Oxford jan 2017
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Manders' analysis of Euclid's proofs in the first four books provides a model for understanding how proofs work in practice, as actual mathematical practice diverges from the ideal of a formal proof being an unbroken chain of logical inferences from explicit axioms. The plausibility that inspecting and manipulating mental representations can reliably convey mathematical information depends on drawings sharing the structure of mathematical objects in an inexact way and being systematically related to formal notation and inference through labeling.
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Incompleteness Theorems: Logical Necessity of Inconsistency
These are slides for video of "Wittgenstein versus Gödel on the Foundations of Logic" Stanford Logic Colloquium on April 23, 2010.
Video can be viewed at:
http://wh-stream.stanford.edu/MediaX/CarlHewittEdit.mp4
The logic(s) of informal proofs (tyumen, western siberia 2019)
The document discusses the difference between formal mathematical proofs and actual mathematical practice. It notes that while a formal proof is an unbroken chain of logical inferences from axioms, in practice mathematicians use informal proofs that are sketches which could be routinely translated into formal proofs. It argues that many mathematical proofs involve actions on mathematical objects other than propositions, and that a full understanding of mathematical reasoning requires acknowledging the role of content and representations in inferences.
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Humans and many other intelligent systems (have to) learn from experience, build
models of the environment from the acquired knowledge, and use these models for
prediction. In philosophy this is called inductive inference, in statistics it is called
estimation and prediction, and in computer science it is addressed by machine
learning.
I will first review unsuccessful attempts and unsuitable approaches towards a
general theory of induction, including Popper’s falsificationism and denial of
confirmation, frequentist statistics and much of statistical learning theory, subjective
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and deductive approaches. I will also debunk some other misguided views, such as
the no-free-lunch myth and pluralism.
I will then turn to Solomonoff’s formal, general, complete, and essentially unique
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and based on the philosophical and technical ideas of Ockham, Epicurus, Bayes,
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approaches, and essentially constitutes a conceptual solution to the induction
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1) Understanding informal proofs is important because most mathematical proofs are informal.
2) Informal proofs rely on both logical form and content.
3) Properly understanding informal proofs requires seeing logic as the study of inferential actions, where content plays a role.
4) This conception accommodates proofs involving actions on objects other than propositions.
5) It explains why mathematics relies on external representations, since representations can be manipulated.
The document argues this view connects logical questions about rigor to the cultural study of mathematical practices.
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This document discusses informal proofs in mathematics. It argues that:
1) A mathematical proof is more than just a recipe for a formal derivation, as it establishes mathematical conclusions through rigorous but informal arguments.
2) Logic should be understood as the general study of inferential actions, as informal proofs involve actions on mathematical objects beyond just propositions.
3) Representations in mathematics are valuable not just for presenting information but for being manipulated, as inferences can be drawn through performing actions on diagrams, graphs, and other representations.
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1. The document discusses the need for a positive account of informal proof in mathematics, as most mathematical proofs are informal. It argues against the view that informal proofs are recipes for formal derivations.
2. The document proposes that logic should be understood more broadly as the general study of inferential actions, as informal proofs often involve actions on mathematical objects beyond propositions. Examples of such actions include diagram manipulation in Euclidean geometry.
3. The document reviews work that may support this broader view of logic in informal proofs, such as studies of reasoning with diagrams in knot theory and using Cayley graphs to prove group theory results.
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These are slides for video of "Wittgenstein versus Gödel on the Foundations of Logic" Stanford Logic Colloquium on April 23, 2010.
Video can be viewed at:
http://wh-stream.stanford.edu/MediaX/CarlHewittEdit.mp4
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and deductive approaches. I will also debunk some other misguided views, such as
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1) A mathematical proof is more than just a recipe for a formal derivation, as it establishes mathematical conclusions through rigorous but informal arguments.
2) Logic should be understood as the general study of inferential actions, as informal proofs involve actions on mathematical objects beyond just propositions.
3) Representations in mathematics are valuable not just for presenting information but for being manipulated, as inferences can be drawn through performing actions on diagrams, graphs, and other representations.
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2. The document proposes that logic should be understood more broadly as the general study of inferential actions, as informal proofs often involve actions on mathematical objects beyond propositions. Examples of such actions include diagram manipulation in Euclidean geometry.
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2) Logics of confirmation that aim to assign probabilities to hypotheses given evidence still rely on initial probability assignments without justification and do not address how these relate to the real world.
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https://yhamami.github.io/mathed-pmp-workshop/program/
Anselm in Context .pptx
Anselm lived from 1033-1109 and was a monk at Bec Abbey and later Archbishop of Canterbury. He wrote philosophical and theological works to reconcile religious teachings with logic and reason. His ontological argument in the Proslogion was intended to provide a unified rational account of God's existence and attributes through a single definition. While the argument has received criticisms, understanding Anselm's broader context and intent helps in properly interpreting his argument rather than viewing it in isolation through a modern logical lens.
