In this paper, I develop a new defense of logicism: one that combines logicism and nominalism. First, I defend the logicist approach from recent criticisms; in particular from the charge that a cruciai principie in the logicist reconstruction of arithmetic, Hume's Principle, is not analytic. In order to do that, I argue, it is crucial to understand the overall logicist approach as a nominalist view. I then indicate a way of extending the nominalist logicist (...) approach beyond arithmetic. Finally, I argue that a nominalist can use the resulting approach to provide a nominalization strategy for mathematics. In this way, mathematical structures can be introduced without ontological costs. And so, if this proposal is correct, we can say that ultimately all the nominalist needs is logic (and, rather loosely, ali the logicist needs is nominalism). (shrink)

One important achievement Rudolf Carnap claimed for his book, The Logical Syntax of Language, was that it effected a synthesis of two seemingly antithetical philosophies of mathematics, logicism and formalism. Reconciling these widely divergent conceptions had been a goal of Carnap’s for several years. But in the years in which Carnap’s synthesis evolved, important intellectual developments influenced the direction of his efforts and, ultimately, the final outcome. These developments were, first of all, the epoch-making theorems proved by Kurt Gödel, which (...) required the abandonment of several theses central to the aims of logicism and formalism. Of far greater significance, in the present context, are the changes in Carnap’s own philosophical outlook, brought about not only by Gödel’s theorems but concurrent discussions within the Vienna Circle as well as his own researches. Consequently, the exact sense in which Carnap attempted the synthesis of logicism and formalism in the Logical Syntax requires careful examination. In what follows below, the evolution of Carnap’s synthesis will be traced, from the first reconciliation he proposed , through the synthesis that appeared with the publication of The Logical Syntax of Language. The aim is to determine which modifications of Carnap’s synthesis were required by Gödel’s theorems, and which were motivated by changes in his own thinking. Although the characteristic theses of both logicism and formalism required profound modifications because of Gödel’s theorems, the philosophical impulses that originally fueled their programs retained much of their former virulence. But the changes in Carnap’s thought that ocurred in the years he was developing his synthesis especially affected his appreciation of the philosophical motivations underwriting the logicist approach, so that much of the philosophical insight that inspired it is lost, and Carnap’s combination of logicism and formalism is a putative synthesis at best. (shrink)

This paper discusses Carnap’s attempts in the late 1920s to provide a formal reconstruction of modern axiomatics.1 One interpretive theme addressed in recent scholarly literature concerns Carnap’s underlying logicism in his philosophy of mathematics from that time, more specifically, his attempt to “reconcile” the logicist approach of reducing mathematics to logic with the formal axiomatic method. For instance, Awodey & Carus characterize Carnap’s manuscript Untersuchungen zur allgemeinen Axiomatik from 1928 as a “large-scale project to reconcile axiomatic definitions with (...) logicism, and transform implicit into explicit definitions.” It is argued that Carnap’s central idea was to balance a Fregean foundational stance with the modern model-theoretic viewpoint introduced in Hilbert’s Grundlagen der Geometrie ). It was also shown in recent literature that Carnap’s attempt to provide a logicist reconstruction of axiomatics is limited in several ways.2 No closer attention, however, has so far been dedicated to some of the details of his proposed reconciliation. (shrink)

The paper discusses Husserl’s criticism of Frege in Philosophy of Arithmetic (1891) and then his later attitude towards logicism as expressed in Logical Investigations (1900-01). In Philosophy of Arithmetic Husserl holds that logicists offer needless and artificial definitions of notions such as equivalence and number. Frege criticized Husserl’s approach in Philosophy of Arithmetic as psychological, thus shifting the focus of the debate away from logicism. However, Frege’s criticism could be seen to lead Husserl to his later transcendental phenomenological concept (...) of correlation and, with it, to a more developed criticism of logicism. In this regard the division of labor between mathematics and philosophy established in Prolegomena to the Logical Investigations (1900) is central. It implies that in Husserl’s view formal methods fail to shed philosophical insight into the essence of mathematics. In his picture the central notions in need of philosophical clarification are, instead of number and quantity, the notions that are central to abstract structural mathematics. Contrary to Frege who can only elucidate the primitive concepts, Husserl engages in detailed description of their constitution. (shrink)

Concerning Harald Wohlrapp’s theories, many fascinating issues arise. I shall concentrate here on aspects especially relevant to the treatment of pro and con argumentation, a type of what has been called conductive argument. Though initially intrigued by my efforts to describe and explore conductive argument, Harald Wohlrapp later concluded that my treatments were seriously flawed and that an alternative approach can serve to replace that problematic and much contested conception. Much of the difference between our approaches concerns what he (...) deems to be a logicistic approach on my part. (shrink)

This article provides the first English translation of Tanabe’s early essay, “The Limit of Logicism in Epistemology: A Critique of the Marburg and Freiburg Schools”. The key notion that the young Tanabe seeks to define in relation to his detailed analyses of contemporary Neo-Kantian epistemology is the notion of “pure experience” presented in Nishida’s philosophy. The general theory of epistemology shared among the thinkers from these two prominent schools of philosophy in early 20 th century Germany aimed to eliminate the (...) empirical residues in Kant’s theory of knowledge while opposing naïve empiricism and the uncritical methodology of positive science. Their “logicistic” approach, according to Tanabe, seems to contradict Nishida’s notion of pure experience, for it cannot allow any vestige of empiricism in its systematic framework, which is specifically designed to ground scientific knowledge. Yet given that the Neo-Kantian configuration of epistemology does not create the object of knowledge, it must face sensation or representational content as its limiting instance. Thus, to ground a Neo-Kantian theory of knowledge while taking account of this limit of logicism involves explaining their understanding of the unity of subject and object in human knowing. For this, Tanabe argues, recourse to Nishida’s notion of pure experience is indispensable. (shrink)

In his doctoral dissertation, O. Chateaubriand favored Dedekind’s analysis of the notion of number; whereas in Logical Forms, he favors a fregean approach to the topic. My aim in this paper is to examine the kind of logicism he defends. Three aspects will be considered: the concept of analysis; the universality of arithmetical properties and their definability; the irreducibility of arithmetical objects.

