only does this demonstrate that we can have emergence without limits, but also provides a way of understanding cases that do involve limits. (shrink)

Normal 0 false false false EN-US X-NONE X-NONE Normal 0 false false false EN-US ZH-TW X-NONE It is commonly believed that impartial utilitarian moral theories have significant demands that we help the global poor, and that the partial virtue ethics of Mencius and Aristotle do not. This ethical partiality found in these virtue ethicists has been criticized, and some have suggested that the partialistic virtue ethics of Mencius and Aristotle are parochial (i.e., overly narrow in their scope of concern). I (...) believe, however, that the ethics of Mencius and Aristotle are both more cosmopolitan than many presume and also are very demanding. In this paper, I argue that the ethical requirements to help the poor and starving are very demanding for the quintessentially virtuous person in Mencius and Aristotle. The ethical demands to help even the global poor are demanding for Mencius’ jun-zi ( 君子 chün-tzu / junzi ) and Aristotle’s megalopsuchos . I argue that both the jun-zi and megalopsuchos have a wide scope of concern for the suffering of poor people. I argue that the relevant virtues of the jun-zi and megalopsuchos are also achievable for many people. The moral views of Mencius and Aristotle come with strong demands for many of us to work harder to alleviate global poverty. . (shrink)

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy (...) of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity, while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic. Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the~$\omega$-rule or to the non-computability and non-continuity of probability assignments. Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments, such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated. In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar. (shrink)

Lévy's Upward Theorem says that the conditional expectation of an integrable random variable converges with probability one to its true value with increasing information. In this paper, we use methods from effective probability theory to characterise the probability one set along which convergence to the truth occurs, and the rate at which the convergence occurs. We work within the setting of computable probability measures defined on computable Polish spaces and introduce a new general theory of effective disintegrations. We use this (...) machinery to prove our main results, which (1) identify the points along which certain classes of effective random variables converge to the truth in terms of certain classes of algorithmically random points, and which further (2) identify when computable rates of convergence exist. Our convergence results significantly generalize earlier results within a unifying novel abstract framework, and there are no precursors of our results on computable rates of convergence. Finally, we make a case for the importance of our work for the foundations of Bayesian probability theory. (shrink)

Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics textbooks: these are aimed squarely at mathematicians; (...) they typically presuppose that the reader has a significant background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are then scattered across disconnected pockets of papers, separated by decades or more. -/- The first aim of Philosophy and Model Theory, then, is to consider the philosophical uses of model theory. On a technical level, we try to show how philosophically significant results connect to one another, and also to state the best version of a result for philosophical purposes. On a philosophical level, we show how similar dialectical situations arise repeatedly, across fragmented debates in varied philosophical areas. -/- The second aim of Philosophy and Model Theory, though, is to consider the philosophy of model theory. Model theory itself is rarely taken as the subject matter of philosophising (contrast this, say, with the philosophy of biology, or the philosophy of set theory). But model theory is a beautiful part of pure mathematics, and worthy of philosophical study in its own right. (shrink)

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\'e to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak K\"onig's (...) Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

Philosophy and Phenomenological Research, Volume 99, Issue 3, Page 732-738, November 2019.

Theorists interested in learning more about any given interpretive approach are often required to navigate a dizzying array of sources, with no clear sense of where to begin. The prose of many primary sources is often steeped in dense and technical argot that novices find intimidating or even impenetrable. Interpretation in Political Theory provide students of political theory a single introductory reference guide to major approaches to interpretation available in the field today. Comprehensive and clearly written, the book includes: A (...) historical and theoretical overview that situates the practice of interpretation within the development of political theory in the twentieth century. Chapters on Straussian esotericism, historical approaches within the Cambridge School of interpretation, materialist approaches associated with Marxism, the critical approaches associated with varieties of feminism, Greimassian semiotics, Foucaultian genealogy, the negative dialectics of Theodor Adorno, deconstruction as exemplified by Jacques Derrida and Paul de Man, and Lacanian psychoanalysis. An exposition of the theoretical and disciplinary background of each approach, the tools and techniques of interpretation it uses, its assumptions about what counts as a relevant text in political theory, and what it considers to be the purpose or objective of reading in political theory. A reading of Thomas Hobbes s Leviathan to illustrate how each approach can be applied in practice. A list of suggestions for further reading that will guide those interested in pursuing more advanced study. An invaluable textbook for advanced undergraduates, graduate students, and even seasoned scholars of political theory interested in learning more about different interpretive approaches. ". (shrink)

