This comprehensive account of the concept and practices of deduction is the first to bring together perspectives from philosophy, history, psychology and cognitive science, and mathematical practice. Catarina Dutilh Novaes draws on all of these perspectives to argue for an overarching conceptualization of deduction as a dialogical practice: deduction has dialogical roots, and these dialogical roots are still largely present both in theories and in practices of deduction. Dutilh Novaes’ account also highlights the deeply human and in fact social nature of deduction, as embedded in actual human practices; as such, it presents a highly innovative account of deduction. The book will be of interest to a wide range of readers, from advanced students to senior scholars, and from philosophers to mathematicians and cognitive scientists.
A Group Theory of Group Theory: Collaborative Mathematics and the ‘Uninvention’ of a 1000-page Proof
Over a period of more than 30 years, more than 100 mathematicians worked on a project to classify mathematical objects known as finite simple groups. The Classification, when officially declared completed in 1981, ranged between 300 and 500 articles and ran somewhere between 5,000 and 10,000 journal pages. Mathematicians have hailed the project as one of the greatest mathematical achievements of the 20th century, and it surpasses, both in scale and scope, any other mathematical proof of the 20th century. The history of the Classification points to the importance of face-to-face interaction and close teaching relationships in the production and transformation of theoretical knowledge. The techniques and methods that governed much of the work in finite simple group theory circulated via personal, often informal, communication, rather than in published proofs. Consequently, the printed proofs that would constitute the Classification Theorem functioned as a sort of shorthand for and formalization of proofs that had already been established during personal interactions among mathematicians. The proof of the Classification was at once both a material artifact and a crystallization of one community’s shared practices, values, histories, and expertise. However, beginning in the 1980s, the original proof of the Classification faced the threat of ‘uninvention’. The papers that constituted it could still be found scattered throughout the mathematical literature, but no one other than the dwindling community of group theorists would know how to find them or how to piece them together. Faced with this problem, finite group theorists resolved to produce a ‘second-generation proof’ to streamline and centralize the Classification. This project highlights that the proof and the community of finite simple groups theorists who produced it were co-constitutive–one formed and reformed by the other.
Mathematical Practice, Crowdsourcing, and Social Machines
The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend.
The Study of Mathematical Practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice – for example the community question-answering system mathoverflow contains around 40,000 mathematical conversations, and polymath collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of “soft” aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal.
Crowdsourced mathematical activity is an example of a “social machine”, a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.
The Role of Testimony in Mathematics
Mathematicians appear to have quite high standards for when they will rely on testimony. Many mathematicians require that a number of experts testify that they have checked the proof of a result p before they will rely on p in their own proofs without checking the proof of p. We examine why this is. We argue that for each expert who testifies that she has checked the proof of p and found no errors, the likelihood that the proof contains no substantial errors increases because different experts will validate the proof in different ways depending on their background knowledge and individual preferences. If this is correct, there is much to be gained for a mathematician from requiring that a number of experts have checked the proof of p before she will rely on p in her own proofs without checking the proof of p. In this way a mathematician can protect her own work and the work of others from errors. Our argument thus provides an explanation for mathematicians’ attitude towards relying on testimony.
Groundwork for a Fallibilist Account of Mathematics
According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then propose a fallibilist account of mathematical justification. I show that the main function of mathematical justification is to guarantee that the mathematical community can correct the errors that inevitably arise from our fallible practices.
What is good mathematics?
Some personal thoughts and opinions on what “good quality mathematics” is and whether one should try to define this term rigorously. As a case study, the story of Szemer´edi’s theorem is presented.
Mathematics, Morally
A source of tension between Philosophers of Mathematics and Mathematicians is the fact that each group feels ignored by the other; daily mathematical practice seems barely affected by the questions the Philosophers are considering. In this talk I will describe an issue that does have an impact on mathematical practice, and a philosophical stance on mathematics that is detectable in the work of practising mathematicians. No doubt controversially, I will call this issue ‘morality’, but the term is not of my coining: there are mathematicians across the world who use the word ‘morally’ to great effect in private, and I propose that there should be a public theory of what they mean by this. The issue arises because proofs, despite being revered as the backbone of mathematical truth, often contribute very little to a mathematician’s understanding. ‘Moral’ considerations, however, contribute a great deal. I will first describe what these ‘moral’ considerations might be, and why mathematicians have appropriated the word ‘morality’ for this notion. However, not all mathematicians are concerned with such notions, and I will give a characterisation of ‘moralist’ mathematics and ‘moralist’ mathematicians, and discuss the development of ‘morality’ in individuals and in mathematics as a whole. Finally, I will propose a theory for standardising or universalising a system of mathematical morality, and discuss how this might help in the development of good mathematics.
Cosmopolitan peace
This chapter explores why, from a cosmopolitan point of view, we should remember some wars, and furthermore how we should remember them. It contrasts itself with remembering war for partial and/or nationalist purposes, and also deals with the particularity problem, on why people of certain countries should remember their past wars.
Commemorating Public Figures–In Favour of a Fictionalist Position
In this article, I discuss the commemoration of public figures such as Nelson Mandela and Yitzhak Rabin. In many cases, our commemoration of such figures is based on the admiration we feel for them. However, closer inspection reveals that most (if not all) of those we currently honour do not qualify as fitting objects of admiration. Yet, we may still have the strong intuition that we ought to continue commemorating them in this way. I highlight two problems that arise here: the problem that the expressed admiration does not seem appropriate with respect to the object and the problem that continued commemorative practices lead to rationality issues. In response to these issues, I suggest taking a fictionalist position with respect to commemoration. This crucially involves sharply distinguishing between commemorative and other discourses, as well as understanding the objects of our commemorative practices as fictional objects.
Objectionable Commemorations, Historical Value, and Repudiatory Honouring
Many have argued that certain statues or monuments are objectionable, and thus ought to be removed. Even if their arguments are compelling, a major obstacle is the apparent historical value of those commemorations. Preservation in some form seems to be the best way to respect the value of commemorations as connections to the past or opportunities to learn important historical lessons. Against this, I argue that we have exaggerated the historical value of objectionable commemorations. Sometimes commemorations connect to biased or distorted versions of history, if not mere myths. We can also learn historical lessons through what I call repudiatory honouring: the honouring of certain victims or resistors that can only make sense if the oppressor(s) or target(s) of resistance are deemed unjust, where no part of the original objectionable commemorations is preserved. This type of commemorative practice can even help to overcome some of the obstacles objectionable commemorations pose against properly connecting to the past.