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)

Hilary Putnam first published the consistency objection against Ludwig Wittgenstein’s account of mathematics in 1979. In 1983, Putnam and Benacerraf raised this objection against all conventionalist accounts of mathematics. I discuss the 1979 version and the scenario argument, which supports the key premise of the objection. The wide applicability of this objection is not apparent; I thus raise it against an imaginary axiomatic theory T similar to Peano arithmetic in all relevant aspects. I argue that a conventionalist can (...) explain the consistency of T and suggest that an analogous explanation can be provided for the consistency of Peano arithmetic. (shrink)

Wittgenstein's views on mathematics are radically original. He criticizes most of the traditional philosophies of mathematics. His views have been subject to harsh criticisms. In this dissertation, I attempt to defend Wittgenstein's philosophy of mathematics from two objections: the objectivity objection and the consistency objection. The first claims that Wittgenstein's account of mathematics is not sufficient for the objectivity of mathematics; the second claims that it is only a partial account of mathematics because it (...) cannot explain the semantic properties of mathematical systems. ;The first chapter outlines Wittgenstein's Philosophy of Mathematics by stressing the differences and similarities with more traditional accounts. The second chapter discusses the objectivity objection. I distinguish epistemic from non-epistemic objectivity and discuss their relation with the issue of realism and anti-realism. I conclude first that either notion of objectivity is insufficient for disparaging anti-realist accounts of mathematics and, second, that Wittgenstein's account is sufficient for both. The third chapter is a detailed discussion of the consistency objection. Some unsuccessful replies are discussed. In the fourth chapter, I reformulate the objection by means of a rather simple axiomatic system that I call theory T. This allows us to see more clearly why the replies considered before are not successful and, furthermore, which view Wittgenstein is actually required to hold in order to reject the objection. In the fifth chapter, I offer a brief outline of Wittgenstein's account of logic and, finally, show how it provides the required answer to the consistency objection. (shrink)

The subject of this investigation is the role of conventions in the formulation of Thomas Reid’s theory of the geometry of vision, which he calls the ‘geometry of visibles’. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reid’s ‘geometry of visibles’ and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject to a choice of conventions regarding the (...) construction and assignment of its various properties, especially metric properties, and this fact undermines the claim for a unique non-Euclidean status for the geometry of vision. Finally, a suggestion is offered for trying to reconcile Reid’s direct realist theory of perception with his geometry of visibles.While Thomas Reid is well-known as the leading exponent of the Scottish ‘common-sense’ school of philosophy, his role in the history of geometry has only recently been drawing the attention of the scholarly community. In particular, several influential works, by N. Daniels and R. B. Angell, have claimed Reid as the discoverer of non-Euclidean geometry; an achievement, moreover, that pre-dates the geometries of Lobachevsky, Bolyai, and Gauss by over a half century. Reid’s alleged discovery appears within the context of his analysis of the geometry of the visual field, which he dubs the ‘geometry of visibles’. In summarizing the importance of Reid’s philosophy in this area, Daniels is led to conclude that ‘there can remain little doubt that Reid intends the geometry of visibles to be an alternative to Euclidean geometry’;1 while Angell, similarly inspired by Reid, draws a much stronger inference: ‘The geometry which precisely and naturally fits the actual configurations of the visual field is a non-Euclidean, two-dimensional, elliptical geometry. In substance, this thesis was advanced by Thomas Reid in 1764...’, 2 The significance of these findings has not gone unnoticed in mathematical and scientific circles, moreover, for Reid’s name is beginning to appear more frequently in historical surveys of the development of geometry and the theories of space., 3Implicit in the recent work on Reid’s ‘geometry of visibles’, or GOV, one can discern two closely related but distinct arguments: first, that Reid did in fact formulate a non-Euclidean geometry, and second, that the GOV is non-Euclidean. This essay will investigate mainly the latter claim, although a lengthy discussion will be accorded to the first. Overall, in contrast to the optimistic reports of a non-Euclidean GOV, it will be argued that there is a great deal of conceptual freedom in the construction of any geometry pertaining to the visual field. Rather than single out a non-Euclidean structure as the only geometry consistent with visual phenomena, an examination of Reid, Daniels, and Angell will reveal the crucial role of geometric ‘conventions’, especially of the metric sort, in the formulation of the GOV. Consequently, while a non-Euclidean geometry is consistent with Reid’s GOV, it is only one of many different geometrical structures that a GOV can possess. Angell’s theory that the GOV can only be construed as non-Euclidean, is thus incorrect. After an exploration of Reid’s theory and the alleged non-Euclidean nature of the GOV, in 1 and 2 respectively, the focus will turn to the tacit role of conventionalism in Daniels’ reconstruction of Reid’s GOV argument, and in the contemporary treatment of a non-Euclidean visual geometry offered by Angell. Finally, in the conclusion, a suggestion will be offered for a possible reconstruction of Reid’s GOV that does not violate his avowed ‘direct realist’ theory of perception, since this epistemological thesis largely prompted his formulation of the GOV. (shrink)

