Portrait of Dr David Corfield

Dr David Corfield

Senior Lecturer in Philosophy

About

Dr David Corfield joined Kent in 2007, having taught and worked in a number of institutions, including the universities of Leeds, Cambridge and Oxford and Tübingen. 

He is the author of Towards a Philosophy of Real Mathematics (Cambridge, 2003) and co-author of Why do people get ill? (Hamish-Hamilton, 2007). In 2020, he published Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy (OUP).

Research interests

David's research interests include historically-informed philosophy of mathematics; the possible roles for Type theory in philosophy; and the philosophy of medicine, in particular the mind-body relation. His current research concerns the new foundational language for mathematics known as 'homotopy type theory'. 

David is co-director of the Centre for Reasoning at Kent and 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 aimed to revive interest in the psychosomatic approach to medicine. He intends to carry out further work in the philosophy of medicine. 

Teaching

David teaches the philosophy of mathematics, science and medicine, and logic.

Publications

Article

  • Corfield, D. (2017). Expressing ‘The Structure of’ in Homotopy Type Theory. Synthese [Online]. Available at: https://doi.org/10.1007/s11229-017-1569-7.
    As a new foundational language for mathematics with its very different idea as to the status of logic, we should expect homotopy type theory to shed new light on some of the problems of philosophy which have been treated by logic. In this article, definite description, and in particular its employment within mathematics, is formulated within the type theory. Homotopy type theory has been proposed as an inherently structuralist foundational language for mathematics. Using the new formulation of definite descriptions, opportunities to express ‘the structure of’ within homotopy type theory are explored, and it is shown there is little or no need for this expression.
  • Corfield, D. (2017). Duality as a category-theoretic concept. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics [Online] 59:55-61. Available at: http://dx.doi.org/10.1016/j.shpsb.2015.07.004.
    In a paper published in 1939, Ernest Nagel described the role that projective duality had played in the reformulation of mathematical understanding through the turn of the nineteenth century, claiming that the discovery of the principle of duality had freed mathematicians from the belief that their task was to describe intuitive elements. While instances of duality in mathematics have increased enormously through the twentieth century, philosophers since Nagel have paid little attention to the phenomenon. In this paper I will argue that a reassessment is overdue. Something beyond doubt is that category theory has an enormous amount to say on the subject, for example, in terms of arrow reversal, dualising objects and adjunctions. These developments have coincided with changes in our understanding of identity and structure within mathematics. While it transpires that physicists have employed the term ‘duality’ in ways which do not always coincide with those of mathematicians, analysis of the latter should still prove very useful to philosophers of physics. Consequently, category theory presents itself as an extremely important language for the philosophy of physics.
  • Krömer, R. and Corfield, D. (2014). The Form and Function of Duality in Modern Mathematics. Philosophia Scientae [Online]:95-109. Available at: http://doi.org/10.4000/philosophiascientiae.976.
  • Corfield, D. (2011). Understanding the Infinite II: Coalgebra. Studies in History and Philosophy of Science A [Online] 42:571-579. Available at: http://dx.doi.org/10.1016/j.shpsa.2011.09.013.
    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, D. (2010). Understanding the Infinite I: Niceness, Robustness, and Realism. Philosophia Mathematica [Online] 18:253-275. Available at: http://dx.doi.org/10.1093/philmat/nkq014.
    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, D. (2010). Varieties of Justification in Machine Learning. Minds and Machines [Online] 20:291-301. Available at: http://dx.doi.org/10.1007/s11023-010-9191-1.
    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, D. (2010). Lautman et la réalité des mathématiques. Philosophiques [Online] 37:95-109. Available at: http://www.er.uqam.ca/nobel/philuqam/philosophiques/index.php?section=sommaire_par_numeros&vol=37&no=1.
    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, D., Schölkopf, B. and Vapnik, V. (2009). Falsificationism and Statistical Learning Theory: Comparing the Popper and Vapnik-Chervonenkis dimensions. Journal for General Philosophy of Science [Online] 40:51-58. Available at: http://dx.doi.org/10.1007/s10838-009-9091-3.
    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, D. (2004). Mathematical Kinds, or Being Kind to Mathematics. Philosophica 74:37-62.
  • Corfield, D. (2001). The Importance of Mathematical Conceptualisation. Studies in History and Philosophy of Science Part A [Online] 32:507-533. Available at: http://dx.doi.org/10.1016/S0039-3681(01)00007-3.
    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

  • Corfield, D. (2020). Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy. [Online]. Oxford, UK: Oxford University Press. Available at: https://global.oup.com/academic/product/modal-homotopy-type-theory-9780198853404.
    For the past century, philosophers working in the tradition of Bertrand Russell - who promised to revolutionise philosophy by introducing the 'new logic' of Frege and Peano - have employed predicate logic as their formal language of choice. In this book, Dr David Corfield presents a comparable revolution with a newly emerging logic - modal homotopy type theory.

