Between December 1855 and March 1856, a public dispute raged, in British national newspapers and locally published pamphlets, between two teachers at the University of Oxford: the mathematical lecturer Francis Ashpitel and Bartholomew Price, the professor of natural philosophy. The starting point for these exchanges was the particularly poor results that had come out of the final mathematics examinations in Oxford that December. Ashpitel, as one of the examiners, stood accused of setting questions that were too difficult (...) for the ordinary student, with the consequence that, in Price’s view, further mathematical study in Oxford – never as robust as in Cambridge – would be discouraged. We examine this short-lived affair, and use it not only to gain insight into the status of mathematical study in Oxford in the mid-nineteenth century, but also to point towards the increasing importance of competitive examinations in British public life at that time. (shrink)

This Oxford Handbook covers the current state of the art in the philosophy of maths and logic in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 newly-commissioned chapters are by established experts in the field and contain both exposition and criticism as well as substantial development of their own positions. Select major positions are represented by two chapters - one supportive and one critical. The book includes a (...) comprehensive bibliography. (shrink)

Generality is a key value in scientific discourses and practices. Throughout history, it has received a variety of meanings and of uses. This collection of original essays aims to inquire into this diversity. Through case studies taken from the history of mathematics, physics and the life sciences, the book provides evidence of different ways of understanding the general in various contexts. It aims at showing how individuals have valued generality and how they have worked with specific types of "general" (...) entities, procedures, and arguments. (shrink)

Contribution to a review symposium on Marc Lange's Because without cause: Non-causal explanation in science and mathematics. Oxford: Oxford University Press, 2017.

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.3 (2003) 426-427 [Access article in PDF] Timothy Smiley, editor. Mathematics and Necessity: Essays in the History of Philosophy. New York: Oxford University Press, 2000. Pp. ix + 166. Cloth, $35.00.Mathematics and Necessity contains essays by M. F. Burnyeat, Ian Hacking, and Jonathan Bennett based on lectures given to the British Academy in 1998. All concern the history of the (...) philosophical treatment of mathematics, necessity, or both, although there is no single thread running through all three essays.Burnyeat focuses on Plato's understanding of mathematics in the Republic, making use of the Timeaus in particular to defend his interpretation. Plato claimed that in the ideal society mathematics should be studied for a full ten years, which many have thought excessive. To explain it, Burnyeat argues that the role of mathematics is pervasive: ethics and politics are mathematical, because in Plato's view mathematics is a constitutive part of ethical understanding. On Burnyeat's interpretation, goodness is part of the objective world, and mathematics is a science of values. The link between mathematics and goodness in the objective world is forged by the concord, harmony, and unity of proportions, which express the goodness of the Divine Craftsman's beneficent design. Proportions are both intrinsically good and the proper study of mathematics.Burnyeat explains why Plato thinks abstract kinematics (the pure non-sensible counterpart to astronomy) and abstract harmonics lead to knowledge of the good. Abstract kinematics concerns the "harmony of the spheres" described in the Republic. Drawing heavily on the Timeaus, Burnyeat argues that the circular motions of the spheres are motions in the intelligence of the world soul and that there are similar motions in human souls. The world soul and human souls are spatial and have circular motions (invisibility and intangibility make them incorporeal). Souls are the most important subject matter of abstract harmonics—the study of good proportions. A soul that understands the various mathematical disciplines and their relations takes on the harmony and hence goodness of the world soul. Such a soul serves as a model for organizing the social world, a model which properly trained rulers can use to shape a community. Burnyeat's well-argued essay is compelling and exciting. [End Page 427]Ian Hacking's views provide a great contrast to Burnyeat's Plato. Hacking conducts a phenomenological inquiry into why some philosophers have become obsessed by mathematics as a source of philosophical inspiration. He holds that mathematics concerns a relatively minor part of human culture and that the influence of mathematics on philosophy is pernicious; he even describes it as an infection.Hacking examines the views of six philosophers: Plato, Leibniz, Mill, Descartes, Lakatos, and Wittgenstein. Hacking's assessment of the six philosophers is at times disputable, but it is also interesting and stimulating, and serves his phenomenological inquiry.Hacking traces philosophy's infection by mathematics to two factors. A mathematical proof can be perspicuous; that is, a proof can be "grasped," which requires seeing directly why something must be true and being able to recapitulate it. A mathematical proof can also anticipate facts about the world. He thinks that these two factors affect us in two ways: they give us the idea of a priori knowledge, and they dazzle us into unreasoned conviction in proofs. One might respond that, setting aside the feeling proofs give us, we are still justified in being impressed by them. Hacking argues that much of what impresses is based on a misunderstanding of their nature, which he attempts to correct.According to Hacking, a mathematical demonstration involves shifts of meaning that analytically bind the proof and the theorem it establishes. They thereby mutually support each other and provide what he calls self-authentication. The correctness of the proof is supported by the "analytified" truth of the theorem and the theorem is considered true because proven.This bootstrapping feature of mathematical proofs supports Hacking's deflationary account of necessity. Hacking suggests that the "must" of logical necessity rests upon our unwillingness to accept empirical counter-examples to the... (shrink)

The Conceptual Roots of Mathematics is a comprehensive study of the foundation of mathematics. J.R. Lucas, one of the most distinguished Oxford scholars, covers a vast amount of ground in the philosophy of mathematics, showing us that it is actually at the heart of the study of epistemology and metaphysics.

