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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 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: 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 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): MüllerHill 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 contextsensitivity of a number of factors.

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Added by: Fenner Stanley TanswellAbstract: The FourColour Theorem (4CT) proof, presented to the mathematical community in a pair of papers by Appel and Haken in the late 1970's, provoked a series of philosophical debates. Many conceptual points of these disputes still require some elucidation. After a brief presentation of the main ideas of Appel and Haken’s procedure for the proof and a reconstruction of Thomas Tymoczko’s argument for the novelty of 4CT’s proof, we shall formulate some questions regarding the connections between the points raised by Tymoczko and some Wittgensteinian topics in the philosophy of mathematics such as the importance of the surveyability as a criterion for distinguishing mathematical proofs from empirical experiments. Our aim is to show that the “characteristic Wittgensteinian invention” (Mühlhölzer 2006) – the strong distinction between proofs and experiments – can shed some light in the conceptual confusions surrounding the FourColour Theorem.
Comment (from this Blueprint): Secco and Pereira discuss the famous proof of the Four Colour Theorem, which involved the essential use of a computer to check a huge number of combinations. They look at whether this constitutes a real proof or whether it is more akin to a mathematical experiment, a distinction that they draw from Wittgenstein.
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.