Contributors are listed in the order in which their talks appear in the conference programme

My paper will consist of a series of linked observations on the aesthetic qualities of mathematics, and the relationship between mathematics and the arts. The topics I will discuss include: mathematics as embodying intelligible beauty; mathematics and music; mathematics and art: perspective and symmetry; the timelessness of mathematics; mathematics and formalism; beauty as richness emerging from simplicity; form and content in mathematics.

In this paper I examine in what sense, if any, it might be meaningful to suggest that we can attribute aesthetic qualities to thought-processes and ideas. I will draw on some of my earlier work on the aesthetic value of ideas in conceptual art, and explore what it means to say that non-perceptual art can be experienced aesthetically. I will then explore the further, related, question of whether apparently non-perceptual entities such as mathematical proofs also permit of aesthetic experience in this sense. Underlying my argument is a re-examination of three closely related issues: first, the notion of aesthetic testimony; second, the possible extension of aesthetic concepts in non-artistic contexts; third, the possibility of a non-sensory kind of aesthetic value.

Although largely neglected by both philosophers of science and of aesthetics, the aesthetic value of mathematical theorems and scientific theories has seen some recent attention. It has also, unsurprisingly, seen some level of scepticism about whether the purported aesthetic value of either is genuinely aesthetic or rather some other species of intellectual pleasure (Zangwill 2001, Todd 2008). I argue that this scepticism is unwarranted, and that these species of aesthetic value are genuine, suggesting an account of their value that has parallels with architecture and design, but which nevertheless has distinctive features of its own.

This paper focuses on the diagrammatic ‘visual proofs’ of ancient Greek mathematics. In previous work, I argue that mathematical knowledge is acquired by perceiving the diagrams ‘dynamically’: by entertaining alternative part-whole integrations, while concurrently imagining motions of the parts. Here I consider implications of this account for the nature of the understanding and beauty of visual proofs. I argue that the relevant kind of understanding is mechanistic understanding, and correlatively, the relevant kind of beauty is functional beauty. I motivate this proposal by considering how the experience of beauty arises in the understanding both of machines and of visual proofs.

Any account of the nature and role of beauty in mathematics must address some fundamental questions. (1) What sort of thing is mathematical beauty? Is it an objective property of mathematical entities, the result of a projection of aesthetic value by mathematicians, or some other kind of thing? (2) What ensures that the taste of mathematicians is disposed to find some properties of mathematical entities, rather than others, aesthetically pleasing? (3) Have the aesthetic preferences of mathematicians evolved in the history of mathematics? (4) Is mathematical beauty linked to the validity, adequacy, or effectiveness of mathematical constructs? (5) If so, is this link necessary or contingent, and how do mathematicians detect it? (6) Can mathematical beauty be used as a diagnostic indicator of good mathematics? In this talk, I will discuss these questions and offer some answers. The answers will be based on the model of the “aesthetic induction” that I have developed for the beauty of scientific theories.

I shall be exploring the thesis that the nature of the pleasurable response of mathematicians to their subject should in many cases be seen not as aesthetic, so much as partaking of the child's wonder and delight at encountering instructive and illuminating novelties|and then understanding them. The usual strategy of representing this experience as aesthetic serves us well, since it serves (at least) two purposes: (i) it depicts mathematicians as people who have access not only to knowledge denied to lesser life forms but also to beauty thus denied (thereby countering the idea that we are soulless robots) and it also (ii) spares us the need to admit to being children at heart|an admission that would gravely compromise our status as adults and Respektspersonen.

Although some philosophers have denied that mathematical objects can literally be beautiful, it is a curious fact that they seem to possess some of the “paradigmatic beauties”—aesthetic properties like simplicity, harmony, elegance, clarity, and simplicity. In this paper, I propose a theory of the unity these beautiful properties and consider whether it sheds light on mathematical beauty. I argue that thinking of beauty as the object of a certain type of emotion—one that is sensitive to the appearance of simplicity, elegance, and the like—makes sense of mathematical beauty.

The notion of fit has been used to characterise beauty in a wide number of discplines, from visual arts to architecture to music. However, the notion of fit has not played a central role in contemporary discussions of beauty in mathematics. This paper attempts to make operational the metaphor of beauty as fit through analysis of several mathematical proofs. We distinguish between two different kinds of fit that proofs can possess, intrinsic and extrinsic, and discuss what these types of fit entail. We also suggest, a bit more provocatively, a connection between fit in mathematics and fit in sexual selection.

