ABSTRACT The idea that two words can be instances of the same word is a central intuition in our conception of language. This fact underlies many of the claims that we make about how we communicate, and how we understand each other. Given this, irrespective of what we think words are, it is common to think that any putative ontology of words, must be able to explain this feature of language. That is, we need to provide criteria of identity for (...) word-types which allow us to individuate words such that it can be the case that two particular word-instances are instances of the same word-type. One solution, recently further developed by Irmak, holds that words are individuated by their history. In this paper, I argue that this view either fails to account for our intuitions about word identity, or is too vague to be a plausible answer to the problem of word individuation. (shrink)

Explaining intuitions in terms of "facts of our natural history" is compatible with rationally trusting them. This compatibilist view is defended in the present paper, focusing upon nomic and essentialist modal intuitions. The opposite, incompatibilist view alleges the following: If basic modal intuitions are due to our cognitive make-up or "imaginative habits" then the epistemologists are left with a mere non-rational feeling of compulsion on the side of the thinker. Intuitions then cannot inform us about modal reality. In contrast, the (...) paper argues that there are several independent sources of justification which make the feeling of compulsion rational: the prima-facie and a priori ones come from the obviousness of our basic modal intuitions and our not being able to imagine things otherwise, others, a posteriori, from the epistemic success of these intuitions. Further, the general scheme of evolutionary learning is reliable, reliability is preserved in the resulting individual's cognitive make-up, and we can come to know this a posteriori. The a posteriori appeal to evolution thus plays a subsidiary role in justification, filling the remaining gap and removing the residual doubt. Explaining modal intuitions is compatible with moderate realism about modality itself. (shrink)

Explaining intuitions in terms of “facts of our natural history” is compatible with rationally trusting them. This compatibilist view is defended in the present paper, focusing upon nomic and essentialist modal intuitions. The opposite, incompatibilist view alleges the following: If basic modal intuitions are due to our cognitive make‐up or “imaginative habits” then the epistemologists are left with a mere non‐rational feeling of compulsion on the side of the thinker. Intuitions then cannot inform us about modal reality. In contrast, the (...) paper argues that there are several independent sources of justification which make the feeling of compulsion rational: the prima‐facie and a priori ones come from the obviousness of our basic modal intuitions and our not being able to imagine things otherwise, others, a posteriori, from the epistemic success of these intuitions. Further, the general scheme of evolutionary learning is reliable, reliability is preserved in the resulting individual's cognitive make‐up, and we can come to know this a posteriori. The a posteriori appeal to evolution thus plays a subsidiary role in justification, filling the remaining gap and removing the residual doubt. Explaining modal intuitions is compatible with moderate realism about modality itself. (shrink)

A basic way of evaluating metaphysical theories is to ask whether they give satisfying answers to the questions they set out to resolve. I propose an account of “third-order” virtue that tells us what it takes for certain kinds of metaphysical theories to do so. We should think of these theories as recipes. I identify three good-making features of recipes and show that they translate to third-order theoretical virtues. I apply the view to two theories—mereological universalism and plenitudinous platonism—and draw (...) out their third-order virtues and vices. One lesson is that there is an important difference between essentially and non-essentially third-order vicious theories. I also argue that if a theory is essentially third-order vicious, it cannot be assessed for more standard “second-order” theoretical virtues and vices, like parsimony. This motivates the idea that third-order virtues are distinct from second-order ones. Finally, I suggest that the relationship between truth, progress, and third-order virtue is more complex than it seems. (shrink)

James Thomson envisaged a lamp which would be turned on for 1 minute, off for 1/2 minute, on for 1/4 minute, etc. ad infinitum. He asked whether the lamp would be on or off at the end of 2 minutes. Use of “internal set theory” (a version of nonstandard analysis), developed by Edward Nelson, shows Thomson's lamp is chimerical; its copy within set theory yields a contradiction. The demonstration extends to placing restrictions on other “infinite tasks” such as Zeno's paradoxes (...) of motion and Kant's First Antinomy. Resolution of such logical-philosophical problems leads to some very general constraints which must be placed upon the syntax of physical theories. In particular, at some scale space and time would appear granular. The suitability of internal set theory for analyzing phenomena is examined, using a paper by Alper and Bridger (1997) to frame the discussion. (shrink)

