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Added by: Fenner Stanley TanswellAbstract:
We present a case study of how mathematicians write for mathematicians. We have conducted interviews with two research mathematicians, the talented PhD student Adam and his experienced supervisor Thomas, about a research paper they wrote together. Over the course of 2 years, Adam and Thomas revised Adam’s very detailed first draft. At the beginning of this collaboration, Adam was very knowledgeable about the subject of the paper and had good presentational skills but, as a new PhD student, did not yet have experience writing research papers for mathematicians. Thus, one main purpose of revising the paper was to make it take into account the intended audience. For this reason, the changes made to the initial draft and the authors’ purpose in making them provide a window for viewing how mathematicians write for mathematicians. We examined how their paper attracts the interest of the reader and prepares their proofs for validation by the reader. Among other findings, we found that their paper prepares the proofs for two types of validation that the reader can easily switch between.

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Added by: Fenner Stanley TanswellAbstract: 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.
Comment (from this Blueprint): The orthodox picture of mathematical knowledge is so individualistic that it often leaves out the mathematician themselves. In this piece, Andersen et al. look at what role testimony plays in mathematical knowledge. They thereby emphasise social features of mathematical proofs, and why this can play an important role in deciding which results to trust in the maths literature.

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Added by: Fenner Stanley TanswellAbstract: This chapter is based on the talk that I gave in August 2018 at the ICM in Rio de Janeiro at the panel on The Gender Gap in Mathematical and Natural Sciences from a Historical Perspective. It provides some examples of the challenges and prejudices faced by women mathematicians during last two hundred and fifty years. I make no claim for completeness but hope that the examples will help to shed light on some of the problems many women mathematicians still face today.
Comment (from this Blueprint): BarrowGreen is a historian of mathematics. In this paper she documents some of the challenges that women faced in mathematics over the last 250 years, discussing many famous women mathematicians and the prejudices and injustices they faced.

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, Contributed by: Quentin PharrPublisher’s Note: Plato's SunLike Good is a revolutionary discussion of the Republic's philosopherrulers, their dialectic, and their relation to the form of the good. With detailed arguments Sarah Broadie explains how, if we think of the form of the good as 'interrogative', we can reconceive those central referencepoints of Platonism in downtoearth terms without loss to our sense of Plato's philosophical greatness. The book's main aims are: first, to show how for Plato the form of the good is of practical value in a way that we can understand; secondly, to make sense of the connection he draws between dialectic and the form of the good; and thirdly, to make sense of the relationship between the form of the good and other forms while respecting the contours of the sungood analogy and remaining faithful to the text of the Republic itself.
Comment: This text is an excellent companion text for reading Plato's Republic  especially Books 5 and 6. It provides clear interpretations of the various metaphors and analogies that Plato presents in those books, and it provides one of the most important new interpretations of Plato's conception of philosopherrulers, the Form of the Good, and philosophical dialectic. This text is primarily for those students who are looking to dive into the relevant debates associated with these books in the Republic. Accordingly, it requires some understanding of some of Plato's other dialogues, as well as some understanding of philosophical and mathematical methodologies as conceived by Plato.

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Added by: Giada FratantonioAbstract: The most central metaphysical question about phenomenal consciousness is that of what constitutes phenomenal consciousness, whereas the most central epistemic question about consciousness is that of whether science can eventually provide an explanation of the phenomenon. Many philosophers have argued that science doesn’t have the means to answer the question of what consciousness is (the explanatory gap) but that consciousness nonetheless is fully determined by the physical facts underlying it (no ontological gap). Others have argued that the explanatory gap in the sciences entails an ontological gap. This position is also known as ‘property dualism’. Here I examine a fourth position, according to which there an ontological gap but no explanatory gap.
Comment: In this paper, the author addresses the socalled "explanatory gap". In a nutshell, the "explanatory gap" refers to the existing difficulty of explaining consciousness in physical terms. The author considers Chalmers's argument which aims to show that there is a metaphysical gap. She argues that the existence of a metaphysical gap does not entail the existence of an explanatory gap, thereby failing to prevent scientists from discovering the nature of consciousness. Good as background reading on the topic of consciousness, its nature, and on whether we can explain in physicalist terms. The first half of the paper is particularly useful, as the author provides a survey of different theories regarding the link between consciousness and the neurological system. In this paper, the author addresses the socalled "explanatory gap". In a nutshell, the "explanatory gap" refers to the existing difficulty of explaining consciousness in physical terms. The author considers Chalmers's argument which aims to show that there is a metaphysical gap. She argues that the existence of a metaphysical gap does not entail the existence of an explanatory gap, thereby failing to prevent scientists from discovering the nature of consciousness. Good as background reading on the topic of consciousness, its nature, and on whether we can explain in physicalist terms. The first half of the paper is particularly useful, as the author provides a survey of different theories regarding the link between consciousness and the neurological system.

