Mary Leng: Homepage

About	Publications	Talks and Conferences	Professional Service	Teaching	Outreach	Full CV

Contact Details

Department of Philosophy

University of Liverpool

7 Abercromby Square

Liverpool
L69 7WY U.K.

Tel: +44 151 794 2769
E-mail: mcleng <at> liv.ac.uk

About

Mary Leng is a Lecturer in Philosophy at the University of Liverpool. Prior to coming to Liverpool, she was a Research Fellow at St John's College, Cambridge (2002-2006), where her main research project was a book defending fictionalism in the philosophy of mathematics (this book, Mathematics and Reality, was published by OUP in April 2010). Her PhD thesis, 'Proof, Practice, and Progress', which defended an anti-Platonist view of mathematics in the light of evidence from mathematical practice, was co-supervised by Prof. James R. Brown and Ian Hacking at the University of Toronto. She has been a visiting fellow in the Department of Logic and Philosophy of Science at the University of California at Irvine (2001), and a visiting junior scholar in residence at the Peter Wall Institute for Advanced Studies at the University of British Columbia (2003).

Leng's main current research interests are in the philosophy of mathematics and science, metaphysics, and the philosophy of logic. She is currently working primarily on the notion of logical consequence, and in particular on alternatives to the standard model theoretic account of this notion.

She is married to the philosopher Martin O'Neill, with whom she has a son, Thomas O'Neill.

Top

Publications

Monograph

Mathematics and Reality (Oxford: OUP, 2010)

This book argues that a naturalist approach to ontology, which sees natural science as the ultimate arbiter on questions of truth and existence, does not (contrary to what is usually assumed) require acceptance of the existence of mathematical objects, even if such objects are indispensably posited in formulating our best scientific theories. It defends a fictionalist understanding of the role of mathematics in science, arguing that the role mathematical posits play in our scientific theories can be explained from a fictionalist perspective that does not assume the existence of the objects posited. This work is discussed (by Mark Balaguer) in the online Stanford Encyclopedia of Philosophy entry on Fictionalism in the Philosophy of Mathematics, and reviewed here by John P. Burgess in Philosophia Mathematica.

From John P. Burgess's review:

"The book is sure to generate considerable discussion: as the most substantial work on nominalism to appear for a decade or so, it demands a prompt response from the antinominalist side if the issue is not to go by default, and as the earliest large-scale treatment of an important type of position, it is likely to be the point of departure in
debates for years to come. Mathematics and Reality belongs on the shelf of every philosopher of mathematics."

Edited Collection

Mathematical Knowledge (Oxford: OUP, 2007), co-edited with Alexander Paseau (Oxford) and Michael Potter (Cambridge)

This collection presents some new perspectives on the epistemology of mathematics, including contributions from mathematicians and psychologists ,as well as philosophers. My contribution includes the sole-authored introductory chapter, and a chapter ("What's There to Know? A Fictionalist Account of Mathematical Knowledge") presenting a fictionalist perspective on mathematical knowledge as knowledge of the consequences of consistent theoretical assumptions. This book is reviewed here by Jeremy Gray for the Mathematical Association of America, and here by Elijah Chudnoff in Mind.

Book Contributions

‘Creation and Discovery in Mathematics’, to appear in John Polkinghorne, ed., Mathematics and its Significance (OUP, 2011)

The John Templeton Foundation's symposium on Mathematics and its Significance, organized as part of its Humble Approach Initiative, brought together philosophers (myself, Michael Detlefsen, the late Peter Lipton, Gideon Rosen, Stewart Shapiro, and Mark Steiner), and mathematicians/scientists (Sir Roger Penrose, Sir John Polkinghorne), to discuss a cluster of questions relating to the nature of mathematics, and its significance. A follow up event involved two additional mathematicians, Marcus du Sautoy and Timothy Gowers. This volume collects together the papers from these two meetings. My contribution argues that the real problem of mathematical discovery involves explaining how we are able to discover the consequences of our mathematical assumptions, rather than the issue of how we are able to discover truths about a realm of mathematical objects.

‘Structuralism and Applied Mathematics’, to appear in Clark Glymour, Wang Wei, and Dag Westerståhl, eds., Proceedings of the 13th International Congress in Logic, Methodology, and Philosophy of Science (King's College Publications, 2010)

It is standardly assumed that structuralist accounts of mathematics have an easier time than do fictionalist accounts in explaining the applications of mathematics. This paper (originally presented as an invited talk at the 2007 Congress in Logic, Methodology, and Philosophy of Science, held in Beijing) argues that structuralists and fictionalists are actually in the same boat.