Unlikely bedfellows or unholy alliance.pptx
Lakatos demonstrates the possibility of philosophical hybridism by combining influences from Hegel, Pólya and Popper in his philosophy of mathematics, which seems unlikely but points to the eclectic nature of the philosophy of mathematical practice. The talk will discuss the tensions and synergies in Lakatos’s thought from his intellectual influences and migration, and illustrate the benefits of philosophical hybridism, including a robustness criterion. It will also discuss prospects for collaboration among various projects under the philosophy of mathematical practices.
Culture Lisbon 30 June 2022.pptx
The document discusses reasons why one may not take a cultural approach to mathematics. It outlines three main reasons: metaphysics, essentialism, and relativism. Under metaphysics, it discusses how naturalism cannot account for concepts essential to science like propositions, validity of arguments, and academic status. Under essentialism, it raises questions around what constitutes a culture and whether the concept of discrete cultures is problematic. And under relativism, it discusses how mathematics is associated with enlightenment values of rationalism that a cultural approach may contradict.
Palimpsest and Bricolage.pptx
This document discusses concepts in mathematics and philosophy that have multiple related meanings developed through history and practice.
It argues that concepts like "proof", "number", and "punishment" present not just one stable definition but a "synthesis of meanings" that has crystallized over time from diverse uses and cannot be fully analyzed by reducing to basic elements. Drawing on Nietzsche and Weber, it views the development of concepts not as revealing an essence but as a process of rationalization acting on a complex unity arising from varied practices.
The document also notes mathematics involves a "colorful mix of proof techniques" and that formalization programs are guided by this diverse landscape of practices rather than aiming for strict definitions. It
We are all in this together
This document summarizes two projects funded by the subject center at the University of Hertfordshire on friendship and collaboration among philosophy students. The first project found that students often see university as a solitary experience due to limited class time and warnings about plagiarism. The second project found that peer support schemes emphasize safety over intellectual risk-taking, and students do not see peers as a learning resource. It recommends that institutions explain the social nature of research, model collaboration, give permission to collaborate, and train students in collaborative skills to address these issues.
Religious language
This document discusses different philosophical approaches to analyzing religious language:
- The Vienna Circle proposed the verification principle, which held that sentences only have meaning if they can be empirically verified, raising issues for religious and ethical claims. Karl Popper proposed the falsification criterion as an alternative.
- Wittgenstein argued that language cannot be reduced to propositions and that different "language games" should be analyzed separately rather than by external criteria. However, religion makes broader truth claims than isolated practices like sports fandom.
- Religious language raises questions about whether cognitive or non-cognitive analyses best capture its nature and claims. Worshippers may interpret it in different cognitive or non-cognitive ways. Philosophers of religion
David hume and the limits of mathematical reason
Talk given at the January 2020 meeting of the Association for the Philosophy of Mathematical Practice (Zurich)
The material-ideal dyad of culture and the revolutionary materialism of pract...
According to Theodore R. Schatzki, theorists who study practices tend to view them as materially present and changing over history, rather than as determined by rigid abstract structures. This amounts to a revolutionary materialist perspective. The author will illustrate this perspective and discuss how it presents dilemmas for philosophers of mathematical practices. They will also argue that culture is better viewed as the expression of values in practices, rather than as distinct anthropological units, in order to understand cultural change and facilitate cross-cultural communication without falling into materialism or post-humanism.
On the unreasonable reliability of mathematical proof
This document summarizes Brendan Larvor's analysis of Jeremy Avigad's paper on the relationship between informal mathematical proofs and formal derivations. Larvor argues that Avigad's observations about how mathematicians write robust informal proofs actually support critics of the "Standard View". However, Avigad's paper fails to clearly define what constitutes an informal proof, oscillating between a "syntactic" view that reduces the Standard View to triviality, and a more promising "semantic" view. Larvor suggests Avigad should adopt the semantic perspective to better account for the value of mathematical logic and how informal proofs demonstrate reliability.
What virtues (zoom with eton june 2020)
The document discusses virtues exhibited by philosophers in the classroom. It notes that in the past, victory in philosophical arguments went to those who spoke with the greatest conviction, even if they lacked subtlety of mind. It questions what virtues academic staff believe they exhibit during live philosophy events and what students actually see. It argues that feeling like one could belong is important for success and happiness in education. The document considers the "signature pedagogy" of philosophy and how virtues like patience, trust, pride, humility, gratitude and forgiveness are needed for thinking together.
Apmp larvor francois
This document discusses different conceptions of culture as it relates to mathematical practices and mathematics education. It argues that there is no agreed upon definition of culture, and outlines some imperfect but commonly used definitions. It also notes a tension between the material and ideal aspects of culture that different conceptions manage differently. The document then examines how the concept of culture has been used and developed in the fields of mathematics education and studies of mathematical practices. It traces the historical and conceptual origins and evolution of these fields, including a social turn towards examining socio-cultural aspects. Ethnomathematics is discussed as having roots in post-colonial criticism and emphasizing different cultural approaches to mathematical learning and practices.