What makes a mathematical proof beautiful? I argue that many, perhaps even most, of the features that we associate with beauty in such proofs can be subsumed to the ideal of explanatory persuasion, which I take to be the essence of mathematical proofs. I adopt a dialogical perspective to raise a functionalist question: what is the point of mathematical proofs? Why do we bother formulating mathematical proofs at all? According to the dialogical perspective, the defining criteria for what counts as a mathematical proof – and in particular, a good mathematical proof – can be explained in terms of the (presumed) ultimate function of a mathematical proof, namely that of convincing an interlocutor that the conclusion of the proof is true (given the truth of the premises) by showing why that is the case. Thus, a proof seeks not only to force the interlocutor to grant the conclusion if she has granted the premises; it seeks also to reveal something about the mathematical concepts involved to the interlocutor so that she also apprehends what makes the conclusion true – its causes, as it were. On this conception of proof, aesthetic considerations may well play an important role, but they will be subsumed to the ideal of explanatory persuasion.

Symmetry plays an important role in some areas of mathematics and has traditionally been regarded as a factor to visual beauty. In this talk I discuss the possibility that symmetry is a feature that contributes to the beauty of mathematical entities. I explore this question using an example from algebraic graph theory. Comparing two diagrams of isomorphic graphs, I argue that we need to refine the question by distinguishing between perceptual and intellectual beauty and by noting that some mathematical symmetries are revealed to us in diagrams while others are hidden.

Nelson Goodman maintains that exemplification – the relation by which a sample refers to what it samples – is a symptom of the aesthetic. Works of art exemplify properties and relations, by providing telling examples of them. The works thereby afford epistemic access to those properties and relations. I will argue that exemplification is also central to mathematics. A proof exemplifies mathematical properties and relations. It makes them manifest. I will urge that aesthetic properties such as elegance and beauty apply to certain proofs because of the ways those proofs prescind from irrelevancies and reveal the core.

In this paper I defend two theses. First, I argue that the aesthetic vocabulary used in discussing mathematics should be taken literally. Secondly, somewhat more tentatively, I argue that mathematics is sometimes an art. I consider some factors in favour of the thesis that mathematics has genuine aesthetic qualities, and reject a contrary argument, due to Zangwill, that since sensory qualities are necessary for aesthetic ones, aesthetic talk in the context of mathematics must be merely metaphorical. Finally I outline how one might argue that mathematics is art, and consider (but reject) some objections.

Regarding the function and signification of aesthetics in mathematics, Poincaré plays as an intellectual authority figure. A vast majority of the works on this question makes an explicit reference to the Poincaré conception of aesthetics in mathematics; in the same fashion, many mathematicians refer to Poincaré when they need to defend the role of aesthetics in their field. The goal of this intervention will be to show that Poincaré's thesis, even if it is coherent and well argued, does require methodological and metaphysical assumptions which need to be taken into account if we want to adopt it.

The main role for aesthetic judgements in mathematics appears to be in the evaluation/appreciation of theorems and proofs, including comparative judgements about the merits of particular proofs. There is,

**John Bell**:*Reflections on Mathematics and Aesthetics*My paper will consist of a series of linked observations on the aesthetic qualities of mathematics, and the relationship between mathematics and the arts. The topics I will discuss include: mathematics as embodying intelligible beauty; mathematics and music; mathematics and art: perspective and symmetry; the timelessness of mathematics; mathematics and formalism; beauty as richness emerging from simplicity; form and content in mathematics.

**Elisabeth Schellekens**:*On the aesthetic value of reasoning*In this paper I examine in what sense, if any, it might be meaningful to suggest that we can attribute aesthetic qualities to thought-processes and ideas. I will draw on some of my earlier work on the aesthetic value of ideas in conceptual art, and explore what it means to say that non-perceptual art can be experienced aesthetically. I will then explore the further, related, question of whether apparently non-perceptual entities such as mathematical proofs also permit of aesthetic experience in this sense. Underlying my argument is a re-examination of three closely related issues: first, the notion of aesthetic testimony; second, the possible extension of aesthetic concepts in non-artistic contexts; third, the possibility of a non-sensory kind of aesthetic value.

