This volume is a window on a period of rich and illuminating philosophical activity that has been rendered generally inaccessible by the supposed "revolution" attributed to "Analytic Philosophy" so-called. Careful exposition and critique is given to every serious alternative account of number and number relations available at the time.

This is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world.

This present collection of (translations of) reviews is intended to help obtain a more balanced picture of the reception and impact of Edmund Husserl’s first book, the 1891 Philosophy of Arithmetic. One of the insights to be gained from this non-exhaustive collection of reviews is that the Philosophy of Arithmetic had a much more widespread reception than hitherto assumed: in the present collection alone there already are fourteen, all published between 1891 and 1895. Three of the (...) reviews appeared in mathematical journals (Jahrbuch über die Fortschritte der Mathematik, Zeitschrift für Mathematik und Physik, and Zeitschrift für mathema- tischen und naturwissenschaftlichen Unterricht), three were published in English journals (The Philosophical Review, The Monist, Mind), two were written by other members of the School of Brentano (Franz Hillebrand and Alois Höfler). Some of the reviews and notices appear to be very superficial, consisting merely of para- phrases (often without references) and lists of topics taken from the table of con- tents, presenting barely acceptable summaries. Others, among which Höfler might be the most significant, engage much more deeply with the topics and problems that Husserl addresses, analyzing his approach in the context of the mathematics of his time and the School of Brentano. (shrink)

This paper outlines a second-philosophical account of arithmetic that places it on a distinctive ground between those of logic and set theory.

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.3 (2001) 454-456 [Access article in PDF] Michael Potter. Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap.New York: Oxford University Press, 2000. Pp. x + 305. Cloth, $45.00. This book tells the story of a remarkable series of answers to two related questions:(1) How can arithmetic be necessary and knowable a priori? [End Page 454](2) What accounts for (...) the applicability of arithmetic to matters of empirical fact?These are philosophical questions, part of a "wider puzzle of explaining the link between experience, language, thought, and the world" (18). But answers to them are heavily constrained by technical results in logic and mathematics, and most of Potter's protagonists are either mathematicians (Frege, Dedekind, Ramsey, Hilbert, Gödel) or mathematically trained philosophers (Kant, Russell, Carnap). (Only Wittgenstein fits neither category.) Potter, whose doctorate was in pure mathematics, is an able guide.It is not hard to think of answers to these questions taken singly; what is hard is answering both at the same time. Formalists can easily explain why arithmetic does not depend for its justification on empirical fact—it is just a game with symbols, not responsible to anything beyond its own rules—but they then have trouble explaining why the world of empirical fact is constrained by this game. Empiricists can explain the applicability of arithmetic in just the same way they explain the applicability of physics, but they then must then explain away the apparent a prioricity and necessity of arithmetic. Kant was the first to propose a plausible answer to both questions: arithmetic is both necessary and empirically applicable because it is grounded in the spatio-temporal form of any possible experience.What unites Frege, Dedekind, Russell, Ramsey, Hilbert, and Carnap is that they reject Kant's answer while continuing to seek an answer to his question: how can arithmetic's necessity be reconciled with its applicability? Potter helpfully classifies their projects by the surrogates they substitute for Kant's appeal to the form of spatio-temporal intuition: Frege and Dedekind appeal to thought, Russell to necessary features of the world, Wittgenstein, Ramsey, and Carnap to language, and Hilbert to the necessary structure of our experience of finitary combinations of objects. (Revealingly, the logicists are not grouped together in Potter's taxonomy.) A chapter (or two) is devoted to each of these thinkers, plus Kant and Gödel.This breadth of coverage does not come at the price of superficiality: Potter's chapters are (by and large) accurate, penetrating, and full of interesting insights. I cannot think of another book that offers such clear explanations of the many mathematical concepts and distinctions one must grasp in order to follow debates in the philosophy of arithmetic. The Russell chapters are enhanced by numerous quotations from archival material. The exposition of Hilbert's programme and the implications for it of Gödel's theorems is the best I have seen. The discussions of Wittgenstein's views on arithmetic and of Carnap's syntactical approach also shine. The chapters on Kant and Frege are somewhat weaker; the discussion of Kant, in particular, would have benefited from some engagement with Michael Friedman's views.But the real contribution of Potter's book is its integration of these nine thinkers into a single coherent story. No one will be surprised that Russell's motivations can only be understood in light of Frege's failures, or that many features of Wittgenstein's cryptic work become clearer when seen in light of the specific problems with Russell's system. But there are also some less obvious payoffs: for example, Hilbert's programme is usefully compared with Russell's regressive method (227-8), and Carnap and Russell are shown to face a common dilemma (269, 286).If there is a central theme to Potter's story, it is "how often we have had to reject an [End Page 455] account not for philosophical reasons but for technical ones" (278). Frege's program founders on Russell's paradox, Russell's... (shrink)

