What is mathematics about? Does the subject-matter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but with some experience of formal (...) logic it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics - Platonism, realism, logicism, formalism, constructivism and structuralism - as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy present-day philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upper-level undergraduate courses and the non-specialist. (shrink)

Most scientific theories are globally inconsistent. Chunk and Permeate is a method of rational reconstruction that can be used to separate, and identify, locally consistent chunks of reasoning or explanation. This then allows us to justify reasoning in a globally inconsistent theory. We extend chunk and permeate by adding a visually transparent way of guiding the individuation of chunks and deciding on what information permeates from one chunk to the next. The visual representation is in the form of bundle diagrams. (...) We then extend the bundle diagrams to include not only reasoning in the presence of inconsistent information or reasoning in the logical sense of deriving a conclusion from premises, but more generally reasoning in the sense of trying to understand a phenomenon in science. This extends the use of the bundle diagrams in terms of the base space and the fibres. We then apply this to a case in physics, that of understanding binding energies in the nucleus of an atom using together inconsistent models: the liquid drop model and the shell model. We draw some philosophical conclusions concerning scientific reasoning, paraconsistent reasoning, the role of logic in science and the unity of science. (shrink)

In this paper, I compare how it is that inconsistencies are handled in mathematics to how they are handled in chemistry. In mathematics, they are very precisely formulated and identified, unlike in chemistry. So the chemists can learn from the precision and the very well-worked out strategies developed by logicians and deployed by mathematicians to cope with inconsistency. Some lessons can also be learned by the mathematicians from the chemists. Mathematicians tend to be intolerant towards inconsistencies. There are some philosophers (...) of chemistry who attribute to chemists, collectively, a more tolerant attitude towards inconsistency, and so, mathematicians can learn from the chemists’ attitude to inform their choice of strategies. (shrink)

The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...) from considering rigour of proof as a fixture, I discuss fixed models, invariant notions and fixed information about objects across theories. There are other fixtures, but it is enough to start with these. (shrink)

In the literature, there have been several methods and definitions for working out whether two theories are “equivalent” or not. In this article, we do something subtler. We provide a means to measure distances between formal theories. We introduce two natural notions for such distances. The first one is that of axiomatic distance, but we argue that it might be of limited interest. The more interesting and widely applicable notion is that of conceptual distance which measures the minimum number of (...) concepts that distinguish two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only. (shrink)

There are three elements in this paper. One is what we shall call ‘the Hungarian project’. This is the collected work of Andréka, Madarász, Németi, Székely and others. The second is Molinini’s philosophical work on the nature of mathematical explanations in science. The third is my pluralist approach to mathematics. The theses of this paper are that the Hungarian project gives genuine mathematical explanations for physical phenomena. A pluralist account of mathematical explanation can help us with appreciating the significance of (...) the Hungarian project. The significance consists in the fruitfulness and spread of the project. The spread is wide because the explanations are written in the very familiar language of first-order logic with identity. For this reason, the explanation is understandable to many mathematicians. Because of the methodology adopted in the Hungarian project, the explanations are fruitful in another sense. In the Hungarian project certain questions are asked that would not be asked with a more usual methodology. The Hungarians are distinguished from other scientists in asking logical and mathematical questions, and these both deepen our understanding of the physical theories and induce further spread to mathematics and philosophy. (shrink)

We answer three questions: 1. Can we give a wholly mathematical explanation of a physical phenomenon? 2. Can we give a wholly mathematical explanation for a whole physical theory? 3. What is gained or lost in giving a wholly, or partially, mathematical explanation of a phenomenon or a scientific theory? To answer these questions we look at a project developed by Hajnal Andréka, Judit Madarász, István Németi and Gergely Székely. They, together with collaborators, present special relativity theory in a three-sorted (...) first-order formal language. (shrink)