‘‘Algebraic’ approaches to the philosophy of mathematics’, in Otávio Bueno and Øystein Linnebo, eds., New Waves in Philosophy of Mathematics (Palgrave Macmillan, 2009)

The New Waves in Philosophy series commissions articles from leading junior researchers, who are asked to reflect on what they take to be the most important new issues in their subject. In my contribution (originally presented in 2008 as an invited talk to the associated conference, at the University of Miami), I argue that Benacerraf's two problems for the philosophy of mathematics (the problem of multiple reductions and the problem of knowledge of abstracta) can be solved by several contemporary philosophical accounts of mathematics, but that these accounts face two new problems of their own (the problem of mixed claims, and the problem of modal notions).

‘Pre-Axiomatic Mathematical Reasoning: An Algebraic Perspective’, in Gila Hanna, Hans Niels Jahnke and Helmut Pulte, eds., Explanation and Proof in Mathematics. (Springer, 2009): 47-58

The 'algebraic' perspective sees mathematical axioms as contextually defining their subject matter. This paper (originally presented as an invited talk at an cross-disciplinary conference, Explanation and Proof in Mathematics (held at the University of Duisburg-Essen), which brought together philosophical and educational perspectives on mathematical explanation and proof) considers what such a perspective can say about pre-axiomatic mathematics.

‘Mathematical Explanation’, in Carlo Cellucci and Donald Gillies, eds., Mathematical Reasoning and Heuristics (King’s College Publications: London, 2005): 167-189

This paper argues that scientific explanations that posit mathematical objects can be explanatory without being true. This work was originally presented as an invited paper to the 2004 conference in Mathematical Reasoning, Heuristics, and the Development of Mathematics at the University of Rome, La Sapienza, where it (along with other papers presented at the conference) received a reply from Sir Michael Dummett. It is further discussed (by Paolo Mancosu) in the Stanford Encyclopedia of Philosophy entry on Mathematical Explanation.

Journal Articles

'Revolutionary Fictionalism: A Call to Arms’, Philosophia Mathematica 13, no. 3 (2005): 277-293

This paper responds to John Burgess's arguments against revolutionary fictionalism, as presented in his paper, 'Mathematics and Bleak House' (Philosophia Mathematica 12, no. 1 (2004)). It argues that revolutionary fictionalists (who hold that mathematical theories should not be understood as bodies of truths even if this is how mathematicians see those theories) fall foul of neither naturalism, nor Carnapian concerns about the meaningfulness of ontological questions, as Burgess claims. The paper is also discussed (along with my 2010 book) in Mark Balaguer's Stanford Encyclopedia of Philosophy entry on Fictionalism in the Philosophy of Mathematics.

‘Platonism and Anti-Platonism: Why Worry?’, International Studies in the Philosophy of Science 19, no. 1 (March 2005): 65-84

Many self-proclaimed scientific realists accept Hilary Putnam's characterization of their view as holding (amongst other things) that "terms in mature scientific theories typically refer ...[and] that the theories accepted in a mature science are typically approximately true" (Putnam, 1975). A consequence of this characterization that is often overlooked (though not, of course, by Putnam himself) is that unless we can dispense with mathematical posits in our mature scientific theories, scientific realism implies realism about the abstract mathematical objects posited by our theories. This paper argues that scientific realists should worry about the fact that their view implies mathematical Platonism, since attention to this consequence of scientific realism raises questions about the efficacy of some arguments for the view. In particular, the 'no miracles' argument is less compelling when one considers the question of whether it would be a miracle for our scientific theories to be predictively successful if the mathematical objects they posit did not exist.

‘Mathematical Practice as a Guide to Ontology’, Logique et Analyse 45, no. 179-180 (2002): 235-248

This paper, originally presented as an invited contribution to the 2002 Perspectives on Mathematical Practice Conference in Brussels, considers whether evidence from mathematical practice can be used to adjudicate between different accounts of the ontology of mathematics. It argues that, while such evidence cannot decide between Platonism and anti-Platonism as such, there are some versions of each view that do have difficulties accounting for the realities of mathematical practice. In particular, Quinean Platonism would seem to require the abandonment of mathematical theories if they are not required by empirical science, but if one looks for confirming cases of this by considering examples of mathematical theories that have been abandoned, one sees that they have typically been abandoned for reasons other than their lack of empirical utility. Furthermore, theories that, on the Quinean perspective, ought to have been abandoned due to empirical disconformation continue to thrive as part of pure mathematics. (In Mathematics and Reality, however, it is argued that the Quinean position can be adapted to survive such criticism.)

‘What’s Wrong with Indispensability? (Or, The Case for Recreational Mathematics)’, Synthese 131, no. 3 (June 2002): 395-417

This paper considers Mark Colyvan's defense of the indispensability argument against criticisms from Penelope Maddy and Elliott Sober. As part of this defense, Colyvan suggests that Quine's labelling of mathematics that does not receive empirical confirmation in science as 'recreational, without ontological rights' should not be interpreted as dismissing that part of mathematics as unimportant. This is fortunate since, this paper argues, a proper, 'modelling' understanding of the role of mathematics in scientific theories suggests that no mathematics receives empirical confirmation, and therefore that all mathematics should be viewed as recreational. Mark Colyvan responds to this paper in his contribution to the volume Mathematical Knowledge (2007).

