Models of Deduction*

In standard model theory, deductions are not the things one models. But in general proof theory, in particular in categorial proof theory, one finds models of deductions, and the purpose here is to motivate a simple example of such models. This will be a model of deductions performed within an abstract context, where we do not have any particular logical constant, but something underlying all logical constants. In this context, deductions are represented by arrows in categories involved in a general adjoint situation. To motivate the notion of adjointness, one of the central notions of category theory, and of mathematics in general, it is first considered how some features of it occur in set-theoretical axioms and in the axioms of the lambda calculus. Next, it is explained how this notion arises in the context of deduction, where it characterizes logical constants. It is shown also how the categorial point of view suggests an analysis of propositional identity. The problem of propositional identity, i.e., the problem of identity of meaning for propositions, is no doubt a philosophical problem, but the spirit of the analysis proposed here will be rather mathematical. Finally, it is considered whether models of deductions can pretend to be a semantics. This question, which as so many questions having to do with meaning brings us to that wall that blocked linguists and philosophers during the whole of the twentieth century, is merely posed. At the very end, there is the example of a geometrical model of adjunction. Without pretending that it is a semantics, it is hoped that this model may prove illuminating and useful. *Since the text of this talk was written in 1999, the author has published several papers about related matters (see ‘Identity of proofs based on normalization and generality’, The Bulletin of Symbolic Logic 9 (2003), 477–503, corrected version available at: http://arXiv.org/math.LO/0208094; other titles are available in the same archive).     

Mathematical Intuition and the Cognitive Roots of Mathematical Concepts

The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition.     

How to be a structuralist all the way down

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down.     

How Galileo dropped the ball and Fermat picked it up

This paper introduces a little-known episode in the history of physics, in which a mathematical proof by Pierre Fermat vindicated Galileo’s characterization of freefall. The first part of the paper reviews the historical context leading up to Fermat’s proof. The second part illustrates how a physical and a mathematical insight enabled Fermat’s result, and that a simple modification would satisfy any of Fermat’s critics. The result is an illustration of how a purely theoretical argument can settle an apparently empirical debate.     

Inner Harmony and the Human Ideal in Republic IV and IX

This paper presents an interpretation of Plato's moral psychology in two books of the Republic that construes Plato as adopting a strong unity for the moral agent. Within this conception reason influences both emotion and action directly. This view is contrasted with the current prevailing interpretation according to which all three parts of the soul have their own reason, feeling, and desire. The latter construal is shown to be both philosophically weak, and less plausible as a historical reconstruction.     

Uncommon Schools: Stanley Cavell and the Teaching of Walden

Thoreau’s Walden is a text that has been misinterpreted in various ways, one consequence of which is a failure to appreciate its significance as a perfectionist and visionary text for education. This paper explores aspects of what might be called its teaching, especially via the kind of teaching that is offered by Stanley Cavell’s commentary, The Senses of Walden. Walden is considered especially in the light of its conception of language as the “father-tongue” and of the ideas of continual rebirth and departure that are associated with this. References to teaching and learning abound in the book, but it is Thoreau’s specific reference to the need for “uncommon schools” that provides a focus for the present discussion. Paul Standish is Professor of Philosophy of Education at the University of Sheffield. His recent books include The Blackwell Guide to Philosophy of Education (2003), co-edited with Nigel Blake, Paul Smeyers and Richard Smith. He is Editor of the Journal of Philosophy of Education and Co-editor of the online Encyclopaedia of Philosophy of Education.     

Have Recommended Book Lists Changed to Reflect Current Expectations for Informational Text in K-3 Classrooms?

Despite both longstanding and recent calls for more informational text in K-3 classrooms, research indicates that narrative text remains in the majority for read alouds, classroom libraries, and instruction, thus limiting children's opportunity to experience the demands of expository text. Because national associations' recommended book lists are frequently proposed to identify books, we analyzed current lists to determine whether they include a higher percentage of expository books than 10 years ago. Our findings show a continuing prevalence of narratives suggesting the need to carefully evaluate books on such lists if the goal is to increase students' experience with expository text.     

