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 approaches to new thinking about the philosophy of 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 inspiring 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. (shrink)

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. (shrink)

The volume includes important criticisms of Bayesian reasoning and also gives an insight into some of the points of disagreement amongst advocates of the ...

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. (shrink)

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.

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

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. (shrink)

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.

In 1908, Henri Poincar? claimed that: ...the mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law, just as experimental facts lead us to the knowledge of a physical law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another. Towards the end of the twentieth century, with many more mathematical (...) facts since discovered, several mathematicians proposed overarching schemes to organise the facts they considered most significant. In this paper, I shall briefly discuss three of these schemes (those of Arnold, Atiyah, and Baez and Dolan), before drawing some philosophical consequences from their attempts. Rather than the kind of claim made by Frege that with the entry of imaginary numbers we reach the ?natural end of the domain of numbers?, we are dealing here with a more open-ended sense of conceptual growth. I shall illustrate this theme by discussing the elaboration of algebraic structures designed to measure symmetry. What emerges is that at any one time mathematicians are operating with a notion very similar to that discussed in the philosophy of science under the heading of ?natural kinds?. We find ?quasi-causal? talk of properties being ?responsible? for a phenomenon, projectability, the transfer of robust mechanisms between domains, and reference to entities not yet fully determined. Debates in philosophy of science prompt further questioning as to the ?naturalness? of these mathematical kinds, whether one should expect them to be dependent on varying human interests, whether there is a distinction between artefactual and real kinds, and whether there is convergence of kinds. I believe that these questions present a wonderful opportunity for a philosophy of mathematics to treat ?real? mathematics, while making powerful points of contact with philosophy of science, and philosophy in general. (shrink)

Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy provides a reasonably gentle introduction to this new logic, thoroughly motivated by intuitive explanations of the need for all of its component parts, and illustrated through innovative applications of the calculus.

Phenomena covered by the term duality occur throughout the history of mathematics in all of its branches, from the duality of polyhedra to Langlands duality. By looking to an “internal epistemology” of duality, we try to understand the gains mathematicians have found in exploiting dual situations. We approach these questions by means of a category theoretic understanding. Following Mac Lane and Lawvere-Rosebrugh, we distinguish between “axiomatic” or “formal” (or Gergonne-type) dualities on the one hand and “functional” or “concrete” (or Poncelet-type) (...) dualities on the other. While the former are often used in the pursuit of a “two theorems by one proof”-strategy, the latter often allow the investigation of “spaces” by studying functions defined on them, which in Grothendieck's terms amounts to the strategy of proving a theorem by working in a dually equivalent framework where the corresponding proof is easier to find. We try to show by some examples that in the first case, dual objects tend to be more ideal (epistemologically more remote) than original ones, while this is not necessarily so in the second case. (shrink)

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 ».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”. (shrink)

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. (shrink)

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

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. (shrink)

Bayesian theory now incorporates a vast body of mathematical, statistical and computational techniques that are widely applied in a panoply of disciplines, from artificial intelligence to zoology. Yet Bayesians rarely agree on the basics, even on the question of what Bayesianism actually is. This book is about the basics e about the opportunities, questions and problems that face Bayesianism today.

Friedman's rich account of the way the mathematical sciences ideally are transformed affords mathematics a more influential role than is common in the philosophy of science. In this paper I assess Friedman's position and argue that we can improve on it by pursuing further the parallels between mathematics and science. We find a richness to the organisation of mathematics similar to that Friedman finds in physics.

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. (shrink)

Weight systems on chord diagrams play a central role in knot theory and Chern-Simons theory; and more recently in stringy quantum gravity. We highlight that the noncommutative algebra of horizontal chord diagrams is canonically a star-algebra, and ask which weight systems are positive with respect to this structure; hence we ask: Which weight systems are quantum states, if horizontal chord diagrams are quantum observables? We observe that the fundamental gl(n)-weight systems on horizontal chord diagrams with N strands may be identified (...) with the Cayley distance kernel at inverse temperature beta=ln(n) on the symmetric group on N elements. In contrast to related kernels like the Mallows kernel, the positivity of the Cayley distance kernel had remained open. We characterize its phases of indefinite, semi-definite and definite positivity, in dependence of the inverse temperature beta; and we prove that the Cayley distance kernel is positive (semi-)definite at beta=ln(n) for all n=1,2,3,... In particular, this proves that all fundamental gl(n)-weight systems are quantum states, and hence so are all their convex combinations. We close with briefly recalling how, under our "Hypothesis H", this result impacts on the identification of bound states of multiple M5-branes. (shrink)

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.