'Phenomenology and Mathematical Practice’, Philosophia Mathematica 10, no. 1 (Jan. 2002): 3-25

This paper takes as its starting point the assumption that it is the role of the philosopher of mathematics to understand the nature of mathematics as it is practised, and not some idealized version of the discipline. This conception of the proper role of philosophy in relation to mathematics, which is now fairly commonplace in the contemporary naturalist tradition, owes its beginnings to Imre Lakatos's influential work, Proofs and Refutations (Cambridge: CUP, 1976). Following Lakatos's example, this work presents case studies of the development of mathematical theorems, paying attention to the kinds of reasoning, proof analysis, and conceptual development that led to the proof of the theorems in question. What is distinctive about these case studies, though, is that they result not from study of the historical development of mathematics, but from observation of the real-time development of mathematical thinking, in the context of two research seminars on the classification of C*-algebras, held by the distinguished mathematician George A. Elliott at the Fields Institute for Research in Mathematical Science in Toronto, Canada. Despite its roots in Lakatos's work, this paper suggests that the process of co-development of theorem and proof observed in these case studies speaks against Lakatos's vision of mathematics as proceeding via a process of proofs and refutations. Brendan Larvor discusses this work as one of three examples of 'dialectical' philosophy of mathematics in his paper, 'What is Dialectical Philosophy of Mathematics?' (Philosophia Mathematica 9, no. 2 (2001): 212-229).

Journal Interview

An interview with Leng, discussing her research and view of the central issues in the philosophy of mathematics, appears (in Mandarin translation) in 'Philosophical Trends', a journal of the Chinese Academy of Social Science.

Top

Talks and Conferences

Leng has given invited papers at conferences, symposia, and seminars worldwide including in Belgium, Canada, China, Denmark, Germany, Italy, Mexico, Spain, and the United States. Full details are available on her CV.

Upcoming engagements include an invited paper at a workshop on the topic "What does it mean to do the history, philosophy and sociology of mathematics in the 21st century?" (UCL, July 30th 2010), and an invited commentary on Professor Ian Hacking's René Descartes Lectures at the Tilburg Centre for Logic and Philosophy of Science (Tilburg, October 6th-8th 2010).

Top

Professional Service

Leng has refereed papers for journals including Australasian Journal of Philosophy, British Journal for the Philosophy of Science, Canadian Journal of Philosophy, Erkenntnis, Mind, Philosophia Mathematica, Philosophical Studies, and Synthese, grant applications for the AHRC, and book proposals for Cambridge University Press and Oxford University Press. For full details, see her CV.

She regularly reviews books and articles for Mathematical Reviews.

She represented the British Society for the Philosophy of Science at the General Committee meeting of the Division of Logic, Methodology and Philosophy of Science of the International Union of History and Philosophy of Science in Beijing in 2007.

External examining appointments have included University College, London (Ph.D., 2008), and Central Queensland University, Australia (MA, 2006).

Top

Teaching

Leng's teaching experience at the University of Liverpool includes Symbolic Logic 1 (Propositional Logic); Symbolic Logic 2 (Predicate Logic); Symbolic Logic 3 (Metalogic and Modal Logic); Philosophy of Mathematics; the first year introductory module, Analysing Philosophical Texts; the third year dissertation; and the MA Main Seminar. She has also supervised dissertations at the MA level, and is currently primary Ph.D. Supervisor to Claudio Ternullo (topic: 'Set Theoretic Platonism and Indeterminacy') and secondary Ph.D. supervisor to John Butterworth (topic: 'The Concept of Argument').

Leng has a particular interest in techniques for teaching logic to students with dyslexia and related learning difficulties, as well as those many students who may be informally described as symbol-phobic. In relation to this, she has developed techniques using coloured symbols and an easily visualised parsing system to help students to parse logical formulae. An example of this technique is available on this sample lecture powerpoint.

Top

Outreach

In 2007-08, Leng organized a series of public lectures entitled Thinking about Mathematics and Science, in which scientists, mathematicians, and philosophers were invited to reflect on broadly philosophical questions arising out of the nature of the sciences.

Leng has also given lectures aimed at a general audience, including a lecture entitled 'Philosophical Modesty', explaining the naturalistic turn in recent philosophy of science, which she gave at the University of British Columbia, and a lecture 'Why do Philosophers worry about Mathematical Knowledge?', which she has given at the University of Oxford and the University of Copenhagen. She also contributes a lecture, 'What is Philosophy?', to the University of Liverpool's 'Talks for Schools and Colleges' programme.

Top