    Homotopy type theory has recently been developed as a new foundational language for mathematics, with a strong philosophical pedigree. Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy offers an introduction to this new language and its modal extension, illustrated through innovative applications of the calculus to language, metaphysics, and mathematics.

    The chapters build up to the full language in stages, right up to the application of modal homotopy type theory to current geometry. From a discussion of the distinction between objects and events, the intrinsic treatment of structure, the conception of modality as a form of general variation to the representation of constructions in modern geometry, we see how varied the applications of this powerful new language can be.
  • Leader, D. and Corfield, D. (2007). Why Do People Get Ill?. [Online]. London: Hamish Hamilton Ltd. Available at: http://www.penguincatalogue.co.uk/lo/general/title.html?catalogueId=null&imprintId=27&titleId=699.
    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, D. (2003). Towards a Philosophy of Real Mathematics. [Online]. Cambridge: Cambridge University Press. Available at: http://assets.cambridge.org/97805218/17226/excerpt/9780521817226_excerpt.pdf.
    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.

Book section

  • Corfield, D. (2017). Reviving the philosophy of geometry. In: Landry, E. ed. Categories for the Working Philosopher. Oxford, UK: Oxford University Press, pp. 18-36. Available at: https://global.oup.com/academic/product/categories-for-the-working-philosopher-9780198748991.
    In the Anglophone world, the philosophical treatment of geometry has fallen on hard times. While in Germany as late as the 1920s there were vibrant discussions concerning the nature of geometry— especially in relation to the direction of its development, the role of intuition and the perception of physical space—the rise of logical empiricism largely brought these to a close. Accounts of mathematics in its totality as uniformly reducible to a language such as set theory have led the recipients of logical empiricist doctrines to ignore the thematic contours of modern mathematics. I argue, however, that geometry in all of its many guises flourishes in contemporary mathematics, and that the term ‘geometry’ itself continues to have an important meaning. One way then for philosophy to come to a better understanding of mathematics is to provide an account of modern geometry, including its development of new forms of space, both for mathematical physics and for arithmetic. I argue that we can find a good starting point for this work by returning to the discussions of Weyl and Cassirer on geometry. With the help of modern interpreters, we see that issues salient to these thinkers are very much relevant to us today. I also propose that an effective way to encompass a great part of modern geometry is via (1, 1)-toposes, also known as homotopy toposes, an important development of category theory. Recently an internal language for such categories has been devised, known as ‘homotopy type theory’, a variant of which captures the ‘cohesiveness’ of geometric spaces. With these tools in place, we can now start to see what an adequate philosophical account of current geometry might look like.
  • Corfield, D. (2017). Homotopy type theory and the vertical unity of concepts in mathematics. In: de Freitas, E., Sinclair, N. and Coles, A. eds. What Is a Mathematical Concept?. Cambridge, UK: Cambridge University Press, pp. 125-142. Available at: https://doi.org/10.1017/9781316471128.
    The mathematician Alexander Borovik speaks of the importance of the `vertical unity' of mathematics. By this he means to draw our attention to the fact that many sophisticated mathematical concepts, even those introduced at the cutting-edge of research, have their roots in our most basic conceptualisations of the world. If this is so, we might expect any truly fundamental mathematical language to detect such structural commonalities. It is reasonable to suppose then that the lack of philosophical interest in such vertical unity is related to the prominence given by philosophers to languages which do not express well such relations. In this chapter, I suggest that we look beyond set theory to the newly emerging homotopy type theory, which makes plain what there is in common between very simple aspects of logic, arithmetic and geometry and much more sophisticated concepts. With this language in mind, new light is thrown on thenature of mathematical concepts with clear benefits for educationalists.
  • Corfield, D. (2012). Narrative and the Rationality of Mathematics. In: Doxiadis, A. and Mazur, B. eds. Circles Disturbed: The Interplay of Mathematics and Narrative. Princeton: Princeton University Press, pp. 244-280. Available at: http://press.princeton.edu/titles/9764.html.
  • Corfield, D. (2009). Projection and Projectability. In: Quinonero-Candela, J., Sugiyama, M., Schwaighofer, A. and Lawrence, N. eds. Dataset Shift in Machine Learning. Cambridge, MA: MIT Press, pp. 29-38.
    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, D. (2006). Some Implications of the Adoption of Category Theory for Philosophy. In: Sica, G. ed. What Is Category Theory?. Monza, Italy: Polimetrica International Scientific Publisher.
  • Corfield, D. (2005). Categorification as a Heuristic Device. In: Cellucci, C. and Gillies, D. eds. Mathematical Reasoning and Heuristics. London: King’s College Publications.
  • Corfield, D. (2002). From Mathematics to Psychology: Lacan’s missed encounters. In: Glynos, J. and Stavirakakis, Y. eds. Lacan and Science. London: Karnac Books, pp. 176-206.
  • Corfield, D. (2002). Argumentation and the Mathematical Process. In: Kampis, G., Kvasz, L. and Stöltzner, M. eds. Appraising Lakatos: Mathematics, Methodology, and the Man. Dordrecht: Kluwer Academic Publishers, pp. 115-138.
  • Williamson, J. (2001). Foundations for Bayesian Networks. In: Corfield, D. and Williamson, J. eds. Foundations of Bayesianism. Dordrecht: Kluwer Academic Publishers Group, pp. 75-115. Available at: http://bookshop.blackwell.co.uk/jsp/id/Foundations_of_Bayesianism/9781402002236.
    Bayesian networks may either be treated purely formally or be given an interpretation. I argue that current foundations are problematic, and put forward new foundations which involve aspects of both the interpreted and the formal approaches.
  • Corfield, D. (2001). Bayesianism in Mathematics. In: Corfield, D. and Williamson, J. eds. Foundations of Bayesianism. Dordrecht: Kluwer Academic Publishers Group, pp. 175-201.
    A study of the possibility of casting plausible matheamtical inference in Bayesian terms.
  • Corfield, D. and Williamson, J. (2001). Introduction: Bayesianism into the 21st Century. In: Corfield, D. and Williamson, J. eds. Foundations of Bayesianism. Dordrecht: Kluwer Academic Publishers Group, pp. 1-16.