David Lewis insists that restrictivist composition must be motivated by and occur due to some intuitive desiderata for a relation R among parts that compose wholes, and insists that a restrictivist’s relation R must be vague. Peter van Inwagen agrees. In this paper, I argue that restrictivists need not use such examples of relation R as a criterion for composition, and any restrictivist should reject a number of related mereological theses. This paper critiques Lewis and van Inwagen (and others) on (...) their respective mereological metaphysics, and offers a Golden Mean between their two opposite extremes. I argue for a novel account of mereology I call Modal Mereology that is an alternative to Classical Mereology. A modal mereologist can be a universalist about the possible composition of wholes from parts and a restrictivist about the actual composition of wholes from parts. I argue that puzzles facing Modal Mereology (e.g., puzzles concerning Cambridge changes and the Problem of the Many, and how to demarcate the actual from the possible) are also faced in similar forms by classical universalists. On my account, restricted composition is rather motivated by and occurs due to a possible whole’s instantiating an actual type. Universalists commonly believe in such types and defend their existence from objections and puzzles. The Modal Mereological restrictivist can similarly defend the existence of such types (adding that such types are the only wholes) from similar objections and puzzles. (shrink)

A semantics for quantified modal logic is presented that is based on Kleene's notion of realizability. This semantics generalizes Flagg's 1985 construction of a model of a modal version of Church's Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church's Thesis and a variant of a modal set theory (...) due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra's generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott's graph model (for the untyped lambda calculus) which witnesses the failure of the stability of non-identity. (shrink)

Philosophy of history is not a subject which has hitherto attracted much attention in this country. Preoccupation with the methods and achievements of the natural sciences, and distaste for the sort of rationale of history as a whole which Hegel and others offered under the title in the early nineteenth century, have served to make most British philosophers accord its problems only the most casual recognition. It is therefore all the more interesting to find an English writer of unusual powers (...) both of argument and expression, who was himself an historian of distinction in his special field, devoting a large part of his philosophical thinking to the problems of historical knowledge and their wider implications. Unfortunately, R. G. Collingwood was not able to begin the major work on philosophy of history, to writing which, as we are told in the introduction to the present book, he had looked forward for twenty-five years, until his powers were seriously affected by bad health, and he did not live to complete it. But though the full results of his thinking on the subject are lost, we are now able, thanks to the skilful work of his friend Professor T. M. Knox, to study what is in effect an extensive interim report on it. Professor Knox has produced The Idea of History largely on the basis of lectures which Collingwood prepared in 1936, supplemented by two essays already published which date from the same period and a certain amount of material from the unfinished Principles of History written in 1939. (shrink)

This article improves two existing theorems of interest to neologicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation E is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only three profiles. The improvements to Fine’s theorem allow for an analysis of (...) the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan’s relative categoricity theorem. (shrink)

Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap. Abstractionists about numbers—those who think natural numbers are objects characterized by abstraction principles—face an acute version of this problem. Many abstraction principles appear to characterize the natural numbers. If each abstraction principle determines its own sort, then there is no single subject-matter of arithmetic—there are too many numbers. That is, unless objects can belong to more than one sort. (...) But if there are multi-sorted objects, there should be cross-sortal identity principles for identifying objects across sorts. The going cross-sortal identity principle, ECIA2 of Cook and Ebert (2005), solves the problem of too many numbers. But, I argue, it does so at a high cost. I therefore propose a novel cross-sortal identity principle, based on embeddings of the induced models of abstracts developed by Walsh (2012). The new criterion matches ECIA2’s success, but offers interestingly different answers to the more controversial identifications made by ECIA2. (shrink)

Eleanor Knox has argued that our concept of spacetime applies to whichever structure plays a certain functional role in the laws. I raise two objections to this inertial functionalism. First, it depends on a prior assumption about which coordinate systems defined in a theory are reference frames, and hence on assumptions about which geometric structures are spatiotemporal. This makes Knox’s account circular. Second, her account is vulnerable to several counterexamples, giving the wrong result when applied to topological (...) quantum field theories and parity- and time-asymmetric theories. I advance an alternative account on which our spacetime concept is a cluster concept. On this view, the notion of metaphysical fundamentality may feature in the cluster, in which case spacetime functionalism may be uninformative in the absence of answers to fundamental metaphysical questions like the substantivalist/relationist debate. (shrink)