Theories and Theories of Truth

Formal theories, as in logic and mathematics, are sets of sentences closed under logical consequence. Philosophical theories, like scientific theories, are often far less formal. There are many axiomatic theories of the truth predicate for certain formal languages; on analogy with these, some philosophers (most notably Paul Horwich) have proposed axiomatic theories of the property of truth. Though in many ways similar to logical theories, axiomatic theories of truth must be different in several nontrivial ways. I explore what an axiomatic theory of truth would look like. Because Horwich’s is the most prominent, I examine his theory and argue that it fails as a theory of truth. Such a theory is adequate if, given a suitable base theory, every fact about truth is a consequence of the axioms of the theory. I show, using an argument analogous to Gödel’s incompleteness proofs, that no axiomatic theory of truth could ever be adequate. I also argue that a certain class of generalizations cannot be consequences of the theory.     

Dispositions, Laws, and Categories

After a short sketch of Lowe’s account of his four basic categories, I discuss his theory of formal ontological relations and how Lowe wants to account for dispositional predications. I argue that on the ontic level Lowe is a pan-categoricalist, while he is a language dualist and an exemplification dualist with regard to the dispositional/categorical distinction. I argue that Lowe does not present an adequate account of disposition. From an Aristotelian point of view, Lowe conflates dispositional predication with hôs epi to poly statements about what is normally or mostly the case.     

What became of Paul Dirac’s classmate?

We examine Paul Dirac’s early life in Bristol and the link with his classmate Herbert Charles Wiltshire. We outline Wiltshire’s subsequent career using archives and the few letters which survive between Dirac and Wiltshire.     

In Defence of Powerful Qualities

The ontology of ‘powerful qualities’ is gaining an increasing amount of attention in the literature on properties. This is the view that the so-called categorical or qualitative properties are identical with ‘dispositional’ properties. The position is associated with C.B. Martin, John Heil, Galen Strawson and Jonathan Jacobs. Robert Schroer (2012) has recently mounted a number of criticisms against the powerful qualities view as conceived by these main adherents, and has also advanced his own (radically different) version of the view. In this paper I have three main aims: firstly, I shall defend the ontology from his critique, arguing that his criticisms do not damage the position. Secondly, I shall argue that Schroer’s own version of the view is untenable. Thirdly, the paper shall serve to clear up some conceptual confusions that often bedevil the powerful qualities view.     

Constructive agents

Brouwer’s ideas of construction, proof, and inquiry in mathematics are more widely applicable. On a well-known philosophical view, intuitionistic logic is a general account of meaning and reasoning for natural language and epistemology. In this brief discussion piece, I go one step further, and discuss how intuitionistic semantics fits with information update and belief revision in agency. In the process, I define a number of new logical systems that give rise to several open problems.     

Immense Multiple Realization

In his latest book Physicalism, or Something near Enough, Jaegwon Kim argues that his version of functional reductionism is the most promising way for saving mental causation. I argue, on the other hand, that there is an internal tension in his position: Functional reductionism does not save mental causation if Kim’s own supervenience argument is sound. My line of reasoning has the following steps: (1) I discuss the supervenience argument and I explain how it motivates Kim’s functional reductionism; (2) I present what I call immense multiple realization, which says that macro-properties are immensely multiply realized in determinate micro-based properties; (3) on that background I argue that functional reductionism leads to a specified kind of irrealism for mental properties. Assuming that such irrealism is part of Kim’s view, which Kim himself seems to acknowledge, I argue that Kim’s position gets the counterfactual dependencies between macro-causal relata wrong. Consequently, his position does not give a conservative account of mental causation. I end the paper by discussing some alternative moves that Kim seems to find viable in his latest book. I argue on the assumption that the supervenience argument is sound, so the discussion provides further reasons to critically reevaluate that argument because it generalizes in deeply problematic ways.     

A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms.     