Edited book

  • Corfield, D. and Williamson, J. eds. (2001). Foundations of Bayesianism. Dordrecht: Kluwer Academic Publishers.
    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.

Review

  • Corfield, D. (2005). Review of Martin H. Krieger, ’Doing Mathematics: Convention, Subject, Calculation, Analogy’. Philosophia Mathematica [Online] 13:106-111. Available at: http://philmat.oxfordjournals.org/cgi/content/extract/13/1/106.
    Singapore: World Scientific Publishing, 2003. Pp. xviii + 454. ISBN 981-238-2003 (cloth); 981-238-2062 (paperback)
  • Corfield, D. (2002). Review of F. William Lawvere and Stephen Schanuel, ’Conceptual Mathematics: A First Introduction to Categories’. Studies in History and Philosophy of Modern Physics 33:359-366.

Thesis

  • Dragulinescu, S. (2018). Grading the Quality of Evidence of Mechanisms.
  • Jansen, C. (2018). Moral Objectivity: Kant, Hume and Psychopathy.
    Moral objectivity is about genuinely better or worse courses of action and states of affairs in
    the moral domain. It seems good to aim at an identification of objective moral justifications
    that is maximally independent of subjectivity (at least if the threat of relativism is to be
    avoided). Having said that, it seems problematic to accept objective discriminations or
    justifications that are devoid of subjectivity. Every account of objective moral justifications
    seems in need of some sort of relationship with naturalistic human minds. How else could
    such justifications enter the universe?
    In this study I build towards arguments for deciding when claims about the status of
    moral objectivity are overambitious. I offer three lines of argument that point to moral
    objectivity being essentially anti-realist and (as such) mind-dependent. The first is grounded
    in Hume's (exclusively psychological) conception of 'reason'. It is paradigmatically well
    illustrated by Kant's philosophy.
    The second and third lines of argument are grounded in research about the nature
    and etiology of psychopathy. The second is about conceptual relativity regarding normative
    judgements about good practical lives. The third is about libertarian freedom over innately
    given components, components crucial to the psychological possibility of taking account of
    others in evaluative decision-making. Due to conceptual and empirical problems about
    (possible worlds of) human nature, which will be laid out, these two lines of argument need
    further conceptual and empirical attention.
    Additional to my constructive theory about the limits of moral objectivity, my study
    contains a critical reflection on methodological aspects of the contemporary meta-ethical
    debate. Overall, my study is a critical call for better reflection on the concept 'reason' and a
    deeper involvement with theoretical claims about human nature.
  • Groves, T. (2015). Let’s Reappraise Carnapian Inductive Logic!.
  • Hibbert, R. (2015). Are There Any Situated Cognition Concepts of Memory Functioning As Investigative Kinds in the Sciences of Memory?.
    This thesis will address the question of whether there are any situated cognition concepts of memory functioning as investigative kinds in the sciences of memory. Situated cognition is an umbrella term, subsuming extended, embedded, embodied, enacted and distributed cognition. I will be looking closely at case studies of investigations into memory where such concepts seem prima facie most likely to be found in order to establish a) whether the researchers in question are in fact employing such concepts, and b) whether the concepts are functioning well – functioning as investigative kinds – and should therefore continue to be employed, or whether something has gone wrong in the practice of the science and they should employ a different kind of concept. An historically situated approach to the case studies will allow me to answer part b) here.

