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.

In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the “axiomatic conception” of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic text books.

This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss a particularly popular approach (...) to resolving indeterminacy, before offering some brief closing reflections. We believe our discussion poses a serious challenge for most philosophical theories of mathematics, since it puts considerable pressure on all views that accept a non-trivial amount of determinacy for even basic arithmetic. (shrink)

A theory T is tight if different deductively closed extensions of T (in the same language) cannot be bi-interpretable. Many well-studied foundational theories are tight, including $\mathsf {PA}$ [39], $\mathsf {ZF}$, $\mathsf {Z}_2$, and $\mathsf {KM}$ [6]. In this article we extend Enayat’s investigations to subsystems of these latter two theories. We prove that restricting the Comprehension schema of $\mathsf {Z}_2$ and $\mathsf {KM}$ gives non-tight theories. Specifically, we show that $\mathsf {GB}$ and $\mathsf {ACA}_0$ each admit different bi-interpretable extensions, (...) and the same holds for their extensions by adding $\Sigma ^1_k$ -Comprehension, for $k \ge 1$. These results provide evidence that tightness characterizes $\mathsf {Z}_2$ and $\mathsf {KM}$ in a minimal way. (shrink)

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)

I defend Putnam’s modal structuralist view of mathematics but reject his claims that Wittgenstein’s remarks on Dedekind, Cantor, and set theory are verificationist. Putnam’s “realistic realism” showcases the plasticity of our “fitting” words to the world. The applications of this—in philosophy of language, mind, logic, and philosophy of computation—are robust. I defend Wittgenstein’s nonextensionalist understanding of the real numbers, showing how it fits Putnam’s view. Nonextensionalism and extensionalism about the real numbers are mathematically, philosophically, and logically robust, but the two (...) perspectives are often confused with one another. I separate them, using Turing’s work as an example. (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)

Conventionalism about mathematics has much in common with two other views: fictionalism and the multiverse view (aka plenitudinous platonism). The three views may differ over the existence of mathematical objects, but they agree in rejecting a certain kind of objectivity claim about mathematics, advocating instead an extreme pluralism. The early parts of the paper will try to elucidate this anti‐objectivist position, and question whether conventionalism really offers a third form of it distinct from fictionalism and the multiverse view. The paper (...) then turns to anti‐objectivism about logic, and suggests that here too conventionalism offers no distinct alternative, and that there are limits on the extent of pluralism/anti‐objectivism. It also argues that these limits can't be used to argue for more extensive objectivity within mathematics, and also that they do allow for more limited pluralism within logic, e.g. as regards resolution of semantic and property‐theoretic paradoxes. (shrink)

Fitch’s paradox shows, from fairly innocent-looking assumptions, that if there are any unknown truths, then there are unknowable truths. This is generally thought to deliver a blow to antirealist positions that imply that all truths are knowable. The present paper argues that a probabilistic version of antirealism escapes Fitch’s result while still offering all that antirealists should care for.

Hilary Putnam observes that a typical competent English speaker who cannot tell an elm tree from a beech tree may nevertheless use the word “elm” to make assertions and ask questions about elm trees. Putnam also observes that scientists may be wrong about the phenomena they investigate, while still being able to use their words to identify and raise research questions about it. This prompts him to ask what “language use” means in these contexts. He proposes two closely related methods (...) for answering this question. The first method is to investigate and clarify the uses of sentences and words in a given linguistic practice from the point of view of a participant in the practice. The second is to explain our applications of ‘is true’ and ‘refers’ to sentences and words whose uses are described in accord with the first method. In this paper I raise several problems for Putnam’s applications of these methods and sketch a different way of applying the methods that avoids the problems. (shrink)

One important role of belief systems is to allow us to represent information about a certain domain of inquiry. This paper presents a formal framework to accommodate such information representation. Three cognitive models to represent information are discussed: conceptual spaces, state-spaces, and the problem spaces familiar from artificial intelligence. After indicating their weakness to deal with partial information, it is argued that an alternative, formulated in terms of partial structures, can be provided which not only captures the positive features of (...) these models, but also accommodates the partiality of information ubiquitous in science and mathematics. (shrink)

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.

