Continuum,name and paradox

The article deals with Cantor’s argument for the non-denumerability of reals somewhat in the spirit of Lakatos’ logic of mathematical discovery. At the outset Cantor’s proof is compared with some other famous proofs such as Dedekind’s recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are “ontologically” safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic paradoxes based on diagonal construction are caused by superficial understanding of what a name is.     

Gratitude, Ressentiment, and Citizenship Education

Patricia White (Stud Philos Educ 18:43–52, 1999) argues that the virtue gratitude is essential to a flourishing democracy because it helps foster universal and reciprocal amity between citizens. Citizens who participate in this reciprocal relationship ought to be encouraged to recognize that “much that people do does in fact help to make communal civic life less brutish, pleasanter and more flourishing.” This is the case even when the majority of citizens do not intentionally seek to make civic life better for others. Were citizens to recognize the appropriateness of gratitude in these situations, the bonds of our democratic communities would be strengthened. In this paper, I examine White’s argument more carefully, arguing that it fails to address adequately the difficulties that arise when we attempt to encourage the virtue of gratitude in our students. To address these difficulties, I turn to an unlikely source for democratic inspiration: Friedrich Nietzsche. In spite of his well-known anti-democratic sentiments, Nietzsche offers democratic citizens insights into the social value of gratitude. I argue that Nietzsche’s ideas resolve the educational difficulties in White’s argument and viably establish gratitude as an important democratic virtue that ought to be cultivated.     

McTaggart and Modern Physics

This paper delves into McTaggart’s metaphysical account of reality without time, and compares and contrasts McTaggart’s account with the account of reality given by modern physics. This comparison is of interest, because there are suggestions from contemporary physics that there is no time at the fundamental level. Physicists and philosophers of physics recognize that we do not have a good understanding of how the world could be such that time is unreal. I argue that, from the perspective of one who is trying to understand modern physics, McTaggart’s metaphysical views do provide some insight into how reality can be timeless at the fundamental level, but the insight that they provide is limited.     

The Pragmatics of Moral Motivation

One of the most prevalent and influential assumptions in metaethics is that our conception of the relation between moral language and motivation provides strong support to internalism about moral judgments. In the present paper, I argue that this supposition is unfounded. Our responses to the type of thought experiments that internalists employ do not lend confirmation to this view to the extent they are assumed to do. In particular, they are as readily explained by an externalist view according to which there is a pragmatic and standardized connection between moral utterances and motivation. The pragmatic account I propose states that a person’s utterance of a sentence according to which she ought to ϕ conveys two things: the sentence expresses, in virtue of its conventional meaning, the belief that she ought to ϕ, and her utterance carries a generalized conversational implicature to the effect that she is motivated to ϕ. This view also makes it possible to defend cognitivism against a well-known internalist argument.     

Epistemological Challenges to Mathematical Platonism

Since Benacerraf’s “Mathematical Truth” a number of epistemological challenges have been launched against mathematical platonism. I first argue that these challenges fail because they unduely assimilate mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism. Very roughly, what I demand is an account of how people’s mathematical beliefs are responsive to the truth of these beliefs. Finally I argue that if we employ a semantic truth-predicate rather than just a deflationary one, there surprisingly turns out to be logical space for a response to the improved challenge where no such space appeared to exist.     

“Cannot” Implies “Not Ought”

I argue for a version of “ought” implies “can”. In particular, I argue that it is necessarily true that if an agent, S, ultima facie ought to do A at T’, then there is a time T* such that S can at T* do A at T’. In support of this principle, I have argued that without it, we cannot explain how it is that, in cases where agents cannot do the best thing, they often ought to do some alternative action – such as get help or do the promised action later; nor can we explain the phenomenon of necessary enablers or the phenomenon of more stringent prima facie obligations overriding less stringent ones in cases where the agent cannot fulfill both.     