**Toby Bryant**:*In defense of the aesthetic value of mathematical-physical theories*Although largely neglected by both philosophers of science and of aesthetics, the aesthetic value of mathematical theorems and scientific theories has seen some recent attention. It has also, unsurprisingly, seen some level of scepticism about whether the purported aesthetic value of either is genuinely aesthetic or rather some other species of intellectual pleasure (Zangwill 2001, Todd 2008). I argue that this scepticism is unwarranted, and that these species of aesthetic value are genuine, suggesting an account of their value that has parallels with architecture and design, but which nevertheless has distinctive features of its own.

**Logan Fletcher:***Mechanistic understanding, visual proof, and mathematical beauty*This paper focuses on the diagrammatic ‘visual proofs’ of ancient Greek mathematics. In previous work, I argue that mathematical knowledge is acquired by perceiving the diagrams ‘dynamically’: by entertaining alternative part-whole integrations, while concurrently imagining motions of the parts. Here I consider implications of this account for the nature of the understanding and beauty of visual proofs. I argue that the relevant kind of understanding is mechanistic understanding, and correlatively, the relevant kind of beauty is functional beauty. I motivate this proposal by considering how the experience of beauty arises in the understanding both of machines and of visual proofs.

**James McAllister**:*Six Questions for Any Account of Mathematical Beauty, with Some Answers*Any account of the nature and role of beauty in mathematics must address some fundamental questions. (1) What sort of thing is mathematical beauty? Is it an objective property of mathematical entities, the result of a projection of aesthetic value by mathematicians, or some other kind of thing? (2) What ensures that the taste of mathematicians is disposed to find some properties of mathematical entities, rather than others, aesthetically pleasing? (3) Have the aesthetic preferences of mathematicians evolved in the history of mathematics? (4) Is mathematical beauty linked to the validity, adequacy, or effectiveness of mathematical constructs? (5) If so, is this link necessary or contingent, and how do mathematicians detect it? (6) Can mathematical beauty be used as a diagnostic indicator of good mathematics? In this talk, I will discuss these questions and offer some answers. The answers will be based on the model of the “aesthetic induction” that I have developed for the beauty of scientific theories.

**Kenneth Manders**:*On Geometricality***Thomas Forster**:*Beauty and Wonder*I shall be exploring the thesis that the nature of the pleasurable response of mathematicians to their subject should in many cases be seen not as aesthetic, so much as partaking of the child's wonder and delight at encountering instructive and illuminating novelties|and then understanding them. The usual strategy of representing this experience as aesthetic serves us well, since it serves (at least) two purposes: (i) it depicts mathematicians as people who have access not only to knowledge denied to lesser life forms but also to beauty thus denied (thereby countering the idea that we are soulless robots) and it also (ii) spares us the need to admit to being children at heart|an admission that would gravely compromise our status as adults and Respektspersonen.

**Nicholas Riggle**:*Aesthetic Properties and Mathematical Beauty*Although some philosophers have denied that mathematical objects can literally be beautiful, it is a curious fact that they seem to possess some of the “paradigmatic beauties”—aesthetic properties like simplicity, harmony, elegance, clarity, and simplicity. In this paper, I propose a theory of the unity these beautiful properties and consider whether it sheds light on mathematical beauty. I argue that thinking of beauty as the object of a certain type of emotion—one that is sensitive to the appearance of simplicity, elegance, and the like—makes sense of mathematical beauty.

**Manya Raman Sundström**:*Beauty as fit in mathematics*The notion of fit has been used to characterise beauty in a wide number of discplines, from visual arts to architecture to music. However, the notion of fit has not played a central role in contemporary discussions of beauty in mathematics. This paper attempts to make operational the metaphor of beauty as fit through analysis of several mathematical proofs. We distinguish between two different kinds of fit that proofs can possess, intrinsic and extrinsic, and discuss what these types of fit entail. We also suggest, a bit more provocatively, a connection between fit in mathematics and fit in sexual selection.