In his Tractatus, Wittgenstein maintained that arithmetic consists of equations arrived at by the practice of calculating outcomes of operations$\Omega ^{n}(\bar {\xi })$defined with the help of numeral exponents. Since$Num$(x) and quantification over numbers seem ill-formed, Ramsey wrote that the approach is faced with “insuperable difficulties.” This paper takes Wittgenstein to have assumed that his audience would have an understanding of the implicit general rules governing his operations. By employing the Tractarian logicist interpretation that theN-operator$N(\bar {\xi })$and recursively (...) defined arithmetic operators$\Omega ^{n}(\bar {\xi })$are not different in kind, we can address Ramsey’s problem. Moreover, we can take important steps toward better understanding how Wittgenstein might have imagined emulating proof by mathematical induction. (shrink)

The traditional view regarding the philosophy of mathematics in the twentieth century is the dogma of three schools: Logicism, Intuitionism and Formalism. The problem with this dogma is not, at least not first and foremost, that it is wrong, but that it is biased and essentially incomplete. 'Biased' because it was formulated by one of the involved parties, namely the logical empiricists - if I see it right - in order to make their own position look more agreeable with Intuitionism (...) and Formalism. 'Essentially incomplete' because there was - and still exists - beside Frege's Logicism, Brouwer's Intuitionism and Hilbert's Formalism at least one further position, namely Husserl's phenomenological approach to the foundations of arithmetic, which is also philosophically interesting. In what follows, I want to do two things: First, I will show that the standard dogma regarding the foundations of mathematics is not only incomplete, but also inaccurate, misleading and basically wrong with respect to the three schools themselves. In doing this I hope to make room for Husserl and his phenomenological approach as a viable alternative in the foundations of arithmetic. Second, I will show how Husserl's phenomenological point of view is a position that fits exactly in between Frege's "logicism", properlyunderstood, and Hilbert's mature proof theory, in which his so called "formalism" turns out to be only a means to an end and not a goal in itself. (shrink)

This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles’ does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV–V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: Russell's logicism (...) does not only contain the claim that mathematics is no more than logic, it also contains the claim that the differences between the various mathematical sciences can be logically justified—and thus, that, contrary to the arithmetization stance, analysis, geometry and mechanics are not merely outgrowths of arithmetic. The second aim of this article is to set out the neglected Russellian theory of quantity. The topic is obviously linked with the first, since the mere existence of a doctrine of magnitude, in a work dated from 1903, is a sign of a distrust vis-à-vis the arithmetization programme. After having shown that, despite the works of Cantor, Dedekind and Weierstrass, many mathematicians at the end of the 19th Century elaborated various axiomatic theories of the magnitude, I will try to define the peculiarity of the Russellian approach. I will lay stress on the continuity of the logicist's thought on this point: Whitehead, in the Principia, deepens and generalizes the first Russellian 1903 theory. (shrink)

In this paper, we describe "metaphysical reductions", in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms and (...) predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Some philosophers have argued that we should accept (something like) them as being logical. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:82 Reviews c:\users\arlene\documents\rj issues\type3501\rj 3501 061 red.docx 2015-07-10 4:07 PM LOGICISM BEYOND PRINCIPIA MATHEMATICA Chris Pincock Philosophy / Ohio State U. Columbus, oh 43210–1365, usa [email protected] William Demopoulos. Logicism and Its Philosophical Legacy. Cambridge: Cambridge U. P., 2013. Pp. xii, 272. isbn: 9781107029804.£60.00; us$104.99 (hb). his book brings together eight previously published essays along with three new essays and a brief introduction. In one way or another, each essay (...) pursues either logicism or some broader implication of logicism that a major figure like Russell or Carnap explored. In a narrow sense, logicism is just the position that grounds the concepts and claims of arithmetic (or all of mathematics) in logic. However, in Demopoulos’ hands, logicism becomes a project of much greater significance. A note added to Chapter 5, the classic 1985 article “Bertrand Russell’s The Analysis of Matter” (co-authored with Michael Friedman), indicates this broader scope and Demopoulos’ own take on logicism. Originally Demopoulos and Friedman had concluded their article with both the observation that there are serious “intellectual tensions produced by logicism’s attempt to account for both pure mathematics and applied mathematics (mathematical physics)” and the pessimistic conclusion that “it appears that we can account for the distinctive character of the one only at the expense of the other” (p. 107). Now Demopoulos indicates that even though “something close to [this] is true of Carnap’s Ramsey-sentence q= Reviews 83 c:\users\arlene\documents\rj issues\type3501\rj 3501 061 red.docx 2015-07-10 4:07 PM reconstruction of the language of science”, “the four previous chapters are an extended argument against” the earlier conjecture. For Demopoulos, the main problem is to make sense of both pure and applied mathematics in something like the original logicist form found in Frege, Russell and Carnap.The solution to this problem arises only when we have pinpointed exactly where these logicist projects failed and how their failures can be overcome. Chapter 1 focuses on Frege and his logicism for arithmetic. Demopoulos’ main contention is that Frege aims to reduce arithmetic to logic in order to demonstrate the autonomy of arithmetic from geometric intuition and empirical experience. Frege’s worry about intuition is not motivated by scepticism or by a concern about the cogency of our knowledge of arithmetic (p. 11).The point of logicism is to free arithmetic from the obscurity associated with intuition and to get a clearer grasp of the generality that is characteristic of arithmetic. (Chapters 8 and 9 develop this point in more detail and use it to motivate a brand of logicism for arithmetic that is taken to improve on current neo-Fregean approaches.) For this reason Hume’s Principle takes a central role in Frege’s reconstruction of arithmetic for it not only establishes the autonomy of arithmetic, but also secures the core use of arithmetic in application, i.e. in counting (p. 19). Unlike Dummett and other commentators Demopoulos does not think that this aspect of Frege’s logicism is flawed. Instead, flaws arise only with the accounts of applied mathematics offered by later figures like Russell and Carnap.The root of these problems is that they extend a particular logicist strategy to other areas of mathematics, such as geometry, and to our scientific knowledge more generally. Chapter 2 considers “Carnap’sThesis”, which Demopoulos summarizes as “the assertion that certain applied mathematical theories are not factual” (p. 28). Although the thesis is clearly central to Carnap throughout his career, Demopoulos ultimately argues for a non-Carnapian basis for the factual/nonfactual distinction.The root of the difference is the kind of criteria of identity that are appropriate for the objects at the heart of the theory: … the factuality of an applied mathematical theory will be shown to turn on the recognition that the criteria of identity appropriate to its objects are empirically constrained in a way that the criteria of identity appropriate to the objects of a nonfactual theory are not. (P. 28) For arithmetic, the natural numbers associated with two sortal concepts are identical just in case the objects that... (shrink)