Reichenbach, Grünbaum, and others have argued that special relativity is based on arbitrary conventions concerning clock synchronizations. Here we present a mathematical framework which shows that this conventionality is almost equivalent to the arbitrariness in the choice of coordinates in an inertial system. Since preferred systems of coordinates can uniquely be defined by means of the Lorentz invariance of physical laws irrespective of the properties of light signals, a special clock synchronization—Einstein's standard synchrony—is selected by this principle. No further restrictions (...) conerning light signal synchronization, as proposed, e.g., by Ellis and Bowman, are required in order to refute conventionalism in special relativity. (shrink)

In this paper I argue that the positivist–conventionalist interpretation of the Restricted Principle of Relativity is flawed, due to the positivists’ own understanding of conventions and their origins. I claim in the paper that, to understand the conventionalist thesis, one has to diambiguate between three types of convention; the linguistic conventions stemming from the fundamental role of mathematical axioms, the conventions stemming from the coordination betweeh theoretical statements and physical, observable facts or entities, and conventions that are made possible by (...) possible revisions to theory. I claim that it is not possible to interpret the Principle of Relativity as based on one of these three types of convention. This renders the conventionalist interpretation of the Principle of Relativity untenable. The paper is part of a larger project that aims to understand the philosophical significance of the Principle of Relativity. (shrink)

In this paper I argue that the positivist–conventionalist interpretation of the Restricted Principle of Relativity is flawed, due to the positivists’ own understanding of conventions and their origins. I claim in the paper that, to understand the conventionalist thesis, one has to diambiguate between three types of convention; the linguistic conventions stemming from the fundamental role of mathematical axioms (conceptual conventions), the conventions stemming from the coordination betweeh theoretical statements and physical, observable facts or entities (coordinative definitions), and conventions that (...) are made possible by possible revisions to theory (the thesis of empirical underdetermination). I claim that it is not possible to interpret the Principle of Relativity as based on one of these three types of convention. This renders the conventionalist interpretation of the Principle of Relativity untenable. The paper is part of a larger project that aims to understand the philosophical significance of the Principle of Relativity. (shrink)

This dissertation makes two primary contributions. The first three chapters develop an interpretation of Carnap's Meta-Philosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tolerance. Most importantly, I suggest, is that Carnap sees philosophy as contiguous with science—as a part of the scientific enterprise—so utilizing the very same methods and subject to the same limitations. I argue that the methodological reforms he suggests for philosophy amount to philosophy as the explication of (...) the concepts of science through the construction and use of suitably robust meta-logical languages. My primary interpretive claim is that Carnap's understanding of logic and mathematics as a set of formal auxiliaries is premised upon this prior analysis of the character of logico-mathematical knowledge, his understanding of its role in the language of science, and the methods used by practicing mathematicians. Thus the Principle of Tolerance, and so Carnap's logical pluralism, is licensed and justified by these methodological insights. This interpretation of Carnap's program contrasts with the popular Deflationary reading as proposed in Goldfarb & Ricketts. The leading idea they attribute to Carnap is a Logocentrism: That philosophical assertions are always made relative to some particular language, and that our choice of syntactical rules for a language are constitutive of its inferential structure and methods of possible justification. Consequently Tolerance is considered the foundation of Carnap's entire program. My third chapter argues that this reading makes Carnap's program philosophically inert, and I present significant evidence that such a reading is misguided. The final chapter attempts to extend the methodological ideals of Carnap's program to the analysis of the ongoing debate between category- and set-theoretic foundations for mathematics. Recent criticism of category theory as a foundation charges that it is neither autonomous from set theory, nor offers a suitable ontological grounding for mathematics. I argue that an analysis of concepts can be foundationally informative without requiring the construction of those concepts from first principles, and that ontological worries can be seen as methodologically unfruitful. (shrink)