In this paper we try to convert the mathematician who calls himself, or herself, “a formalist” to a position we call “meth-odological pluralism”. We show how the actual practice of mathe-matics fits methodological pluralism better than formalism while preserving the attractive aspects of formalism of freedom and crea-tivity. Methodological pluralism is part of a larger, more general, pluralism, which is currently being developed as a position in the philosophy of mathematics in its own right.1 Having said that, henceforth, in this (...) paper, we abbreviate “methodological pluralism” with “pluralism”. (shrink)

I present a comparative and holistic method for qualitatively measuring sound ecological practice in chemistry. I consider chemicals developed and used by man from cradle to grave, that is, from the moment they are extracted from the earth, biomass, water or air, to their transportation, purification, mixing and elaboration in a factory, to their distribution by means of the market, to waste products both from the factory, packaging, transportations and by the consumer. I divide the locations of the ‘life’ of (...) the chemical into four spatio-temporal areas accordingly. I then use the ‘instituional compass’ method to determine a qualitative reading of the ecological soundness of the practice, where practice means the research, the adoption by industry and the distribution at scale on the market. The qualitative reading is in the form of an arrow on a trisected circle. The arrow holistically represents a table of data. The data can be economic, social or environmental. The arrow has a measurement: a degree and a length. The degrees, represent qualities spaning through: harmony, discipline and excitement. The length represents the importance, momentum or amplitude with which the quality is present. We use the compass method to compare the same product over time, or inter-substitutable, chemicals developed in different places, using different equipment or processes. In the conclusion, I discuss objectivity and science as they apply to the compass. (shrink)

We are told that there are standards of rigour in proof, and we are told that the standards have increased over the centuries. This is fairly clear. But rigour has also changed its nature. In this paper we as-sess where these changes leave us today.1 To motivate making the new assessment, we give two illustra-tions of changes in our conception of rigour. One, concerns the shift from geometry to arithmetic as setting the standard for rig-our. The other, concerns the notion (...) of effective proof or compu-tations. To make the assessment, we look at one motivation for increasing the rigour of a mathematical argument: explicitness and honesty. We then present a standard of rigour by means of a characterisation developed with reference to what we call ‘an account-proof’. We evaluate the standard with reference to the motivation. With the analysis we discover that, regardless of the motivations for rigour, the standard is almost never met, and that the motivations are not all satisfied. It follows that, in some sense, the motivations have misfired. The misfiring suggests to us that we re-assess our notion of rigour. We think of rigour as a relative term. Moreover, the standard for rigour depends on philosophical underpinnings. The strength of the argument of this paper rests on the plausibility of our selection of motivations and on the plausibility of our standard. (shrink)

I present a formal language that imposes a structure on processes in macro-chemistry. Each symbol in the language invites a type of analysis that is carried out either by looking into the semantics if the language or by looking at the context. Every formal language has assumptions underlying it. The assumptions made in developing the formal language are meant to help with conceptual analysis by inviting certain types of question.

In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the “axiomatic conception” of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic text books.

Hilbert’s axiomatic approach to the sciences was characterized by a dynamic methodology tied to scientific and mathematical fields under investigation. In particular, it is an analytic art for choosing axioms but, at the same time, it has to include dynamically synthetic procedures and meta-theoretical reflections. Axioms have to be useful, or capture something, or help as part of explanations. The Andréka-Németi group use several formal axiomatic theories together to re-capture, predict, recover or explain the phenomena of special relativity, general relativity (...) and classical kinematics. Their scientific methodology is close to Hilbert’s conceptions of how science should be done ideally. In this paper, we compare Formica’s reading of Hilbert’s axiomatic method to the method used by the Andréka-Németi group. (shrink)

Dispositions are a Teacher's Greatest Strength will fuel and reignite your classroom practice. Focusing on 13 dispositions specific to teaching, this book encourages educators to identify, reflect, and develop their dispositions, attitudes, and self-awareness to flourish in the profession. Emphasizing pedagogical knowledge and skills, this text serves as affirmation of a teacher's commitment to challenging, complex and rewarding work. It invites educators to consider what a unique privilege it is to teach--to dive into reading, creating space, and embracing dispositions as (...) a teacher's greatest strength. Each chapter focuses on one of 13 teaching dispositions--such as curiosity, adaptability, gratitude, resilience, and courage--and offers: Definitions and contexts for the disposition of focus. Concrete applications for teachers to practice and develop dispositions with reader-friendly examples and practical strategies. "Pause and Reflect" sections with questions and space for professional reflection. This book serves as a love letter to educators and administrators in K-12, higher education faculty, and pre-service programs and students by reminding teachers of the significant work they do, and by putting dispositions at the forefront of their daily work. (shrink)