Deflationists about truth embrace the positive thesis that the notion of truth is useful as a logical device, for such purposes as blanket endorsement, and the negative thesis that the notion doesn’t have any legitimate applications beyond its logical uses, so it cannot play a significant theoretical role in scientific inquiry or causal explanation. Focusing on Christopher Hill as exemplary deflationist, the present paper takes issue with the negative thesis, arguing that, without making use of the notion of truth conditions, (...) we have little hope for a scientific understanding of human speech, thought, and action. For the reference relation, the situation is different. Inscrutability arguments give reason to think that a more-than-deflationary theory of reference is unattainable. With respect to reference, deflationism is the only game in town. (shrink)

In this article, I present a novel account of a priori warrant, which I then use to examine the relationship between a priori and a posteriori warrant in mathematics. According to this account of a priori warrant, the reason that a posteriori warrant is subordinate to a priori warrant in mathematics is because processes that produce a priori warrant are reliable independent of the contexts in which they are used, whereas this is not true for processes that produce a posteriori (...) warrant. Following this, I show how this difference explains why a priori warranting processes, such as proof (and mathematical intuition) can, and a posteriori warranting processes cannot, definitively establish mathematical results. I then show why, in the event of a conflict between a priori and a posteriori warranting processes, the former undermine, and cannot be undermined by, the latter. Finally, I explain the connection between apriority and mathematical necessity, but do so in a way that allows for the existence of contingent a priori knowledge. (shrink)

While anti-exceptionalism about logic is now a popular topic within the philosophy of logic, there’s still a lack of clarity over what the proposal amounts to. currently, it is most common to conceive of AEL as the proposal that logic is continuous with the sciences. Yet, as we show here, this conception of AEL is unhelpful due to both its lack of precision, and its distortion of the current debates. Rather, AEL is better understood as the rejection of certain traditional (...) properties of logic. The picture that results is not of one singular position, but rather a cluster of often connected positions with distinct motivations, understood in terms of their rejection of clusters of the various traditional properties. In order to show the fruitfulness of this new conception of AEL, we distinguish between two prominent versions of the position, metaphysical and epistemological AEL, and show how the two positions need not stand or fall together. (shrink)

I propose that we approach the epistemology of modality by putting modal metaphysics first and, specifically, by investigating the metaphysics of essence. Following a prominent Neo-Aristotelian view, I hold that metaphysical necessity depends on the nature of things, namely their essences. I further clarify that essences are core properties having distinctive superexplanatory powers. In the case of natural kinds, which is my focus in the paper, superexplanatoriness is due to the fact that the essence of a kind is what causes (...) all the many properties and behaviors that are typically shared by all the instances of the kind. Accordingly, we know what is necessarily true of kinds by knowing what is essential to them in the sense of actually playing such causal-explanatory roles. Modal reasoning aimed at discovering metaphysical necessity thus proceeds via essentialist deduction: we move from essentialist truths to reach necessary truths. (shrink)

This work deals with obstacles hindering a metaphysics of laws of nature in terms of dispositions, i.e., of fundamental properties that are causal powers. A recent analysis of the principle of least action has put into question the viability of dispositionalism in the case of classical mechanics, generally seen as the physical theory most easily amenable to a dispositional ontology. Here, a proper consideration of the framework role played by the least action principle within the classical image of the world (...) allows us to build a consistent metaphysics of dispositions as charges of interactions. In doing so we develop a general approach that opens the way towards an ontology of dispositions for fundamental physics also beyond classical mechanics. (shrink)

In this article, I discuss Hawthorne'€™s contextualist solution to Benacerraf'€™s dilemma. He wants to find a satisfactory epistemology to go with realist ontology, namely with causally inaccessible mathematical and modal entities. I claim that he is unsuccessful. The contextualist theories of knowledge attributions were primarily developed as a response to the skeptical argument based on the deductive closure principle. Hawthorne uses the same strategy in his attempt to solve the epistemologist puzzle facing the proponents of mathematical and modal realism, but (...) this problem is of a different nature than the skeptical one. The contextualist theory of knowledge attributions cannot help us with the question about the nature of mathematical and modal reality and how they can be known. I further argue that Hawthorne'€™s account does not say anything about a priori status of mathematical and modal knowledge. Later, Hawthorne adds to his account an implausible claim that in some contexts a gettierized belief counts as knowledge. (shrink)