Discursive psychology as an alternative perspective on mathematics teacher knowledge

Research on mathematics teacher knowledge, including work on mathematical knowledge for teaching, draws heavily on Shulman’s categories of teacher knowledge. These categories have been adopted, developed and modified by mathematics education researchers. This approach has led to some valuable insights. In this paper, I draw on discursive psychology to develop a critique of this work. This critique highlights some of the unstated assumptions of much research inspired by Shulman’s work, including, in particular, a representational view of knowledge and argues that the resulting theories do not reflect the discourses of knowledge that arise in mathematics classrooms. These ideas are illustrated with discussion of two examples, with the aim of showing how discursive psychology can offer an alternative perspective.     

Basic skills versus conceptual understanding in mathematics education: The case of fraction division

The distinction between conceptual understanding and basic skills is as old as mathematics education research itself. It still remains a central issue for many disputes. In this paper, building upon professor Hung-Hsi Wu's rejection of the distinction, I explore three possible accounts of it: (a) conceptual understanding first, (b) explaining the distinction away and emphasizing “procedural-understanding” instead, and finally (c) treating understanding and procedural skill as two separate, irreducible, complementary components. In contrast to Wu who favors the second account, I argue that as far as mathematics teaching is concerned the third view is the preferable one     

Informational Semantics and Frege Cases

One of the most important objections to information-based semantic theories is that they are incapable of explaining Frege cases. The worry is that if a concept’s intentional content is a function of its informational content, as such theories propose, then it would appear that coreferring expressions have to be synonymous, and if this is true, it’s difficult to see how an agent could believe that a is F without believing that b is F whenever a and b are identical. I argue that this appearance is deceptive. If we heed the distinction between the analog and digital contents of a signal, it is actually possible to reconstruct something akin to Frege’s sense/reference distinction in purely information-theoretic terms. This allows informational semanticists to treat coreferring expressions as semantically distinct and to solve Frege cases in the same way that Frege did—namely, by appealing to the different contents of coreferring expressions.     

Visual Experience: Cognitive Penetrability and Indeterminacy

This paper discusses a counterexample to the thesis that visual experience is cognitively impenetrable. My central claim is that sometimes visual experience is influenced by the perceiver’s beliefs, rendering her experience’s representational content indeterminate. After discussing other examples of cognitive penetrability, I focus on a certain kind of visual experience— that is, an experience that occurs under radically nonstandard conditions—and show that it may have indeterminate content, particularly with respect to low-level properties such as colors and shapes. I then explain how this indeterminacy depends on the perceiver’s beliefs or thoughts. Finally, I attempt to generalize the case and show how other sorts of visual experiences can also be penetrated by beliefs and, hence, be indeterminate.     

Kant, Spinoza, and the Metaphysics of the Ontological Proof

This paper provides an interpretation and evaluation of Spinoza’s highly original version of the ontological proof in terms of the concept of substance instead of the concept of perfection in the first book of his Ethics. Taking the lead from Kant’s critique of ontological arguments in the Critique of Pure Reason, the paper explores the underlying ontological and epistemological presuppositions of Spinoza’s proof. The main topics of consideration are the nature of Spinoza’s definitions, the way he conceives of the relation between a substance and its essence, and his conception of existence. Once clarity is shed upon these fundamental issues, it becomes possible to address the proof in its own terms. It is then easy to see that Kant’s objections miss their target and that the same is true of those advanced by another of the ontological argument’s most famous critics, Bertrand Russell. Finally, several interpretations of Spinoza’s proof are proposed and critically evaluated; on all of them, the argument turns out to be either invalid or question-begging.     

TRACING FOUCAULT'S ‘POWER’ IN THE MATHEMATICS CLASSROOM

In this paper I discuss some reflections on power and how its workings may be traced in the mathematics classroom. I explore some work of Michael Foucault on the nature of power and attempt to use his analytics of power to consider the formation and organisation of interactions in mathematics classrooms. I offer this as a strategy to start to (re)examine the practices of teachers and children in this arena. I move on to explore a number of concerns including the ‘regulation’ of teachers’ and children's practices: how power, in this sense, might legitimate certain classroom actions and limit others.     