Doxastic Voluntarism and the Function of Epistemic Evaluations

Control of our own beliefs is allegedly required for the truth of epistemic evaluations, such as “S ought to believe that p”, or “S ought to suspend judgment (and so refrain from any belief) whether p”. However, we cannot usually believe or refrain from believing at will. I agree with a number of recent authors in thinking that this apparent conflict is to be resolved by distinguishing reasons for believing that give evidence that p from reasons that make it desirable to believe that p whether or not p is true. I argue however that there is a different problem, one that becomes clearer in light of this solution to the first problem. Someone’s approval of our beliefs is at least often a non-evidential reason to believe, and as such cannot change our beliefs. Ought judgments aim to change the world. But ‘ought to believe’ judgments can’t do that by changing the belief, if they don’t give evidence. So I argue that we should instead regard epistemic ought judgments as aimed mainly at influencing assertions that express the belief and other actions based on the belief, in accord with recent philosophical claims that we have epistemic norms for assertion and action.     

The discovery of the principles of reinforcement,extinction, generalization,and differentiation of conditional reflexes in Pavlov’s laboratories

The discovery of reinforcement, extinction, generalization, and differentiation with the conditional reflex method in Pavlov’s laboratories is described. Modern American introductory texts show that contemporary understanding of the experimental work on conditioning in Pavlov’s laboratories is derived from a 1927 English translation of Pavlov’s lectures on the conditional reflexes. The lectures present the discoveries topically, not chronologically. In contrast, this article presents a chronological account of the contributions of S. G. Vul’fson, I. F. Tolochinov, and B. P. Babkin, which led to the conceptualization of reinforcement and extinction, and the work of V. N. Boldyrev and N. A. Kashereninova, which led to the formulation of the concepts of generalization and differentiation. This historical approach avoids giving the impression that the development of the Pavlovian paradigm was a highly systematic pursuit.     

Putnam’s account of apriority and scientific change: its historical and contemporary interest

In the 1960s and 1970s, Hilary Putnam articulated a notion of relativized apriority that was motivated to address the problem of scientific change. This paper examines Putnam’s account in its historical context and in relation to contemporary views. I begin by locating Putnam’s analysis in the historical context of Quine’s rejection of apriority, presenting Putnam as a sympathetic commentator on Quine. Subsequently, I explicate Putnam’s positive account of apriority, focusing on his analysis of the history of physics and geometry. In the remainder of the paper, I explore connections between Putnam’s account of relativized a priori principles and contemporary views. In particular, I situate Putnam’s account in relation to analyses advanced by Michael Friedman, David Stump, and William Wimsatt. From this comparison, I address issues concerning whether a priori scientific principles are appropriately characterized as “constitutive” or “entrenched”. I argue that these two features need to be clearly distinguished, and that only the constitutive function is essential to apriority. By way of conclusion, I explore the relationship between the constitutive function a priori principles and entrenchment.     

Peirce’s Philosophy of Mathematical Education: Fostering Reasoning Abilities for Mathematical Inquiry

I articulate Charles S. Peirce’s philosophy of mathematical education as related to his conception of mathematics, the nature of its method of inquiry, and especially, the reasoning abilities required for mathematical inquiry. The main thesis is that Peirce’s philosophy of mathematical education primarily aims at fostering the development of the students’ semeiotic abilities of imagination, concentration, and generalization required for conducting mathematical inquiry by way of experimentation upon diagrams. This involves an emphasis on the relation between theory and practice and between mathematics and other fields including the arts and sciences. For achieving its goals, the article is divided in three sections. First, I expound Peirce’s philosophical account of mathematical reasoning. Second, I illustrate this account by way of a geometrical example, placing special emphasis on its relation to mathematical education. Finally, I put forth some Peircean philosophical principles for mathematical education.     