The statements traditionally labelled “necessary,” among them the valid theorems of mathematics and logic, are identified as “those whose truth is independent of experience.” The “truth” of a necessary statement has to be independent of the truth or falsity of experiential statements; a necessary statement can be neither confirmed nor refuted by empirical tests.The admission of genuinely necessary statements presents the empiricist with a troublesome problem. For an empiricist may be defined, in terms of the current idiom, as one (...) who adheres to some version, however “weak,” of a principle of verifiability. One, that is, who claims that no statement can have cognitive meaning unless its truth depends, however indirectly, upon the truth of experiential statements; unless it can be provisionally confirrned or refuted by empirical tests. Allowing that some necessary statements have cognitive meaning, then, would be to provide a prima facie case against the validity of the principle of verifiability. (shrink)

Distinctively mathematical explanations (DMEs) explain natural phenomena primarily by appeal to mathematical facts. One important question is whether there can be an ontic account of DME. An ontic account of DME would treat the explananda and explanantia of DMEs as ontic structures and the explanatory relation between them as an ontic relation (e.g., Pincock 2015, Povich 2021). Here I present a conventionalist account of DME, defend it against objections, and argue that it should be considered ontic. Notably, if indeed it (...) is ontic, the conventionalist account seems to avoid a convincing objection to other ontic accounts (Kuorikoski 2021). (shrink)

Conventionalism about mathematics claims that mathematical truths are true by linguistic convention. This is often spelled out by appealing to facts concerning rules of inference and formal systems, but this leads to a problem: since the incompleteness theorems we’ve known that syntactic notions can be expressed using arithmetical sentences. There is serious prima facie tension here: how can mathematics be a matter of convention and syntax a matter of fact given the arithmetization of syntax? This challenge has (...) been pressed in the literature by Hilary Putnam and Peter Koellner. In this paper I sketch a conventionalist theory of mathematics, show that this conventionalist theory can meet the challenge just raised , and clarify the type of mathematical pluralism endorsed by the conventionalist by introducing the notion of a semantic counterpart. The paper’s aim is an improved understanding of conventionalism, pluralism, and the relationship between them. (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and (...) mathematics. It argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

The article is devoted to the applicability of Wittgenstein’s following the rule in the context of his philosophy of mathematics to real mathematical practice. It is noted that in «Philosophical Investigations» and «Remarks on the Foundations of Mathematics» Wittgenstein resorted to the analysis of rather elementary mathematical concepts, accompanied also by the inherent ambiguity and ambiguity of his presentation. In particular, against this background, his radical conventionalism, the substitution of logical necessity with the «form of life» of (...) the community, as well as the inadequacy of the representation of arithmetic rules by a language game are criticized. It is shown that the reconstruction of the Wittgenstein concept of understanding based on the Fregian division of meaning and referent goes beyond the conceptual framework of Wittgenstein language games. (shrink)

In the mid twentieth century, logical positivists and many other philosophers endorsed a simple equation: something was necessary just in case it was analytic just in case it was a priori. Kripke’s examples of a posteriori necessary truths showed that the simple equation is false. But while positivist-style inferentialist approaches to logic and mathematics remain popular, there is no inferentialist account of necessity a posteriori. I give such an account. This sounds like an anti-Kripkean project, but it is not. (...) Some of Kripke’s remarks even suggest this kind of approach. This inferentialist approach reinstates neither the simple equation nor pure conventionalism about necessity a posteriori. But it does lead to something near enough, a type of impure conventionalism. In recent years, metaphysically heavyweight approaches to modality have been popular, while other approaches have lagged behind. The inferentialist, impure conventionalist theory of necessity I describe aims to provide a metaphysically lightweight option in modal metaphysics. (shrink)

According to Wittgenstein, mathematical propositions are rules of grammar, that is, conventions, or implications of conventions. So his position can be regarded as a form of conventionalism. However, mathematical conventionalism is widely thought to be untenable due to objections presented by Quine, Dummett and Crispin Wright. It has also been argued that only an implausibly radical form of conventionalism could withstand the critical implications of Wittgenstein’s rule-following considerations. In this article I discuss those objections to conventionalism (...) and argue that none of them is convincing. (shrink)

