Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that arithmetical (...) knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology. (shrink)

This is a review of Mind and Supermind. By KEITH FRANKISH. (Cambridge: Cambridge University Press, 2004. Pp xiv + 255. Price £45.00 (US $75.00). ISBN 0521 812038 (hardback).

Semantic interpretations of both natural and formal languages are usually taken to involve the specification of a domain of entities with respect to which the sentences of the language are to be evaluated. A question that has received much attention of late is whether there is unrestricted quantification, quantification over a domain comprising absolutely everything there is. Is there a discourse or inquiry that has absolute generality? After framing the debate, this article provides an overview of the main arguments for (...) and against the possibility of unrestricted quantification, highlighting some of the broader implications of the debate. (shrink)

The prevailing approach to the problem of the ontological status of mathematical entities such as numbers and sets is to ask in what sense it is legitimate to ascribe a reference to abstract singular terms; those expressions of our language which, taken at face value, denote abstract objects. On the basis of this approach, neo‐Fregean Abstractionists such as Hale and Wright have argued that abstract singular terms may be taken to effect genuine reference towards objects, whereas nominalists such as Field (...) have asserted that these apparent ontological commitments should not be taken at face value. In this article I argue for an intermediate position which upholds the legitimacy of ascribing a reference to abstract singular terms in an attenuated sense relative to the more robust ascription of reference applicable to names denoting concrete entities. In so doing I seek to clear up some confusions regarding the ramifications of such a thin notion of reference for ontological claims about mathematical objects. (shrink)

Computationalism holds that our grasp of notions like ‘computable function’ can be used to account for our putative ability to refer to the standard model of arithmetic. Tennenbaum's Theorem has been repeatedly invoked in service of this claim. I will argue that not only do the relevant class of arguments fail, but that the result itself is most naturally understood as having the opposite of a reference-fixing effect — i.e., rather than securing the determinacy of number-theoretic reference, Tennenbaum's Theorem points (...) towards a sense in which portions of our computational vocabulary should be regarded as model-relative. (shrink)

SummaryMany contemporary philosophers of language believe that realist metaphysics and a beha‐viouristic approach to language are incompatible, debate centring on which is to be given up. In this paper I argue that no incompatibility has been shown to exist. In the first section I attempt to give both a characterization of, and an argument for, behaviourism. Then I attempt to characterize realism more generally than is often done, evaluating the work of Dummett, Quine, Putnam and Wittgenstein, as recently interpreted, in (...) that light. Finally I attempt to sketch a behaviouristic justification of classical predicate logic and offer some programmatic remarks on behaviouristic accounts of mathematics and natural science.RésuméBien des philosophes contemporains du langage, estimant qu'une métaphysique réaliste et une approche behavioriste sont incompatibles, centrent le débat sur la question «lequel des deux doit‐il être sacrifité?» Je montre dans ce papier qu'on n'a jamais pu prouver l'existence d'une incompatibilité. Dans la première partie, je tente de caractériser et de défendre le behaviorisme. Puis j'essaie de caractériser le réalisme plus généralement que cela n'a été souvent fait et évalue dans cette perspective les oeuvres de Dummett, de Quine, de Putnam et de Wittgenstein à la lumière de leurs interprétations récentes. Finalement, je propose une justification behavioriste de la classique logique des prédicats et quelques remarques programmatiques quant à une approche behavioriste des mathématiques et des sciences naturelles.ZusammenfassungManche zeitgenössische Philosophen glauben, dass eine realistische Metaphysik nicht mit einer behavioristischen Betrachtungsweise der Sprache vereinbar sei, wobei sich der Streit um die Frage dreht, welche der beiden Haltungen aufgegeben werden müsse. Ich argurnentiere, dass die Unvereinbarkeit nicht wirklich nachgewiesen wurde. Im ersten Abschnitt versuche ich, den Beha‐viorismus genauer zu charakterisieren und zu seinen Gunsten zu argumentieren. Ich versuche dann, eine allgemeinere Charakterisierung des Realismus zu geben, als das üblich ist, indem ich die Werke von Dummett, Quine, Putnam und Wittgenstein, wie sie neuerdings unter diesen Gesichts‐punkt interpretiert werden, beurteile. lch versuche schliesslich, eine behavioristische Rechtferti‐gung der klassischen Prädikatenlogik zu skizzieren, und lege auch einige programmatische Bemerkungen zur behavioristischen Behandlung von Mathematik und Naturwissenschaft vor. (shrink)