Moore’s Paradox is not just another pragmatic paradox

One version of Moore’s Paradox is the challenge to account for the absurdity of beliefs purportedly expressed by someone who asserts sentences of the form ‘p & I do not believe that p’ (‘Moorean sentences’). The absurdity of these beliefs is philosophically puzzling, given that Moorean sentences (i) are contingent and often true; and (ii) express contents that are unproblematic when presented in the third-person. In this paper I critically examine the most popular proposed solution to these two puzzles, according to which Moorean beliefs are absurd because Moorean sentences are instances of pragmatic paradox; that is to say, the propositions they express are necessarily false-when-believed. My conclusion is that while a Moorean belief is a pragmatic paradox, it is not just another pragmatic paradox, because this diagnosis does not explain all the puzzling features of Moorean beliefs. In particularly, while this analysis is plausible in relation to the puzzle posed by characteristic (i) of Moorean sentences, I argue that it fails to account for (ii). I do so in the course of an attempt to formulate the definition of a pragmatic paradox in more precise formal terms, in order to see whether the definition is satisfied by Moorean sentences, but not by their third-person transpositions. For only an account which can do so could address (ii) adequately. After rejecting a number of attempted formalizations, I arrive at a definition which delivers the right results. The problem with this definition, however, is that it has to be couched in first-person terms, making an essential use of ‘I’. Thus the problem of accounting for first-/third-person asymmetry recurs at a higher order, which shows that the Pragmatic Paradox Resolution fails to identify the source of such asymmetry highlighted by Moore’s Paradox.     

Normative Naturalism and the Relativised <Emphasis Type="Italic">A Priori</Emphasis>

Summary  In this paper I address some shortcomings in Larry Laudan’s normative naturalism. I make it clear that Laudan’s rejection of the “meta-methodology thesis”, or MMT is unnecessary, and that a reformulated version MMT can be sustained. I contend that a major difficulty that attends Laudan’s account is his contention that a naturalistic philosophy of science cannot accommodate any a priori justification of methodological rules, and consider what sort of naturalism might best replace Laudan’s. To do this, I discuss Michael Friedman’s account of a relativised a priori and show that it is consistent with naturalistic philosophy of science and that it can help form the basis of a plausible normative naturalism. In particular, this discussion shows that Laudan’s rejection of any a priori justification of methodological rules is unjustified and inconsistent with scientific practice. Finally, I point the way to a version of normative naturalism that includes MMT and accounts for the role of constitutive a priori principles within science.     

Question authority: in defense of moral naturalism without clout

Metaethicists of all stripes should read and learn from Richard Joyce’s book The Evolution of Morality. This includes moral realists, despite Joyce’s own nihilism. Joyce thinks that moral obligations, prohibitions, and the like are myths. But that is just a bit of a rich, broad account of moral attitudes and practices, the bulk of which can comfortably be accepted by realists. In fact, other than nihilism itself, there’s only one claim of Joyce’s which realists must reject. I argue that that claim ought to be rejected, and reply to Joyce’s argument to the contrary. The result is that—aside from nihilism and one rejectable claim—realists are free to take from Joyce whatever they like.     

Contra Cartwright: Structural Realism, Ontological Pluralism and Fundamentalism About Laws

In this paper I argue against Nancy Cartwright’s claim that we ought to abandon what she calls “fundamentalism” about the laws of nature and adopt instead her “dappled world” hypothesis. According to Cartwright we ought to abandon the notion that fundamental laws (even potentially) apply universally, instead we should consider the law-like statements of science to apply in highly qualified ways within narrow, non-overlapping and ontologically diverse domains, including the laws of fundamental physics. For Cartwright, “laws” are just locally applicable refinements of a more open-ended concept of capacities. By providing a critique of the dappled world approach’s central notion of open ended capacities and substituting this concept with an account of properties drawn from recent writing on the subject of structural realism I show that a form of fundamentalism is viable. I proceed from this conclusion to show that this form of fundamentalism provides a superior reading of case studies, such as the effective field theory program (EFT) in quantum field theory, than the “dappled world” view. The case study of the EFT program demonstrates that ontological variability between theoretical domains can be accounted for without altogether abandoning fundamentalism or adopting Cartwright’s more implausible theses.     

Hell Despite Vagueness: A Response to Sider

Ted Sider argues that a binary afterlife is inconsistent with a proportionally just God because no just criterion for placing persons in such an afterlife exists. I provide a possible account whereby God can remain proportionally just and allow a binary afterlife. On my account, there is some maximum amount of people God can allow into Heaven without sacrificing some greater good. God gives to all people at least their due but chooses to allow some who do not deserve Heaven to enter out of grace. Although this model implies a precise cutoff between those who enter Heaven and those who do not, I have argued that there is a precise point where God best serves justice and some greater good. Although God’s actions may appear arbitrary and ‘whimsically generous,’ it is merely because we are ignorant of the precise cutoff point that best serves his purposes.     