[EDIT: This book will be published open access. Production is taking longer than expected but I will post the whole book sometime this summer.] One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory (...) power of science by expressing conceptual rules, rules which allow the transformation of empirical descriptions. Mathematics should not be thought of as describing, in any substantive sense, an abstract realm of eternal mathematical objects, as traditional platonists have thought. In Rules to Infinity, this view, which I call mathematical normativism, is updated with contemporary philosophical tools, and it is argued that normativism is compatible with mainstream semantic theory. This allows the normativist to accept that there are mathematical truths, while resisting the platonistic idea that there exist abstract mathematical objects that explain such truths. Furthermore, Rules to Infinity defends a particular account of the distinction between scientific explanations that are in some sense distinctively mathematical – those that explain natural phenomena in some uniquely mathematical way – and those that are only standardly mathematical, and it lays out desiderata for any account of this distinction. Normativism is compared with other prominent views in the philosophy of mathematics such as neo-Fregeanism, fictionalism, conventionalism, and structuralism. Rules to Infinity serves as an entry point into debates at the forefront of philosophy of science and mathematics, and it defends novel positions in these debates. (shrink)

This paper defends a reading of Wittgenstein’s philosophy of mathematics in the Lectures on the Foundation of Mathematics as a radical conventionalist one, whereby our agreement about the particular case is constitutive of our mathematical practice and ‘the logical necessity of any statement is a direct expression of a convention’ (Dummett 1959, p. 329). -/- On this view, mathematical truths are conceptual truths and our practices determine directly for each mathematical proposition individually whether it is true or false. (...) Mathematical truths are thus not consequences of a prior adoption of a convention or rules as orthodox conventionalism has it. -/- The goal of the paper is not merely exegetical, however, and argues that radical conventionalism can withstand some of the most difficult objections that have been brought forward against it, including those of Dummett himself, and thus that radical conventionalism has been prematurely excluded from consideration by philosophers of mathematics. (shrink)

Considered in its historical context, conventionalism is quite different from the way in which it has been caricatured in more recent philosophy of science, that is, as a conservative philosophy that allows the preservation of theories through arbitrary ad hoc stratagems. It is instead a liberal outgrowth of Comtean positivism, which broke with the Reidian interpretation of the Newtonian tradition in France and defended a role for hypotheses in the sciences. It also has roots in the social contract political (...) philosophy of Renouvier, who explicitly drew the analogy between conventions in political life and the conventional acceptance of hypotheses in the sciences, and conceived a philosophy that permits scientists to set aside foundational worries and explore new ideas. Although Poincaré and Renouvier may have hesitated to accept certain then recent developments in mathematics and the sciences such as non-Euclidean geometries, this conservatism cannot necessarily be attributed to their conventionalism. It may instead reflect the engineering background they shared with Comte, which emphasizes practical applications. Although Renouvier and Poincaré may have seen no practical use for these new ideas, unlike Comte they did not prohibit others from pursuing them, reflecting conventionalism’s more liberal attitude toward recent developments in the sciences. (shrink)

Quine's arguments for the indeterminacy of translation demonstrate the existence and help to explain the rationale of restraints upon what we can say and understand. In particular they show that there are logical truths to which there are no intelligible alternatives. Thus the standard view that the truths of logic and mathematics differ from "synthetic" statements in being true solely by virtue of linguistic convention--Which requires for its plausibility the existence of intelligible alternatives to our present logical truth--Is opposed (...) directly, And not by the espousal of "a more thorough pragmatism". This raises problems about possibility and conceptual novelty. (shrink)

According to Mark Steiner, Wittgenstein’s intense work in the philosophy of mathematics during the early 1930s brought about a distinct turning point in his philosophy. The crux of this transition, Steiner contends, is that Wittgenstein came to see mathematical truths as originating in empirical regularities that in the course of time have been hardened into rules. This interpretation, which construes Wittgenstein’s later philosophy of mathematics as more realist than his earlier philosophy, challenges another influential interpretation which reads Wittgenstein (...) as moving in the opposite direction, from a more realist toward a less realist position. Both of these readings, I argue here, tend to overlook the crucial role of conventionalism in shaping Wittgenstein’s later philosophy. I show that during the transition period, Wittgenstein was first strongly attracted to conventionalism, but then, upon discovery of the rule following paradox, came to realize its weaknesses. Did this realization mark the end of the liaison with conventionalism and a wholehearted acceptance of the empirical regularities account? Illustrating Wittgenstein’s ongoing ambivalence toward conventionalism and pointing to his novel understanding of this position, I answer this question in the negative. (shrink)

