Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured previous (...) discussions of these projects, and presents clear, concise accounts of a dozen strategies for nominalistic interpretation of mathematics, thus equipping the reader to evaluate each and to compare different ones. The authors also offer critical discussion, rare in the literature, of the aims and claims of nominalistic interpretation, suggesting that it is significant in a very different way from that usually assumed. (shrink)

This is the first systematic survey of modern nominalistic reconstructions of mathematics, and for this reason alone it should be read by everyone interested in the philosophy of mathematics and, more generally, in questions concerning abstract entities. In the bulk of the book, the authors sketch a common formal framework for nominalistic reconstructions, outline three major strategies such reconstructions can follow, and locate proposals in the literature with respect to these strategies. The discussion is presented with admirable precision and clarity, (...) and should be accessible even to readers with only minimal background in logic and mathematics. There will be many who will turn directly to these pages and use them as a brief manual on the state of the art of nominalism in mathematics. But the most intriguing parts of this elegant book—at least in my view—are the introduction and the conclusion, where the authors examine the significance of reconstructive nominalism. (shrink)

Peter Abelard (1079–1142 ce) was the most wide‐ranging philosopher of the twelfth century. He quickly established himself as a leading teacher of logic in and near Paris shortly after 1100. After his affair with Heloise, and his subsequent castration, Abelard became a monk, but he returned to teaching in the Paris schools until 1140, when his work was condemned by a Church Council at Sens. His logical writings were based around discussion of the “Old Logic”: Porphyry's Isagoge, aristotle'S Categories and (...) On Interpretation and boethius'S textbook on topical inference. They comprise a freestanding Dialectica (“Logic”; probably c.1116), a set of commentaries (known as the Logica [Ingredientibus], c. 1119) and a later (c. 1125) commentary on the Isagoge (Logica Nostrorum Petititoni Sociorum or Glossulae). In a work Abelard called his Theologia, issued in three main versions (between 1120 and c.1134), he attempted a logical analysis of trinitarian relations and explored the philosophical problems surrounding God's claims to omnipotence and omniscience. The Collationes (“Debates,” also known as “Dialogue between a Christian, a Philosopher and a Jew”; probably c.1130) present a rational investigation into the nature of the highest good, in which the Christian and the Philosopher (who seems to be modeled on a philosopher of pagan antiquity) are remarkably in agreement. The unfinished Scito teipsum (“Know thyself,” also known as the “Ethics”; c.1138) analyses moral action. (shrink)

It would seem that Geach's claim is that the relation expressed by 'is identical with' is like the relation expressed by 'is better than', at least in one respect. If x and y are people, it may turn out that x is a better golfer than y, while y is a better poet than x. If we merely say that x is better than y, we fail to specify the respect in which we hold x to be the better. Another (...) way to put this is to say that we have not made clear "in relation to what" x is the better. We can call the relation expressed by 'is better than' a "relative relation," since an ordered pair of things may stand in that relation relative to one thing but not relative to another thing. Geach seems to think that identity is a relative relation. (shrink)

There are things we routinely say that may strike us as literally false but that we are nonetheless reluctant to give up. This might be something mundane, like the way we talk about the sun setting in the west, or it could be something much deeper, like engaging in talk that is ostensibly about numbers despite believing that numbers do not literally exist. Rather than regard such behaviour as self-defeating, a "fictionalist" is someone who thinks that this kind of discourse (...) is entirely appropriate, even helpful, so long as we treat what is said as a useful fiction, rather than as the sober truth. "Fictionalism" can be broadly understood as a view that uses a notion of pretense or fiction in order to resolve certain puzzles or problems that otherwise do not necessarily have anything to do with literature or fictional creations. Within contemporary analytic philosophy, fictionalism has been on the scene for well over a decade and has matured during that time, growing in popularity. There are now myriad competing views about fictionalism and consequently the discussion has branched out into many more subdisciplines of philosophy. Yet there is widespread disagreement on what philosophical fictionalism actually amounts to and about how precisely it ought to be pursued. This volume aims to guide these discussions, collecting some of the most up-to-date work on fictionalism and tracing the view's development over the past decade. After a detailed discussion in the book's introductory chapter of how philosophers should think of fictionalism and its connection to metaontology more generally, the remaining chapters provide readers with arguments for and against this view from leading scholars in the fields of epistemology, ethics, metaphysics, philosophy of science, philosophy of language, and others. (shrink)