Breathing fresh air into the philosophy of mathematics Content Type Journal Article DOI 10.1007/s11016-010-9470-8 Authors Marco Panza, IHPST, 13, rue du Four, 75006 Paris, France Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.

This chapter discusses the significance of mathematics in natural theology. It suggests that the existence of an independent noetic realm of mathematics should encourage an openness to the possibility of further metaphysical riches to be explored. Engagement with mathematics is only a part of our mental experience. In itself it can give just a hint of what might be meant by the spiritual. The realm of the divine is yet more distant still, but just as arithmetic may (...) have led our ancestors to begin an exploration that would eventually lead them into the riches of higher mathematics, so taking mathematical reality seriously might be the start of a journey which will lead to greater discoveries about the scope and nature of reality. In this way, mathematics has a modest preparatory part to play in the approach to natural theology, and may also contribute to natural theology through its role as a component in a more complex form of encounter with reality. (shrink)

Critical notice of C. Pincock's Mathematics and Scientific Representation (2012).

Mathematical practitioners in seventeenth-century London formed a cohesive knowledge community that intersected closely with instrument-makers, printers and booksellers. Many wrote books for an increasingly numerate metropolitan market on topics covering a wide range of mathematical disciplines, ranging from algebra to arithmetic, from merchants’ accounts to the art of surveying. They were also teachers of mathematics like John Kersey or Euclid Speidell who would use their own rooms or the premises of instrument-makers for instruction. There was a high degree of (...) interdependency even beyond their immediate milieu. Authors would cite not only each other, but also practitioners of other professions, especially those artisans with whom they collaborated closely. Practical mathematical books effectively served as an advertising medium for the increasingly self-conscious members of a new emerging professional class. Contemporaries would talk explicitly of ‘the London mathematicians’ in distinction to their academic counterparts at Oxford or Cambridge. The article takes a closer look at this metropolitan knowledge culture during the second half of the century, considering its locations, its meeting places and the mathematical clubs which helped forge the identity of its practitioners. It discusses their backgrounds, teaching practices and relations to the London book trade, which supplied inexpensive practical mathematical books to a seemingly insatiable public. (shrink)

The Conceptual Roots of Mathematics is a comprehensive study of the foundation of mathematics. Lucas, one of the most distinguished Oxford scholars, covers a vast amount of ground in the philosophy of mathematics, showing us that it is actually at the heart of the study of epistemology and metaphysics.

We argue that the set of humanly known mathematical truths (at any given moment in human history) is finite and so recursive. But if so, then given various fundamental results in mathematical logic and the theory of computation (such as Craig’s in J Symb Log 18(1): 30–32(1953) theorem), the set of humanly known mathematical truths is axiomatizable. Furthermore, given Godel’s (Monash Math Phys 38: 173–198, 1931) First Incompleteness Theorem, then (at any given moment in human history) humanly known mathematics (...) must be either inconsistent or incomplete. Moreover, since humanly known mathematics is axiomatizable, it can be the output of a Turing machine. We then argue that any given mathematical claim that we could possibly know could be the output of a Turing machine, at least in principle. So the Lucas-Penrose (Lucas in Philosophy 36:112–127, 1961; Penrose, in The Emperor’s new mind. Oxford University Press, Oxford (1994)) argument cannot be sound. (shrink)

(No abstract is available for this citation).

Annalisa Coliva (Int J Study Skept 10(3–4):346–366, 2020) asks, “Are there mathematical hinges?” I argue here, against Coliva’s own conclusion, that there are. I further claim that this affirmative answer allows a case to be made for taking the concept of a hinge to be a useful and general-purpose tool for studying mathematical practice in its real complexity. Seeing how Wittgenstein can, and why he would, countenance mathematical hinges additionally gives us a deeper understanding of some of his latest thoughts (...) on mathematics. For example, a view of how mathematical hinges relate to Wittgenstein’s well-known river-bed analogy enables us to see how his way of thinking about mathematics can account nicely for a “dynamics of change” within mathematical research—something his philosophy of mathematics has been accused of missing (e.g., by Robert Ackermann (Wittgenstein’s city, The University of Massachusetts Press, Amherst, 1988) and Mark Wilson (Wandering significance: an essay on conceptual behavior, Oxford University Press, Oxford, 2006). Finally, the perspective on mathematical hinges ultimately arrived at will be seen to provide us with illuminating examples of how our conceptual choices and theories can be ungrounded but nevertheless the right ones (in a sense to be explained). (shrink)