In this critical notice, I argue that the semantic view championed by Thompson no logical advantage over the syntactic view of theories, especially in the area of interpretation. Each weakness of the syntactic view has a corresponding weakness in the semantic view. In principle the two are not different in power, but it is sometimes better to adopt one rather than the other, for practical reasons. I agree with Thompson that many issues in the philosophy of biology can be illuminated (...) by the semantic view, but that other things, especially deriving specific prediction, are best understood with the syntactic view. (shrink)

This paper is concerned with connections between scientific and metaphysical realism. It is not difficult to show that scientific realism, as expounded by Psillos (1999) clearly qualifies as a kind of metaphysical realism in the sense of Putnam (1980). The statement of scientific realism therefore must not only deal with underdetermination and the dynamics of scientific theories but also answer the semantic challenges to metaphysical realism. As will be argued, the common core of these challenges is the proposition that a (...) (metaphysical) realist semantics leads to semantic agnosticism in the sense that we are unable to grasp the proper meanings and referents of our linguistic expressions. Having established this, I will focus more specifically on the question of whether scientific realism—in its state-of-the-art account—has the resources to make reference to scientific concepts intelligible such that the semantic challenges can be answered. (shrink)

The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features which instead explains the (...) attributes them to the mathematical practice of representing numbers using more concrete tokens, such as sets, strokes and so on. (shrink)

Externalism holds that the content of our utterances and thoughts are determined partly by the environment. Here, I offer an argument which suggests that externalism is incompatible with a natural view about ontological commitment--namely, the Quinean view that such commitments are fixed by the range of the variables in your theory. The idea in brief is that if Oscar mistakenly believes that water = XYZ, the externalist ontologically commits Oscar to two waterish kinds, whereas the Quinean commits him to one (...) such kind (albeit a metaphysically impossible kind). The penultimate section addresses a variety of objections to the argument. (shrink)

Although Kripke’s skepticism leads to the conclusion that meaning does not exist, his argument relies upon the supposition that more than one interpretation of words is consistent with linguistic evidence. Relying solely on metaphors, he assumes that there is a multiplicity of possible interpretations without providing any strict proof. In his book The Taming of the True, Neil Tennant pointed out that there are serious obstacles to this thesis and concluded that the skeptic’s nonstandard interpretations are “will o’ wisps.” In (...) this paper, contra Tennant, I demonstrate how to construct alternative interpretations of the language of algebra. These constructed interpretations avoid Tennant’s objections and are shown to be interdefinable with the standard interpretation. Kripke’s skepticism is, as it were, an incarnate demon. In contrast, it is currently uncertain whether the same technique is generally applicable to the construction of an alternative interpretation of natural language. However, the reinterpretation of those aspects of natural language that directly relate to numbers seems to be a promising candidate for the development of nonstandard interpretations of natural language. (shrink)

Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.

The Three Papers comprising this series, together with my earlier [34] also published in this journal, constitute an attempt to set out the major issues in the theoretical domain of reduction and to develop a general theory of theory reduction. The fourth paper, [34], though published separately from this trio, is integral to the presentation and should be read in conjunction with these papers. Even so, the presentation is limited in scope – roughly, to intertheoretic reduction among empirical theories – (...) and informal in presentation – not least because a satisfying formal account of theories has yet to be offered. And despite the length, the treatment is still condensed; often corroborating and/or intuitively helpful detail has had to be consigned to footnotes or omitted. I approach the problem from within my own naturalistic realist philosophy of science and formal analysis of abstract hierarchy in theory. The sources for the former are [25], [29], [31], and [32] and those for the latter essentially [27] and [30]. Hierarchical notions played a significant role in the already published [34]. (shrink)