**Catarina Dutilh-Novaes**:*Beauty, explanation, and persuasion in mathematical proofs*What makes a mathematical proof beautiful? I argue that many, perhaps even most, of the features that we associate with beauty in such proofs can be subsumed to the ideal of explanatory persuasion, which I take to be the essence of mathematical proofs. I adopt a dialogical perspective to raise a functionalist question: what is the point of mathematical proofs? Why do we bother formulating mathematical proofs at all? According to the dialogical perspective, the defining criteria for what counts as a mathematical proof – and in particular, a good mathematical proof – can be explained in terms of the (presumed) ultimate function of a mathematical proof, namely that of convincing an interlocutor that the conclusion of the proof is true (given the truth of the premises) by showing why that is the case. Thus, a proof seeks not only to force the interlocutor to grant the conclusion if she has granted the premises; it seeks also to reveal something about the mathematical concepts involved to the interlocutor so that she also apprehends what makes the conclusion true – its causes, as it were. On this conception of proof, aesthetic considerations may well play an important role, but they will be subsumed to the ideal of explanatory persuasion.

**Irina Starikova**:*Symmetry and beauty in mathematics*Symmetry plays an important role in some areas of mathematics and has traditionally been regarded as a factor to visual beauty. In this talk I discuss the possibility that symmetry is a feature that contributes to the beauty of mathematical entities. I explore this question using an example from algebraic graph theory. Comparing two diagrams of isomorphic graphs, I argue that we need to refine the question by distinguishing between perceptual and intellectual beauty and by noting that some mathematical symmetries are revealed to us in diagrams while others are hidden.

**Catherine Elgin**:*Elegance, Exemplification, and Mathematics*Nelson Goodman maintains that exemplification – the relation by which a sample refers to what it samples – is a symptom of the aesthetic. Works of art exemplify properties and relations, by providing telling examples of them. The works thereby afford epistemic access to those properties and relations. I will argue that exemplification is also central to mathematics. A proof exemplifies mathematical properties and relations. It makes them manifest. I will urge that aesthetic properties such as elegance and beauty apply to certain proofs because of the ways those proofs prescind from irrelevancies and reveal the core.

**Adam Rieger**:*Mathematics and Beauty*In this paper I defend two theses. First, I argue that the aesthetic vocabulary used in discussing mathematics should be taken literally. Secondly, somewhat more tentatively, I argue that mathematics is sometimes an art. I consider some factors in favour of the thesis that mathematics has genuine aesthetic qualities, and reject a contrary argument, due to Zangwill, that since sensory qualities are necessary for aesthetic ones, aesthetic talk in the context of mathematics must be merely metaphorical. Finally I outline how one might argue that mathematics is art, and consider (but reject) some objections.

**Caroline Jullien**:*On the aesthetics in Poincaré's philosophy of mathematics*Regarding the function and signification of aesthetics in mathematics, Poincaré plays as an intellectual authority figure. A vast majority of the works on this question makes an explicit reference to the Poincaré conception of aesthetics in mathematics; in the same fashion, many mathematicians refer to Poincaré when they need to defend the role of aesthetics in their field. The goal of this intervention will be to show that Poincaré's thesis, even if it is coherent and well argued, does require methodological and metaphysical assumptions which need to be taken into account if we want to adopt it.

**Cain Todd**:*Fitting Feelings: the possibility of aesthetic judgements in mathematics*The main role for aesthetic judgements in mathematics appears to be in the evaluation/appreciation of theorems and proofs, including comparative judgements about the merits of particular proofs. There is,

*prima facie*, a number of obstacles to interpreting literally the various aesthetic claims made in the context of mathematics. The one I will focus on in the first part of the talk concerns the role of reason-giving in the justification of aesthetic judgements. There is reason to think that aesthetic judgements in mathematics, unlike in other paradigmatic aesthetic contexts, can be reduced to non-aesthetic judgements, concerning for example judgements about the properties of simplicity, symmetry, or significance of a theorem of proof. This would also help to explain the relatively limited aesthetic vocabulary used in mathematics. Rather than engaging in what I take to be a hopeless conceptual-semantic dispute about whether such properties are truly aesthetic, the second part of my talk will be concerned with showing*how*theaesthetic judgements operative in mathematics are expressions of aesthetic feelings that serve a genuine cognitive and epistemic function. I will suggest a naturalistic account of these aesthetic-epistemic feelings. More specifically, I will suggest that they are all determinate variations of what I will call the ‘feeling of fittingness’ which, I will suggest, is an experiential manifestation of certain sub-personal processes of representing and assessing a) explanatory coherence, b) simplicity, and c) salience, which together constitute what will be broadly referred to as ‘cognitive consonance’.