In lieu of an abstract, here is a brief excerpt of the content:American Journal of Philology 125.1 (2004) 140-144 [Access article in PDF] C. J. Tuplin and T. E. Rihll, eds. Science and Mathematics in Ancient Greek Culture. Foreword by Lewis Wolpert. Oxford: Oxford University Press, 2002. xvi + 379 pp. 21 black-and white ills. 3 tables. Cloth, $80. It has become something of a truism to say that, whatever their ambitions for abstraction, scientists remain profoundly caught (...) up in the life of their culture. Yet, that is a truism only at a certain level. No one is surprised to hear that Euclid had to eat and drink just like everyone else and relied on royal patronage for the leisure to pursue his studies, but it may occasion surprise to learn that the shape of his number theory should owe something to the particular way in which his eye fell upon a multiplication table. That Galen was influenced by his teachers is not news; but that his commitment to a particular school of philosophy could cause him to see and not see certain things when he was conducting dissections—here is something of interest. The papers in this volume, which were first delivered at a conference in Liverpool in 1996, have it as their common task to show how ancient forms of life influenced the pursuit and acquisition of knowledge on the part of scientists like Euclid and Galen.Heading the collection is a study by Andrew Barker of some idiosyncratic features of Greek harmonic theory. Archytas, Aristotle, and Ptolemy each attempted to characterize the physical basis of the two opposite sound qualities, oxusand barus. As Aristotle himself observed, the terms are metaphors drawn from the tactile realm, and trouble arose because in that realm they are heterogeneous opposites, i.e., the proper opposites of "sharp" and "heavy" are "dull" and "light." Barker uncovers a corresponding uncertainty in the texts about whether these sound qualities should be analyzed as something more like weight or something more like sharpness. Thus, an excessive trust in the descriptive adequacy of ordinary language terms led these thinkers to frustration.Harry Hine's paper on earthquakes and vulcanology deals with ancient knowledge of two phenomena that elicited the attention of scientist and non-scientist alike. Like Barker, Hine draws our attention to a terminological deficiency: although there was a fairly elaborate vocabulary for seismoi, neither Greek nor Latin possessed a generic noun that could properly be translated as [End Page 140] "volcano." This deficiency seems to be correlated with the fact that earthquakes form a regular part of the meteorological curriculum, while volcanoes usually received independent treatment in works such as the Aetna. Particularly intriguing is Hine's suggestion that the opinion of locals, especially sailors, may have been the source of certain scientific theses about volcanoes, e.g., that Aetna's behavior was closely correlated with the weather.The difference between ancient astronomy as practiced and as it was popularly believed to operate is the subject of Alan Bowen's paper. His central question is, when did astronomers first set out to make predictions of eclipses? The surprising answer is that, despite popular beliefs to the contrary, such prediction was never practiced by astronomers in a serious way before Ptolemy. Along with scrutiny of the misunderstanding in Herodotus, Diodorus, and Pliny, the paper offers thorough analysis of the most well-documented "prediction," that of a lunar eclipse that took place the night before the battle of Pydna in 167 B.C.E., which was supposedly foreseen by the Roman (!) general C. Sulpicius Gallus. Bowen casts doubts upon the entire tradition, showing how adherence to an ideology of the well-rounded commander led the sources to inflate his achievement: while some reports have Gallus simply assuaging his troops' fears at the event, others have him explaining its causes or even foretelling it. Bowen's skepticism is well founded, and one must regard these supposed feats of eclipse-prediction as a kind of science fiction, prescient yet false all the same.R. Hanna's subject is the format and function of the parapegma calendar first devised by... (shrink)

DPhil dissertation, University of Oxford.

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an area, (...) D—the challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism. (shrink)

In his book on The Mathematics of Great Amateurs Coolidge starts the chapter on Bolzano saying that he included Bolzano because it seemed interesting to him ‘that a man who was a remarkable pulpit orator, only removed from his chair for his political opinions, should have thought so far into the deepest problems of a science which he never taught in a professional capacity’ [Coolidge, 1990, p. 195]. In fact, considering Bolzano's poor health and his enormous productivity in his (...) ‘professional areas’, the results, the breadth and the extent of this ‘great amateur's’ thoughts on mathematics, and its proper scientific presentation are simply astonishing. Even this new selection of more than 650 printed pages of text, well edited and translated by Steve Russ, represents only a part of Bolzano's legacy on mathematical topics: the complete German edition of Bolzano's writings, the Bernard Bolzano-Gesamtausgabe edited by J. Berg, E. Morscher, B. van Rootselaar, and others is projected to include over 120 volumes of which more than 36 volumes will contain his mathematical works. About 25 volumes have been published so far in this mathematical subseries including Bolzano's extensive Miscellanea Mathematica .To most readers of this journal, Bolzano will be well-known for his contributions to logic—his ‘logic of variation’ is viewed as a precursor of the modern semantic conception of logical consequence—and his contributions to the foundations of calculus, especially his ‘purely analytic proof’ of the intermediate-value theorem and the Bolzano-Weierstrass theorem, or his early discovery of a counterintuitive function which is continuous in its domain, but nowhere-differentiable.Despite his lifelong interest in the foundations of mathematics, Bolzano studied theology and taught philosophy of religion in his native city, Prague, from 1805 until his dismissal in 1820. Part of Bolzano's duties …. (shrink)