Dasgupta poses a serious challenge to realism about natural properties. He argues that there is no acceptable explanation of why natural properties deserve the value realists assign to them and are consequently absent of value. In response, this paper defines and defends an alternative non-explanatory account of normativity compatible with realism. Unlike Lewis and Sider, who believe it is sufficient to defend realism solely on realist terms, I engage with the challenge on unfriendly grounds by revealing a tu quoque. Dasgupta (...) and anti-realists face a similar challenge to that directed against realism: one that not only undermines the objection to realism by legitimising non-explanatory normativity but leaves them facing a significant dilemma. (shrink)

Super-Humeans argue that the most parsimonious ontology of the natural world compatible with our best physical theories consists exclusively of particles and the distance relations between them. This paper argues by contrast that Super-Humean reduction goes insufficiently far, by showing there to be a more parsimonious ontology compatible with physics: Ultimate-Humeanism. This novel view posits an ontology consisting solely of the particles and distance relations required for the existence of a single brain. Super-Humeans impose conditions on what counts as an (...) ontology of the natural world to avoid their view slipping into this kind of ontology, but these conditions are arbitrarily imposed and once this is exposed, Super-Humeans face a dilemma. Either they can embrace Ultimate-Humeanism as the minimal ontology of the natural world compatible with physics, or (more likely) they can rethink the methodology that got them there. Overall, this paper argues that Super-Humeanism currently lacks principled motivation, outlines a framework for naturalistic ontological reductions, and exposes the consequences of unchecked adherence to a simplicity-driven methodology. (shrink)

In some musical genres, professional performers play live shows many times a week. Arduous touring schedules bring encounters with wildly diverse audiences across many different performance ecologies. We investigate the kinds of creativity involved in such repeated live performance, kinds of creativity that are quite unlike songwriting and recording, and examine the central factors that influence musicians’ wellbeing over the course of a tour. The perspective of the professional musician has been underrepresented in research on relations between music and wellbeing, (...) with little attention given to the experience of touring. In this case study, we investigate influences on positive and negative performance experiences for the four professional musicians of Australian pop/rock band Cloud Control. Geeves conducted intensive cognitive ethnographic fieldwork with Cloud Control members over a two-week national Australian tour for their second album, Dream Cave (2013). Adapting a Grounded Theory approach to data analysis, we found the level of wellbeing musicians reported and displayed on tour to be intimately linked to their creative performance experiences through the two emergent, overarching and interdependent themes of Performance Headspace (PH) and Connection with Audience (CA). We explore these themes in detail and provide examples to demonstrate how PH and CA can feed off each other in virtuous ways that positively shape musicians’ wellbeing, or loop in vicious ways that negatively shape musicians’ wellbeing. We argue that their creative practice, in thus re-enacting musical performance afresh in each venue’s distinctive setting, emerges within unique constraints each night, and is in a sense a co-creation of the crowd and the band. (shrink)

Leibniz’s views on modality are among the most discussed by his interpreters. Although most of the discussion has focused on Leibniz’s analyses of modality, this essay explores Leibniz’s grounding of modality. Leibniz holds that possibilities and possibilia are grounded in the intellect of God. Although other early moderns agreed that modal truths are in some way dependent on God, there were sharp disagreements surrounding two distinct questions: (1) On what in God do modal truths and modal truth-makers depend? (2) What (...) is the manner(s) of dependence by which modal truths and modal truth-makers depend on God? Very roughly, Leibniz’s own answers are: (1) God’s intellect and (2) a form of ontological dependence. The essay first distinguishes Leibniz’s account from two nearby (and often misunderstood) alternatives found in Descartes and Spinoza. It then examines Leibniz’s theory in detail, showing how, on his account, God’s ideas provide both truth-makers for possibilities and necessities and an ontological foothold for those truth-makers, thereby explaining modal truths. Along the way, it suggests several refinements and possible amendments to Leibniz’s grounding thesis. It then defends Leibniz against a pair of recent objections by Robert Merrihew Adams and Andrew Chignell that invoke the early work of Kant. I conclude that whereas Leibniz’s alternative avoids collapsing into yet another form of Spinozism, the alternatives proposed by Adams, Chignell, and the early Kant do not. (shrink)