Syntax-directed discovery in mathematics

It is shown how mathematical discoveries such as De Moivre's theorem can result from patterns among the symbols of existing formulae and that significant mathematical analogies are often syntactic rather than semantic, for the good reason that mathematical proofs are always syntactic, in the sense of employing only formal operations on symbols. This radically extends the Lakatos approach to mathematical discovery by allowing proof-directed concepts to generate new theorems from scratch instead of just as evolutionary modifications to some existing theorem. The emphasis upon syntax and proof permits discoveries to go beyond the limits of any prevailing semantics. It also helps explain the shortcomings of inductive AI systems of mathematics learning such as Lenat's AM, in which proof has played no part in the formation of concepts and conjectures.     

Reconsidering the miracle argument on the supposition of transient underdetermination

In this paper, I will show that the Miracle Argument is unsound if one assumes a certain form of transient underdetermination. For this aim, I will first discuss and formalize several variants of underdetermination, especially that of transient underdetermination, by means of measure theory. I will then formalize a popular and persuasive form of the Miracle Argument that is based on “use novelty”. I will then proceed to the proof that the miracle argument is unsound by means of a mathematical example. Finally, I will expose two hidden presuppositions of the Miracle Argument that make it so immensely though deceptively persuasive.     

Why Euclid’s geometry brooked no doubt: J. H. Lambert on certainty and the existence of models

J. H. Lambert proved important results of what we now think of as non-Euclidean geometries, and gave examples of surfaces satisfying their theorems. I use his philosophical views to explain why he did not think the certainty of Euclidean geometry was threatened by the development of what we regard as alternatives to it. Lambert holds that theories other than Euclid’s fall prey to skeptical doubt. So despite their satisfiability, for him these theories are not equal to Euclid’s in justification. Contrary to recent interpretations, then, Lambert does not conceive of mathematical justification as semantic. According to Lambert, Euclid overcomes doubt by means of postulates. Euclid’s theory thus owes its justification not to the existence of the surfaces that satisfy it, but to the postulates according to which these “models” are constructed. To understand Lambert’s view of postulates and the doubt they answer, I examine his criticism of Christian Wolff’s views. I argue that Lambert’s view reflects insight into traditional mathematical practice and has value as a foil for contemporary, model-theoretic, views of justification.     

How to have a radically minimal ontology

In this paper I further elucidate and defend a metaontological position that allows you to have a minimal ontology without embracing an error-theory of ordinary talk. On this view ‘there are Fs’ can be strictly and literally true without bringing an ontological commitment to Fs. Instead of a sentence S committing you to the things that must be amongst the values of the variables if it is true, I argue that S commits you to the things that must exist as truthmakers for S if it is true. I rebut some recent objections that have been levelled against this metaontological view.     

Mathematical Method and Proof

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.     

Decoding the language of the heart: Developing a physiology of inclusion

Constructs such as homeostasis and fight/flight have supported a scientific approach to physiology that has yielded a vast database of obvious heuristic value. Yet in spite of its value, these constructs have tended to create a mind-set that unwittingly supports what this article has labeled a “physiology of exclusion.” Reinforced by the philosophy of René Descartes, this perspective has led investigators to focus on isolated or separate animal organisms that are reflexively wired for self-preservation. It has created a mind-set in which both research investigators and the public at large tend to view the human body as either in a steady state of vigilance, maximally prepared for fight/flight, or in a state of quiescence. Assumptions of the solitary body, and solitary man wired to react for “self” preservation, has made it difficult to incorporate a growing body of evidence that indicates that social support and loving relationships are conducive to good health. It also has made it difficult for investigators to fully understand why human loneliness is a major cause of premature death. This article delineates these trends and offers a new construct, one that suggests that a “physiology of inclusion” be added to the prevailing view of a “physiology of exclusion.” Recent cardiovascular research is cited to help underscore the potential heuristic value of this new physiological construct.     