Where do preferences come from?

Rational choice theory analyzes how an agent can rationally act, given his or her preferences, but says little about where those preferences come from. Preferences are usually assumed to be fixed and exogenously given. Building on related work on reasons and rational choice (Dietrich and List, Nous, forthcoming), we describe a framework for conceptualizing preference formation and preference change. In our model, an agent’s preferences are based on certain ‘motivationally salient’ properties of the alternatives over which the preferences are held. Preferences may change as new properties of the alternatives become salient or previously salient properties cease to be salient. Our approach captures endogenous preferences in various contexts and helps to illuminate the distinction between formal and substantive concepts of rationality, as well as the role of perception in rational choice.     

Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya

Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings. In this report, I first summarize the literature examining students’ and teachers’ inverse function meanings. I then provide my theoretical perspective, including my use of the terms understanding and meaning and my operationalization of productive inverse function meanings. I describe a conceptual analysis of ways students may reorganize their limited inverse function meanings into productive meanings via reasoning about relationships between covarying quantities. I then present one pre-service teacher’s activity in a semester long teaching experiment to characterize how her quantitative, covariational, and bidirectional reasoning supported her in reorganizing her limited inverse function meanings into more productive meanings. I describe how this reorganization required her to reconstruct her meanings for various related mathematical ideas. I conclude with research and pedagogical implications stemming from this work and directions for future research.     

Analyticity and the Deviant Logician: Williamson’s Argument from Disagreement

One way to discredit the suggestion that a statement is true just in virtue of its meaning is to observe that its truth is the subject of genuine disagreement. By appealing to the case of the unorthodox philosopher, Timothy Williamson has recast this response as an argument foreclosing any appeal to analyticity. Reconciling Quine’s epistemological holism with his treatment of the ‘deviant logician’, I show that we may discharge the demands of charitable interpretation even while attributing trivial semantic error to Williamson’s philosophers. Williamson’s effort to generalize the argument from disagreement therefore fails.     

Uncertain agency models with multi-dimensional incomplete information based on confidence level

This paper discusses a principal–agent problem with multi-dimensional incomplete information between a principal and an agent. Firstly, how to describe the incomplete information in such agency problem is a challenging issue. This paper characterizes the incomplete information by uncertain variable, because it has been an appropriate tool to depict subjective assessment and model human uncertainty. Secondly, the relevant literature often used expected-utility-maximization to measure the two participators’ goals. However, Ellsberg paradox indicates that expected utility criterion is not always appropriate to be regarded as decision rule. For this reason, this paper presents another decision rule based on confidence level. Instead of expected-utility-maximization, the principal’s aim is to maximize his potential income under the acceptable confidence level, and the agent’s aim depends on whether he has private information about his efforts. According to the agent’s different decision rules, three classes of uncertain agency (UA) models and their respective optimal contracts are presented. Finally, a portfolio selection problem is studied to demonstrate the modeling idea and the viability of the proposed UA models.     

‘There are great alterations in the geometry of late’. The rise of Isaac Newton’s early Scottish circle†

This paper traces the rise of three Scottish mathematicians – Colin Campbell, John Craig, and David Gregory – to become key figures in the dissemination and promotion of Newton’s mathematical ideas and natural philosophy in the 1680s. Two medical men – Archibald Pitcairne and his former student George Cheyne – both likewise captivated by the Principia, played minor roles in the story of Newton’s mathematics, while at the same time promoting the concept of mathematical medicine derived from his philosophical thought. Drawing on contemporary correspondence and previously unpublished papers, it considers how these men contributed to the scholarly perception of Newton and how, conversely, Newton used his increasing influence in order to encourage their work, most notably obtaining for Gregory the vacant chair in astronomy at Oxford in 1691.