I will analyse some properties of abduction that are essential from a logical standpoint. When dealing with the so-called ‘inferential problem’, I will opt for the more general concepts of input and output instead of those of premisses and conclusions, and show that in this framework two consequences can be derived that help clarify basic logical aspects of abductive reasoning: (i) it is more natural to accept the ‘multimodal’ and ‘context-dependent’ character of the inferences involved, (ii) inferences are not merely (...) conceived of in the terms of the process leading to the ‘generation of an output’ or to the proof of it, as in the traditional and standard view of deductive proofs, but rather, from this perspective abductive inferences can be seen as related to logical processes in which input and output fail to hold each other in an expected relation, with the solution involving the modification of inputs, not that of outputs. I will also describe that if we wish to naturalize the logic of the abductive processes and its special consequence relation, we should refer to the following main aspects: ‘optimization of situatedness’, ‘maximization of changeability’ of both input and output, and high ‘information-sensitiveness’. (shrink)

This talk surveys a range of positions on the fundamental metaphysical and epistemological questions about elementary logic, for example, as a starting point: what is the subject matter of logic—what makes its truths true? how do we come to know the truths of logic? A taxonomy is approached by beginning from well-known schools of thought in the philosophy of mathematics—Logicism, Intuitionism, Formalism, Realism—and sketching roughly corresponding views in the philosophy of logic. Kant, Mill, Frege, Wittgenstein, Carnap, Ayer, Quine, and Putnam (...) are among the philosophers considered along the way. (shrink)

For some time now, academic philosophers of mathematics have concentrated on intramural debates, the most conspicuous of which has centered on Benacerraf's epistemological challenge. By the late 1980s, something of a consensus had developed on how best to respond to this challenge. But answering Benacerraf leaves untouched the more advanced epistemological question of how the axioms are justified, a question that bears on actual practice in the foundations of set theory. I suggest that the time is ripe for philosophers of (...) mathematics to turn outward, to take on a problem of real importance for mathematics itself. (shrink)

Naturalism in philosophy is sometimes thought to imply both scientific realism and a brand of mathematical realism that has methodological consequences for the practice of mathematics. I suggest that naturalism does not yield such a brand of mathematical realism, that naturalism views ontology as irrelevant to mathematical methodology, and that approaching methodological questions from this naturalistic perspective illuminates issues and considerations previously overshadowed by (irrelevant) ontological concerns.

My goal here is to explore the relationship between pure and applied mathematics and then, eventually, to draw a few morals for both. In particular, I hope to show that this relationship has not been static, that the historical rise of pure mathematics has coincided with a gradual shift in our understanding of how mathematics works in application to the world. In some circles today, it is held that historical developments of this sort simply represent changes in fashion, or in (...) social arrangements, governments, power structures, or some such thing, but I resist the full force of this way of thinking, clinging to the old school notion that we have gradually learned more about the world over time, that our opinions on these matters have improved, and that seeing how we reached the point we now occupy may help us avoid falling back into old philosophies that are now no longer viable. In that spirit, it seems to me that once we focus on the general question of how mathematics relates to science, one observation is immediate: the march of the centuries has produced an amusing reversal of philosophical fortunes. Let me begin there. In the beginning, that is, in Plato, mathematical knowledge was sharply distinguished from ordinary perceptual belief about the world. According to Plato’s metaphysics, mathematics is the study of eternal and unchanging abstract Forms,1 while science is uncertain and changeable opinion about the world of mere becoming. Indeed, in Plato’s lights, of the two, only mathematics deserves to be called ‘knowledge’ at all! Of course if sense perception cannot give us knowledge, if mathematics is not about perceivable things, then Plato owes us an account of how we ordinary humans achieve this wonderful insight into the properties of the abstract world of Forms. Plato’s answer is that we do not actually acquire mathematical knowledge at all; rather, we recollect it from a time before birth, when our souls, unencumbered by physical bodies, were free to commune with the Forms, and not just the mathematical ones, either – also Truth, Beauty, Justice, the Good, and so on.2 Whatever appeal this position may have held for the ancient Greeks, it will not begin to satisfy a contemporary, scientifically minded philosopher.. (shrink)

One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers.