Arsenic and phosphorus are group 15 elements with similar chemical properties. Is it possible that arsenate could replace phosphate in some of the chemicals that are required for life? Phosphate esters are ubiquitous in biomolecules and are essential for life, from the sugar phosphates of intermediary metabolism to ATP to phospholipids to the phosphate backbone of DNA and RNA. Some enzymes that form phosphate esters catalyze the formation of arsenate esters. Arsenate esters hydrolyze very rapidly in aqueous solution, which makes (...) it improbable that phosphorous could be completely replaced with arsenic to support life. Studies of bacterial growth at high arsenic:phosphorus ratios demonstrate that relatively high arsenic concentrations can be tolerated, and that arsenic can become involved in vital functions in the cell, though likely much less efficiently than phosphorus. Recently Wolfe‐Simon et al. 1 reported the isolation of a microorganism that they maintain uses arsenic in place of phosphorus for growth. Here, we examine and evaluate their data and conclusions. (shrink)

This collection of recent articles by leading scholars is designed to illuminate one of the greatest and most influential philosophical books of all time. It includes incisive commentary on every major theme and argument in the Meditations, and will be valuable not only to philosophers but to historians, theologians, literary scholars, and interested general readers.

This paper criticises a line of argument adopted by peter winch, Karl popper, And others, To the effect that the course of human history cannot be predicted. On this view it is impossible to predict in a particularly detailed way certain events ('original acts') on which important social developments depend. We analyze the argument, Showing that one version fails: original acts are in principle predictable in the relevant way. A cogent version is presented; this requires a special definition for 'original (...) act'. But, We claim, Social developments do not depend on original acts so defined. We argue separately for the possibility of a person, Or a scientific community, Predicting his or its own original acts. (shrink)

We take another look at “Just-So” solar neutrino oscillations, characterizing them by the energy Eπ/2 at which the distance-varying angle is π/2, instead of by the usual Δm 2 . The rising spectrum recently observed by SuperKamiokande is consistent with an Eπ/2 ∼6–9 MeV and marginally with 48 MeV. The pp neutrinos must then be reduced to one-half the standard solar model prediction, and 7Be neutrinos must make up a significant part of the SAGE and GALLEX gallium signal. For Eπ/2 (...) close to 9 and 48 MeV, the 7Be neutrinos will also show a large seasonal variation, emphasizing the importance of direct measurements of the 7Be neutrinos. (shrink)

Kants Vernunftkritik hat das Wissen zugunsten des Glaubens aufgehoben. Angeregt vor allem durch F.H. Jacobi, wird das Verhältnis von Glauben und Vernunft in der Philosophie nach Kant erneut zu einem zentralen Thema. Zur Entscheidung stehen die Fragen, ob der Glaube das Fundament von Wissen sein kann, ob der Glaube eine Grenze der Vernunft markiert oder ob eine absolut ges.

Humans can learn complex functional relationships between variables from small amounts of data. In doing so, they draw on prior expectations about the form of these relationships. In three experiments, we show that people learn to adjust these expectations through experience, learning about the likely forms of the functions they will encounter. Previous work has used Gaussian processes—a statistical framework that extends Bayesian nonparametric approaches to regression—to model human function learning. We build on this work, modeling the process of learning (...) to learn functions as a form of hierarchical Bayesian inference about the Gaussian process hyperparameters. (shrink)

This book provides practical and research-based chapters that offer greater clarity about the particular kinds of teacher reflection that matter and avoids talking about teacher reflection generically, which implies that all kinds of reflection are of equal value.

