David Corfield - Academia.edu (original) (raw)

Papers by David Corfield

Cambridge University Press eBooks, Apr 24, 2003

Studies in History and Philosophy of Modern Physics, Jun 1, 2002

Studies in History and Philosophy of Modern Physics, 2002

The MIT Press eBooks, Dec 12, 2008

The problem of dataset shift can be viewed in the light of the more general problems of induction... more 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

This chapter shows how the problem of dataset shift has been addressed by different philosophical... more This chapter shows how the problem of dataset shift has been addressed by different philosophical schools under the concept of “projectability.” When philosophers tried to formulate scientific reasoning with the resources of predicate logic and a Bayesian inductive logic, it became evident how vital background knowledge is to allow us to project confidently into the future, or to a different place, from previous experience. To transfer expectations from one domain to another, it is important to locate robust causal mechanisms. An important debate concerning these attempts to characterize background knowledge is over whether it can all be captured by probabilistic statements. Having placed the problem within the wider philosophical perspective, the chapter turns to machine learning, and addresses a number of questions: Have machine learning theorists been sufficiently creative in their efforts to encode background knowledge? Have the frequentists been more imaginative than the Bayesi...

病気の原因傾聴の大切さストレスが犯人？病気になるタイミング言葉と信念病気のもつ意味身体が返事をするとき心臓二つの身体、ひとつの身体同一化免疫系ガン正常に潜む健康リスクセ... more 病気の原因傾聴の大切さストレスが犯人？病気になるタイミング言葉と信念病気のもつ意味身体が返事をするとき心臓二つの身体、ひとつの身体同一化免疫系ガン正常に潜む健康リスクセラピーの効力医師の求めるもの

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2017

In a paper published in 1939, Ernest Nagel described the role that projective duality had played ... more 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.

What is a Mathematical Concept?

The mathematician Alexander Borovik speaks of the importance of the 'vertical unity' of mathemati... more 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.

Modal Homotopy Type Theory

A further innovation is the introduction of an intensional type theory. Here one need not reduce ... more A further innovation is the introduction of an intensional type theory. Here one need not reduce equivalence to mere identity. How two entities are the same tells us more than whether they are the same. This is explained by the homotopical aspect of HoTT, where types are taken to resemble spaces of points, paths, paths between paths, and so on. This allows us to rethink Russell’s definite descriptions. Mathematicians use a ‘generalized the’ in situations where it appears that they might be talking about a multiplicity of instances, but there is essentially a unique instance. Taken together with the ‘univalence axiom’ there results a language in which anything that can be said of a type can be said of an equivalent type. This allows homotopy type theory to become the language of choice for a structuralist, avoiding the need for any kind of abstraction away from multiple instantiations.

Mathematical Reasoning and Heuristics, 2005

I was drawn into the philosophy of mathematics by the writings of Lakatos, Brunschvicg and Lautma... more I was drawn into the philosophy of mathematics by the writings of Lakatos, Brunschvicg and Lautman. When I first started to read the mainstream English-language philosophy of mathematics literature, I was immediately struck by its almost complete lack of interest in what I ...

Synthese, 2018

The original article has been corrected. The article is published with Open Access but was missin... more The original article has been corrected. The article is published with Open Access but was missing Open Access information. This has been added.

Circles Disturbed

This chapter examines the rationality of mathematical practice in relation to narrative. It begin... more This chapter examines the rationality of mathematical practice in relation to narrative. It begins with a discussion of Alasdair MacIntyre's account of rational enquiry, Three Rival Versions of Moral Enquiry, and how this might translate to scientific and mathematical enquiry. It then considers the telos of mathematical enquiry, along with rival claims to truth as the aim of mathematics. The chapter argues that to be fully rational, mathematicians must embrace narrative as a basic tool for understanding the nature of their discipline and research. It also calls for the partial validity of a pre-Enlightenment epistemology of mathematics as a craft whose advance is made possible only through a certain discipleship.

The mathematician Alexander Borovik speaks of the importance of the 'vertical unity' of m... more 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.

Towards a Philosophy of Real Mathematics

Philosophia Mathematica, 2005

Have you ever wondered why people get ill when they do? How does the mind affect the body? Why do... more 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 ...