A Virtue-Based Defense of Mathematical Apriorism

Mathematical apriorists usually defend their view by contending that axioms are knowable a priori, and that the rules of inference in mathematics preserve this apriority for derived statements—so that by following the proof of a statement, we can trace the apriority being inherited. The empiricist Philip Kitcher attacked this claim by arguing there is no satisfactory theory that explains how mathematical axioms could be known a priori. I propose that in analyzing Ernest Sosa’s model of intuition as an intellectual virtue, we can construct an “intuition–virtue” that could supply the missing explanation for the apriority of axioms. I first argue that this intuition–virtue qualifies as an a priori warrant according to Kitcher’s account, and then show that it could produce beliefs about mathematical axioms independent of experience. If my argument stands, this paper could provide insight on how virtue epistemology could help defend mathematical apriorism on a larger scale.     

Beweislastverteilung und Intuitionen in philosophischen Diskursen

Allocating the burden of proof and intuitions in philosophical disputes.– This paper criticises the view that in philosophical disputes the onus probandi rests on those who advance a position that contradicts our basic intuitions. Such a rule for allocating the burden of proof may be an adequate reconstruction of everyday justification, but is unreasonable in the area of philosophy. In philosophy it is not only difficult to determine the plausibility of a proposition, at the same time contradictory claims may be equally plausible. – In contrast to such common sense proposals I try to show that in philosophical disputes the burden of proof does not depend on the material content of speech acts. A speaker simply bears the burden of proof for a proposition p if he has asserted that p and has agreed to justify it. This revised version was published online in August 2006 with corrections to the Cover Date.     

What Experiences Do Expository Books on Recommended Book Lists Offer to K–2 Students?

Teachers can use expository texts to teach academic vocabulary, content knowledge, text structure, and text features. National associations' recommended book lists are often used to identify books for classrooms. Previously we identified expository texts on these lists from 2001–2002 and 2011–2012. The current study explored instructional possibilities these expository books afford. More recent books have more content, academic vocabulary, and supplementary information. However, many earlier and recent books lack text features such as page numbers, tables of contents, glossaries, and indexes. Teachers will need to supplement these books with other texts to provide experiences enabling proficiency with exposition.     

The Role of Notations in Mathematics

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic notations, notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways.

     

Historicism,Entrenchment, and Conventionalism

W. V. Quine famously argues that though all knowledge is empirical, mathematics is entrenched relative to physics and the special sciences. Further, entrenchment accounts for the necessity of mathematics relative to these other disciplines. Michael Friedman challenges Quine’s view by appealing to historicism, the thesis that the nature of science is illuminated by taking into account its historical development. Friedman argues on historicist grounds that mathematical claims serve as principles constitutive of languages within which empirical claims in physics and the special sciences can be formulated and tested, where these mathematical claims are themselves not empirical but conventional. For Friedman, their conventional, constitutive status accounts for the necessity of mathematics relative to these other disciplines. Here I evaluate Friedman’s challenge to Quine and Quine’s likely response. I then show that though we have reason to find Friedman’s challenge successful, his positive project requires further development before we can endorse it.     

Zombies and the Case of the Phenomenal Pickpocket

A prevailing view in contemporary philosophy of mind is that zombies are logically possible. I argue, via a thought experiment, that if this prevailing view is correct, then I could be transformed into a zombie. If I could be transformed into a zombie, then surprisingly, I am not certain that I am conscious. Regrettably, this is not just an idiosyncratic fact about my psychology; I think you are in the same position. This means that we must revise or replace some important positions in the philosophy of mind. We could embrace radical skepticism about our own consciousness, or maintain the complete and total infallibility of our beliefs about our own phenomenal experiences. I argue that we should actually reject the logical possibility of zombies.     

熟读经典,培养中医思维

师古为效今,熟读历代著名医家的经典医籍,亦是为当今之新中医培养中医思维观而用。通过阐述了熟读典籍的重要意义,以及从典籍中吸取何种精华为我所用,探讨了当今之新中医如何发展创新,试图探索一条复兴中医之路。