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

Penelope Maddy has defended a modified version of mathematical platonism that involves the perception of some sets. Frederick Suppe has developed a conclusive reasons account of empirical knowledge that, when applied to the sets of interest to Maddy, yields that we have knowledge of these sets. Thus, Benacerraf's challenge to the platonist to account for mathematical knowledge has been met, at least in part. Moreover, it is argued that the modalities involved in Suppe's conclusive reasons account of knowledge can be (...) handled without recourse to either laws of nature or possible worlds, and that this approach is preferable. (shrink)

Ante rem structuralism is the doctnne that mathematics descubes a realm of abstract (structural) universab. According to its proponents, appeal to the exutence of these universab provides a source distinctive insight into the epistemology of mathematics, in particular insight into the so-called 'access problem' of explaining how mathematicians can reliably access truths about an abstract realm to which they cannot travel andfiom which they recave no signab. Stewart Shapiro offers the most developed version of this view to date. Through an (...) examination of Shapiro's proposed structuralist epistemology for mathematics I argue that ante rem structuralism faib to provide the ingredients for a satisfactory resolution of the access problem for infinite structures (whether small or large). (shrink)

Philosophers have recently expressed interest in accounting for the usefulness of mathematics to science. However, it is certainly not a new concern. Putnam and Quine have each worked out an argument for the existence of mathematical objects from the indispensability of mathematics to science. Were Quine or Putnam to disregard the applicability of mathematics to science, he would not have had as strong a case for platonism. But I think there must be ways of parsing mathematical sentences which account for (...) applicability of mathematics and also do not require us to believe in entities we have no evidence for, other than through reading these sentences literally. We will explore a particular way to interpret sentences of arithmetic which promises to account for their applicability without bringing in metaphysics not also brought in by science. The investigation will be limited to the arithmetic of cardinal numbers. The general strategy is to argue for the analogy between arithmetic and science, rather than to argue for one case having a particular characteristic independently of the other. (shrink)

A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the Continuum Hypothesis, by relying on the existence of a plurality of set-theoretical universes except for a single one, i.e., the well-known set-theoretical universe V associated with the cumulative hierarchy of sets. The multiverse approach has some varying versions of the general concept of multiverse yet my intention is to primarily (...) address ontological multiversism as advocated, for instance, by Hamkins or Väätänen, precisely for the reason that they proclaim, to the one or the other extent, ontological preoccupations for the introduction of respective multiverse theories. Taking also into account Woodin’s and Steel’s multiverse versions, I take up an argumentation against multiversism, and in a certain sense against platonism in mathematical foundations, mainly on subjectively founded grounds, while keeping an eye on Clarke-Doane’s concern with Benacerraf’s challenge. I note that even though the paper is rather technically constructed in arguing against multiversism, the non-negligible philosophical part is influenced to a certain extent by a phenomenologically motivated view of the matter. (shrink)

I argue on the basis of an example, Fourier theory applied to the problem of vibration, that Field's program for nominalizing science is unlikely to succeed generally, since no nominalistic variant will provide us with the kind of physical insight into the phenomena that the standard theory supplies. Consideration of the same example also shows, I argue, that some of the motivation for mathematical fictionalism, particularly the alleged problem of cognitive access, is more apparent than real.

In this paper I argue three things: (1) that the interactionist view underlying Benacerraf's (1973) challenge to mathematical beliefs renders inexplicable the reliability of most of our beliefs in physics; (2) that examples from mathematical physics suggest that we should view reliability differently; and (3) that abstract mathematical considerations are indispensable to explanations of the reliability of our beliefs.

Metaphysical views typically draw some distinction between reality and appearance, endorsing realism about some subject matters and antirealism about others. There are different conceptions of how best to construe antirealist theories. A simple view has it that we are antirealists about a subject matter when we believe that this subject matter fails to obtain. This paper discusses an alternative view, which I will call the fundamentality-based conception of antirealism. We are antirealists in this sense when we think that the relevant (...) matter fails to be constitutive of fundamental reality. The following discussion will not rely on any particular understanding of fundamental reality, covering conceptions based on grounding, naturalness and truthmaking, to name three salient ones. This paper argues that there are serious issues with fundamentality-based metaphysics. It will be argued that: the fundamentality-based approach shapes and restricts our realist and antirealist views in unsatisfying ways, that it is unable to handle the conflicting facts that lie across the envisaged ‘layers’ of the metaphysically structured world, and that the methodological reasons for adopting the fundamentality-based approach fail. The paper will conclude with a diagnosis of the discussed issues, identifying a common source. (shrink)

