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Added by: Fenner Stanley TanswellAbstract: In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice. In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education.
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Added by: Jamie CollinAbstract: The goal of the research programme I describe in this article is a realist epistemology for arithmetic which respects arithmetic's special epistemic status (the status usually described as a prioricity) yet accommodates naturalistic concerns by remaining funda- mentally empiricist. I argue that the central claims which would allow us to develop such an epistemology are (i) that arithmetical truths are known through an examination of our arithmetical concepts; (ii) that (at least our basic) arithmetical concepts are accurate mental representations of elements of the arithmetical structure of the inde- pendent world; (iii) that (ii) obtains in virtue of the normal functioning of our sensory apparatus. The first of these claims protects arithmetic's special epistemic status relative, for example, to the laws of physics, the second preserves the independence of arithmetical truth, and the third ensures that we remain empiricists.
Comment: Useful as a primary or secondary reading in an advanced undergraduate course epistemology (in a section on a priori knowledge) or an advanced undergraduate course on philosophy of mathematics. This is not an easy paper, but it is clear. It is also useful within a teaching context, as it provides a summary of the influential neo-Fregean approach to mathematical knowledge.
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Added by: Jamie CollinSummary: Surveys the opposition between views of mathematics which take mathematics to represent a independent mathematical reality and views which take mathematical axioms to define or circumscribe their subject matter; and defends the latter view against influential objections.
Comment: A very clear and useful survey text for advanced undergraduate or postgraduate courses on metaphysics or philosophy of mathematics.
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Added by: Jamie CollinPublisher's Note: Mary Leng offers a defense of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focussed on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories. Leng, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized.
Comment: This book presents the most developed account of mathematical fictionalism. The book, or chapters from it, would provide useful further reading in advanced undergraduate or postgraduate courses on metaphysics or philosophy of mathematics.
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Added by: Sara PeppeAbstract: This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.
Comment: Previous knowledge both on Platonism in philosophy of mathematics and scientific realism is needed. Essential paper for advanced courses of philosophy of science.
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Added by: Jamie CollinSummary: Defends an account of mathematical knowledge in which mathematical knowledge is a kind of modal knowledge. Leng argues that nominalists should take mathematical knowledge to consist in knowledge of the consistency of mathematical axiomatic systems, and knowledge of what necessarily follows from those axioms. She defends this view against objections that modal knowledge requires knowledge of abstract objects, and argues that we should understand possibility and necessity in a primative way.
Comment: This would be useful in an advanced undergraduate course on metaphysics, epistemology or philosophy of logic and mathematics. This is not an easy paper, but Leng does an excellent job of making clear some difficult ideas. The view defended is an important one in both philosophy of logic and philosophy of mathematics. Any reasonably comprehensive treatment of nominalism should include this paper.
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Added by: Jamie CollinPublisher's Note: Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view - realism - is assessed and finally rejected in favour of another - naturalism - which attends less to metaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in the assessment of candidates for axiomatic status in set theory. Maddy's clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.
Comment: Good further reading in advanced undergraduate or postgraduate courses on metaphysics, naturalism or philosophy of mathematics. Sections from the book - for instance, the chapters in Part II on indispensability considerations in scientific and mathematical practice - could be profitably read on their own. These sections may also be of interest in philosophy of science courses, as they provide a careful analysis of scientific practice (as it relates to what scientists take themselves to be ontologically committed to).
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Added by: Jamie CollinSummary: A clear introduction to mathematical naturalism and its Quinean roots; developing and defending Maddy's own naturalist philosophy of mathematics. Maddy claims that the Quinian ignores some nuances of scientific practice that have a bearing on what the naturalist should take to be the real scientific standards of evidence. Historical studies show that scientists sometimes do not take themselves to be committed to entities that are indispensably quantified over in their best scientific theories, hence the Quinian position that naturalism dictates that we are committed to entities that are indispensably quantified over in our best scientific theories is incorrect.
Comment: Good primary reading in advanced undergraduate or postgraduate courses on metaphysics, naturalism or philosophy of mathematics. This would serve well both as a clear and fairly concise introduction to Quinean naturalism and to the indispensability argument in the philosophy of mathematics.
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Added by: Franci Mangraviti and Viviane FairbankAbstract: Val Plumwood charged classical logic not only with the invalidity of some of its laws, but also with the support of systemic oppression through naturalization of the logical structure of dualisms. In this paper I show that the latter charge - unlike the former - can be carried over to classical mathematics, and I propose a new conception of inconsistent mathematics - queer incomaths - as a liberatory activity meant to undermine said naturalization.
Comment: available in this Blueprint
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Added by: Fenner Stanley TanswellAbstract: 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.
Comment (from this Blueprint): In this paper, Martin and Pease look at how mathematics happens online, emphasising how this embodies the picture of mathematics given by Polya and Lakatos, two central figures in philosophy of mathematical practice. They look at multiple venues of online mathematics, including the polymath projects of collaborative problem-solving, and mathoverflow, which is a question-and-answer forum. By looking at the discussions that take place when people are doing maths online, they argue that you can get rich new kinds of data about the processes of mathematical discovery and understanding. They discuss how online mathematics can become a “social machine”, and how this can open up new ways of doing mathematics.