It has widely been assumed, by philosophers, that our first-person preferences regarding pleasurable and painful experiences exhibit a bias toward the future (positive and negative hedonic future-bias), and that our preferences regarding non-hedonic events (both positive and negative) exhibit no such bias (non-hedonic time-neutrality). Further, it has been assumed that our third-person preferences are always time-neutral. Some have attempted to use these (presumed) differential patterns of future-bias—different across kinds of events and perspectives—to argue for the irrationality of hedonic future-bias. This (...) paper experimentally tests these descriptive hypotheses. While as predicted we found first-person hedonic future-bias, we did not find that participants were time-neutral in all other conditions. Hence, the presumed asymmetry of hedonic/non-hedonic and first/third-person preferences cannot be used to argue for the irrationality of future-bias, since no such asymmetries exist. Instead, we develop a more fine-grained approach, according to which three factors—positive/negative valence, first/third-person, and hedonic/non-hedonic—each independently influence, but do not determine, whether an event is treated in a future-biased or time-neutral way. We discuss the upshots of these results for the debate over the rationality of future-bias. (shrink)

In this paper, I present and defend a novel account of doubt. In Part 1, I make some preliminary observations about the nature of doubt. In Part 2, I introduce a new puzzle about the relationship between three psychological states: doubt, belief, and confidence. I present this puzzle because my account of doubt emerges as a possible solution to it. Lastly, in Part 3, I elaborate on and defend my account of doubt. Roughly, one has doubt if and only if (...) one believes one might be wrong; I argue that this is superior to the account that says that one has doubt if and only if one has less than the highest degree of confidence. (shrink)

Philosophers commonly say that beliefs come in degrees. Drawing from the literature, I make precise three arguments for this claim: an argument from degrees of confidence, an argument from degrees of firmness, and an argument from natural language. I show that they all fail. I also advance three arguments that beliefs do not come in degrees: an argument from natural language, an argument from intuition, and an argument from the metaphysics of degrees. On the basis of these arguments, I conclude (...) that beliefs do not come in degrees. (shrink)

In their article, Thom Scott‐Phillips, Stefaan Blancke, and Christophe Heintz do a commendable job summarizing the position and misunderstandings of “cultural attraction theory” (CAT). However, they do not address a longstanding problem for the CAT framework; that while it has an encompassing theory and some well‐worked out case studies, it lacks tools for generating models or empirical hypotheses of intermediate generality. I suggest that what the authors diagnose as misunderstandings are instead superficial interpretive errors, resulting from researchers who have attempted (...) to extract generalizable hypotheses from CAT and bring them into contact with the analytical and inferential models of contemporary cultural evolutionary research. (shrink)

Aesthetic reasons are reasons to do and think various things. For example, it makes sense to wonder if a tree stump on the lawn was left there for environmental rather than aesthetic reasons, or for no reason at all. Aesthetic considerations of this kind are often contrasted with non-aesthetic reasons—such as moral or epistemic reasons. For example, they seem connected to pleasure-in-experience in a distinctive way that differs from paradigmatic moral reasons. Relatedly, the authority of aesthetic reasons has often been (...) thought to involve less of an “external demand” upon us than in the other cases. In this chapter, I suggest that such distinctiveness and modesty coheres well with an anti-realist treatment that views them as non-objective in nature. I then go on to consider an alternative, more robustly realist conception of aesthetic reasons. (shrink)

Frege-Geach worries about embedding and composition have plagued metaethical theories like emotivism, prescriptivism and expressivism. The sharpened point of such criticism has come to focus on whether negation and inconsistency have to be understood in descriptivist terms. Because they reject descriptivism, these theories must offer a non-standard account of the meanings of ethical and normative sentences as well as related semantic facts, such as why certain sentences are inconsistent with each other. This paper fills out such a solution to the (...) negation problems, following some of the original emotivist ideas about the interplay of interests in conversation. We communicate both to share information and coordinate our actions, and we use distinctively normative language like deontic ‘must’ and ‘may’ to negotiate what people are to do. The kinds of disagreement involved in such negotiation can illuminate the issues with negation and inconsistency. This paper outlines a dynamic semantic system in which these ideas can bear fruit, developing the scorekeeping model of conversation. The result is clarification about what Frege-Geach worries can mean for nondescriptive semantics. (shrink)

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and (...) proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim. (shrink)

This paper argues against both conceptual and linguistic analysis as sources of a priori knowledge. Whether such knowledge is possible turns on the nature of concepts. The paper's chief contention is that none of the main views about what concepts are can underwrite the possibility of such knowledge.

Some moral realists have defended moral realism on the basis of the purported fact that moral facts figure as components in some good explanations of non-moral phenomena. In this paper I explore the relationship between theism and this sort of explanationist defense of moral realism. Theistic explanations often make reference to moral facts, and do so in a manner which is ineliminable in an important respect – remove the moral facts from those explanations, and they suffer as a result. In (...) this respect theistic moral explanations seem to differ from the sorts of moral explanations typically offered by moral explanationists. (shrink)

Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...) to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered. (shrink)

This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.