Russell's philosophy is rightly described as a programme of reduction of mathematics to logic. Now the theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim is more far-reaching. The fact that such a theory of geometry was (...) sustained by Russell compels us to question the meaning of logicism: how is it possible to reconcile Russell's global reductionist standpoint with his local defence of the specificities of geometry? * This paper was first presented at the conference ‘Qu'est ce que la géométrie aux époques modernes et contemporaines?’, organized by the Universität Köln and the Archives Poincaré. I would like to thank Philippe Nabonnand for having enlightened me about the issues relative to projective geometry. I would like also to thank Nicholas Griffin, Brice Halimi, Bernard Linsky, Marco Panza, Ivahn Smadja for their helpful discussions. Many thanks also to the two anonymous referees for their useful suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for (...) granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)

This paper introduces and evaluates two contemporary approaches of neo-logicism. Our aim is to highlight the differences between these two neo-logicist programmes and clarify what each projects attempts to achieve. To this end, we first introduce the programme of the Scottish school – as defended by Bob Hale and Crispin Wright1 which we believe to be a..

Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternative consistent principle governing value-ranges. Given some natural assumptions (...) regarding what an acceptable neo-logicistic theory of value-ranges might look like, successfully implementing this alternative approach is impossible. (shrink)

Peirce's realism is a sophisticated realism inherited from the Avicennian Scotistic tradition, which may be briefly characterized by its opposition to metaphysical realism (Platonism) and various forms of nominalism. In this chapter, I consider how Peirce's realism fits his approach to mathematics, which is often presented as a somewhat incoherent mixture of Platonistic and conceptualistic elements. Without denying these, I claim that Peirce's subtle position not only helps to clear up some of these so-called inconsistencies but offers many insights (...) for contemporary ways of dealing with the mathematical aspects of the problem of universals. (shrink)

A reproach has been done many times to post-modernism: its picking up mathematical notions or results, mostly by misrepresenting their real content, in order to strike the readers and obtaining their assent only by impressing them . In this paper I intend to point out that although Alain Badiou’s approach to philosophy starts with taking distance both from analytic philosophy and from French post-modernism, the categories that he uses for labelling logicism, formalism and intuitionism do not reflect the real (...) content of the foundational schools. Hence, a re-thinking from him would be required about them, otherwise he would risk the same reproaches as post-modernists. (shrink)

In this paper, we present a formal recipe that Frege followed in his magnum opus “Grundgesetze der Arithmetik” when formulating his definitions. This recipe is not explicitly mentioned as such by Frege, but we will offer strong reasons to believe that Frege applied it in developing the formal material of Grundgesetze. We then show that a version of Basic Law V plays a fundamental role in Frege’s recipe and, in what follows, we will explicate what exactly this role is and (...) explain how it differs from the role played by extensions in his earlier book “Die Grundlagen der Arithmetik”. Lastly, we will demonstrate that this hitherto neglected yet foundational aspect of Frege’s use of Basic Law V helps to resolve a number of important interpretative challenges in recent Frege scholarship, while also shedding light on some important differences between Frege’s logicism and recent neo-logicist approaches to the foundations of mathematics. (shrink)

I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can (...) be thought of as a sort of logicism for the logical expressivist---a person who believes that the purpose of logical language is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. ;I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of "innate" and "a priori," I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. (shrink)

Any division between scientific practice and a metalevel of the methods and goals of science is largely a false dichotomy. Since a priori, foundationist or logicist approaches to normative principles have proven unequal to the task of representing actual scientific practice, methodologies of science must be abstracted from episodes in the history of science. Of course, it is possible that such characteristics could prove universal and constant across various eras. But, case studies show that they are not in anything (...) beyond the strictures applied to everyday, commonsense reasoning (e.g., a requirement of noncontradiction in a deductive argument). Hence, even if some presently-on-offer methodology or description of past scientific practice were adequate, it need not remain so for current (‘frontier’) areas of science. For this reason, it is important to examine recent episodes in, say, high-energy physics. Results from case studies of several episodes in that field are used to argue that successful practice leads scientists to countenance essential changes in the methodological framework at the levels of the criteria employed in judging theories (i.e., what counts for an explanation and what are canons of rationality) and of the goals of science. *Partial support for this research was provided by the History and Philosophy of Science Program of the National Science Foundation under grants Nos. SES-8606472 and SES-8705469. A preliminary version of this paper was given at an HPS seminar at King's College, London University in May 1988. Helpful comments and useful criticisms were made by several colleagues, especially Ernan McMullin, Heinz Post and Simon Saunders (none of whom are to be held responsible for or necessarily even in agreement with the views expressed here.). (shrink)