In broadest terms, this thesis is concerned to answer the question of whether the view that arithmetic is analytic can be maintained consistently. Lest there be much suspense, I will conclude that it can. Those who disagree claim that accounts which defend the analyticity of arithmetic are either unable to give a satisfactory account of the foundations of mathematics due to the incompleteness theorems, or, if steps are taken to mitigate incompleteness, then the view loses the ability to account (...) for the applicability of mathematics in the sciences. I will show that this criticism is not successful against every view whereby arithmetic is analytic by showing that the brand of "conventionalism" about mathematics that Rudolf Carnap advocated in the 1930s, especially in Logical Syntax of Language, does not suffer from these difficulties. There, Carnap develops an account of logic and mathematics that ensures the analyticity of both. It is based on his famous "Principle of Tolerance", and so the major focus of this thesis will to defend this principle from certain criticisms that have arisen in the 80 years since the book was published. I claim that these criticisms all share certain misunderstandings of the principle, and, because my diagnosis of the critiques is that they misunderstand Carnap, the defense I will give is of a primarily historical and exegetical nature. Again speaking broadly, the defense will be split into two parts: one primarily historical and the other argumentative. The historical section concerns the development of Carnap's views on logic and mathematics, from their beginnings in Frege's lectures up through the publication of Logical Syntax. Though this material is well-trod ground, it is necessary background for the second part. In part two we shift gears, and leave aside the historical development of Carnap's views to examine a certain family of critiques of it. We focus on the version due to Kurt Gödel, but also explore four others found in the literature. In the final chapter, I develop a reading of Carnap's Principle - the `wide' reading. It is one whereby there are no antecedent constraints on the construction of linguistic frameworks. I argue that this reading of the principle resolves the purported problems. Though this thesis is not a vindication of Carnap's view of logic and mathematics tout court, it does show that the view has more plausibility than is commonly thought. (shrink)

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

This paper has two main purposes: first to compare Wittgenstein's views to the more traditional views in the philosophy of mathematics; second, to provide a general outline for a Wittgensteinian reply to two objections against Wittgenstein's account of mathematics: the objectivity objection and the consistency objections, respectively. Two fundamental thesmes of Wittgenstein's account of mathematics title the first two sections: mathematical propositions are rules and not descritpions and mathematics is employed within a form of life. Under (...) each heading, I examine Wittgenstein's rejection of alternative views. My aim is to make clear the differences and to suggest some similarities. As will become clear, Wittgenstein often rejects opposing views for the same or similar reasons. This comparison will provide the necessary background for better understanding Wittgenstein's philosophy of mathematics, for appreciating its many unappreciated advantages and, finally, for defending a conventionalist account of mathematics. (shrink)

The thesis argues, in lie main, for both a negative and positive agenda to Wittgenstein's rule-following remarks in both his Philosophical Investigations and Remarks on the foundations of Mathematics. The negative agenda is a sceptical agenda, different than as conceived by Kripke, that is destructive of a realist account of rules and contends that the correct application of a rule is not fully determined in an understanding of the rule. In addition to these consequences, this negative agenda opens Wittgenstein (...) to Dummett's charge of radical conventionalism. These negative consequences are left unresolved by Kripke's sceptical solution and, notably, are wrongly assessed by those that dissent from a sceptical reading. The positive agenda builds on these negative considerations arguing that although there is no determination in the understanding of a rule of what will count as a correct application in so far unconsidered situations, we are still able to follow a rule correctly. This seems to involve an epistemic leap, from an underdetermined understanding to a determinate application, and, in respect of this appearance, involves what Wittgenstein calls following a rule "blindly" in an epistemic sense. Developing this view, of following a rule blindly, involves developing an account of an alternative rational response to rule instruction, one that need not involve a role for interpreting or inferring, but all the same allows for correctness in rule application in virtue of enabling agreement in rule application. (shrink)

From a metaphysical point of view, it is important clearly to see the ontological difference between what is studied in mathematics and mathematical physics, respectively. In this respect, the paper is concerned with the vectors of classical physics. Vectors have both a scalar magnitude and a direction, and it is argued that neither conventionalism nor wholesale anti‐conventionalism holds true of either of these components of classical physical vectors. A quantification of a physical dimension requires the discovery of (...) ontological order relations among all the determinate properties of this dimension, as well as a conventional definition that connects the number one and mathematical unit vectors to determinate spatiotemporal physical entities. One might say that mathematics deals with numbers and vectors, but mathematical physics with scalar quantities and vector quantities, respectively. The International System of Units distinguishes between basic and derived scalar quantities; if a similar distinction should be introduced for the vector quantities of classical physics, then duration in directed time ought to be chosen as the basic vector quantity. The metaphysics of physical vectors is intimately connected with the metaphysics of time. From a philosophical‐historical point of view, the paper revives W. E. Johnson's distinction ‘determinates‐determinables’ and Hans Reichenbach's notion of ‘coordinative definition’. (shrink)