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Added by: Fenner Stanley TanswellAbstract: The low visibility and specialised languages of mathematical work pose challenges for the ethnographic study of communication in mathematics, but observation-based study can offer a real-world grounding to questions about the nature of its methods. This paper uses theoretical ideas from linguistic pragmatics to examine how mutual understandings of diagrams are achieved in the course of conference presentations. Presenters use shared knowledge to train others to interpret diagrams in the ways favoured by the community of experts, directing an audience’s attention so as to develop a shared understanding of a diagram’s features and possible manipulations. In this way, expectations about the intentions of others and appeals to knowledge about the manipulation of objects play a part in the development and communication of concepts in mathematical discourse.
Comment (from this Blueprint): McCallum is an ethnographer and artist, who in this piece explores the way in which mathematicians use diagrams in conference presentations, especially in knot theory. She emphasises that there are a large number of ways that diagrams can facilitate communication and understanding. The diagrams are dynamic in many way, and she shows how the way in which a speaker interacts with the diagram (through drawing, erasing, labelling, positioning, emphasising etc.) is part of explaining the mathematics it represents.
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Added by: Fenner Stanley TanswellAbstract: The story of a team of female African-American mathematicians who served a vital role in NASA during the early years of the U.S. space program.
Comment (from this Blueprint): This film depicts a historical biopic of African American female mathematicians working at NASA in the 1960s, focusing on the story of Katherine Johnson. In it, the plot depicts struggles with racism and sexism, as well as the impacts of the move from human calculation to the use of computers.
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Added by: Fenner Stanley TanswellAbstract: Despite increasing rates of women researching in math-intensive fields, publications by female authors remain underrepresented. By analyzing millions of records from the dedicated bibliographic databases zbMATH, arXiv, and ADS, we unveil the chronological evolution of authorships by women in mathematics, physics, and astronomy. We observe a pronounced shortage of female authors in top-ranked journals, with quasistagnant figures in various distinguished periodicals in the first two disciplines and a significantly more equitable situation in the latter. Additionally, we provide an interactive open-access web interface to further examine the data. To address whether female scholars submit fewer articles for publication to relevant journals or whether they are consciously or unconsciously disadvantaged by the peer review system, we also study authors’ perceptions of their submission practices and analyze around 10,000 responses, collected as part of a recent global survey of scientists. Our analysis indicates that men and women perceive their submission practices to be similar, with no evidence that a significantly lower number of submissions by women is responsible for their underrepresentation in top-ranked journals. According to the self-reported responses, a larger number of articles submitted to prestigious venues correlates rather with aspects associated with pronounced research activity, a well-established network, and academic seniority.
Comment (from this Blueprint): Mihaljević and Santamaría here use large-scale quantitative research methods to investigate the gender gap in contemporary mathematics. I’ve recommended reading the introduction and conclusion in order to see what they were doing and what they found out, but the rest of the paper is worth looking at if you want more detailed methods and results.
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Added by: Fenner Stanley TanswellAbstract: Prominent mathematician William Thurston was praised by other mathematicians for his intellectual generosity. But what does it mean to say Thurston was intellectually generous? And is being intellectually generous beneficial? To answer these questions I turn to virtue epistemology and, in particular, Roberts and Wood's (2007) analysis of intellectual generosity. By appealing to Thurston's own writings and interviewing mathematicians who knew and worked with him, I argue that Roberts and Wood's analysis nicely captures the sense in which he was intellectually generous. I then argue that intellectual generosity is beneficial because it counteracts negative effects of the reward structure of mathematics that can stymie mathematical progress.
Comment (from this Blueprint): In this paper, Morris looks at ascriptions of intellectual generosity in mathematics, focusing on the mathematician William Thurston. She looks at how generosity should be characterised, and argues that it is beneficial in counteract some of the negative effects of the reward structure of mathematics.
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Added by: Fenner Stanley TanswellAbstract:
We investigate the truth conditions of knowledge ascriptions for the case of mathematical knowledge. The availability of a formalizable mathematical proof appears to be a natural criterion:
(*) X knows that p is true iff X has available a formalizable proof of p.
Yet, formalizability plays no major role in actual mathematical practice. We present results of an empirical study, which suggest that certain readings of (*) are not necessarily employed by mathematicians when ascribing knowledge. Further, we argue that the concept of mathematical knowledge underlying the actual use of “to know” in mathematical practice is compatible with certain philosophical intuitions, but seems to differ from philosophical knowledge conceptions underlying (*).
Comment (from this Blueprint): Müller-Hill is interested in the question of when mathematicians have mathematical knowledge and to what extent it relies on the formalisability of proofs. In this paper, she undertakes an empirical investigation of mathematicians’ views of when mathematicians know a theorem is true. Amazingly, while they say that they believe proofs have an exact definition and that the standards of knowledge are invariant, when presented with various toy scenarios, their judgements seem to suggest systematic context-sensitivity of a number of factors.
Comment (from this Blueprint): While this paper by Hamami & Morris is not a necessary reading, it provides a fairly broad overview of the practical turn in mathematics. Since it was aimed at mathematics educators, it is a very accessible piece, and provides useful directions to further reading beyond what is included in this blueprint.