I discuss Skolem's own ideas on his ‘paradox’, some classical disputes between Skolemites and Antiskolemites, and the underlying notion of ‘informal mathematics’, from a point of view which I hope to be rather unusual. I argue that the Skolemite cannot maintain that from an absolute point of view everything is in fact denumerable; on the other hand, the Antiskolemite is left with the onus of explaining the notion of informal mathematical knowledge of the intended model of set theory. 1 conclude (...) that one must take seriously the embodiment of the uncountable into countable languages, as a symptom of some specific features which characterize mathematical languages with respect to other languages. (shrink)

This article indicates some ways in which the fantasy of the brain in the vat has been used in thought experiments to discuss important philosophical problems. The first has to do with scepticism about the external world. The second has to do with Hilary Putnam’s arguments for the indeterminacy of reference and his rejection of metaphysical realism. The third issue to which the brain in the vat is relevant has to do with the difference between broad and narrow content of (...) beliefs and Putnam’s challenging assertion that reference is not ‘in the head’. Though there are brief suggestions for dealing with these problems and saving metaphysical realism, the main purpose is to indicate the relevance of the fantasy of the brain in the vat to discussion of these issues. (shrink)

The dream of a community of philosophers engaged in inquiry with shared standards of evidence and justification has long been with us. It has led some thinkers puzzled by our mathematical experience to look to mathematics for adjudication between competing views. I am skeptical of this approach and consider Skolem's philosophical uses of the Löwenheim-Skolem Theorem to exemplify it. I argue that these uses invariably beg the questions at issue. I say ?uses?, because I claim further that Skolem shifted his (...) position on the philosophical significance of the theorem as a result of a shift in his background beliefs. The nature of this shift and possible explanations for it are investigated. Ironically, Skolem's own case provides a historical example of the philosophical flexibility of his theorem. Our suspicion ought always to be aroused when a proof proves more than its means allow it. Something of this sort might be called ?a puffed-up proof?. Ludwig Wittgenstein, Remarks on the foundations of mathematics (revised edition), vol. 2, 21. (shrink)

Philosophers often assume that we can somehow quantitatively measure how good things are for people. But what does such talk mean? And what are the measures? In *Weighing Goods* John Broome offers one treatment of these questions. In his later *Weighing Lives* he offers a different treatment. This article discusses both positions but advocates a third. But while the three positions disagree about matters of meaning, they agree about the form of the measures. Roughly speaking, they are such that the (...) goodness of a lottery for an individual is equal to its expected goodness for her. (shrink)

This article discusses the argument we cannot have knowledge of abstract entities because they are not part of the causal order. The claim of this article is that the argument fails because of equivocation. Assume that the “causal order” is concerned with contingent facts involving time and space. Even if the existence of abstract entities is not contingent and does not involve time or space it does not follow that no truths about abstract entities are contingent or involve time or (...) space. I argue that it is the latter which is required to obtain the desired conclusion. (shrink)

This paper is concerned with the use of logic to solve philosophical problems. Such use of logic goes counter to the prevailing empiricist tradition in analytic circles. Specifically, model-theoretic tools are applied to three fundamental issues in the philosophy of logic and mathematics, namely, to the issue of the existence of mathematical entities, to the dispute between first- and second-order logic and to the definition of analyticity.

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980 ). Therefore, (...) it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988 ), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970 ), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non - isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995 ). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985 )], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994 ). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs. (shrink)

This paper develops a novel version of mathematical fictionalism and defends it against three objections or worries, viz., (i) an objection based on the fact that there are obvious disanalogies between mathematics and fiction; (ii) a worry about whether fictionalism is consistent with the fact that certain mathematical sentences are objectively correct whereas others are incorrect; and (iii) a recent objection due to John Burgess concerning “hermeneuticism” and “revolutionism”.