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)

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)

Few if any distinctions are more easily recognisable and assented to than that between _objects_, that is, things which are some ways, and that which they are, that is, _ways for objects to be_ (‘ways of being’ for short). In this paper I present an argument designed to show that this distinction is indeterminate in the sense that the truth-conditions of predicational sentences leave open what should count as an object and a way of being. The bulk of the argument (...) is inspired by the celebrated permutation argument advanced by Quine, Wallace, Putnam and others. (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)

In the Introduction to the Critique of Pure Reason Kant raises a famous question: how is metaphysics possible as a science? Kant posed this question for his predecessors in early modern philosophy. I raise this question anew for the resurgence of metaphysics within analytic philosophy. I begin by dividing the question of the possibility of metaphysics into separate questions about its semantic and epistemic possibility, and translate them into contemporary terms as: (1) Why do terms in metaphysical theories refer? (2) (...) How do we have knowledge in metaphysics? I then argue that the inflationary conception of metaphysics cannot explain the semantic possibility of metaphysics and, consequently, cannot explain its epistemic possibility. I then argue, more briefly, that a deflationary conception cannot satisfactorily answer the Kantian questions either. The critical path alone remains open. (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)

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)

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.

In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it. Unlike Quine's, this argument begins with the simple observation that there are different candidate interpretations of the predicate ‘could have been the case’. This is analogous to the observation that there are different candidate interpretations of the predicate ‘is a member of’. The argument then infers that the search for metaphysical necessities is misguided in much the way the ‘set-theoretic pluralist’ claims that the (...) search for the true axioms of set theory is. We show that the obvious responses to this argument fail. However, a new response has emerged that purports to prove, from higher-order logical principles, that metaphysical possibility is the broadest kind of possibility applying to propositions, and is to that extent special. We distill two lines of reasoning from the literature, and argue that their import depends on premises that any ‘modal pluralist’ should deny. Both presuppose that there is a unique typed hierarchy, which is what the modal pluralist, in the context of higher-order logic, should disavow. In other words, both presupposes that there is a unique candidate for what higher-order claims could mean. We consider the worry that, in a higher-order setting, modal pluralism faces an insuperable problem of articulation, collapses into modal monism, is vulnerable to the Russell-Myhill paradox, or even contravenes the truism that there is a unique actual world, and argue that these worries are misplaced. We also sketch the bearing of the resulting ‘Higher-Order Pluralism’ on the theory of content. One theme of the discussion is that, if Higher-Order Pluralism is correct, then there is no fixed metatheory from which to characterize higher-order reality. (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)

The corpus of physical theory is a paradigm of knowledge. The evolution of modern physical theory constitutes the clearest exemplar of the growth of knowledge. If the development of physical theory does not constitute an example of progress and growth in what we know about the Universe nothing does. So anyone interested in the theory of knowledge must be interested consequently in the evolution and content of physical theory. Crucial to the conception of physics as a paradigm of knowledge is (...) the way in which physical theory provides explanations of a vast diversity of natural phenomena on the basis of a very few fundamental principles. A central problem for the epistemologist is therefore what is theoretical explanation in physics? Here we can get good insight from what Redhead has said. Indeed one could agree with almost everything Redhead says and simply endorse much of his careful and extensive defence of the covering law account of explanation in the physical sciences at least as an ideal. However I shall, I fear, try the reader's patience by extending some of the considerations he introduced and raising those issues where we disagree, especially in the important area of statistical explanation. (shrink)

The paper argues against an intuitive reading of the Stone-von Neumann theorem as a categoricity result, thereby pointing out that this theorem does not entail any model-theoretical difference between the theories that validate it and those that don't.