The problem of the theoretical reconstruction of the process of scientific paradigm change is by no means a new one in the philosophy and sociology of science. Nevertheless, one cannot say that its investigation has reached the point at which an overwhelming majority of specialists would agree at least about exactly how and in what directions it is necessary to move forward. Notwithstanding this circumstance, one can specify a certain set of basic questions that are recognized as such by the (...) scholarly community of specialists in the field of the sociology and philosophy of science. This set includes such problems as the interaction of the empirical and the theoretical, of the rational and the irrational, of the individual and the social and, of course, the problem of the interaction of cognitive and social factors in the process of paradigm change. The last problem is a kind of focus, at which various approaches converge in order to engage in harsh criticism of one another. Despite the fact that everyone agreed long ago that the logico-empiricistic model of the development of scientific knowledge is inadequate and one-sided, the approaches proposed during the last ten to fifteen years-especially those belonging to the "strong program of the sociology of science"-are also subject to serious criticism. The observer cannot but form the impression that researchers, disillusioned with the extremes of the logicist approach, have turned to a no less one-sided and extreme sociocultural approach that leads to a boundless relativism that is extremely vulnerable in respect of methodology . In particular, the latter approach relies on the well-known Kuhn-Feyerabend thesis concerning the incommensurability of successive paradigms, which presents in a provocatively irrational light the behavior of specialists in the mathematically most developed and rationally most elaborated fields of scientific knowledge. (shrink)

This paper looks at the history of the problem of individuation from Plato to Whitehead. Part I takes as its point of departure Reiner Wiehl’s interpretation of the different meanings of “abstract” in the metaphysics of Alfred North Whitehead and arrives at a corresponding taxonomy of different ways things can be called concrete. Part II compares the way philosophers in different periods understand the relation between thought and intuition. The view mostly associated with ancient philosophy is that thought and sense-perception (...) target different kinds of objects. The view mostly associated with modern philosophy (although it was introduced by the Stoics) is that thought and sense-perception are different ways of targeting the same objects. These differences have specific consequences for theories of individuation, which are assessed historically in Part III and then applied to Whitehead’s difficult texts in part IV. (shrink)

This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present (...) a generic form of the Bad Company objection and introduce an epistemic issue connected to this general problem that will be the focus of the rest of the paper. In the third section, we present an alternative approach within philosophy of mathematics, a view that emerges from Hilbert's Grundlagen der Geometrie (1899, Leipzig: Teubner; Foundations of geometry (trans.: Townsend, E.). La Salle, Illinois: Open Court, 1959.). We will argue that Bad Company-style worries, and our concomitant epistemic issue, also affects this conception and other foundationalist approaches. In the following sections, we then offer various ways to address our epistemic concern, arguing, in the end, that none resolves the issue. The final section offers our own resolution which, however, runs against the foundationalist spirit of the Scottish neo-logicist program. (shrink)

Frege's logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the "neologicist" approach of Hale and Wright. Less attention has been given to Frege's extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the (...) extension of a concept C, together with extra-logical axioms governing such a predicate, and show that arithmetic can be obtained in such a framework. As a philosophical payoff, we investigate the status of the so-called Hume's Principle and its connections to the root of the contradiction in Frege's system. (shrink)

In the late nineteenth and early twentieth centuries the need to rethink the status of space and time which Kant considered to be a priori forms of sensibility was prompted by the emergence of new approaches to the methodology of scientific cognition. In neo-Kantian interpretation these cognitive forms acquire a special epistemological status, manifesting themselves in theoretical research as “pre-given” foundations of knowledge. It seems necessary to conduct a comparative analysis of two interconnected neo-Kantian concepts, of Hermann Cohen and Ivan (...) Lapshin. Studying Kant’s philosophy since his student days, Lapshin gradually came to the conclusion that the need to clarify and develop Kant’s transcendental method was dictated by the development of scientific knowledge. Indeed, the works of the Russian neo-Kantian contain echoes and polemical adjustments of Cohen’s spatio-temporal ideas. Our study has revealed common epistemological attitudes in Cohen and Lapshin: the wish to improve elements of Kantian philosophy, adjusting them to prove the possibility of scientific-theoretical cognition and of overcoming psychologism and developing the logicistic approach to the critique of cognition. Each of the two authors developed their own “mechanism” of reducing space and time to a range of intellectual procedures for the construction of the object of knowledge. In Cohen’s account space and time pre-establish the language of observation and found all scientific-theoretical work. Lapshin, on the other hand, in discussing the formal and substantive features of these categories (the introduction of “axioms” of time, the need to specify the concepts of time and space through other categories), notes that their use in scientific judgment implies an epistemological givenness of the concept of the object. This I see as a variant of solving one and the same epistemological task. I submit that Lapshin worked out an independent concept of space and time as cognitive forms that are congruent with the spirit of European neo-Kantianism and contemporary science. (shrink)

This paper offers a short and biased overview of the history of discussion and controversy about the role of different forms of representation in intelligent agents. It repeats and extends some of the criticisms of the `logicist' approach to AI that I first made in 1971, while also defending logic for its power and generality. It identifies some common confusions regarding the role of visual or diagrammatic reasoning including confusions based on the fact that different forms of representation (...) may be used at different levels in an implementation hierarchy. This is contrasted with the way in the use of one form of representation (e.g. pictures) can be {\em controlled} using another (e.g. logic, or programs). Finally some questions are asked about the role of metrical information in biological visual systems. (shrink)

The following note has on purpose to introduce interested students to logicism. Our objective is not to show any new interpretation or thesis about logicism or its rebirth between the 60s and 80s of the last century. What we will do is systematically show the evolution of logicism from Frege to Russell-Whitehead, with greater emphasis on this latest development, and approach some problems that arise within that movement, for example: The logical paradoxes and the principle of intuitive comprehension, the (...) impredicative definitions and the semantic paradoxes, the Ramified Hierarchy and real numbers, the axiom of reducibility and the impossibility of reducing mathematics to logic, etc. (shrink)

Planlama --- bir amaca ulaşmak üzere bir aksiyonlar bütünü tasarlamak --- yapay zekadaki en temel problemlerden biridir. Bu yazıda, robotikte planlama konusuna mantıkçı (logicist) yaklaşım ele alınmaktadır. [Planning --- devising a plan of action to reach a given goal --- is a fundamental problem in AI. This paper reviews the logicist approach to planning in robotics.].

