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Philosophy

David Corfield works in the philosophy of science and mathematics. His interests range from probability theory and physics to psychology and medicine, and he looks both to formal methods and historical narrative to understand disciplinary rationality. He is co-director of the Centre for Reasoning at Kent. He is also one of the three owners of the blog The n-category Café, where the implications for philosophy, mathematical and physical of the exciting new language of higher-dimensional category theory are discussed. In 2007 David published *Why Do People Get Ill?* (co-authored with Darian Leader), which aims to revive interest in the psychosomatic approach to medicine. He intends to carry out further work in the philosophy of medicine.

Also view these in the Kent Academic Repository

Books

Leader, Darian and Corfield, David (2007)
Why Do People Get Ill?
Hamish Hamilton Ltd, London, 384 pp. ISBN 978-0241143162.

Abstract

Have you ever wondered why people get ill when they do? How does the mind affect the body? Why does modern medicine seem to have so little interest in the unconscious processes that can make us fall ill? And what, if anything, can we do about it? "Why Do People Get Ill?" lucidly explores the relationship between our minds and our bodies. Containing remarkable case studies, cutting-edge research and startling new insights into why we fall ill, this intriguing and thought-provoking book should be read by anyone who cares about their own health and that of other people.

Corfield, David (2003)
Towards a Philosophy of Real Mathematics.
Cambridge University Press, 300 pp. ISBN 9780521035255.

Abstract

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically, and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of new ways to think philosophically about mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme, and the ways in which new concepts are justified. His highly original book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines, and points clearly to the ways in which this can be done.

Articles

Corfield, David (2011)
Understanding the Infinite II: Coalgebra.
Studies in History and Philosophy of Science A, 42 (4). pp. 571-579.

Abstract

In this paper we give an account of the rise and development of coalgebraic thinking in mathematics and
computer science as an illustration of the way mathematical frameworks may be transformed. Originating
in a foundational dispute as to the correct way to characterise sets, logicians and computer scientists
came to see maximizing and minimizing extremal axiomatisations as a dual pair, each necessary to represent
entities of interest. In particular, many important infinitely large entities can be characterised in
terms of such axiomatisations. We consider reasons for the delay in arriving at the coalgebraic framework,
despite many unrecognised manifestations occurring years earlier, and discuss an apparent asymmetry
in the relationship between algebra and coalgebra.

Corfield, David (2010)
Varieties of Justification in Machine Learning.
Minds and Machines, 20 (2). pp. 291-301. ISSN 0924-6495.

Abstract

Forms of justification for inductive machine learning techniques are discussed and classified into four types. This is done with a view to introduce some of these techniques and their justificatory guarantees to the attention of philosophers, and to initiate a discussion as to whether they must be treated separately or rather can be viewed consistently from within a single framework.

Corfield, David (2010)
Understanding the Infinite I: Niceness, Robustness, and Realism.
Philosophia Mathematica, 18 (3). pp. 253-275. ISSN 0031-8019.

Abstract

This paper treats the situation where a single mathematical construction satisfies a multitude of interesting mathematical properties. The examples treated are all infinitely large entities. The clustering of properties is termed 'niceness' by the mathematician Michiel Hazewinkel, a concept we compare to the 'robustness' described by the philosopher of science William Wimsatt. In the final part of the paper, we bring our findings to bear on the question of realism which concerns not whether mathematical entities exist as abstract objects, but rather whether the choice of our concepts is forced upon us.

Corfield, David (2010)
Lautman et la réalité des mathématiques.
Philosophiques, 37 (1). pp. 95-109. ISSN 0316-2923.

Abstract

This paper examines Lautman’s claim that the reality of mathematics is to be addressed through the “realisation of dialectical ideas”. This is done in the context of two examples treated by Lautman himself. The question is raised as to whether we might better describe dialectical ideas as mathematical ones, especially now that we have mathematical means to approach these ideas at the right level of generality. For example, category theory, unknown to Lautman, can describe the idea of duality very thoroughly. It is argued that the instances given by Lautman of the realisation of dialectical ideas outside of mathematics and mathematical physics are rather slight, leading us to conclude that the ideas he so brilliantly describes are immanent to mathematical practice, rather than belonging to “an ideal reality, superior to mathematics”.
Cet article examine la thèse de Lautman selon laquelle la réalité des mathématiques doit être approchée par la « réalisation des idées dialectiques ». Pour ce faire, nous reprenons deux exemples que Lautman a lui-même traités. La question est de savoir si on peut ou non mieux décrire les idées dialectiques comme mathématiques, particulièrement maintenant que les moyens mathématiques d’approcher ces idées au niveau de généralisation appropriée existent. Ainsi, la théorie des catégories, inconnue de Lautman, peut donner une description très approfondie de l’idée de dualité. Je soumets de plus que les instances, données par Lautman, de la réalisation des idées dialectiques en dehors des mathématiques et de la physique mathématique sont assez maigres, ce qui suggère fortement que les idées qu’il a décrites si admirablement sont immanentes à la pratique des mathématiques, au lieu d’appartenir à « une réalité idéale, supérieure aux mathématiques ».

Corfield, David and Schölkopf, Bernhard and Vapnik, Vladimir (2009)
Falsificationism and Statistical Learning Theory: Comparing the Popper and Vapnik-Chervonenkis dimensions.
Journal for General Philosophy of Science, 40 (1). pp. 51-58. ISSN 0925-4560.