The replacement, under totalitarian regimes, of multiple sources of information with a single information monopoly confers an indeterminacy on the concepts of truth, fact, objectivity, and reality. From a pragmatist perspective, these words can then no longer mean exactly what they mean to speakers accustomed to freedom of discussion and inquiry. This corruption of discourse is detailed, e.g., in Arthur Koestler's Darkness at Noon, where criteria for belief?formation are ultimately completely divorced from the objects of belief. Like George Orwell, Koestler (...) postulates a coherent totalitarian perspective, in which the objects of belief are no longer relevant to belief?formation. The present article argues that, once the connection between belief?formation and the object of belief is severed, one can no longer have a coherent body of belief. There is also no evidence that this connection was ever severed in the mind of Koestler's historical protagonist Bukharin, who thus differs from his counterpart Rubashov in Koestler's novel. On the other hand, Koestler's critics are at fault for exaggerating the contrast between Rubashov and the historical Bukharin. The latter's confession at the Moscow Trials can be best understood as a testimony to the epistemically and linguistically corruptive process detailed in Koestler's novel. (shrink)

In "Sider on Existence" (Noužs, 2007), David Liebesman and Matti Eklund argue that my "indeterminacy argument", according to which quantifiers are never vague, clashes with my "naturalness argument", according to which quantifiers "carve at the joints". There is, I argue, no outright inconsistency. But Liebesman and Eklund have shown that my arguments are not as independent as it may have appeared. The best defense of the indeterminacy argument is via the naturalness argument.

There are two doctrines for which Quine is particularly well known: the doctrine of ontological commitment and the inscrutability thesis—the thesis that reference and quantification are inscrutable. At first glance, the two doctrines are squarely at odds. If there is no fact of the matter as to what our expressions refer to, then it would appear that no determinate commitments can be read off of our best theories. We argue here that the appearance of a clash between the two doctrines (...) is illusory. The reason that there is no real conflict is not simply that in determining our theories’ ontological commitments we naturally rely on our home language but also (and more importantly) that ontological commitment is not intimately tied to objectual quantification and a reference-first approach to language. Or so we will argue. We conclude with a new inscrutability argument which rests on the observation that the notion of objectual quantification, when properly cashed out, deflates. (shrink)

The doctrines of scientific realism have enjoyed a close and enduring, if not always harmonious, association with Tarski's semantic conception of truth and theories of formal semantics generally. From its inception Tarski's theory received unqualified support from some realists, like Karl Popper, who saw it as legitimizing the use of semantic notions in epistemology and the philosophy of science.

Nussbaum misconstrues the difference between Plato and Aristotle over what is real for a debate over a conception of truth. She seems to mistake Aristotle's arguments against Plato' version of realism as an argument against realism per se, though the texts do not permit such a reading. She claims Aristotle is convinced that realism involves a fatal “failure of reference,” yet she produces not a single text where Aristotle is even remotely concerned about such a failure of reference given the (...) commitments of realism. And nowhere is the crucial question of the relationship between Aristotle's antirealism and his method of appearances explicitly addressed or resolved.Nussbaum offers us a fashionable Aristotle. I have argued that, far from being attractive and obviously right on a deep and recent metaphysuical debate, Nussbaum's Aristotle is confused and inconsistent and thus that it is a good thing the texts do not support such a characterization. (shrink)

Inscrutability arguments threaten to reduce interpretationist metasemantic theories to absurdity. Can we find some way to block the arguments? A highly influential proposal in this regard is David Lewis’ ‘ eligibility ’ response: some theories are better than others, not because they fit the data better, but because they are framed in terms of more natural properties. The purposes of this paper are to outline the nature of the eligibility proposal, making the case that it is not ad hoc, but (...) instead flows naturally from three independently motivated elements; and to show that severe limitations afflict the proposal. In conclusion, I pick out the element of the eligibility response that is responsible for the limitations: future work in this area should therefore concentrate on amending this aspect of the overall theory. (shrink)