In To be is to be the object of a possible act of choice the authors defended Boolos’ thesis that plural quantification is part of logic. To this purpose, plural quantification was explained in terms of plural reference, and a semantics of plural acts of choice, performed by an ideal team of agents, was introduced. In this paper, following that approach, we develop a theory of concepts that—in a sense to be explained—can be labeled as a theory of logical (...) concepts. Within this theory, we propose a new logicist approach to natural numbers. Then, we compare our logicism with Frege’s traditional logicism. (shrink)

Mathematics is one of the most successful human endeavors—a paradigm of precision and objectivity. It is also one of our most puzzling endeavors, as it seems to deliver non-experiential knowledge of a non-physical reality consisting of numbers, sets, and functions. How can the success and objectivity of mathematics be reconciled with its puzzling features, which seem to set it apart from all the usual empirical sciences? This book offers a short but systematic introduction to the philosophy of mathematics. Readers are (...) introduced to all of the classical approaches to the field, including logicism, formalism, intuitionism, empiricism, and structuralism. The book also contains accessible introductions to some more specialized issues, such as mathematical intuition, potential infinity, the iterative conception of sets, and the search for new mathematical axioms. The exposition is always closely informed by ongoing research in the field and sometimes draws on the author’s own contributions to this research. This means that Gottlob Frege—a German mathematician and philosopher widely recognized as one of the founders of analytic philosophy—figures prominently in the book, both through his own views and his criticism of other thinkers. (shrink)

Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core (...) of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism. (shrink)

Hilbert’s unpublished 1917 lectures on logic, analyzed here, are the beginning of modern metalogic. In them he proved the consistency and Post-completeness (maximal consistency) of propositional logic -results traditionally credited to Bernays (1918) and Post (1921). These lectures contain the first formal treatment of first-order logic and form the core of Hilbert’s famous 1928 book with Ackermann. What Bernays, influenced by those lectures, did in 1918 was to change the emphasis from the consistency and Post-completeness of a logic to its (...) soundness and completeness: a sentence is provable if and only if valid. By 1917, strongly influenced by PM, Hilbert accepted the theory of types and logicism -a surprising shift. But by 1922 he abandoned the axiom of reducibility and then drew back from logicism, returning to his 1905 approach of trying to prove the consistency of number theory syntactically. (shrink)

This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

Hilbert's axiomatic approach was an optimistic take over on the side of the logical foundations. It was also a response to various restrictive views of mathematics supposedly bounded by the reaches of epistemic elements in mathematics. A complete axiomatization should be able to exclude epistemic or ontic elements from mathematical theorizing, according to Hilbert. This exclusion is not necessarily a logicism in similar form to Frege's or Dedekind's projects. That is, intuition can still have a role in mathematical reasoning. (...) Nevertheless, this role is to be given a structural orientation with the help of explications of the underlying logic of axiomatization. (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)

A central part of Frege's logicism is his reconstruction of the natural numbers as equivalence classes of equinumerous concepts or classes. In this paper, I examine the relationship of this reconstruction both to earlier views, from Mill all the way back to Plato, and to later formalist and structuralist views; I thus situate Frege within what may be called the “rise of pure mathematics” in the nineteenth century. Doing so allows us to acknowledge continuities between Frege's and other approaches, but (...) also to understand better the motivation and the significance of his innovations, as well as their limits. (shrink)