In the late nineteenth and early twentieth centuries, philosophy of mathematics was a relatively new subject divided into a variety of dynamically opposed research programs. Edmund Husserl’s Philosophie der Arithmetik joined the ensuing debate by offering a unique phenomenological perspective on the psychological origins of arithmetical concepts. Husserl’s theory, apparently going against the grain of what was to become the dominant Platonist and extensionalist force majeure, was destined to be misunderstood, its purpose and arguments sometimes willfully misrepresented. As a (...) result, despite its brilliant down to earth discussions of the nature of mathematical ideas, Husserl’s text has remained obscure, virtually unacknowledged except as an illustration of how not to explain the principles of arithmetic by mainstream analytic movements in the philosophy of logic and mathematics. (shrink)

Officially, for Kant, judgments are analytic iff the predicate is "contained in" the subject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierarchy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of (...) containment: genera are contained in the species formed from them. Kant's thesis then amounts to the claim that no concept hierarchy conforming to division rules can express truths like '7+5=12.' Kant is correct. Operation concepts ( ) bear two relations to number concepts: and are inputs, is output. To capture both relations, hierarchies must posit overlaps between concepts that violate the exclusion rule. Thus, such truths are synthetic. (shrink)

This volume contains an English translation of Edmund Husserl’s first major work, the Philosophie der Arithmetik, (Husserl 1891). As a translation of Husserliana XII (Husserl 1970), it also includes the first chapter of Husserl’s Habilitationsschrift (Über den Begriff der Zahl) (Husserl 1887) and various supplementary texts written between 1887 and 1901. This translation is the crowning achievement of Dallas Willard’s monumental research into Husserl’s early philosophy (Husserl 1984) and should be seen as a companion to volume V of the (...) Husserliana: Collected Works series (Husserl 1994b), which already contained selected translations from Hua XII. As Willard re- marks on the inner cover of the volume, it is “a window on a period of rich and illuminating philosophical activity”, to which I wholeheartedly agree. Willard’s two volumes of translations open this window on the beginnings of Husserl’s philosophy for the English-speaking world. This earliest period of Husserl’s philosophy has often been unjustly ignored and Willard provides the English reader with an excellent starting point. Husserl’s first steps into phenomenology and the philosophy of logic and mathematics contain many promising seeds that will flower later on, in the Logische Untersuchungen and beyond. (shrink)

This paper demonstrates that Edmund Husserl’s frequently overlooked 1890 manuscript, “On the Logic of Signs,” when closely investigated, reveals itself to be the hermeneutical touchstone for his seminal 1891 Philosophy of Arithmetic. As the former comprises Husserl’s earliest attempt to account for all of the different kinds of signitive experience, his conclusions there can be directly applied to the latter, which is focused on one particular type of sign; namely, number signs. Husserl’s 1890 descriptions of motivating and replacing (...) signs will be respectively employed to clarify his 1891 understanding of the authentic and inauthentic presentations of numbers via number signs. Moreover, his schematic classification of replacement-signs in Semiotic will illuminate the reasons why he believed the number system to be necessary for the operation of replacing number signs. (shrink)

It is argued that the finitist interpretation of wittgenstein fails to take seriously his claim that philosophy is a descriptive activity. Wittgenstein's concentration on relatively simple mathematical examples is not to be explained in terms of finitism, But rather in terms of the fact that with them the central philosophical task of a clear 'ubersicht' of its subject matter is more tractable than with more complex mathematics. Other aspects of wittgenstein's philosophy of mathematics are touched on: his view (...) that mathematical propositions are 'grammatical propositions' (and so, Strictly speaking, Not genuine propositions at all) and his view that in mathematics 'everything is algorithm, Nothing meaning'. His views on consistency and his anti-Foundationalism are linked with this central thesis. (shrink)

External obstacles to properly understanding Wittgenstein’s philosophy of mathematics are not lacking, either. For one thing, there is the piecemeal way that his mathematical manuscripts have been made available. The editors of Remarks on the Foundations of Mathematics write that “the time has not yet come to print the whole of Wittgenstein’s MSS on these... topics”. One wonders what sorts of reasons there could be for that editorial choice.