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Added by: Fenner Stanley TanswellAbstract: This article discusses the role of diagrams in mathematical reasoning in the light of a case study in analysis. In the example presented certain combinatorial expressions were first found by using diagrams. In the published proofs the pictures were replaced by reasoning about permutation groups. This article argues that, even though the diagrams are not present in the published papers, they still play a role in the formulation of the proofs. It is shown that they play a role in concept formation as well as representations of proofs. In addition we note that 'visualization' is used in two different ways. In the first sense 'visualization' denotes our inner mental pictures, which enable us to see that a certain fact holds, whereas in the other sense 'visualization' denotes a diagram or representation of something.
Comment (from this Blueprint): In this paper, Carter discusses a case study from free probability theory in which diagrams were used to inspire definitions and proof strategies. Interestingly, the diagrams were not present in the published results making them dispensable in one sense, but Carter argues that they are essential in the sense that their discovery relied on the visualisation supplied by the diagrams.

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Added by: Fenner Stanley TanswellAbstract:
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.
Comment (from this Blueprint): Cheng is a mathematician working in Category Theory. In this article she complains about traditional philosophy of mathematics that it has no bearing on real mathematics. Instead, she proposes a system of “mathematical morality” about the normative intuitions mathematicians have about how it ought to be.

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Added by: Jamie CollinPublisher's Note: Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings. A Structural Account of Mathematics will be required reading for anyone working in this field. generally put forward, which he maintains have led to serious misunderstandings.
Comment: This book, or chapters from it, would provide useful further reading on nominalism in courses on metaphysics or the philosophy of mathematics. The book does a very good job of summarising and critiquing other positions in the debate. As such individual chapters on (e.g.) mathematical structuralism, Platonism and Field and Balaguer's respective developments of fictionalism could be helpful. The chapter on his own contructibility theory is also a good introduction to that position: shorter and less technical than his earlier (1991) book Constructibility and Mathematical Existence, but longer and more developed than his chapter on Nominalism in the Oxford Handbook of the Philosophy of Mathematics and Logic.

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Added by: Jamie CollinSummary: Introduction to mathematical nominalism, with special attention to Chihara's own development of the position and the objections of John Burgess and Gideon Rosen. Chihara provides an outline of his constructibility theory, which avoids quantification over abstract objects by making use of contructibility quantifiers which instead of making assertions about what exists, make assertions about what sentences can be constructed.
Comment: This chapter would be a good primary or secondary reading in a course on philosophy of mathematics or metaphysics. Chihara is very good at conveying difficult ideas in clear and concise prose. It is worth noting however that, despite the title, this is not really an introduction to nominalism generally but to Chihara's own (important) development of a nominalist philosophy of mathematics / metaphysics.

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Added by: Fenner Stanley TanswellAbstract: The aim of this article is to investigate speciﬁc aspects connected with visualization in the practice of a mathematical subﬁeld: lowdimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in lowdimensional topology, justiﬁcations can be based on sequences of pictures. Three theses will be defended. First, the representations used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in lowdimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two and threedimensional space and motor agency. This imagination allows recognizing the transformations which connect diﬀerent pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a speciﬁc practice: this form of reasoning is subjectmatter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in lowdimensional topology.
Comment (from this Blueprint): De Toffoli and Giardino look at proof practices in lowdimensional topology, and especially a proof by Rolfsen that relies on epistemic actions on a diagrammatic representation. They make the case that the many diagrams are used to trigger our manipulative imagination to make inferential moves which cannot be reduced to formal statements without loss of intuition.

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Added by: Fenner Stanley TanswellAbstract: 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, wellfunctioning 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.
Comment (from this Blueprint): De Toffoli makes a strong case for the importance of mathematical practice in addressing important issues about mathematics. In this paper, she looks at proof and justification, with an emphasis on the fact that mathematicians are fallible. With this in mind, she argues that there are circumstances under which we can have mathematical justification, despite a possibility of being wrong. This paper touches on many cases and questions that will reappear later across the Blueprint, such as collaboration, testimony, computer proofs, and diagrams.