Frege's main contributions to logic and the philosophy of mathematics are, on the one hand, his introduction of modern relational and quantificational logic and, on the other, his analysis of the concept of number. My focus in this paper will be on the latter, although the two are closely related, of course, in ways that will also play a role. More specifically, I will discuss Frege's logicist reconceptualization of the natural numbers with the goal of clarifying two aspects: the (...) motivations for its core ideas; the step-by-step development of these ideas, from Begriffsschrift through Die Grundlagen der Arithmetik and Grundgesetze der Arithmetik to Frege's very last writings, indeed even beyond those, to a number of recent "neo-Fregean" proposals for how to update them. One main development, or break, in Frege's views occurred after he was informed of Russell's antinomy. His attempt to come to terms with this antinomy has found some attention in the literature already. It has seldom been analyzed in connection with earlier changes in his views, however, partly because those changes themselves have been largely ignored. Nor has it been discussed much in connection with Frege's basic motivations, as formed in reaction to earlier positions. Doing both in this paper will not only shed new light on his response to Russell's antinomy, but also on other aspects of his views. In addition, it will provide us with a framework for comparing recent updates of these views, thus for assessing the remaining attraction of Frege's general approach. I will proceed as follows: In the first part of the paper (§1.1 and §1.2), I will consider the relationship of Frege's conception of the natural numbers to earlier conceptions, in particular to what I will call the "pluralities conception", thus bringing into sharper focus his core ideas and their motivations. In the next part (§2.1 and §2.2), I will trace the order in which these ideas come up in Frege's writings, as well as the ways in which his position gets modified along the way, both before and after Russell's antinomy.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:360 HUME'S NATURALI Z EP PHILOSOPHY In "Epistemology Naturalized," Quine claimed that the failure of reductive-foundationalist attempts in epistemology, after the model of Carnap' s Aufbau, must lead to a redefinition of epistemology's task. Instead of setting out to reconstruct the whole fabric of our knowledge from absolute data through deductive operations, we should investigate how human subjects derive their knowledge of nature from sensory inputs. Thus epistemology is (...) to be conceived as a branch of natural science, cognate with psychology, not as the study of the foundations of science any more. In the present paper, I will first outline Quine's position and his related interpretation of Hume's philosophical enterprise. Then, I will try to assess the relevance and fairness of Quine's account and consider in what measure it allows us to understand the novelty of Hume's conception of philosophy, since Hume also advocated a nonfoundationalist conception of philosophy which bears some striking analogy to naturalized epistemology. Finally, I will suggest which consequences are to be drawn from this unusual view of Hume, especially as regards the self-reference of philosophy. Quine's position in the matter of epistemology is well known. For him, under its traditional form and program, epistemology used to deal with the foundations of science, be it mathematics or the science of nature. This foundationalist approach had two connected aspects, one which Quine calls conceptual; the other he calls doctrinal. As Quine 361 puts it "the conceptual studies are concerned with clarifying concepts by defining them, some in terms of others" (Quine, p. 69). In the same manner, the doctrinal studies attempt to establish laws "by proving them, some on the basis of others" (Quine, p. 69). Derivative concepts will be defined in terms of the clearer ones and complex laws will be proved from primitive and obvious ones. Thus the traditional or classical conception of epistemology involves simultaneously a theory of concepts or meanings and a theory of truth. These two branches (Quine speaks of a duality of structure or bifurcation) converge so as to provide an analysis of the way our knowledge of nature -- and our knowledge in general — stems from and is built upon our sensory experience which validates it. In so characterizing epistemology, Quine simply restates in his own terms the f oundationalist and reductivist program of positivist epistemology, which itself tied in with the logicist program in mathematics. Then Quine suggests that Hume dealt with both sides of this program, the conceptual and the doctrinal. To begin with, Hume developed what might be called a theory of concepts or meanings in terms of our sensory data. Such is the point of his theory of ideas which consists in tracing them back to the original impressions they are derived from. In spite of its limitations, which hinge on the lack of an approach in terms of statements, Hume's phenomenalism does the job of a theory of meaning and concepts. Hume also attempted to give an answer to the question of the truth of our knowledge. In this case again, his phenomenalist stand allows him to account for the truth of our singular statements concerning our perceptions, even though this result is achieved at the cost of identifying objects and sensory data. 362 However, when Hume comes to general statements and singular statements concerning future events, his attempt fails. This is not surprising. According to Quine, if it were possible to improve on Hume's theory of meaning and concepts, especially through the use of contextual definition techniques, by contrast, all the attempts to logically derive the truth of our knowledge from true premises have failed. In matter of induction, we are not one step further than Hume was. "The most modest of generalizations about observable traits will cover more cases than its utterer can have had occasions actually to observe. The hopelessness of grounding natural science upon immediate experience in a firmly logical way was acknowledged" (Quine, p. 74). Or "the Humean predicament is the human predicament." For Quine, foundationalist attempts of the sort Hume engaged in involve two cardinal principles of empiricism which remain unchallenged. One of these principles is to the effect... (shrink)

Alfred North Whitehead (1861-1947), a mathematician and logician by training, is the author of highly original works at the crossroads of science and philosophy, exploring the nature of the world around us and its temporal flow. Convinced that everyday terms distort reality, Whitehead invented or borrowed terms more appropriate to his project. The word "Process", which gives its title to his most famous work Process and Reality (1929), is central to his thinking. Process introduces his vision of nature as a (...) succession of crystallizations, each of which proves the finite granularity of time: the instant does not exist. It also implies a confrontation with the theory of relativity and quantum physics. Whitehead's perspective allows for the occurrence of creative novelties, but for which the world necessarily cooperates with a deity, the role of which is examined in this book's last chapter. In Alfred North Whitehead, Philosopher of Time, author Rémy Lestienne mixes biographical elements and intellectual advances, from logicism to philosophies of nature. A compelling introduction to Whitehead's demanding work, this book deciphers a metaphysical landscape often considered too abstract to be approachable, but points out the philosopher's imperfections with respect to the scientific advances of our time. (shrink)

There are two possible strategies for investigating questions on logic and mathematics. First, one can adopt the pattern recommended by the phenomenologists, which consists in looking for the actual essences of logic and mathematics in order to relate both fields. The second approach, adopted in this paper, starts with a historical review of the foundational standpoints. I will then try to extract on this base some insights on how logic and mathematics are mutually related. In particular, I am interested (...) in the concept of logic suggested by the development of the foundational debate. Yet one more preliminary remark is here in order. I will take into account the classical foundational positions: logicism, intuitionism, and formalism, as well as their later modifications and changes.1 Of course, this, so to speak, Foundational Trinity, does not exhaust the complete map of the foundations and philosophy of mathematics. For instance, I entirely neglect views wich might be exemplified by Lakatos, Dieudonné or Hersh, who, roughly speaking, radically deny that the study of logic and its relation to mathematics provides any interesting problem. I do not feel competent myself to judge this matter from the point of view of working mathematicians. On the other hand, philosophers are traditionally interested in the nature of logic and its links with other fields. Consequently, I regard my problem rather as a way of understanding what logic is, than whether it is important for mathematics or not. (shrink)

We propose a critique ofnormativism, defined as the idea that human thinking reflects a normative system against which it should be measured and judged. We analyze the methodological problems associated with normativism, proposing that it invites the controversial “is-ought” inference, much contested in the philosophical literature. This problem is triggered when there are competing normative accounts (the arbitration problem), as empirical evidence can help arbitrate between descriptive theories, but not between normative systems. Drawing on linguistics as a model, we propose (...) that a clear distinction between normative systems and competence theories is essential, arguing that equating them invites an “is-ought” inference: to wit, supporting normative “ought” theories with empirical “is” evidence. We analyze in detail two research programmes with normativist features – Oaksford and Chater's rational analysis and Stanovich and West's individual differences approach – demonstrating how, in each case, equating norm and competence leads to an is-ought inference. Normativism triggers a host of research biases in the psychology of reasoning and decision making: focusing on untrained participants and novel problems, analyzing psychological processes in terms of their normative correlates, and neglecting philosophically significant paradigms when they do not supply clear standards for normative judgement. For example, in a dual-process framework, normativism can lead to a fallacious “ought-is” inference, in which normative responses are taken as diagnostic of analytic reasoning. We propose that little can be gained from normativism that cannot be achieved by descriptivist computational-level analysis, illustrating our position with Hypothetical Thinking Theory and the theory of the suppositional conditional. We conclude that descriptivism is a viable option, and that theories of higher mental processing would be better off freed from normative considerations. (shrink)