This paper joins the voices warning against hasty transference of legal concepts of presumption to other kinds of argumentation, especially to deliberation about future acts and policies. Comparison of the pragmatics which respectively constitute the illocutionary acts of accusing and proposing reveals important differences in the ways presumptions prompt accusers and proposers to undertake probative responsibilities and, also, points to corresponding differences in their probative duties. This comparison has theoretically important implication regarding the norms governing persuasive argumentation. The paper is (...) based on a broadly Gricean account of speech acts. (shrink)

Policy Review, the organ of the conservative Heritage Foundation, devoted their winter 1984 issue to lamenting the failures of the Reagan administration. Publisher M. Stan ton Evans complained that “This has been essentially another Ford Administration, … not much different from any other Republican administration in our lifetime. While the other Senator from Colorado, arch-conservative William Armstrong noted that Reagan had ‘managed to polarize the country over budget cuts that didn't happen.” “He cut the budget,” bemoaned Armstrong, “enough to make (...) the special interests and the press mad, but not enough” to restructure the government. For his part, National Conservative Political Action Committee (NCPAC) chair Terry Donlan has complained of the administration failure to move on its social agenda. (shrink)

The debate between externalists and internalists in epistemology can be viewed as a disagreement about whether there are epistemic rights without corresponding duties or obligations. Taking an epistemic right to believe P as an authorization to not only accept P as true but to use P as a positive reason for accepting other propositions, the debate is about whether there are unjustified justifiers. It is about whether there are propositions that provide for others what nothing need provide for them—viz., reasons (...) for thinking them true. (shrink)

We investigate the expressive power of fragments of first-order logic that are defined in terms of prefixes. The main result establishes a strict hierarchy among these fragments over the signature consisting of a single binary relation. It implies that for each prefix p, there is a sentence in prenex normal form with prefix p, over a single binary relation, such that for all sentences θ in prenex normal form, if θ is equivalent to , then p can be embedded in (...) the prefix of θ. This strengthens a theorem of Walkoe. (shrink)

Externalism about knowledge commits one to a modest form of contextualism: whether one knows depends (or may depend) on circumstances (context) of which one has no knowledge. Such modest contextualism requires the rejection of the KK Principle (If S knows that P, then S knows that S knows that P) - something most people would want to reject anyway - but it does not require (though it is compatible with) a rejection of closure. Radical contextualism, on the other hand, goes (...) a step farther and relativizes knowledge not just to the circumstances of the knower, but to the circumstances of the person attributing knowledge. I reject this more radical form of contextualism and suggest that it confuses (or that it can, at least, be avoided by carefully distinguishing) the relativity in what S is said to know from the relativity in whether S knows what S is said to know. (shrink)

According to the causal interpretation of quantum mechanics, one can precisely define the state of an individual particle in a many-body system by its position, momentum, and spin. It is shown in the EPR spin experiment that the quantum torque brings about an instantaneous change in the state of one of the particles when the other undergoes a local interaction, but that such a transfer of “information” cannot be extracted by any experiment subject to the laws of quantum mechanics.

Melnyk complains that my account of the way reasons explain behaviour cannot be extended to cover novel behaviours. I admit that I did not extend it, but deny that it is not extendible. This, indeed, is what Chapter 6 of Dretske (1988) was all about. Noordhof finds faults with my account and claims there is another account (partial supervenience) that does a better job. I acknowledge one of the defects—a defect I was aware of when I wrote the book‐but deny (...) that the partial supervenience of content on intrinsic properties represents a better theory of the explanatory role of content. (shrink)

Though one believes that P is true, one can have reasons for thinking it false. Yet, it seems that one cannot know that P is true and (still) have reasons for thinking it false. Why is this so? What feature of knowledge (or of reasons) precludes having reasons or evidence to believe (true) what you know to be false? If the connection between reasons (evidence) and what one believes is expressible as a probability relation, it would seem that the only (...) satisfactory explanation of this fact is that when one knows that P is true, the reasons or evidence one has in support of P are such as to confer upon P the probability of 1. It is shown by an application of Bayes' Theorem that any value smaller than 1 would permit having reasons to believe what one knows to be false. Hence, it would seem that knowledge requires conclusive reasons to believe (if reasons or evidence is required at all). (shrink)

During the past decade there has been a very effective campaign against any explanation of remembering whose basic concept is that of a causally mediating trace. This paper attempts to provide such an explanation by presenting an explicit deductive argument for the existence of the memory trace. The conclusion is shown to follow from reasonable, empirical assumptions of which the most interesting is a spatiotemporal contiguity thesis. Set-theoretic techniques are used to provide a framework of analysis and probabilistic definitions of (...) some causal notions, as that of a causal chain, are presented. (shrink)