Otto Neurath’s empiricist methodology of economics and his contributions to political economy have gained increasing attention in recent years. We connect this research with contemporary debates regarding the epistemological status of thought experiments by reconstructing Neurath’s utopias as linchpins of thought experiments. In our three reconstructed examples of different uses of utopias/dystopias in thought experiments we employ a reformulation of Häggqvist’s model for thought experiments and we argue that: (1) Our reformulation of Häggqvist’s model more adequately complies with many uses (...) of thought experiments, especially with the open-ended discussions of utopias and dystopias in thought experiments. (2) As a strict logical empiricist, Neurath is committed to a strictly empiricist account of thought experiments. John Norton’s empiricist argument view can indeed account for the justifications of empirical beliefs and genuine discoveries targeted by scientific utopianism in three distinct (yet connected) ways, all of which Neurath already contemplated: (2.I) Dealing with utopias and thought experiments on a regular basis increases creativity and inventiveness. (2.II) Particular ways of presenting knowledge facilitate scientific discovery and social progress. (2.III) The use of utopias in thought experiments can prompt conceptual change and allow access to new phenomena. We conclude by highlighting that, even though thought experiments support a positive attitude for exploring new social possibilities, Neurath points out that active decisions are unavoidable. The exploration of alternatives and the awareness of a need for decisions in policy discussion avert a technocratic outlook in social science. (shrink)

Can there be objects that are ‘thin’ in the sense that very little is required for their existence? A number of philosophers have thought so. For instance, many Fregeans believe it suffices for the existence of directions that there be lines standing in the relation of parallelism; other philosophers believe it suffices for a mathematical theory to have a model that the theory be coherent. This article explains the appeal of thin objects, discusses the three most important strategies for articulating (...) and defending the idea of such objects, and outlines some problems that these strategies face. (shrink)

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. (shrink)

The goal of this paper is to raise a novel objection to Lewis’s modal realist epistemology. After reformulating his modal epistemology, I shall argue that his view that we have necessary knowledge of the existence of counterparts ends up with an absurdity. Specifically, his analogy between mathematical knowledge and modal knowledge leads to an unpleasant conclusion that one’s counterpart exists in all possible worlds. My argument shows that if Lewis’s modal realism is true, we cannot know what is possible. Conversely, (...) if we can know what is possible, his modal realism is false. In the remainder of the paper, I shall consider and block possible objections to my argument. (shrink)

In the introduction to his Realism, mathematics and modality, and in earlier papers included in that collection, Hartry Field offered an epistemological challenge to platonism in the philosophy of mathematics. Justin Clarke-Doane Truth, objects, infinity: New perspectives on the philosophy of Paul Benacerraf, 2016) argues that Field’s challenge is an illusion: it does not pose a genuine problem for platonism. My aim is to show that Clarke-Doane’s argument relies on a misunderstanding of Field’s challenge.

Platonism in the philosophy of mathematics is the doctrine that there are mathematical objects such as numbers. John Burgess and Gideon Rosen have argued that that there is no good epistemological argument against platonism. They propose a dilemma, claiming that epistemological arguments against platonism either rely on a dubious epistemology, or resemble a dubious sceptical argument concerning perceptual knowledge. Against Burgess and Rosen, I show that an epistemological anti- platonist argument proposed by Hartry Field avoids both horns of their dilemma.

Penelope Maddy's original solution to the dilemma posed by Benacerraf in his 'Mathematical Truth' was to reconcile mathematical empiricism with mathematical realism by arguing that we can perceive realistically construed sets. Though her hypothesis has attracted considerable critical attention, much of it, in my view, misses the point. In this paper I vigorously defend Maddy's account against published criticisms, not because I think it is true, but because these criticisms have functioned to obscure a more fundamental issue that is well (...) worth addressing: in general -- and not only in the mathematical domain -- empiricism and realism simply cannot be reconciled by means of an account of perception anything like Maddy's. But because Maddy's account of perception is so plausible, this conclusion raises the specter of the broader incompatibility of realism and empiricism, which contemporary philosophers are frequently at pains to forget. (shrink)