In (2001), (2003), and elsewhere, Ted Sider presents two arguments concerning the existential quantifier which are justly central to the recent discussion of metaontology. What we will call Sider's indeterminacy argument is an attempted reductio of the suggestion that the existential quantifier might be semantically indeterminate. What we will call Sider's naturalness argument is an argument for the claim that the semantic value of the existential quantifier is the most eligible existence-like meaning there is, à la David Lewis' eligibility theory (...) of meaning. We will argue that these arguments cannot be jointly maintained: Sider must give up at least one. Before arguing this, we will present Sider's two arguments in a bit more detail, and discuss their relationship. A few remarks on the broader significance of our conclusions are in order at the outset. One may think that since one successful argument for a given conclusion is sufficient the point that Sider's arguments cannot both work is purely academic. But we think that Sider's two arguments at bottom reflect different ways of thinking about metaphysics. Moreover, we think that it is only the naturalness argument that promises to deliver all that Sider wants when it comes to metaontology, and it is only the indeterminacy argument that promises to deliver all that Sider wants when it comes to ontology. (shrink)

I argue here for a number of ways that modern computational science requires a change in the way we represent the relationship between theory and applications. It requires a switch away from logical reconstruction of theories in order to take surface mathematical syntax seriously. In addition, syntactically different versions of the same theory have important differences for applications, and this shows that the semantic account of theories is inappropriate for some purposes. I also argue against formalist approaches in the philosophy (...) of science and for a greater role for perceptual knowledge rather than propositional knowledge in scientific empiricism. (shrink)

In a recent article G. H. Merrill has defended realism against an argument devised by Hilary Putnam. My first aim is to show that Merrill's defence is inadequate. I shall also argue that the proper conclusion of Putnam's argument is somewhat different from the conclusion Putnam himself offers.

J.D. Collier sees Putnam as arguing that metaphysical realism is false.' He sees the argument as proceeding from the background assumption that metaphysical realism has the consequence that truth is 'radically non-epistemic', so that 'an [epistemically] ideal theory could be radically wrong about the world' [3, p. 413]. But, according to Collier, Putnam argues that 'an ideal theory satisfying all of our methodological and theoretical constraints cannot be false' [3, p. 413]. Collier attempts to defend metaphysical realism against this Putnamian (...) attack by distinguishing 'between our ability to state a theory and our ability to conceive its possibility' [3, p. 413]. I will show that this defence is unsuccessful. (shrink)

Constructivism undermines realism by arguing that experience is mediated by concepts, and that there is no direct way to examine those aspects of objects that belong to them independently of our conceptualizations; perception is theory-laden. To defend realism one has to show first that perception relates us directly with the world without any intermediary conceptual framework. The result of this direct link is the nonconceptual content of experience. Second, one has to show that part of the nonconceptual content extracted from (...) the environment correctly represents features of mind independent objects. With regard to the first condition, I have argued elsewhere that a part of visual processing, which I call “perception,” is theory-neutral and nonconceptual. In this paper, facing the second demand, I argue that a part of the nonconceptual content of perception presents properties that are the properties of mind independent objects. I claim first that nonconceptual content is the appropriate level of analysis of the issue of realism since it avoids the main problems besetting various types of analysis of the issue at the level of beliefs about the world. Then I claim that a subset of the nonconceptual content presents features of objects in the environment as they really are. (shrink)

This paper contributes to an ongoing debate regarding the cognitive processes involved when one person predicts a target person's behavior and/or attributes a mental state to that target person. According to simulation theory, a person typically performs these tasks by employing some part of her brain as a simulation of what is going on in a corresponding part of the brain of the target person. I propose a general intuitive analysis of what 'simulation' means. Simulation is a particular way of (...) using one process to acquire knowledge about another process. What distinguishes simulation from other ways of acquiring knowledge is that simulation requires, for its non-accidental success, that the simulating process reflect significant aspects of the simulated process. This conceptual work is of independent philosophical interest, but it also enables me to argue for two conclusions that are of great significance to the debate about mental simulation theory. First, I argue that, in order to stake a non-trivial claim, simulation theory must hold that mental simulation involves what I call concretely similar processes. Second, I argue for the surprising conclusion that a significant class of cases that simulation theorists have claimed as intuitive cases of simulation do not actually involve simulation, after all. I close by sketching an alternative account that might handle these problematic cases. (shrink)