Biogea is a mixture of poetry, philosophy, science, and biography exemplary of the style that has made Michel Serres one of the most extraordinary thinkers of his age. His philosophical and poetic inquiry sings in praise of earth and life, what he names singularly as Biogea. In these times when species are disappearing, when catastrophic events such as earthquakes and tsunamis impale the earth, Serres wonders if anyone “worries about the death pangs of the rivers.” And for Serres, one can (...) ask the same question of philosophy as the humanities increasingly find themselves in need of defenders. Today, all living organisms discover themselves part of this Biogea. “Today we have other neighbors, constituents of the Biogea: the sea, my lover; our mother, the Earth, becomes our daughter; this beautiful breeze which inspires the spirit, a spiritual mistress; our light friends, the fresh and flowing waters.”. (shrink)

ABSTRACTMetaphors frequently link negative affect with darkness and associations of this type have been established in several experimental paradigms. Given the ubiquity and strength of these associations, people who prefer dark to light may be more prone to negative emotional experiences and symptoms. A five study investigation couches these ideas in a new theoretical framework and then examines them. Across studies, 1 in 4 people preferred the perceptual concept of dark over the perceptual concept of light. These dark-preferring people scored (...) higher in neuroticism and experienced greater depressive feelings in daily life. Moreover, dark preferences shared a robust relationship with depressive symptoms as well as generalised anxiety symptoms. The results provide novel insights into negative affectivity and extend conceptual metaphor theory in a way that is capable of making individual difference predictions. (shrink)

This article takes issue with two prominent views in the current debate around epistemic partiality in friendship. Strong views of epistemic partiality hold that friendship may require biased beliefs in direct conflict with epistemic norms. Weak views hold that friendship may place normative expectations on belief formation but in a manner that does not violate these norms. It is argued that neither view succeeds in explaining the relationship between epistemic norms and friendship norms. Weak views inadvertently endorse a form of (...) motivated reasoning, failing to resolve the normative clash they seek to avoid. Strong views turn out to be incoherent once we consider the question of whether the requirement to form an epistemically partial belief is independent of whether the belief in question would be true. It is then argued that an epistemology of friendship should recognise the special role that understanding plays in friendship. On this view, friendship normatively requires understanding the truth about our friends. This entails that epistemic partiality, far from being a requirement, is in fact at odds with good friendship. (shrink)

It is important for the materialist historian, in the most rigorous way possible, to differentiate the construction of a historical state of affairs from what one customarily calls its "reconstruction." The "reconstruction" in empathy is one- dimensional. "Construction" presupposes "destruction." Almost fourteen years after the death of Jacques Derrida, the least one can say is that his inheritance is as contested and fraught with rivalries, rejections, and appropriations as at the time of the flowering of Deconstruction in American universities in (...) the seventies and eighties. A halt was observed in 1988-89 after the posthumous revelations about Paul de Man's past in Belgium and Derrida's embattled defense of his friend. Today, in France, one often hears that "Derrida a fait l'Ecole mais n'a pas fait ecole," meaning that Derrida passed the entrance examination of the prestigious Ecole Normale Superieure where he taught for a long time, where he met Paul Celan, Louis Althusser, Michel Foucault and many other luminaries, whereas he never had a real "school" in France-no real institutional backing beyond the various Parisian places at which he taught later, or those that he founded like the College de Philosophie; his following constituted of young philosophers, only a few of whom became university professors in their turn and disseminated his teachings. However, the mechanisms of power within the French university remained closed to Derrida until the end; besides, he wanted to prevent the stereotyped reproduction currently observed from master to disciple. (shrink)

Giving directions to a tourist, sharing the latest news with a colleague, describing a place to a friend after a trip, and telling someone that we have met a common acquaintance at a party are all...

