Hume: Science, Logic, and Mathematics

This second edition of Historical Dictionary of Hume's Philosophy contains a chronology, an introduction, and an extensive bibliography. The dictionary section has over 100 cross-referenced entries covering key terms, as well as brief discussions of Hume's major works and of some of his most important predecessors, contemporaries, and successors.

In this paper I aim to defend one version at least of Hume’s dictum: roughly, the idea that possibility is determined by ontology through something like independent variation. My defence is broadly pragmatic, in the sense that adherence to something like Hume’s dictum delivers at least three benefits. The first benefit is that, through Hume’s dictum, a physical theory’s ontology delimits a range of possibilities, that I call kinematical possibilities, which serves as a sufficiently permissive notion of possibility to sustain (...) something like an intensional semantics for its claims, and a sufficiently demanding notion of supervenience to sustain plausible claims of inter-theoretic reduction and theoretical equivalence. The second benefit is that Hume’s dictum allows us to work backwards from a range of kinematical possibilities to an ontology. This is especially useful when aiming to glean an interpretation of a physical theory, since often we are more confident that we have arrived at the right space of possibilities than that we have arrived at the right ontology. The third benefit is that Hume’s dictum —- at least the version I aim to defend here —- may be applied to physical theories more or less as we find them, and therefore we can practice something resembling ontology without having to force our theories into some Procrustean bed, such as a first-order language. (shrink)

This collection contains fourteen critical essays on Hume's *A Treatise of Human Nature*, plus an Introduction: 1 The Association of Ideas in Hume’s Treatise (John P. Wright), 2 Methodizing Hume’s Metaphysics (Donald L. M. Baxter), 3 Hume on Belief (Jennifer Smalligan Marǔsić), 4 “All the Logic I think Proper to Employ”: Hume’s Rules by which to Judge of Causes and Effects (Hsueh Qu), 5 Imagining the Unseen: The External World of Hume’s Treatise (Angela Coventry), 6 The Updating Problem for Hume’s (...) Account of Belief in Personal Identity (Annemarie Butler), 7 Experimental Philosophy, Blind Submission, and Hume’s Other Sceptical Principles (Miren Boehm), 8 How to Read Book 2 of the Treatise (Katharina Paxman), 9 Hume on Curiosity: The Conclusion to Treatise Book 2? (Emilio Mazza), 10 Hume’s Geography of Feeling in A Treatise of Human Nature (Don Garrett), 11 Some Vexations about Character in Hume’s Treatise (Elizabeth S. Radcliffe), 12 Hume on Promising and Self-Obligation (Rachel Cohon), 13 No Men Are Created Equal: Rank, Passions, and Virtues in Hume’s Treatise (Margaret Watkins), 14 The Eighteenth-Century British Reception of Hume’s A Treatise of Human Nature (Mark G. Spencer and Mikko Tolonen). -/- . (shrink)