This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and (...) geometry and the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer. (shrink)

Brouwer and Husserl both aimed to give a philosophical account of mathematics. They met in 1928 when Husserl visited the Netherlands to deliver his Amsterdamer Vorträge. Soon after, Husserl expressed enthusiasm about this meeting in a letter to Heidegger, and he reports that they had long conversations which, for him, had been among the most interesting events in Amsterdam. However, nothing is known about the content of these conversations; and it is not clear whether or not there were any other (...) exchanges between them. The following is an attempt at some remarks on Husserl’s Philosophy of Arithmetic from an intuitionistic, by which I mean a Brouwerian, point of view. I will confine myself specifically to some intuitionistic musings on Husserl’s analysis of the concept of finite number, which he developed in his Habilitationsschrift of 1887, “On the Concept of Number,” and presented again in 1891 in the first four chapters of the Philosophy of Arithmetic. It will turn out that the intuitionistic account of number is very different from Husserl’s. For instance, they strongly disagree about the number 2. On the other hand, the roots of the projects of Husserl and Brouwer have much in common. Since this fact makes their disagreement pressing and lends it much of its significance, I will begin there. (shrink)

In this essay I revisit Kant's much-criticized views on arithmetic. In so doing I make a case for the claim that his theory of arithmetic is not in fact subject to the most familiar and forceful objection against it, namely that his doctrine of the dependence of arithmetic on time is plainly false, or even worse, simply unintelligible; on the contrary, Kant's doctrine about time and arithmetic is highly original, fully intelligible, and with qualifications due to (...) the inherent limitations of his conceptions of arithmetic and logic, defensible to an important extent. (edited). (shrink)

In the third Logical Investigation Husserl presents an integrated theory of wholes and parts based on the notions of dependency, foundation ( Fundierung ), and aprioricity. Careful examination of the literature reveals misconceptions regarding the meaning and scope of the central axis of this theory, especially with respect to its proper context within the development of Husserl's thought. The present paper will establish this context and in the process correct a number of these misconceptions. The presentation of mereology in the (...) Logical Investigations will be shown to originate largely from Husserl's implicit self-criticism of his prior views on the unity of a whole presented in his first work, Philosophy of Arithmetic. (shrink)