As is well known, Carnap's conventionalism was a rejection to Kant's view ofmathematics and was fully developed in his Logische Syntax der Sprache.The purpose of this article is to step back to Der Logische Aufbau der Weltto show that the Logical Syntax of Language is an attempt to solve difficultiesfound in the earlier construction. I first clarify the notion of conventionalism, whichplays a central role in the application of mathematics to the reconstruction of empiricalknowledge. By not strictly (...) distinguishing between the intuitive notion and thetopological concept of dimension, Carnap is led to a construction which is highlyquestionable. To illustrate the constructive method developed in the Aufbauand some of its inherent difficulties, I consider the computational aspects of theconstruction of phenomenological space via the mathematical concept of dimension.Contrary to Carnap's conventionalism, a dual nature of mathematical statements isbrought into existence by his logical reconstruction. So, if Carnap wants to retainhis mathematics as devoid of content, he must make a clear-cut distinction betweenanalytic and synthetic statements. Thus the natural follow-up to the Aufbau isthe Logical Syntax of Language. (shrink)

Doctoral programs at U.S. universities play a critical role in the development of human resources both in the United States and abroad. This volume reports the results of an extensive study of U.S. research-doctorate programs in five broad fields: physical sciences and mathematics, engineering, social and behavioral sciences, biological sciences, and the humanities. Research-Doctorate Programs in the United States documents changes that have taken place in the size, structure, and quality of doctoral education since the widely used 1982 editions. (...) This update provides selected information on nearly 4,000 doctoral programs in 41 subdisciplines at 274 doctorate-granting institutions. This volume also reports the results of the National Survey of Graduate Faculty, which polled a sample of faculty for their views on the scholarly quality of program faculty and the effectiveness of doctoral programs in preparing research scholars/scientists. This much-anticipated update of such an essential reference will be useful to education administrators, university faculty, and students seeking authoritative information on doctoral programs. (shrink)

W. V. Quine famously argues that though all knowledge is empirical, mathematics is entrenched relative to physics and the special sciences. Further, entrenchment accounts for the necessity of mathematics relative to these other disciplines. Michael Friedman challenges Quine’s view by appealing to historicism, the thesis that the nature of science is illuminated by taking into account its historical development. Friedman argues on historicist grounds that mathematical claims serve as principles constitutive of languages within which empirical claims in physics (...) and the special sciences can be formulated and tested, where these mathematical claims are themselves not empirical but conventional. For Friedman, their conventional, constitutive status accounts for the necessity of mathematics relative to these other disciplines. Here I evaluate Friedman’s challenge to Quine and Quine’s likely response. I then show that though we have reason to find Friedman’s challenge successful, his positive project requires further development before we can endorse it. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

Geometry was a main source of inspiration for Carnap’s conventionalism. Taking Poincaré as his witness Carnap asserted in his dissertation Der Raum (Carnap 1922) that the metrical structure of space is conventional while the underlying topological structure describes "objective" facts. With only minor modifications he stuck to this account throughout his life. The aim of this paper is to disprove Carnap's contention by invoking some classical theorems of differential topology. By this means his metrical conventionalism turns out to (...) be indefensible for mathematical reasons. This implies that the relation between to-pology and geometry cannot be conceptualized as analogous to the relation between the meaning of a proposition and its expression in some language as logical empiricists used to say. (shrink)

Geometry was a main source of inspiration for Carnap's conventionalism. Taking Poincaré as his witness, Carnap asserted in his dissertation Der Raum that the metrical structure of space is conventional while the underlying topological structure describes ‘objective’ facts. With only minor modifications he stuck to this account throughout his life. The aim of this paper is to disprove Carnap's contention by invoking some classical theorems of differential topology. It is shown that his metrical conventionalism is indefensible for mathematical (...) reasons. This implies that the relation between topology and geometry cannot be conceptualized as analogous to the relation between the meaning of a proposition and its expression in some language as the logical empiricists used to say. (shrink)