Abstract

We compare Karl Popper's ideas concerning the falsifiability of a theory with similar notions from the part of statistical learning theory known as VC-theory. Popper's notion of the dimension of a theory is contrasted with the apparently very similar VC-dimension. Having located some divergences, we discuss how best to view Popper's work from the perspective of statistical learning theory, either as a precursor or as aiming to capture a different learning activity.

Corfield, David (2004)
Mathematical Kinds, or Being Kind to Mathematics.
Philosophica, 74 (2). pp. 37-62. ISSN 0379-8402.

Corfield, David (2001)
The Importance of Mathematical Conceptualisation.
Studies in History and Philosophy of Science Part A, 32 (3). pp. 507-533.

Abstract

Mathematicians typically invoke a wide range of reasons as to why their research is valuable. These reveal considerable differences between their personal images of mathematics. One of the most interesting of these concerns the relative importance accorded to conceptual reformulation and development compared with that accorded to the achievement of concrete results. Here I explore the conceptualists' claim that the scales are tilted too much in favour of the latter. I do so by taking as a case study the debate surrounding the question as to whether groupoids are significantly more powerful than groups at capturing the symmetry of a mathematical situation. The introduction of groupoids provides a suitable case as they score highly according to criteria relating to theory-building rather than problem-solving.
Several of the arguments for the adoption of the groupoid concept are outlined, including claims as to its capacity for reformulating existing theory, its ability to measure symmetry more systematically, and its ‘naturalness’. This last notion is given an extensive treatment.

Book Sections

Corfield, David (2012)
Narrative and the Rationality of Mathematics.
In: Doxiadis, Apostolos and Mazur, Barry Circles Disturbed: The interplay of mathematics and narrative. Princeton University Press, pp. 244-280. ISBN 9780691149042.

Corfield, David (2009)
Projection and Projectability.
In: Quinonero-Candela, Joaquin and Sugiyama, Masashi and Schwaighofer, Anton et al. Datast Shift in Machine Learning. MIT Press, pp. 29-38. ISBN 978-0-262-17005-5.

Abstract

The problem of dataset shift can be viewed in the light of the more general problems of induction, in particular the question of what it is about some objects' features or properties which allow us to project correlations confidently to other times and other places.

Corfield, David (2006)
Some Implications of the Adoption of Category Theory for Philosophy.
In: Sica, Giandomenico What is Category Theory? Polimetrica International Scientific Publisher. ISBN 9788876990311.

Corfield, David (2005)
Categorification as a Heuristic Device.
In: Cellucci, C. and Gillies, Donald Mathematical Reasoning and Heuristics. King's College Publications. ISBN 9781904987079.

Corfield, David (2002)
Argumentation and the Mathematical Process.
In: Kampis, George and Kvasz, Ladislav and Stöltzner, Michael Appraising Lakatos: Mathematics, Methodology, and the Man. Kluwer Academic Publishers, Dordrecht, pp. 115-138. ISBN 9781402002267.

Corfield, David (2002)
From Mathematics to Psychology: Lacan’s missed encounters.
In: Glynos, Jason and Stavirakakis, Yannis Lacan and Science. Karnac Books, London, pp. 176-206. ISBN 9781855759213.

Corfield, David (2001)
Bayesianism in Mathematics.
In: Corfield, David and Williamson, Jon Foundations of Bayesianism. Kluwer Academic Publishers Group, Dordrecht, pp. 175-201. ISBN 9781402002236.

Abstract

A study of the possibility of casting plausible matheamtical inference in Bayesian terms.

Corfield, David and Williamson, Jon (2001)
Introduction: Bayesianism into the 21st Century.
In: Corfield, David and Williamson, Jon Foundations of Bayesianism. Kluwer Academic Publishers Group, Dordrecht, pp. 1-16. ISBN 9781402002236.

Reviews

Corfield, David (2005)
Martin H. Krieger. Doing Mathematics: Convention, Subject, Calculation, Analogy. Singapore: World Scientific Publishing, 2003. Pp. xviii + 454. ISBN 981-238-2003 (cloth); 981-238-2062 (paperback).
Review of: Doing Mathematics: Convention, Subject, Calculation, Analogy. by Krieger, M. Philosophia Mathematica, 13. pp. 106-111.

Corfield, David (2002)
Conceptual mathematics: a first introduction to categories.
Review of: Conceptual Mathematics by Lawvere, F. William and Schanuel, Stephen. Studies in History and Philosophy of Modern Physics, 33 (2). pp. 359-366. ISSN 1355-2198.

Edited Books

Corfield, David and Williamson, Jon (2001)
Foundations of Bayesianism.
Kluwer Academic Publishers, 428 pp. ISBN 978-1402002236.

Abstract

Foundations of Bayesianism is an authoritative collection of papers addressing the key challenges that face the Bayesian interpretation of probability today. Some of these papers seek to clarify the relationships between Bayesian, causal and logical reasoning. Others consider the application of Bayesianism to artificial intelligence, decision theory, statistics and the philosophy of science and mathematics. The volume includes important criticisms of Bayesian reasoning and also gives an insight into some of the points of disagreement amongst advocates of the Bayesian approach. The upshot is a plethora of new problems and directions for Bayesians to pursue. The book will be of interest to graduate students or researchers who wish to learn more about Bayesianism than can be provided by introductory textbooks to the subject. Those involved with the applications of Bayesian reasoning will find essential discussion on the validity of Bayesianism and its limits, while philosophers and others interested in pure reasoning will find new ideas on normativity and the logic of belief.