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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 (*).
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Added by: Fenner Stanley TanswellAbstract: The Four-Colour 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 Four-Colour 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.
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Added by: Fenner Stanley TanswellAbstract: 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.
Comment (from this Blueprint): Steingart is a sociologist who charts the history and sociology of the development of the extremely large and highly collaborative Classification Theorem. She shows that the proof involved a community deciding on shared values, standards of reliability, expertise, and ways of communicating. For example, the community became tolerant of so-called “local errors” so long as these did not put the main result at risk. Furthermore, Steingart discusses how the proof’s text is distributed across a wide number of places and requires expertise to navigate, leaving the proof in danger of uninvention if the experts retire from mathematics.
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Added by: Fenner Stanley TanswellAbstract: 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.
Comment (from this Blueprint): Tao is a mathematician who has written extensively about mathematics as a discipline. In this piece he considers what counts as “good mathematics”. The opening section that I’ve recommended has a long list of possible meanings of “good mathematics” and considers what this plurality means for mathematics. (The remainder details the history of Szemerédi’s theorem, and argues that good mathematics also involves contributing to a great story of mathematics. However, it gets a bit technical, so only look into it if you’re particularly interested in the details of the case.)
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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.