For several decades, Carnap's philosophy of mathematics used to be either dismissed or ignored. It was perceived as a form of linguistic conventionalism and thus taken to rely on the bankrupt notion of truth by convention. However, recent scholarship has revealed a more subtle picture. It has been forcefully argued that Carnap is not a linguistic conventionalist in any straightforward sense, and that supposedly decisive objections against his position target a straw man. This raises two questions. First, how (...) exactly should we characterise Carnap's actual philosophy of mathematics? Secondly, is his position an attractive alternative to established views? I will tackle these issues by looking at Carnap's response to the incompleteness theorems. Drawing on arguments put forward by Gödel and Beth, I show that some crucial aspects of Carnap's positive account have remained underdeveloped. Suggestions on what a full evaluation of Carnap's position requires are made. (shrink)

In his comments, Daniel Nicholls succeeds in saying more than a few things that I had scarcely realized about the ways in which I write and, therefore, of what I tend to take for granted. He sees in what I write a capacity ‘to utilize the “obvious” whilst at the same time saying something about it.’ Not every philosopher would take that as a compliment. Many philosophers and philosophies have quite other pretensions – to transcend the illusions of common thought (...) and perception towards a higher plane of metaphysical philosophy – or of physics and mathematics. I find that Nicholls’s remark both compliments and complements my work, however. The use of everyday language and example to disturb that everyday world to ‘bring it into high relief’ as Nicholls puts it – is a significant part of philosophical method. To juxtapose the esoteric language of metaphysics (being-in-itself, being-for-itself, being-for-others) with the everyday (what there is, living for oneself, living for others) is to haunt the everyday with a metaphysical imagination even as it reveals what metaphysics had erased from the coinage of common words. (I am thinking, of course, of Derrida’s remarks about Anatole France’s satirical view of philosophical language as coinage from which all marking has been erased). (shrink)

This book presents a new nominalistic philosophy of mathematics: semantic conventionalism. Its central thesis is that mathematics should be founded on the human ability to create language – and specifically, the ability to institute conventions for the truth conditions of sentences. This philosophical stance leads to an alternative way of practicing mathematics: instead of “building” objects out of sets, a mathematician should introduce new syntactical sentence types, together with their truth conditions, as he or she develops (...) a theory. Semantic conventionalism is justified first through criticism of Cantorian set theory, intuitionism, logicism, and predicativism; then on its own terms; and finally, exemplified by a detailed reconstruction of arithmetic and real analysis. Also included is a simple solution to the liar paradox and the other paradoxes that have traditionally been recognized as semantic. And since it is argued that mathematics is semantics, this solution also applies to Russell’s paradox and the other mathematical paradoxes of self-reference. In addition to philosophers who care about the metaphysics and epistemology of mathematics or the paradoxes of self-reference, this book should appeal to mathematicians interested in alternative approaches. (shrink)

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both (...) disciplines. (shrink)

The economist Paul Samuelson acknowledged that he was a disciple of Edwin Bidwell Wilson (1879–1964), an American polymath who was a protégé of Josiah Willard Gibbs. Wilson’s influence on the development of sciences in America has been relatively neglected, as he mostly acted behind the scenes of academia at the organizational and pedagogical fronts. At the basis of his activism were original ideas about the foundations of mathematics and science. This essay reconstructs Wilson’s career and foundational discussions, which evolved (...) as he reacted to David Hilbert’s mathematics, Bertrand Russell’s logic, Henri Poincaré’s conventionalism, Karl Pearson’s statistics, Charles Sanders Peirce’s inference theory, and Lawrence Henderson’s work in social sciences. In brief, after having started his professional life as a mathematician at Yale University (1902–1907), Wilson marginalized himself from mathematics and joined other academic communities and fields, working in mathematical physics at MIT (1907–1922) and in statistics, social science, and economics at Harvard University (1922–retirement). He defined mathematics as a language, meaning by this that mathematics and science were reciprocally indispensable, with the former providing structure and the latter meaning. In an American exceptionalist spirit, Wilson also meant to suggest that homegrown mathematical and scientific traditions should prevail over European ones. (shrink)

_The Universal_ proposes a radically new philosophical system that moves from ontology to ethics. Drawing on the work of De Beauvoir, Sartre, and Le Doeuff, among others, and addressing a range of topics from the Asian sex trade to late capitalism, quantum gravity, and Merleau-Ponty's views on cinema, Dorothea Olkowski stretches the mathematical, political, epistemological, and aesthetic limits of continental philosophy and introduces a new perspective on political structures. Straddling a course between formalism and conventionalism, Olkowski develops the concept (...) of an ontological unconscious that arises from our "sensible" relation to the world-the information we absorb and emit that affects our encounters with the environment and others. In this "realm of the senses," or the field of vulnerability defined by our experience with pleasure and pain, Olkowski is able to rethink the space-time relations put forth by Irigaray's notion of the "interval," Bergson's "recollection," Merleau-Ponty's idea of the "flesh," and Deleuze's "plane of immanence." This aesthetic sense is shared by all humankind and nonhuman entities in the organic and inorganic world. The sensible universal can be applied to categories of pure and practical reason; experiential binaries of male-female and subject-object; and issues of autonomy, moral laws, and the regulation of perception. (shrink)