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Added by: Fenner Stanley TanswellAbstract: During the 1970s and 1980s, a team of Automated Theorem Proving researchers at the Argonne National Laboratory near Chicago developed the Automated Reasoning Assistant, or AURA, to assist human users in the search for mathematical proofs. The resulting hybrid humans+AURA system developed the capacity to make novel contributions to pure mathematics by very untraditional means. This essay traces how these unconventional contributions were made and made possible through negotiations between the humans and the AURA at Argonne and the transformation in mathematical intuition they produced. At play in these negotiations were experimental practices, nonhumans, and nonmathematical modes of knowing. This story invites an earnest engagement between historians of mathematics and scholars in the history of science and science studies interested in experimental practice, material culture, and the roles of nonhumans in knowledge making.
Comment (from this Blueprint): Dick traces the history of the AURA automated reasoning assistant in the 1970s and 80s, arguing that the introduction of the computer system led to novel contributions to mathematics by unconventional means. Dick’s emphasis is on the AURA system as changing the material culture of mathematics, and thereby leading to collaboration and even negotiations between the mathematicians and the computer system.

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Added by: Fenner Stanley TanswellPublisher’s Note: 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.
Comment (from this Blueprint): This book by Dutilh Novaes recently won the coveted Lakatos Award. In it, she develops a dialogical account of deduction, where she argues that deduction is implicitly dialogical. Proofs represent dialogues between Prover, who is aiming to establish the theorem, and Skeptic, who is trying to block the theorem. However, the dialogue is both partially adversarial (the two characters have opposite goals) and partially cooperative: the Skeptic’s objections make sure that the Prover must make their proof clear, convincing, and correct. In this chapter, Dutilh Novaes applies her model to mathematical practice, and looks at the way social features of maths embody the ProverSkeptic dialogical model.

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Added by: Fenner Stanley TanswellAbstract: In this paper we first explore how Wittgenstein’s philosophy provides a conceptual tools to discuss the possibility of the simultaneous existence of culturally different mathematical practices. We will argue that Wittgenstein’s later work will be a fruitful framework to serve as a philosophical background to investigate ethnomathematics (Wittgenstein 1973). We will give an overview of Wittgenstein’s later work which is referred to by many researchers in the field of ethnomathematics. The central philosophical investigation concerns Wittgenstein’s shift to abandoning the essentialist concept of language and therefore denying the existence of a universal language. Languages—or ‘language games’ as Wittgenstein calls them—are immersed in a form of life, in a cultural or social formation and are embedded in the totality of communal activities. This gives rise to the idea of rationality as an invention or as a construct that emerges in specific local contexts. In the second part of the paper we introduce, analyse and compare the mathematical aspects of two activities known as string figuremaking and sand drawing, to illustrate Wittgenstein’s ideas. Based on an ethnomathematical comparative analysis, we will argue that there is evidence of invariant and distinguishing features of a mathematical rationality, as expressed in both string figuremaking and sand drawing practices, from one society to another. Finally, we suggest that a philosophicalanthropological approach to mathematical practices may allow us to better understand the interrelations between mathematics and cultures. Philosophical investigations may help the reflection on the possibility of culturally determined ethnomathematics, while an anthropological approach, using ethnographical methods, may afford new materials for the analysis of ethnomathematics and its links to the cultural context. This combined approach will help us to better characterize mathematical practices in both sociological and epistemological terms.
Comment (from this Blueprint): Francois and Vandendriessche here present a later Wittgensteinian approach to “ethnomathematics”: mathematics practiced outside of mainstream Western contexts, often focused on indigenous or tribal groups. They focus on two case studies, stringfigure making and sanddrawing, in different geographic and cultural contexts, looking at how these practices are mathematical.

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Added by: Berta Grimau, Contributed by: Matt ClemensPublisher's Note: What is mathematics about? Does the subjectmatter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but with some experience of formal logic it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics  Platonism, realism, logicism, formalism, constructivism and structuralism  as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy presentday philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upperlevel undergraduate courses and the nonspecialist.
Comment: This book provides an introduction to the philosophy of mathematics. No previous mathematical skills/knowledge required. Suitable for undergraduate courses on philosophy of mathematics.
Comment (from this Blueprint): In this paper, Andersen et al. track the genesis of a maths research paper written in collaboration between a PhD student and his supervisor. They track changes made to sequential drafts and interview the two authors about the motivations for them, and show how the edits are designed to engage the reader in a mathematical narrative on one level, and prepare the paper for different types of validation on another level.