The Scottish philosopher David Hume (1711-1776) is widely regarded as the greatest and most significant English-speaking philosopher and often seen as having had the most influence on the way philosophy is practiced today in the West. His reputation is based not only on the quality of his philosophical thought but also on the breadth and scope of his writings, which ranged over metaphysics, epistemology, morals, politics, religion, and aesthetics. The Handbook's 38 newly commissioned chapters are divided into six parts: Central (...) Themes; Metaphysics and Epistemology; Passion, Morality and Politics; Aesthetics, History, and Economics; Religion; Hume and the Enlightenment; and After Hume. The volume also features an introduction from editor Paul Russell and a chapter on Hume's biography. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Hume’s Natural Philosophy and Philosophy of Physical Science by Matias SlavovKrisztián PeteMatias Slavov. Hume’s Natural Philosophy and Philosophy of Physical Science. London: Bloomsbury Academic, 2020. Pp. 216. Hardcover. ISBN 9781350087866, £95.Although the relationship between Hume and Newton is a recurring theme in the Hume literature, Matias Slavov’s book does not seek to contribute to the debate between the traditional (Hume imitated Newton’s natural philosophy) and the critical (Hume (...) intended his “science of Man” as the foundation of all other sciences) approaches regarding the nature of this relation. The book is intended to be a summary of Hume’s natural philosophy—or rather an aggregation of his comments on natural philosophical topics—pursued by exploring and analysing themes and issues that were prominent in the natural philosophy of the early modern period. In this respect it is a pioneering undertaking.Of course, Slavov does not claim in his book that Hume is a natural philosopher. “[F]irst and foremost... [h]e is not; for his main objective is to establish a new science of human nature” (1). But Slavov does claim that Hume’s natural philosophical views can be understood and valued in themselves and in the philosophical-scientific context of his time as well. The book is not a discussion of the natural philosophical dimensions of the “science of human nature,” but rather, of the natural philosophical views which are reconcilable with his main objective. In this respect, the book is not a methodological approach to Hume’s supposed natural philosophy, but a synthesis of certain elements of Hume’s philosophy that can feature in a consistent natural philosophy. Although his arguments are generally indirect and are more about what Hume seems to be committed to and what follows from his epistemological position rather than his actual positions, Slavov’s conclusions seem mostly convincing.Thus, Slavov’s point is not that the empiricist method is compatible in every detail with Newtonian mechanics, but rather that Hume seems to be committed more to a Cartesian natural philosophy. Slavov does not write about the possibility and significance of applying natural philosophical methods to moral philosophy; rather, he wants to build a complete philosophy of nature around some of Hume’s basic ideas to “fill the gap for a book on Hume’s relation to natural philosophy and philosophy of physical science” (2).The book has two “equally important aims” (ix): to shed more light onto Hume’s relationship to natural philosophy, and to demonstrate that physics and philosophy have overlapping domains. Hume was hardly concerned with physics, so Slavov’s strategy of defining natural philosophy as an overlap between physics and philosophy (chapter 1) does, to some extent, explain the lack of textual evidence in this area. The definition of natural philosophy chosen by Slavov is also intended to ensure [End Page 170] that these two goals are intertwined, since according to Slavov’s definition, natural philosophy is “a grey area between philosophy and physics” (12).Personally, I do not feel comfortable with this definition because it is not informative enough; it says that natural philosophy is more than modern physics (mathematized natural science), but it is not quite philosophy. While the first implication is very agreeable, and a number of scholars have addressed the issue, to assess the second implication, we would need to know exactly what philosophy is in this context. Perhaps the best candidate is metaphysics, even if Slavov keeps the reader somewhat in the dark on this point, despite the fact that chapter 2 is intended to establish precisely that point. Yet, the metaphysics of this period was usually (with the possible exception of Spinoza) used to support natural scientific explanations. I believe that natural philosophy is more than a “grey area” (9, 12, 22), or a contact zone; it is a complex enterprise that evolved in different forms with different emphases: Cartesian mechanics still sought to understand nature, pursuing the Aristotelian ideal of knowledge, albeit rejecting Aristotelian methods, while empiricists increasingly envisaged a more instrumentalist role for natural explanations. Newton himself was caught between the two.In this context it is intriguing whether Hume carries forward Berkeley... (shrink)

Se puede considerar que Berkeley y Hume son antecedentes filosóficos del Formalismo Matemático. Ambos sostienen una visión instrumentalista y no-realista de las matemáticas. En la conferencia se explora las diferencias y similitudes de ambos autores, así como el porque se les puede considerar ser antecesores del Formalismo.

This chapter examines what Hume and Reid had to say about what Reid called our intellectual powers: sensation, conception, perception, memory, abstraction, judgement, and reasoning. In the process it examines their opposed views on the nature of mind, on the representation of space and the spatiality of mental content, on temporal experience and the metaphysics of time, on the conception of non-existent objects, and on conceivability and possibility. The chapter critically examines what each had to say in his own defence (...) and what Reid had to say in opposition to Hume, and conjectures what Hume might have said in response to Reid. (shrink)

This M.A. thesis explores the intricate Problem of Induction, contrasting three seminal approaches: Hume's habit-centric view, Reichenbach's emphasis on the Principle of Uniformity of Nature, and Strawson's belief in the innate rationality of induction. While Hume's perspective lays the groundwork for Kant's a priori and Cleve's a posteriori validation, Reichenbach and Salmon present pragmatic justifications, underscoring the methodological and probabilistic underpinnings of inductive reasoning, specifying epistemological ignorance as a guidance for the optimality criteria. Strawson, challenging prevailing notions, posits that induction, (...) anchored by prior probabilities and evidence, is inherently rational, obviating the need for external validation. The study integrates concepts of frequentism, conditionalization, and probabilistic laws to develop the truth-conducive nuance of induction. The research culminates by confronting the dual challenges of quantitative and profound scepticism, championing a holistic approach. Therefore, the study aims to enrich the discourse on the epistemic foundations of the Problem of Induction, particularly its implications for scientific inquiry and the laws of nature. (shrink)