This is a volume of specially commissioned essays of analytical philosophy, on topics of current interest in ethics and the philosophy of logic and language. Among the topics discussed are the making of wicked promises, G. E. Moore's early ethical views, as well as indexicals, tense, indeterminism, conventionalism in mathematics, and identity and necessity. The essays are all by former students of Casimir Lewy, until recently Reader in Philosophy at the University of Cambridge and an exponent of a (...) particularly thoroughgoing form of philosophical analysis. Together, they represent some of the best work in these areas at present, and express what may be described as a characteristic 'Cambridge' voice. (shrink)

This is a volume of specially commissioned essays of analytical philosophy, on topics of current interest in ethics and the philosophy of logic and language. Among the topics discussed are the making of wicked promises, G. E. Moore's early ethical views, as well as indexicals, tense, indeterminism, conventionalism in mathematics, and identity and necessity. The essays are all by former students of Casimir Lewy, until recently Reader in Philosophy at the University of Cambridge and an exponent of a (...) particularly thoroughgoing form of philosophical analysis. Together, they represent some of the best work in these areas at present, and express what may be described as a characteristic 'Cambridge' voice. (shrink)

This paper has two main purposes: first, to compare Wittgenstein's views to the more traditional views in the philosophy of mathematics; second, to provide a general outline for a Wittgensteinian reply to these two objections. Two fundamental themes of Wittgenstein's account of mathematics title the following two sections: mathematical propositions are rules and not descriptions and mathematics is employed within a form of life. Under each heading, I examine Wittgenstein's rejection of alternative views. My aim is to (...) make clear the differences and too suggest some similarities. As will become soon clear, Wittgenstein often rejects opposing views for the same or similar reasons. This comparison will provide the necessary background for better understanding Wittgenstein's philosophy of mathematics, for appreciating its many unappreciated advantages and, finally, for defending a conventionalist account of mathematics. (shrink)

In this paper, the connection between logicism and the principle of tolerance in Carnap’s philosophy of logic and mathematics is to be presented in terms of the history of its development. Such development is conditioned by two lines of criticism to Carnap’s attempt to combine Logicism and Conventionalism, the first of which comes from Gödel, the second from Alfred Tarski. The presentation will take place in three steps. First, the Logicism of Carnap before the publication of The Logical (...) Syntax of Language will be analyzed in its main features. Second, the relationship between Logicism and the Principle of Tolerance at the time of the publication of Logical Syntax, and thirdly, in the so-called semantical phase of Carnap’s thought will be examined. On the basis of the two preceding points, we will then critically analyze two interpretations of the possibility to maintain a philosophical position like that of Carnap nowadays. (shrink)

Provability, Computability and Reflection.

Having defined a notion of homology for paired graphs, Métayer ([Ma]) proves a homological correctness criterion for proof nets, and states that for any proof net \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $G$\end{document} there exists a Jordan-Hölder decomposition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathsf H}_0(G)$\end{document}. This decomposition is determined by a certain enumeration of the pairs in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $G$\end{document}. We correct his (...) proof of this fact and show that there exists a 1-1 correspondence between these Jordan-Hölder decompositions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathsf H}_0(G)$\end{document} and the possible ‘construction-orders’ of the par-net underlying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $G$\end{document}. (shrink)

Two families of mathematical methods lie at the heart of investigating the hierarchical structure of genetic variation in Homo sapiens: /diversity partitioning/, which assesses genetic variation within and among pre-determined groups, and /clustering analysis/, which simultaneously produces clusters and assigns individuals to these “unsupervised” cluster classifications. While mathematically consistent, these two methodologies are understood by many to ground diametrically opposed claims about the reality of human races. Moreover, modeling results are sensitive to assumptions such as preexisting theoretical commitments to certain (...) linguistic, anthropological, and geographic human groups. Thus, models can be perniciously reified. That is, they can be conflated and confused with the world. This fact belies standard realist and antirealist interpretations of “race,” and supports a pluralist conventionalist interpretation. (shrink)