Each person has a special relation to his or her own well-being. This rough thought, which can be sharpened in different ways, is supposed to substantially count against objectivist theories on which one can intrinsically benefit from, or be harmed by, factors that are independent of one’s desires, beliefs, and other attitudes. It is often claimed, contra objectivism, that one cannot be _alienated_ from one’s own interests, or that improvements in a person’s well-being must _resonate_ with that person. However, I (...) argue that every theory of well-being must allow that we can be alienated from our own well-being, and that sophisticated objectivists can accept and make use of a resonance constraint against their opponents. (shrink)

A movement dedicated to applying neuroscience to traditional philosophical problems and using philosophical methods to illuminate issues in neuroscience began about twenty-five years ago. Results in neuroscience have affected how we see traditional areas of philosophical concern such as perception, belief-formation, and consciousness. There is an interesting interaction between some of the distinctive features of neuroscience and important general issues in the philosophy of science. And recent neuroscience has thrown up a few conceptual issues that philosophers are perhaps best trained (...) to deal with. After sketching the history of the movement, we explore the relationships between neuroscience and philosophy and introduce some of the specific issues that have arisen. (shrink)

Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.

Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used (...) several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project. (shrink)

Four-dimensionalists think that we persist over time by having different temporal parts at each of the times at which we exist. Eternalists think that all times are equally real. Deprivationists think that death is an evil for the one who dies because it deprives them of something. I argue that four-dimensionalist eternalism, conjoined with a standard deprivationist account of the evil of death, has surprising implications for what we should think about the evil of death. In particular, given these assumptions, (...) we will lack any grounds for thinking that death is an evil for some individuals for whom we would antecedently expect it to be an evil, namely those individuals who cease to exist at death. Alternatively, we will only have some grounds for thinking that death is an evil for certain individuals for whom we might antecedently be more inclined to think death is not an evil, namely those individuals who survive death, in the sense that they continue to exist after death. (shrink)

Communitarians like Alasdair MacIntyre, Charles Taylor, and Michael Sandel, defend what we may call the ‘social constitution thesis.’ This is the view that participation in society makes us what we are. This claim, however, is ambiguous. In an attempt to shed some light on it and to better understand the impact its truth would have on our beliefs regarding autonomy, I offer four possible ways it could be understood and four corresponding senses of individual independence and autonomy. I also indicate (...) what senses liberals can accept that we are socially constituted and in what sense I take communitarians to argue we are socially constituted. (shrink)

Chapter in Wittgenstein at the Movies, an in-depth explorations of two experimental films on Wittgenstein: Derek Jarman's Wittgenstein and Péter Forgács' Wittgenstein Tractatus.

A broad overview of the NeoKantian movement in Germany, written as an introduction to a series of essays about that movement. -/- .

This short paper is a "quick and dirty" introduction for non-philosophers (with some background in propositional logic) to Jaegwon Kim's famous supervenience argument against non-reductive physicalism (also known as the exclusion problem). It motivates the problem of mental causation, introduces Kim's formulation of the issue centered around mind-body supervenience, presents the argument in deductive form, and makes explicit why Kim concludes that vindicating mental causation demands a reduction of mind.

Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.

The Proper Functionist account of warrant – like many otherexternalist accounts – is vulnerable to certain Gettier-style counterexamples involving accidentally true beliefs. In this paper, I briefly survey the development of the account, noting the way it was altered in response to such counterexamples. I then argue that Alvin Plantinga's latest amendment to the account is flawed insofar as it rules out cases of true beliefs which do intuitively strike us as knowledge, and that a conjecture recently put forward by (...) Thomas Crisp is also defective. I conclude by presenting my own suggestion as to how the account can be made less vulnerable to counterexamples involving accidentally true beliefs. Although I stay within the confines of Proper Functionism here, I think that my proposal (modulo a few details) could be attached to other externalist accounts of warrant as well. -/- . (shrink)

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...) naïf, il semblerait que la première soit pure et la seconde impure. Des objections à cette vue naïve sont ici considérées et réfutées. Concernant la preuve euclidienne, la question relève de la logique, notamment de la définissabilité arithmétique de l’addition en termes de successeur et de divisibilité telle que démontrée par Julia Robinson, tandis qu’en ce qui concerne la preuve topologique, la question relève de la sémantique, notamment pour tout ce qui touche au problème de savoir ce qui est « inclus » dans le contenu d’énoncés particuliers.A proof is pure, roughly, if it draws only on what is « close » or « intrinsic » to the statement being proved. The infinitude of prime numbers, a classical theorem of arithmetic, is a rich case study for philosophical investigation of purity. Two different proofs of this result are considered, namely the classical Euclidean proof and a more recent « topological » proof by Furstenberg. Naively the former would seem to be pure and the latter to be impure. Objections to these naive views are considered and met. In the case of the former the issue rests on logical matters, specifically the arithmetic definability of addition in terms of successor and divisibility shown by Julia Robinson, while in the case of the latter the issue rests on semantic matters, specifically with respect to what is « contained » in the content of particular statements. (shrink)