David Hume is the greatest figure in the empiricist tradition in philosophy and was a particular source of inspiration for the logical positivists (see logical positivism). Hume was born in 1711 and entered Edinburgh University at the age of 12. After graduating, he had a varied career in commerce, diplomacy, as a librarian, and as a writer of history. Twice he was secretary to General St Clair and on one occasion set off with him on an expedition to drive the (...) French out of Canada. Forced back by the wind, it was decided instead to make a brief incursion on the coast of Brittany. This failed, for the expedition had maps of Canada, but none of Brittany. During a three‐year stay in France he wrote his major philosophical work Treatise of Human Nature. It appeared in 1739 and, in his own words, “fell dead‐born from the Press.” His Inquiry Concerning Human Understanding, which was intended to be a more popular version of book I of his Treatise, was published in 1748 but was only a little more favorably received. His six‐volume History of England, on the other hand, was a great success. He died in 1776, never having succeeded in his ambition to be appointed to a professorship of philosophy in Scotland, due to his religious skepticism. (shrink)

The aim of this paper is to examine how much Hume knew about astronomy, in order to understand the reasons for his acceptance of Copernicanism. My contention is that Hume’s positive reception of the Copernican system arises at least from the importance that he gives to three features that he attributes to the Copernican system: beauty, simplicity and uniformity. I also give some evidence that Hume had first-hand knowledge of some sections of Galileo’s Dialogo sopra i due massimi sistemi del (...) mondo tolemaico e copernicano (1632), where the “sole proofs” of the Copernican system are said to be found. (shrink)

A Critical Introduction to the Metaphysics of Modality examines the eight main contemporary theories of possibility behind a central metaphysical topic. Covering modal skepticism, modal expressivism, modalism, modal realism, ersatzism, modal fictionalism, modal agnosticism, and the new modal actualism, this comprehensive introduction to modality places contemporary debates in an historical context. Beginning with a historical overview, Andrea Borghini discusses Parmenides and Zeno; looks at how central Medieval authors such as Aquinas, and Buridan prepared the ground for the Early Modern radical (...) views of Leibniz, Spinoza, and Hume and discusses advancements in semantics in the later-half of the twentieth century a resulted in the rise of modal metaphysics, the branch characterizing the past few decades of philosophical reflection. Framing the debate according to three main perspectives - logical, epistemic, metaphysical- Borghini provides the basic concepts and terms required to discuss modality. With suggestions of further reading and end-of-chapter study questions, A Critical Introduction to the Metaphysics of Modality is an up-to-date resource for students working in contemporary metaphysics seeking a better understanding of this crucial topic. (shrink)

Hume’s Law that one cannot derive an “ought” from an “is” has often been deemed to bear a significance that extends far beyond logic. Repeatedly, it has been invoked as posing a serious threat to views about normativity: naturalism in metaethics and positivism in jurisprudence. Yet in recent years, a puzzling asymmetry has emerged: while the view that Hume’s Law threatens naturalism has largely been abandoned (due mostly to Pigden’s work, see e.g. Pigden 1989), the thought that Hume’s Law is (...) a serious challenge to positivism has only grown in prominence. Our main aim is to establish that Hume’s Law is not a threat to positivism or naturalism. First, we connect extensive, but unfortunately siloed, discussions of this issue. Second, we show that Hume’s Law is not a serious threat to naturalism or positivism, for the gap between logic and such theses is very hard to bridge in a way that would make Hume’s Law able to bear this significance. Finally, we emphasize an implication of our discussion: it undermines one of the main “dialectical tributaries” in jurisprudence (Toh 2018). (shrink)