    Along the way, I will argue for a way of construing scientific research that I call the dynamic framework account, an account of (im)maturity for science, a variety of conceptual role semantics with respect to scientific concepts, and the historically situated case study-based method I will employ in answering the central question. My conclusions, and the way I reach them, constitute contributions to debates about situated cognition particularly, and to philosophy of science more generally, as well as recommendations for scientific practice.
  • Wilde, M. (2015). Causing Problems: The Nature of Evidence and the Epistemic Theory of Causality.
    The epistemic theory of causality maintains that causality is an epistemic relation, so that causality is taken to be a feature of the way an agent represents the world rather than an agent-independent or non-epistemological feature of the world. The objective of this essay is to cause problems for the epistemic theory of causality. This is not because I think that the epistemic theory is incorrect. In fact, I spend some time arguing in favour of the epistemic theory of causality. Instead, this essay should be regarded as something like an exercise in stress testing. The hope is that by causing problems for a particular version of the epistemic theory, the result will be a more robust version of that theory.

    My gripe is with a particular version of the epistemic theory of causality, a version that is articulated with the help of objective Bayesianism. At first sight, objective Bayesianism looks like a plausible theory of rational belief. However, I argue that it is committed to a certain theory of evidence, a theory of evidence that recent work in epistemology has shown to be incorrect. In particular, objective Bayesianism maintains that evidence is perfectly accessible in a certain sense. But evidence just is not so perfectly accessible, according to recent developments in epistemology. However, this is not the end of the line for the epistemic theory of causality. Instead, I propose an epistemic theory of causality that dispenses with the assumption that evidence is perfectly accessible in the relevant sense.

Forthcoming

  • Hoctor, J. (2019). The Phenomenology of Twinship: An Investigation into the Exceptional Intersubjective Capacities Found in Twin-Twin Social Interactions.
    The overall aim of this thesis is to describe the exceptional intersubjective capacities we find in cases of twin-twin social interaction. Phenomenological approaches to intersubjectivity and empathy provide rich, varying and often competing conceptual resources for such a project, however, the chief focus of these accounts up until this point has been to describe the intersubjective capacities found in the social interactions between single-born persons. Thus, there is a lack of phenomenological literature that is explicitly concerned with outlining intersubjectivity in twins. Yet, recent literature in the sciences of mind as well as first person accounts from twins and their observers, point to important and unique variances in the manifestation of intersubjectivity between twins when compared to the intersubjective capacities of singletons.
    In essence, I contend prior phenomenological accounts are underpinned by a concept of passive synthesis or operative intentionality that is too narrow. Instead, I argue that if we are to fully appreciate twins and their social interactions without pathologising twinship, we need to expand these concepts to account for cases of exceptional mutual understanding (EMU) we find between them. In short, I argue that in twins (particularly in monozygotic twins), a more robust passive synthesis or a novel operative intentionality enables the kinds of EMU we find in their relations. Put differently, twins are highly attuned to one another's contextualised expressive bodily phenomena, which means they can directly experience greater aspects of their co-twin's mental and emotional life. This novel operative intentionality initially manifests in gestation as a result of a reciprocal and transformative influencing or coupling of each twin's body schema, and continues to develop and form the basis of their interactions throughout their respective lives. This means the primary and secondary intersubjective capacities of twins are highly developed when compared to single-born persons, and this allows them to rapidly exploit the implicit and nuanced narratives they have about each other to immediately grasp one another in the here and now.
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