Matheamtical cutlures (bath june 15)
This document summarizes three conferences on mathematical cultures organized by Dr. Brendan Larvor of the University of Hertfordshire. The first conference focused on diversity in mathematical cultures and included talks on topics like mathematics in ancient China, the Russian tradition of mathematics, and mathematical practices in Brazilian engineering schools. The second conference centered on values in mathematics, with talks addressing concepts like purity, explanation, rigor and generality. The third conference covered additional topics related to mathematical cultures, such as mathematics in fiction, morality and mathematics, and mathematics education for indigenous communities in Western Canada.
Philosophy and history (lodge07)
The document discusses different conceptions of history and its relationship to philosophy. It argues that focusing solely on formal logic leaves out important non-logical aspects of mathematics, including its social and historical dimensions. A new epistemology of mathematics should recognize the temporal nature of rationality and appeal to intellectual and social contexts rather than only general principles. Such an approach follows Bernard Williams' conception of history of philosophy as philosophically motivated rather than relying on rigid distinctions between the two fields.
Manifesto
Higher education aims to teach students values through reasoned discussion rather than just giving students what they want, as every model of education reflects assumptions about human nature and should be judged accordingly. While students may not fully understand higher education's purpose, its goal is to provide a common space for developing reason and imparting values that are necessary for further economic and career aims imposed by outside forces.
Is it irrational to be moved by the
We can feel genuine sadness and emotions in response to fictional stories and characters, even though we know they are not real. This seems paradoxical, as we do not truly believe anyone has suffered. Hume argued these emotions are pleasurable and come from our engagement with the author's artistry. However, Radford cannot reconcile how we can feel involuntary emotions for fictional things we know don't exist. He concludes our emotional responses to fiction involve an inconsistency.
Titanic is a terrible film, so utterly
This document provides a critique of the film Titanic in 3 paragraphs. It argues that Titanic is a terrible film that promotes crass stereotypes and implausible characters. It claims the film tries to combine three different stories but they do not enhance each other and instead undermine the film. It also asserts the film insults the intelligence of viewers and promotes a simplistic "good vs. bad" political message about America that is patronizing to American audiences. The document suggests analyzing popular culture like films can provide philosophical insights and questions why philosophers do not typically engage in this type of critique.
Thought and feeling
The document discusses the shift from a mystical view of the soul to a rational view of the self that occurred during the Enlightenment. It established a contradiction where the mind is used to understand the world but is also an object of investigation. Later, Karen Armstrong's work is discussed and how she describes two different modes of understanding - mythos and logos. Mythos deals with meaning and the intuitive while logos is rational and scientific. Both are necessary but modern Western society has come to only value logos, discounting mythos, and leading to fundamentalism emerging as a new phenomenon rather than a return to old ways of understanding.
Thinking with shapes
This document discusses how shapes and structures play an important role in mathematical practice and notation. It examines seven cases where mathematical concepts have been extended from their original domains through an analogy to structure or shape. These include extending the factorial function to real numbers, defining exponentials of complex numbers and matrices, extending trigonometric functions to complex arguments, introducing pseudo-inverses of non-invertible matrices, defining derivatives of non-differentiable functions, extending derivatives to vector spaces, and taking unions and intersections of partitions. These extensions are argued to be unique to mathematics and reveal its distinctive epistemological status.
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Oxford jan 2017
- 1. Does Manders’ on Euclid I-IV offer a model for etc.. Brendan Larvor www.herts.ac.uk/philosophy
- 2. What’s the problem? How do proofs prove in practice? “The ultimate standard of proof is a formal proof, which is nothing other than an unbroken chain of logical inferences from an explicit set of axioms. While this may be the mathematical ideal of proof, actual mathematical practice generally deviates significantly from the ideal.” Thomas Hales Dense Sphere Packings: A Blueprint For Formal Proof, 2012, p.x
- 3. What’s the problem? How do proofs prove in practice?
- 4. Manders on Euclid I-IV Co-exact information
- 5. Leitgeb rehabilitates Gödel What do you see? Let me draw it for you…
- 6. Feferman And so on… the Cantor-Bernstein theorem
- 7. A conjecture A general condition for picture-reasoning In both Leitgeb and De Toffoli & Giardino, the plausibility of the claim that inspections and manipulations of mental entities might be mathematically reliable depends on the facts that: a)it is easy to draw a diagram that shares (or in the case of the natural numbers, indicates) the structure of the mathematical object, b)the information thus displayed is not exact (in Manders’ sense, i.e. is not metrical) and c)it is possible to label the diagram and thereby relate it systematically to syntactic, semantic, algebraic or logical notation and inference.