The great vision of logical empiricism brought up a hope for ultimate solving the problem of understanding our knowledge. Starting, for the first time in the history of empiricism, from a point of view typical for mathematicians not empiricists, i.e. from the program of logicism, the philosophers of Vienna Circle aimed at creating a consistent philosophy which makes possible both mathematics and empirical knowledge. Today, after more than a half century, their efforts seem to be completely futile. The topic of (...) the paper is to find out the reasons why the program failed. It seems that the deepest reason for the failure lies in the philosophy of mathematics adopted by logical empiricists - namely logicism. The main problem of the contemporary methodology is to find a remedy for this ilness. the paper suggests some ideas in this direction. One of them is more serious approach to the philosophy of mathematics. Another is the recognition of the crucial role epistemic and other values play in scientific research. (shrink)

Anthony Price's recent book presents a contextualist approach to practical rationality. Price develops his proposal in four chapters. In the first one , he outlines a contextual account of the validity of practical inferences. This chapter deals with logicism. Logicism assumes that ‘there is a form of rationality within practical thinking that connects with the logical validity of a practical entailment’ . Price argues that although the principles of logic are ‘invariant and universal’ , their relevance in evaluating a (...) practical inference is constrained by contextual factors related to the teleology of practical thinking and by the functional role of intentions in guiding our behaviour.In the second chapter , Price argues that the content of A's remark to B – ‘I must put in another 20p’, while both stand together on an underground platform in front of a dispenser – must be interpreted contextually, i.e. as referring to a particular end of A for which she needs to put in …. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Die Philosophische Logik Gottlob Freges. Ein Kommentar, mit den Texten des Vorworts zu Grundgesetze der Arithmetik und der Logischen Untersuchungen I–IVLeila HaaparantaWolfgang Künne. Die Philosophische Logik Gottlob Freges. Ein Kommentar, mit den Texten des Vorworts zu Grundgesetze der Arithmetik und der Logischen Untersuchungen I–IV. RoteReihe 30. Frankfurt: V. Klostermann, 2010. Pp. 840. Paper, €29.80.Frege’s thought has been a permanent point of interest among philosophers for several decades. In (...) the late sixties a few philosophers, such as Christian Thiel and Ignacio Angelelli, published studies on Frege’s logic, semantics, and metaphysics. In 1973 Michael Dummett’s work on Frege’s philosophy of language appeared, which held the stage for a long time. It presented Frege as a predecessor, even as a representative, of the analytic tradition. New approaches came up in the late 1970s and early 1980s, such as Hans Sluga’s work in 1980 and other studies, which paid special attention to Frege’s German background, where Kant played a significant role. Several scholars, including Michael Dummett, started to dig into Frege’s background. Positions varied; some studied Frege’s background in order to show that there is nothing that could help us understand his philosophical insights; others found the origins of his philosophical and logical ideas precisely in his background and philosophical environment, in such philosophers as Leibniz, Kant, Trendelenburg, and Lotze.After a few somewhat more quiet years, Frege scholarship expanded after 2000. A number of books have appeared that discuss Frege in the analytic context and also seek [End Page 507] to find more details about his philosophical sources. Wolfgang Künne’s work is a most welcome contribution to contemporary scholarship in the field. Lothar Kreiser’s Gottlob Frege: Leben—Werk—Zeit (2001) has already been available, and interpretations of Frege’s philosophy as well as editions and translations of his books contain short biographies. Still, because of its special composition, Künne’s work serves to fill a gap in the scholarship and significantly contributes to what is known about Frege’s logic and philosophy.The volume consists of three parts. First, there is an extensive biography (about 50 pages) of Frege that places Frege’s thought in its historical context and also contextualizes those of his articles that are included in the work. The main projects Frege pursued, most notably the project of logicism, receive proper attention in the first part of the volume. The four texts that are included in the volume are the Preface of the Grundgesetze der Arithmetik I (1893) and the three logical investigations that were published during Frege’s lifetime, namely “Der Gedanke” (1918), “Die Verneinung” (1919), and “Gedankengefüge” (1923), which were originally published in Beiträge zur Philosophie des deutschen Idealismus. Künne has also included a posthumously published text, “Logische Allgemeinheit,” in the volume, as that fragment, which was not written before 1923, was intended to be a fourth logical investigation. The five texts written by Frege that are included in the edition discuss topics that we would now call philosophical logic, such as meaning, reference, and truth, although they also study themes that we would include in ontology and the philosophy of mind. The title of the volume is thus sufficiently motivated.Frege’s texts take up about 100 pages of the volume, while the commentary is about 600 pages. The book also contains an appendix on Bolzano and Frege, where Künne pays attention to similarities between the two logicians and philosophers but also shows his skepticism about the thesis that Frege had read Bolzano. Apart from this position, Künne prefers to read Frege’s texts closely and inform his reader of historical and philological matters rather than take part in any heated debates concerning interpretations; his own positions become visible only indirectly, through the extensive footnotes. The work also contains a very useful bibliography and indexes of names and subjects.The book is extremely rich. If one, for example, wishes to know more about the views of Frege’s German contemporaries, about the standpoints that Russell or Wittgenstein had on Fregean topics, or about the arguments of Frege scholars, one can find... (shrink)