Underdetermination of theory choice claims that empirical evidence fails to provide sufficient grounds for choosing a theory over its rivals. We explore the epistemological and methodological significance of this thesis by utilising a classificatory scheme to situate three arguments that purport to establish its plausibility. Proponents of these three arguments, W.V.O Quine, John Earman, and Kyle Stanford, use different premises to arrive at the conclusion that theory choice is empirically underdetermined and their classification along the proposed schema brings out the (...) variety in underdetermination arguments and the historical trajectory of the thesis. Although the epistemological significance of underdetermination—it is seen as undermining the doctrine of scientific realism—is widely discussed in the literature, Quine understood the acceptance of the thesis as interrogating the attitude one is justified in adopting towards rival theories. We argue that sticking with one’s own theory in the face of underdetermination leads to a distinct type of disagreement, which we present as the methodological significance of underdetermination. The examination of the methodological significance of underdetermination allows us to propose weak underdetermination as a philosophically interesting variant of underdetermination. (shrink)

Scientific change has two important dimensions: conceptual change and structural change. In this paper, I argue that the existence of conceptual change brings serious difficulties for scientific realism, and the existence of structural change makes structural realism look quite implausible. I then sketch an alternative account of scientific change, in terms of partial structures, that accommodates both conceptual and structural changes. The proposal, however, is not realist, and supports a structuralist version of van Fraassen’s constructive empiricism (structural empiricism).

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)

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)

This paper discusses the relevance of supertask computation for the determinacy of arithmetic. Recent work in the philosophy of physics has made plausible the possibility of supertask computers, capable of running through infinitely many individual computations in a finite time. A natural thought is that, if supertask computers are possible, this implies that arithmetical truth is determinate. In this paper we argue, via a careful analysis of putative arguments from supertask computations to determinacy, that this natural thought is mistaken: supertasks (...) are of no help in explaining arithmetical determinacy. (shrink)

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)

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)

I discuss and try to evaluate the argument about constructible sets made by Putnam in ‘ ”Models and Reality”, and some of the counterarguments directed against it in the literature. I shall conclude that Putnam’s argument, while correct in substance, nevertheless has no direct bearing on the philosophical question of unintended models of set theory.

In an earlier paper, I claimed that one version of Putnam's model-theoretic argument against realism turned on a subtle, but philosophically significant, mathematical mistake. Recently, Luca Bellotti has criticized my argument for this claim. This paper responds to Bellotti's criticisms.

PurposeThis paper aims to establish that a certain sort of mathematical pluralism is true. MethodsThe paper proceeds by arguing that that the best versions of mathematical Platonism and anti-Platonism both entail the relevant sort of mathematical pluralism. Result and ConclusionThis argument gives us the result that mathematical pluralism is true, and it also gives us the perhaps surprising result that mathematical Platonism and mathematical pluralism are perfectly compatible with one another.

In this paper, I outline and defend a novel approach to alethic pluralism, the thesis that truth has more than one metaphysical nature: where truth is, in part, explained by reference, it is relational in character and can be regarded as consisting in correspondence; but where instead truth does not depend upon reference it is not relational and involves only coherence. In the process, I articulate a clear sense in which truth may or may not depend upon reference: this involves (...) distinguishing semantic denotation from pragmatic speaker reference and claiming that there may or may not exist a metasemantic connection between these two notions. Finally, I argue that reference is not in general inscrutable—that this metasemantic connection does exist in the case of our ordinary discourse about present macroscopic concrete objects—but that it is in pure mathematics, where reference cannot be secured, and which therefore plays no role in accounting for truth. In this manner, alethic pluralism is upheld. (shrink)

ABSTRACT Is it possible to effect singular reference to mathematical objects in the abstractionist framework? I will argue that even if mathematical expressions pass the relevant syntactic and inferential tests to qualify as singular terms, that does not mean that their semantic function is to refer to a particular object. I will defend two arguments leading to this claim: the permutation argument for the referential indeterminacy of mathematical terms, and the argument from the semantic idleness of the terms introduced by (...) abstraction principles. (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 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)

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)