Given the difficulty of characterizing the quandary introduced in Hume’s Appendix to the Treatise, coupled with the alleged “underdetermination” of the text, it is striking how few commentators have considered whether Hume addresses and/or redresses the problem after 1740—in the first Enquiry, for example. This is not only unfortunate, but ironic; for, in the Appendix, Hume mentions that more mature reasonings may reconcile whatever contradiction(s) he has in mind. I argue that Hume’s 1746 letter to Lord Kames foreshadows a subtle, (...) but significant, shift in Hume’s reasonings regarding the relevance of “real connexions”; that the Enquiry of 1748 provides evidence for this shift; and that this shift obviates Hume’s second thoughts by reconciling the contradiction that he had in mind. In short, Hume’s letter to Kames and Enquiry supply the retrodictive keys to a systematically satisfactory account. (shrink)

This article centers on Hume’s position on the intelligibility of natural philosophy. To that end, the controversy surrounding universal gravitation shall be scrutinized. It is very well-known that Hume sides with the Newtonian experimentalist approach rather than with the Leibnizian demand for intelligibility. However, what is not clear is Hume’s overall position on the intelligibility of natural philosophy. It shall be argued that Hume declines Leibniz’s principle of intelligibility. However, Hume does not eschew intelligibility altogether; his concept of causation itself (...) stipulates mechanical intelligibility. (shrink)

This book contextualizes David Hume's philosophy of physical science, exploring both Hume's background in the history of early modern natural philosophy and its subsequent impact on the scientific tradition.

The last few decades have witnessed intense debates in Hume scholarship concerning Hume’s account of causation. At the core of the “old–new Hume” debate is the question of whether causation for Hume is more than mere regularity, in particular, whether Hume countenances necessary connections in mind-independent nature. This chapter assesses this debate against the background of Hume’s “foundational project” in the Treatise. The question of the role and import of Hume’s account of the idea of cause is examined and compared (...) with Hume’s treatment of other ideas. (shrink)

The introduction considers the state of scholarship on empiricism as a philosophical and historical category, particularly as it pertains to experimental philosophy. It concludes that empiricism properly understood is a rich category encompassing epistemic, semantic, methodological, experimental, and moral elements. Its richness makes it a suitable lens through which to account for actual historical complexity. The introduction relates the category to the work of Sir Isaac Newton, who influenced all of empiricism’s elements.

Up till this day one cannot find much scholarship which situates Hume in the context of early modern natural philosophy. Tamás Demeter's new book, David Hume and the Culture of Scottish Newtonianism, does a spectacular job in filling this gap. His monograph is the most comprehensive pursuit to understand Hume's place in the Newtonian tradition of natural philosophy. Demeter specifies Hume's place both in the context of Newtonian moral philosophy and Newtonian chemistry and physiology.

Given the sharp distinction that follows from Hume’s Fork, the proper epistemic status of propositions of mixed mathematics seems to be a mystery. On the one hand, mathematical propositions concern the relation of ideas. They are intuitive and demonstratively certain. On the other hand, propositions of mixed mathematics, such as in Hume’s own example, the law of conservation of momentum, are also matter of fact propositions. They concern causal relations between species of objects, and, in this sense, they are not (...) intuitive or demonstratively certain, but probable or provable. In this article, I argue that the epistemic status of propositions of mixed mathematics is that of matters of fact. I wish to show that their epistemic status is not a mystery. The reason for this is that the propositions of mixed mathematics are dependent on the Uniformity Principle, unlike the propositions of pure mathematics. (shrink)

Though it is well known that Hume composed his Treatise and the first version of his essay On Miracles during his stay in La Flèche, this fact has not received the attention it surely deserves. What may have induced a gentleman from Edinburgh to bury himself in the library of a Jesuit college? While every historian of early modern philosophy is quick to credit La Flèche with having provided Descartes’s education, Burton’s amazement that Hume himself never alludes to this unique (...) circumstance is still suggestive. I shall argue that what attracted Hume’s attention were the scholastic items contained in every good Jesuit library. (shrink)