What should a Quinean naturalist say about moral and mathematical truth? If Quine’s naturalism is understood as the view that we should look to natural science as the ultimate ‘arbiter of truth’, this leads rather quickly to what Huw Price has called ‘placement problems’ of placing moral and mathematical truth in an empirical scientific world-view. Against this understanding of the demands of naturalism, I argue that a proper understanding of the reasons Quine gives for privileging ‘natural science’ as authoritative when (...) it comes to questions of truth and existence also apply to other stable and considered elements of our inherited world-view, including, arguably, our firmly held mathematical and moral beliefs. If so, then the ‘thin’ mathematical and moral realisms of Penelope Maddy and T. M. Scanlon, respectively, are vindicated. We do not need to shoehorn mathematical and moral truths into the pushings and pullings of our empirical scientific world-view; for the busy sailor adrift on Neurath’s boat, mathematical and moral truths already have their place. (shrink)

This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply (...) the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism. (shrink)

Charles Peirce's diagrammatic logic — the Existential Graphs — is presented as a tool for illuminating how we know necessity, in answer to Benacerraf's famous challenge that most ‘semantics for mathematics’ do not ‘fit an acceptable epistemology’. It is suggested that necessary reasoning is in essence a recognition that a certain structure has the particular structure that it has. This means that, contra Hume and his contemporary heirs, necessity is observable. One just needs to pay attention, not merely to individual (...) things but to how those things are related in larger structures, certain aspects of which relations force certain other aspects to be a certain way. (shrink)

In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...) localized a faculty to register them. We defend the perception of necessity against such Humeanism, drawing on examples from mathematics. (shrink)

This paper puts forward and defends an account of mathematical truth, and in particular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerrafs work on informal rigour. Kreisel defends the view that axioms are arrived at by a rigorous examination of our informal notions, as opposed to being stipulated or arrived at by trial and error. This view is then supplemented by a (...) Fregean account of the objectivity and our knowledge of abstract objects. It is then argued that the resulting view faces no insurmountable metaphysical or epistemic obstacles. (shrink)

Several philosophers have suggested that some scientific explanations work not by virtue of describing aspects of the world’s causal history and relations, but rather by citing mathematical facts. This paper investigates what mathematical facts could be in order for them to figure in such “distinctively mathematical” scientific explanations. For “distinctively mathematical explanations” to be explanations by constraint, mathematical language cannot operate in science as representationalism or platonism describes. It can operate as Aristotelian realism describes. That is because Aristotelian realism enables (...) mathematics to apply to the physical world without a morphism’s mediation. (shrink)

This paper argues that in mathematical practice, conjectures are sometimes confirmed by “Inference to the Best Explanation” as applied to some mathematical evidence. IBE operates in mathematics in the same way as IBE in science. When applied to empirical evidence, IBE sometimes helps to justify the expansion of scientists’ ontological commitments. Analogously, when applied to mathematical evidence, IBE sometimes helps to justify mathematicians' in expanding the range of their ontological commitments. IBE supplements other forms of non-deductive reasoning in mathematics, avoiding (...) obstacles sometimes faced by enumerative induction or hypothetico-deductive reasoning. Both platonist and non-platonist interpretations of mathematics ought to accommodate explanation in mathematics and ought to recognize IBE in mathematics, though these interpretations disagree on the ontological commitments that mathematicians ought to have. This paper offers an inductive account of why mathematical IBE tends to lead to mathematical truths. (shrink)

In this essay, I seek to demonstrate the interplay of philosophical voices – particularly, that of a platonist voice and a community-agreement-view voice – that drives Wittgenstein’s rule-following dialectic forward; and I argue that each voice succumbs to a particular form of dialectical oscillation that renders its response to the problem of rule-following philosophically inadequate. Finally, I suggest that, by seeing and taking stock of the dilemma in which these responses to the skeptical problem are caught, we can come to (...) appreciate Wittgenstein’s own view of what might constitute a proper a response to the so-called problem of rule-following. This view can be preliminarily characterized by saying that Wittgenstein’s aim is to dissolve the temptation to philosophically rebut the skeptical challenge posed by the rule-following dialectic, an aim he achieves by revealing the semantic emptiness of the apparent sentences that raise the skeptical problem. (shrink)