L’hypothèse développée ici est que la « meilleure amie » évoquée par Socrate à la fin du chapitre n’est autre que son âme. L’exemple ainsi donné par Socrate de sa maîtrise de soi face à l’attrait exercé par une femme à la beauté « plus forte que les paroles » s’adresse non à cette dernière, mais aux assistants. On comprend ainsi pourquoi ce chapitre prend place dans une série d’entretiens où Socrate prodigue ses conseils à ceux qui « désirent les (...) belles choses » . Cette interprétation plaide en faveur de l’unité du livre III des Mémorables.The hypothesis exposed here is that the « better friend » called up by Socrates in the last sentence of the chapter is just his own soul. By giving a such example of self-control in face of an appealing woman whose beauty was « stronger than words », Socrates is not addressing this woman but his audience. We thus understand why this chapter is located in a series of conversations in which Socrates is giving advices to people who « desire beautiful things » . The thesis of the unity of Book III of Memorabilia is supported by this interpretation. (shrink)

Contemporary humanitarianism is held in a permanent and tense relationship with the warlike, destructive, and exclusionary strategies of the states which dominate the planet. On the one hand, a politics of the "clenched fist" champion of both holy and just wars, exemplary sanctions, lightning raids and surgical strikes; in other words, the technical arsenal of a police force acting globally on an ad hoc basis and according to the friend /enemy relation, following the principles of partisan fidelity and the (...) vendetta. On the other hand, occupying the place of social politics at the same global scale is a spectacular humanism, manifesting itself through an ensemble of private organisations whose role is to keep the survivors alive, treating them as nameless victims, held at a distance for the sole purpose of salving the conscience of the powerful. However, the displaced and refugee populations are responding to this situation by developing diverse forms of, generally illegal, action and by so doing, are exacerbating the tensions inherent in the humanitarian field. (shrink)

Using data from the 1979 to 1998 waves of the National Longitudinal Survey of Youth, the author explores how gender, family, and class alter the impact of self-employment on earnings. Fixed-effect regression results show that while self-employment positively influences men’s earnings, not all women similarly benefit. Professionals receive the same self-employment earnings premium, regardless of gender. However, self-employment in nonprofessional occupations negatively affects women’s earnings, with wives and mothers incurring the greatest penalties. The high concentration of nonprofessional self-employed women in (...) child care accounts for much of these penalties. Results are robust despite inclusion of controls for human capital and labor supply, job characteristics, occupational and industrial gender segregation, and demographic characteristics. The compensating differentials argument, that women with greater family responsibilities trade earnings for the family-friendly aspects of self-employment, is discussed in light of these findings. While this argument may explain women’s returns to nonprofessional self-employment, it is less persuasive for interpreting women’s returns to professional self-employment. (shrink)

Is it possible for someone to be a sceptic on their own deathbed? Such behaviour would be a paradox for philosophers and a scandal for common sense. Attacks came first from philosophers, mainly Beattie, which proved to be popular. But there was no change in Hume's mood. He was still joking with Adam Smith a few days before his death and died in a lucid and serene state of mind. After his death, it was proved by the friends of religion (...) that it was impossible to be both a sceptic and an honest man. (shrink)

In his review of Charles Spearman’s The Nature of ‘Intelligence’ (1923), R. G. Collingwood launched an attack upon psychometrics that was expanded in his Essay on Metaphysics (1940). Although underrated by friend and foe alike, Collingwood’s critique identified a number of defects in the thinking of psychometricians that subsequently became entrenched. However, his main complaint was that psychology generally (and, by implication, psychometrics) was a ‘fashionable scientific fraud’. This charge was inspired by his more general views on logic and (...) metaphysics, which, however, as I argue, are logically unsustainable. Ironically, other elements of his philosophy – his ‘fallacy of calculation’ and concept of ‘scale of forms’ – are relevant to psychometrics and tip the scales in favour of his otherwise unwarranted charge. (shrink)