The subject of this essay-based dissertation is Hume’s natural philosophy. The dissertation consists of four separate essays and an introduction. These essays do not only treat Hume’s views on the topic of natural philosophy, but his views are placed into a broader context of history of philosophy and science, physics in particular. The introductory section outlines the historical context, shows how the individual essays are connected, expounds what kind of research methodology has been used, and encapsulates the research contributions of (...) the essays. The first essay treats Newton’s experimentalist methodology in gravity research and its relation to Hume’s causal philosophy. It is argued that Hume does not see the relation of cause and effect as being founded on a priori reasoning, similar to the way in which Newton criticized non-empirical hypotheses about the causal properties of gravity. Contrary to Hume’s rules of causation, the universal law does not include a reference either to contiguity or succession, but Hume accepts it in interpreting the force and the law of gravity instrumentally. The second article considers Newtonian and non-Newtonian elements in Hume more broadly. He is sympathetic to many prominently Newtonian themes in natural philosophy, such as experimentalism, critique of hypotheses, inductive proof, and the critique of Leibnizian principles of sufficient reason and intelligibility. However, Hume is not a Newtonian philosopher in many respects: his conceptions regarding space and time, the vacuum, the specifics of causation, the status of mechanism, and the reality of forces differ markedly from Newton’s related conceptions. The third article focuses on Hume’s Fork and the proper epistemic status of propositions of mixed mathematics. It is shown that the epistemic status of propositions of mixed mathematics, such as those concerning laws of nature, is that of matters of fact. The reason for this is that the propositions of mixed mathematics are dependent on the Uniformity Principle. The fourth article analyzes Einstein’s acknowledgement of Hume regarding special relativity. The views of the scientist and the philosopher are juxtaposed, and it is argued that there are two common points to be found in their writings, namely an empiricist theory of ideas and concepts and a relationist ontology regarding space and time. (shrink)

Berkeley, and Hume following him, rejected the doctrine that lines, e.g., are infinitely divisible. They both thought that the notion of infinite divisibility lead to paradoxical results. To avoid these paradoxes, they adopted the view that a line is composed of finitely many minimal parts. In addition, Berkeley argued, successfully it seems, that all the standard geometrical proofs for infinite divisibility are, in fact, question‐begging.

Central aspects of Hume’s proposed “system of the sciences” as described in the Treatise are modeled on Newton’s Principia. But, as recent scholarship has suggested, Hume’s Treatise also bears a deeply subversive message with respect to Newtonian science. This chapter offers a revised overview of what Hume takes from Newton and what he rejects: The first part of the chapter argues that in the Treatise Hume adopts a version of Newton’s “analytic and synthetic method” for philosophy, thereby placing a distinctively (...) Newtonian form of explanatory reduction at the center of his own philosophical method. The second part of the chapter, on the other hand, shows that many of the most important aspects of Hume’s argument in Book 1 of the Treatise can be understood as critical of core conceptual and ontological commitments of Newton’s mechanics as developed in the Principia. (shrink)

Einstein acknowledged that his reading of Hume influenced the development of his special theory of relativity. In this article, I juxtapose Hume’s philosophy with Einstein’s philosophical analysis related to his special relativity. I argue that there are two common points to be found in their writings, namely an empiricist theory of ideas and concepts, and a relationist ontology regarding space and time. The main thesis of this article is that these two points are intertwined in Hume and Einstein.

First published in 1985, D. M. Armstrong's original work on what laws of nature are has continued to be influential in the areas of metaphysics and philosophy of science. Presenting a definitive attack on the sceptical Humean view, that laws are no more than a regularity of coincidence between stances of properties, Armstrong establishes his own theory and defends it concisely and systematically against objections. Presented in a fresh twenty-first-century series livery, and including a specially commissioned preface written by Marc (...) Lange, illuminating its continuing importance and relevance to philosophical enquiry, this influential work is available for a new generation of readers. (shrink)

Argues for an objective protomoral normativity in terms of what an adaptation is for, without falling victim to Hume's Law, open-question arguments, queerness arguments, and internalism/externalism debates. Also provides a general strategy for naturalizing objective moral normativity which is likewise proof against the usual-suspect objections.

In “A neglected reply to Prior’s dilemma” Beall [2012] presents a Weak Kleene framework where Prior’s dilemma for Hume’s no-ought-fromis thesis fails. It fails in the framework because addition, the inference rule that one of its horns relies on, is invalid. In this paper, we show that a more general result is necessary for the viability of Beall’s proposal – a result, which implies that Hume’s thesis holds in the proposed framework. We prove this result and thus show that Beall’s (...) proposal is indeed viable. (shrink)

