Andersen, Line Edslev, Johansen, Mikkel Willum, Kragh Sørensen, Henrik. Mathematicians Writing for Mathematicians
2021, Synthese, 198(26): 6233-6250.
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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.
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.2021, Synthese, 199(1): 859-870.-
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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.Barrow-Green, June. Historical Context of the Gender Gap in Mathematics2019, in World Women in Mathematics 2018: Proceedings of the First World Meeting for Women in Mathematics, Carolina Araujo et al. (eds.). Springer, Cham.-
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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): Barrow-Green 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.Broadie, Sarah. Plato’s Sun-Like Good: Dialectic in the Republic2021, Cambridge University Press-
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, Contributed by: Quentin PharrPublisher’s Note:
Plato's Sun-Like Good is a revolutionary discussion of the Republic's philosopher-rulers, 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 re-conceive those central reference-points of Platonism in down-to-earth 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 sun-good 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 philosopher-rulers, 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.Brogaard, Berit. The Status of Consciousness in Nature2015, In Steven Miller (ed.), The Constitution of Phenomenal Consciousness: Toward a science and theory. John Benjamins Publishing Company.-
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Added by: Giada Fratantonio
Abstract: 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 so-called "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 so-called "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.Carter, Jessica. Diagrams and Proofs in Analysis2010, International Studies in the Philosophy of Science, 24(1): 1-14.-
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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.Cheng, Eugenia. Mathematics, Morally2004, Cambridge University Society for the Philosophy of Mathematics.-
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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.Chihara, Charles. A Structural Account of Mathematics2004, Oxford: Oxford University Press.-
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Added by: Jamie Collin
Publisher'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.Chihara, Charles. Nominalism2005, in The Oxford Hanbook of Philosophy of Mathematics and Logic, ed. S. Shapiro. New York: Oxford University Press.-
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Added by: Jamie Collin
Summary: 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.De Toffoli, Silvia. Groundwork for a Fallibilist Account of Mathematics2021, The Philosophical Quarterly, 71(4).-
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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, 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.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.Can’t find it?Contribute the texts you think should be here and we’ll add them soon!
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