Pavel Tichy (1936-1994) was a Czech philosopher who originally studied and worked at Charles University in Prague, and spent the second half of his life in New Zealand as a political refugee. Early in his career he invented intensional logic, simultaneously with Richard Montague, but published his version in 1971, slightly after Montague's 1970 papers, and has never been recognised for this achievement. But this was only the beginning of his work. He developed a highly original theory of semantics called (...) Transparent Intensional Logic, which is the basis of an important research program based in the Czech Republic and Slovenia. He published a wide range of original work in semantics, philosophy of logic and language, philosophy of science, and metaphysics; but despite the indisputable quality of this work, he has gained little contemporary recognition. This article provides a brief introduction to his work, focussing mainly on basic ideas of his intensional semantics and his theory of 'constructions'. (shrink)

This paper is guided by, and begins to make plausible, the idea that there can be a naturalistic metaphilosophy, i.e., an inquiry that takes philosophy as an object of study in something like the way that contemporary (naturalistic) philosophy of science takes science as an object of study. The paper’s more specific goal is to ventilate certain provocative speculations concerning the character of philosophy’s cognitive achievement, especially over time. But this more specific goal will be approached indirectly, through addressing in (...) a preliminary way the question what the proper relationship is between philosophy and the study of philosophy’s history. (shrink)

In this paper I begin by extending two results of Solovay; the first characterizes the possible Turing degrees of models of True Arithmetic (TA), the complete first-order theory of the standard model of PA, while the second characterizes the possible Turing degrees of arbitrary completions of P. I extend these two results to characterize the possible Turing degrees of m-diagrams of models of TA and of arbitrary complete extensions of PA. I next give a construction showing that the conditions Solovay (...) identified for his characterization of degrees of models of arbitrary completions of PA cannot be dropped (I showed that these conditions cannot be simplified in the paper. (shrink)

David Albert has recently argued that classical electromagnetic theory (EM) is not time reversal invariant (non-TRI), while David Malament rejects this argument and maintains the orthodox result, that EM is TRI. Both Albert's and Malament's arguments are analysed, and both are found wanting in certain respects. It is argued here that the result really depends on the choice of theoretical ontology choosen to interpret EM theory, and there is more than one plausible choice. Albert and Malament have choosen different plausible (...) ontologies; but neither shows that their choice of interpretation is definitive. Deeper principles about this choice are examined. The extension to EM theory with magnetic monopoles is also examined. It is concluded that, despite certain flaws in his account, Albert's analysis does reveal serious problems in the orthodox account, which Malament's response does not adequately address. (shrink)

The analysis of the reversibility of quantum mechanics depends upon the choice of the time reversal operator for quantum mechanical states. The orthodox choice for the time reversal operator on QM states is known as the Wigner operator, T*, where * performs complex conjugation. The peculiarity is that this is not simply the unitary time reversal operation, but an anti-unitary operator, involving complex conjugation in addition to ordinary time reversal. The alternative choice is the Racah operator, which is simply ordinary (...) time reversal, T. Orthodox treatments hold that it is either logically or empirically necessary to adopt the Wigner operator, and the Racah operator has received little attention. The basis for this choice is analysed in detail, and it is concluded that all the conventional arguments for rejecting the Racah operator and adopting the Wigner operator are mistaken. The additional problem of whether the deterministic part of quantum mechanics should be judged to be reversible or not is also considered. The adoption of the Racah operator for time reversal appears prima facie to entail that quantum mechanics is irreversible. However, it is concluded that the real answer to question depends upon the choice of interpretation of the theory. In any case, the conventional reasons for claiming that quantum mechanics is reversible are incorrect. (shrink)