Hsueh M. Qu has recently argued that Hume’s famed ‘Separability Principle’ from the Treatise entangles him in a contradiction. Qu offers a modified principle as a solution but also argues that the mature Hume would not have needed to avail himself of it, given that Hume’s arguments in the first Enquiry do not depend on this principle in any form. To the contrary, I show that arguments in the first Enquiry depend on this principle, but I agree with Qu that (...) Qu’s solution to Hume’s quandary frees him of the contradiction. Next, I compare Qu’s solution to Hume’s original position. By analysing the divergent forms of 'Hume’s Dictum’ that follow from them, I show that Qu’s solution and Hume’s original position have significantly different consequences in a range of domains, including Hume’s modality. Generally, Qu’s solution fits better with Hume’s other commitments—even though Hume often fails to recognize it—thereby increasing its plausibility. (shrink)

One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently (...) large domains. In this paper, we raise a worry for this response to the Bad Company objection. We argue, perhaps surprisingly, that it requires very strong assumptions about the range of the second-order quantifiers; assumptions that the abstractionist should reject. (shrink)

According to the neo-Fregean abstractionism, numerical expressions of the form ‘the number of Fs’, introduced by Hume’s Principle, should be read as purportedly referential singular terms. I will explore the prospects of a version of abstractionism in which such expressions have presuppositional content, as in Strawson’s account. I will argue that the thesis that ‘the number of Fs’ semantically presupposes the existence of a number is inconsistent with the required ‘modest’ stipulative character of the truth of Hume’s Principle: since Hume’s (...) Principle is true and provably presupposes that numbers exist, what it presupposes is also true; and so numbers exist. This, however, means that numbers are conjured into existence by a direct stipulation. (shrink)

In this paper we use proof-theoretic methods, specifically sequent calculi, admissibility of cut within them and the resultant subformula property, to examine a range of philosophically-motivated deontic logics. We show that for all of those logics it is a (meta)theorem that the Special Hume Thesis holds, namely that no purely normative conclusion follows non-trivially from purely descriptive premises (nor vice versa). In addition to its interest on its own, this also illustrates one way in which proof theory sheds light on (...) philosophically substantial questions. (shrink)

Commentators have rightly focused on the reasons why Hume maintains that the conclusions of skeptical arguments cannot be believed, as well as on the role these arguments play in Hume’s justification of his account of the mind. Nevertheless, Hume’s interpreters should take more seriously the question of whether Hume holds that these arguments are demonstrations. Only if the arguments are demonstrations do they have the requisite status to prove Hume’s point—and justify his confidence—about the nature of the mind’s belief-generating faculties. (...) In this paper, I treat Hume’s argument against the primary/secondary quality distinction as my case study, and I argue that it is intended by Hume to be a demonstration of a special variety. (shrink)

According to the Bad Company objection, the fact that Frege’s infamous Basic Law V instantiates the general definitional pattern of higher-order abstraction principles is a good reason to doubt the soundness of this sort of definitions. In this paper I argue against this objection by showing that the definitional pattern of abstraction principles – as extrapolated from §64 of Frege’s Grundlagen– includes an additional requirement (which I call the specificity condition) that is not satisfied by the Basic Law V while (...) is satisfied by other higher-order abstractions such as Hume’s Principle. I also show that the failure of this additional requirement in the case of Basic Law V is engendered by an essential feature of Frege’s conception of logic and thus that Frege himself should not have regarded the Basic Law V as a definition by abstraction. (shrink)

It’s well known that it’s possible to extract, from Frege’s Grudgesetze, an interpretation of second-order Peano Arithmetic in the theory  HP2, whose sole axiom is Hume’s principle. What’s less well known is that, in Die Grundlagen Der Arithmetic §82–83 Boolos (2011), George Boolos provided a converse interpretation of HP2 in PA2 . Boolos’ interpretation can be used to show that the Frege’s construction allows for any model of PA2 to be recovered from some model of HP2. So the space (...) of possible arithmetical universes is precisely characterized by Hume’s principle. -/- In this paper, I show that a large class of second-order theories admit characterization by an abstraction principle in this sense. The proof makes use of structural abstraction principles, a class of abstraction principles of considerable intrinsic interest, and categories of interpretations in the sense of Visser (2003). (shrink)

Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. (...) Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle. (shrink)

The paper formulates and proves a strengthening of ‘Frege’s Theorem’, which states that axioms for second-order arithmetic are derivable in second-order logic from Hume’s Principle, which itself says that the number of Fs is the same as the number ofGs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. ‘Finite Hume’s Principle’ also suffices for (...) the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Frege’s definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed. (shrink)