This book explicates Leibnizian analysis as a search for conditions of intelligibility, and reconsiders his use of principles and methods as well as his account of truth in this way. Via careful reading of well-known, lesser known, and previously unedited texts, it gives a more accurate picture of his philosophical intentions, as well as the relevance of his project to contemporary debate. Two case studies are included, one concerning logic and the other arithmetic; they illustrate a theory of intelligibility that (...) takes as its central notion "possibility for thought", a notion which allows Leibniz to escape certain traps of psychologism, the pseudo-ontology of empiricism, and the empty forms of logicism, and suggests new approaches for contemporary philosophy. "In this remarkable study, Grosholz and Yakira offer a fresh interpretive and conceptual angle on Leibniz's metaphysics. [...] this study deserves high marks for its subtlety, novelty, and creative insight into Leibniz's modes of inquiry as well as for its philosophical acumen." Annals of Science. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Husserl and Mathematics by Mirja HartimoAndrea StaitiMirja Hartimo. Husserl and Mathematics. Cambridge: Cambridge University Press, 2021. Pp. 214. Hardback, $99.99.Mirja Hartimo has written the first book-length study of Husserl's evolving views on mathematics that takes his intellectual context into full consideration. Most importantly, Hartimo's historically informed approach to the topic benefits from her extensive knowledge of Husserl's library. Throughout the book, she provides references to texts and (...) articles that Husserl read, marked, and annotated, although regrettably, very few of these sources are quoted explicitly in his published works. The overall picture that emerges from Hartimo's book shows that Husserl continued to read and engage with mathematics throughout his career, and, even though extensive discussions of mathematical issues are not frequent in his later works, some of the ideas he develops, for instance, in The Crisis of the European Sciences and Transcendental Phenomenology, can be read against the backdrop of the developments of mathematics in his time.Chapter 1 sets the method for the rest of the book. Hartimo emphasizes the importance of a methodological device that takes center stage in Husserl's work from the 1920s onward: Besinnung. Hartimo claims that Besinnung is a method to reactivate the "goals and purposes" (21) that originally motivated scientists engaging in a particular line of research. According to Husserl, as specialized research progresses and becomes standard practice, such original goals and purposes might be blurred by other concerns. When applied to mathematics, Besinnung thus amounts to a reflection "on what mathematicians should do, on what the genuine goal of their activities should be" (21–22). Hartimo argues that this kind of approach to mathematics comes close to Penelope Maddy's naturalism, which takes its departure from what mathematicians are actually doing and then proceeds to inquire into the ontological status of mathematical objects and theories. Hartimo identifies particularly in David Hilbert, a colleague of Husserl's at Göttingen, a vivid embodiment of what Husserl considered to be exemplary work oriented toward the ideal goal of mathematics.In chapters 2 and 3, Hartimo expands upon Husserl's conception of what the true goal of mathematics should be. She does so, first, by addressing the classical topic of Husserl's psychologism in his first (and only) work entirely devoted to mathematics, The Philosophy of Arithmetic, and Frege's well-known critical review of it. In Hartimo's rendition, Husserl criticizes Frege for his attempt to provide formal definitions of basic notions, such as equality. For Husserl, such basic notions are intuitively available and require no definition—rather, they should be put to work to clarify vague notions. Frege, in return, attacks what he takes to be Husserl's wishy-washy psychological account of numbers, which revolves around the concrete act of collecting items and abstracting from their specific contents. Ultimately, Hartimo argues, Husserl takes Frege's criticism to heart and in his later Prolegomena no longer pursues a psychological foundation of number, but rather emphasizes the distinction between the subjective and the objective sides of mathematics, with philosophers studying the former and mathematicians focusing on the latter. While Hartimo's verdict on the philosophical yield of the debate is in line with the standard interpretation of Frege's review as prompting Husserl's antipsychologistic turn, she makes a remark that might be worthy of further discussion: "While for Husserl the debate was about logicism, Frege shifts the debate to be about psychologism and here commentators have followed Frege's example" (50). It would be interesting to imagine what a Fregean reply to Husserl's view about the uselessness of formal definitions for basic notions like equality might look like. In any case, Hartimo's reconstruction makes it clear that, however motivated, Husserl's rejection of psychologism does not amount to an acceptance of logicism.In chapter 3, Hartimo puts forward two of her central claims: (1) Husserl was a structuralist about mathematics; and (2) for Husserl, the goal of modern mathematics is encapsulated in the concept of definiteness. According to (1), mathematics is about the formal structures of systems, which can be isolated and compared. Definiteness, in turn, is the property of systems... (shrink)

Historically, the fallacies have been neglected as objects of systematic study. Yet, since Hamblin's famous criticism of the state of fallacy theory, a substantial literature has been produced. A large portion of this literature is the work of Douglas Walton and John Woods. This paper will deal directly with the criticism of that work which has been advanced by van Eemeren and Grootendorst, particularly the complaints found in their writings of 1992, concerning the disunification of the fallacies and the exemplaristic (...) approach of Woods and Walton's theories. It proposes a unification of the theories of Woods and Walton with that of van Eemeren and Grootendorst, and suggests that such a unification could be advantageous to both theories, and highly interesting for fallacy theory in general. (shrink)

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific (...) issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. (shrink)

The paper focuses on Gottlob Frege’s so called Context Principle (CP hereafter), which counts as one of the most controversial points of his philosophy. Due to its importance and centrality in Frege’s thought, a detailed discussion of the principle requires a detailed analysis of almost all aspects of his philosophy. Obviously, such a task cannot be successfully accomplished here. Thus I limit myself to address only two questions concerning the CP: what role does the principle play (in Grundlagen) and how (...) can we interpret it. Addressing the first problem is required in order to address the second. Most authors interpreted CP from the perspective of Frege’s later distinction between sense and reference, which I will call the ‘semantic interpretation’. Although I accept this perspective as valuable and important, I will initially inverse the action and I will try to approach CP, and generally Grundlagen, in a more natural way, contextually, namely setting them in the initial logicist plan of the Begriffschrift. Finally, I will try to provide an interpretation concerning the alleged conflict between CP and Frege’s compositionality thesis such that they could coherently stay together. (shrink)