Representationalism and the perspectival character of perceptual experience

Perceptual experiences inform us about objective properties of things in our environment. But they also have perspectival character in the sense that they differ phenomenally when objects are viewed from different points of view. Contemporary representationalists hold, at a minimum, that phenomenal character supervenes on representational content. Thus, in order to account for perspectival character, they need to indentify a type of representational content that changes in appropriate ways with the perceiver’s point of view. Many representationlists, including Shoemaker and Lycan, argue that such contents are best construed in terms of mind-dependent properties. Other representationalists, including Tye and Dretske, hold that these contents involve only mind-independent properties. Susanna Schellenberg has recently developed an account of perceptual experience that would serve these latter representationalists extremely well. She suggests that we can do justice to the perspectival character of perceptual experience by appeal to representations of a certain type of relational properties, so-called ‘situation-dependent properties.’ In this paper, I critically engage with Schellenberg’s proposal in order to show how mind-independent representationalists can explain perspectival character. I argue that appeal to situation-dependent properties is problematic. I then show that mind-independent representationalists can account for perspectical character by means of scenario contents in Christopher Peacocke’s sense.     

Confucius and Aristotle on the Goods of Friendship

This essay discusses the goods of friendship as they are articulated by Confucius, Mencius, and Aristotle. It is argued that since Confucius and Mencius tend to conceive personal relationships in hierarchical terms, they do not directly address the goods of symmetrical friendships. Using Aristotle’s account of friendship, I argue that friendship is necessary for the cultivation of virtue outside the family. This is supported by discussing the virtues of generosity, trust, and wisdom as they develop within family life and then are refined in friendships. Lastly, as Confucius, Mencius, and Aristotle agree that the good friendship is necessarily a virtuous one, I consider what value aesthetic friendships have.     

Curves in Gödel-Space: Towards a Structuralist Ontology of Mathematical Signs

I propose an account of the metaphysics of the expressions of a mathematical language which brings together the structuralist construal of a mathematical object as a place in a structure, the semantic notion of indexicality and Kit Fine’s ontological theory of qua objects. By contrasting this indexical qua objects account with several other accounts of the metaphysics of mathematical expressions, I show that it does justice both to the abstractness that mathematical expressions have because they are mathematical objects and to the element of concreteness that they have because they are also used as signs. In a concluding section, I comment on the pragmatic element that has entered ontology by way of the notion of indexicality and use it to give an answer to a question Stewart Shapiro has recently posed about the status of meta-mathematics in the structuralist philosophy of mathematics.     

Reichenbach and Weyl on apriority and mathematical applicability

I examine Reichenbach’s theory of relative a priori and Michael Friedman’s interpretation of it. I argue that Reichenbach’s view remains at bottom conventionalist and that one issue which separates Reichenbach’s account from Kant’s apriorism is the problem of mathematical applicability. I then discuss Hermann Weyl’s theory of blank forms which in many ways runs parallel to the theory of relative a priori. I argue that it is capable of dealing with the problem of applicability, but with a cost.     

New directions for nominalist philosophers of mathematics

The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous influence of Quine’s “Indispensability Argument”, which challenged the nominalist to come up with an explanation of how science could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to enlarge the target of their investigations to include the many uses mathematicians make of concepts such as structures and models to advance pure mathematics. I shall illustrate my reasons for admonishing nominalists to strike out in these new directions by using Hartry Field’s nominalistic view of mathematics as a model of a philosophy of mathematics that was developed in just the sort of way I argue one should guard against. I shall support my reasons by providing grounds for rejecting both Field’s fictionalism and also his deflationist account of mathematical knowledge—doctrines that were formed largely in response to the Indispensability Argument. I shall then give a refutation of Mark Balaguer’s argument for his thesis that fictionalism is “the best version of anti-realistic anti-platonism”.