Metaphysics as theoretical framework for more empirically oriented research in science and in the humanities seems to be either ignored or regarded with great suspicion at the present time. Natural scientists, for example, by and large employ an instrumentalist approach to the study of the laws of nature. Their aim is to deal, not with things in themselves (the Kantian noumena) but with their empirical manifestation (Kantian phenomena) via tentative hypotheses subject to empirical verification. In the humanities, there is deep (...) suspicion of so-called “meta-narratives” that offer a comprehensive vision of reality on the grounds that their authors are consciously or unconsciously trying to control the thinking and behavior of others. Yet, as Colin Gunton points out, human beings, thus faced with an enormous diversity of options in virtually every area of human life, end up making important decisions simply on the basis of momentary personal preference. In this article, I first review the classical philosophical understanding of the relation between the One and Many in which the One is conceived as ontologically prior to the Many as their necessary principle of unity and intelligibility. I then propose a new paradigm for the relation between the One and the Many in which the Many by their dynamic interrelation from moment to moment initially co-create and then sustain the One not as a higher-order individual entity but as their conjoint energy-source and ongoing field of activity. The One can then be readily integrated into a hierarchical ordered system of fields within fields at all levels of existence and activity within nature. In the final part of the article, I indicate how this new field-oriented understanding of the relation between the One and the Many may help to solve controversial issues in natural science and Christian systematic theology. (shrink)

Hume’s criticisms of divine causation are insufficient because he does not respond to important philosophical positions that are defended by those whom he closely read. Hume’s arguments might work against the background of a Cartesian definition of body, or a Malebranchian conception of causation, or some defenses of occasionalism. At least, I will not here argue that they succeed or fail against those targets. Instead, I will lay out two major deficiencies in his arguments against divine causation. I call these (...) “deficiencies” because Hume does not adequately address live positions. This does not mean, of course, that there are not problems with these views or that Hume could not have given strong arguments against them. Rather, Hume’s arguments, which can seem comprehensive to the twenty-first century reader, are in fact not so. For the deficiencies discussed in this essay, I point to writers from Hume’s near context (many of whom we know he read carefully) who held the views not discussed, and I provide reasons why Hume seems not to have entertained these possibilities. I won’t distinguish between accidentally overlooking and actively ignoring. By drawing attention to these three deficiencies, I have two goals. The first is to demonstrate the diversity of seventeenth and early eighteenth century views on divine causation, especially among philosophers in England, Scotland, and Ireland, who are often ignored by philosophers today who, like Hume, focus on continental Cartesians while ignoring British dualists and vitalists. The second goal is to make today’s readers aware of shortcomings in Hume’s arguments to encourage productively contextual readings of Hume and his contemporaries. (shrink)

A critical review of Michael Ridge's book Impassioned Belief.

This critical study aims mainly to do two things: (i) throw some cold water on the claim that supervenience can be used to formulate a doctrine of non-reductive physicalism, and (ii) rebut an argument for physicalism offered (separately) by David Papineau and Barry Loewer. -/- The title alludes to the following lyric from "Mary Poppins", and was intended to hint that there is less to supervenience than meets the eye: -/- It's supercalifragilisticexpialidocious Even though the sound of it is something (...) quite atrocious If you say it loud enough, you'll always sound precocious Supercalifragilisticexpialidocious. (shrink)

Review of Dowek, Gilles, Computation, Proof, Machine, Cambridge University Press, Cambridge, 2015. Translation of Les Métamorphoses du calcul, Le Pommier, Paris, 2007. Translation from the French by Pierre Guillot and Marion Roman.

When mathematicians think of the philosophy of mathematics, they probably think of endless debates about what numbers are and whether they exist. Since plenty of mathematical progress continues to be made without taking a stance on either of these questions, mathematicians feel confident they can work without much regard for philosophical reflections. In his sharp–toned, sprawling book, David Corfield acknowledges the irrelevance of much contemporary philosophy of mathematics to current mathematical practice, and proposes reforming the subject accordingly.

Review of M. Audi, The Interpretation of Quantum Mechanics.

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late nineteenth century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis (in the sense of the infinitesimal calculus) received much attention in the nineteenth century. They helped instigate what Hans Hahn called a “crisis of intuition”, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this “crisis” as (...) follows: Mathematicians had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof... (p. 67) Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called “logicization”, “when the discipline requires nothing but purely logical fundamental concepts and propositions for its development.” On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals to visual representations. This would be a radical reformation of past practice, necessary, according to its advocates, for avoiding “unclarities and mistakes” like the one exposed by Peano. (shrink)

This collection of essays explores what makes modern mathematics ‘modern’, where ‘modern mathematics’ is understood as the mathematics done in the West from roughly 1800 to 1970. This is not the trivial matter of exploring what makes recent mathematics recent. The term ‘modern’ (or ‘modernism’) is used widely in the humanities to describe the era since about 1900, exemplified by Picasso or Kandinsky in the visual arts, Rilke or Pound in poetry, or Le Corbusier or Loos in architecture (a building (...) by the latter graces the cover of this book’s dust jacket). (shrink)

We review Solomon Feferman's 1998 essay collection In The Light of Logic (Oxford University Press).