Unraveling all the mysteries of the khipu--the knotted string device used by the Inka to record both statistical data and narrative accounts of myths, histories, and genealogies--will require an understanding of how number values and relations may have been used to encode information on social, familial, and political relationships and structures. This is the problem Gary Urton tackles in his pathfinding study of the origin, meaning, and significance of numbers and the philosophical principles underlying the practice of arithmetic among (...) Quechua-speaking peoples of the Andes. Based on fieldwork in communities around Sucre, in south-central Bolivia, Urton argues that the origin and meaning of numbers were and are conceived of by Quechua-speaking peoples in ways similar to their ideas about, and formulations of, gender, age, and social relations. He also demonstrates that their practice of arithmetic is based on a well-articulated body of philosophical principles and values that reflects a continuous attempt to maintain balance, harmony, and equilibrium in the material, social, and moral spheres of community life. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.3 (2001) 454-456 [Access article in PDF] Michael Potter. Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap.New York: Oxford University Press, 2000. Pp. x + 305. Cloth, $45.00. This book tells the story of a remarkable series of answers to two related questions:(1) How can arithmetic be necessary and knowable a priori? [End Page 454](2) What accounts for (...) the applicability of arithmetic to matters of empirical fact?These are philosophical questions, part of a "wider puzzle of explaining the link between experience, language, thought, and the world" (18). But answers to them are heavily constrained by technical results in logic and mathematics, and most of Potter's protagonists are either mathematicians (Frege, Dedekind, Ramsey, Hilbert, Gödel) or mathematically trained philosophers (Kant, Russell, Carnap). (Only Wittgenstein fits neither category.) Potter, whose doctorate was in pure mathematics, is an able guide.It is not hard to think of answers to these questions taken singly; what is hard is answering both at the same time. Formalists can easily explain why arithmetic does not depend for its justification on empirical fact—it is just a game with symbols, not responsible to anything beyond its own rules—but they then have trouble explaining why the world of empirical fact is constrained by this game. Empiricists can explain the applicability of arithmetic in just the same way they explain the applicability of physics, but they then must then explain away the apparent a prioricity and necessity of arithmetic. Kant was the first to propose a plausible answer to both questions: arithmetic is both necessary and empirically applicable because it is grounded in the spatio-temporal form of any possible experience.What unites Frege, Dedekind, Russell, Ramsey, Hilbert, and Carnap is that they reject Kant's answer while continuing to seek an answer to his question: how can arithmetic's necessity be reconciled with its applicability? Potter helpfully classifies their projects by the surrogates they substitute for Kant's appeal to the form of spatio-temporal intuition: Frege and Dedekind appeal to thought, Russell to necessary features of the world, Wittgenstein, Ramsey, and Carnap to language, and Hilbert to the necessary structure of our experience of finitary combinations of objects. (Revealingly, the logicists are not grouped together in Potter's taxonomy.) A chapter (or two) is devoted to each of these thinkers, plus Kant and Gödel.This breadth of coverage does not come at the price of superficiality: Potter's chapters are (by and large) accurate, penetrating, and full of interesting insights. I cannot think of another book that offers such clear explanations of the many mathematical concepts and distinctions one must grasp in order to follow debates in the philosophy of arithmetic. The Russell chapters are enhanced by numerous quotations from archival material. The exposition of Hilbert's programme and the implications for it of Gödel's theorems is the best I have seen. The discussions of Wittgenstein's views on arithmetic and of Carnap's syntactical approach also shine. The chapters on Kant and Frege are somewhat weaker; the discussion of Kant, in particular, would have benefited from some engagement with Michael Friedman's views.But the real contribution of Potter's book is its integration of these nine thinkers into a single coherent story. No one will be surprised that Russell's motivations can only be understood in light of Frege's failures, or that many features of Wittgenstein's cryptic work become clearer when seen in light of the specific problems with Russell's system. But there are also some less obvious payoffs: for example, Hilbert's programme is usefully compared with Russell's regressive method (227-8), and Carnap and Russell are shown to face a common dilemma (269, 286).If there is a central theme to Potter's story, it is "how often we have had to reject an [End Page 455] account not for philosophical reasons but for technical ones" (278). Frege's program founders on Russell's paradox, Russell's... (shrink)

In this book a non-realist philosophy of mathematics is presented. Two ideas are essential to its conception. These ideas are (i) that pure mathematics--taken in isolation from the use of mathematical signs in empirical judgement--is an activity for which a formalist account is roughly correct, and (ii) that mathematical signs nonetheless have a sense, but only in and through belonging to a system of signs with empirical application. This conception is argued by the two authors and is critically discussed (...) by three philosophers of mathematics. (shrink)

_The Foundations of Arithmetic_ is undoubtedly the best introduction to Frege's thought; it is here that Frege expounds the central notions of his philosophy, subjecting the views of his predecessors and contemporaries to devastating analysis. The book represents the first philosophically sound discussion of the concept of number in Western civilization. It profoundly influenced developments in the philosophy of mathematics and in general ontology.

This paper ties together three threads of discussion about the following question: in accepting a system of axioms S, what else are we thereby warranted in accepting, on the basis of accepting S? First, certain foundational positions in the philosophy of mathematics are said to be epistemically stable, in that there exists a coherent rationale for accepting a corresponding system of axioms of arithmetic, which does not entail or otherwise rationally oblige the foundationalist to accept statements beyond the (...) logical consequences of those axioms. Second, epistemic stability is said to be incompatible with the implicit commitment thesis, according to which accepting a system of axioms implicitly commits the foundationalist to accept additional statements not immediately available in that theory. Third, epistemic stability stands in tension with the idea that in accepting a system of axioms S, one thereby also accepts soundness principles for S. We offer a framework for analysis of sets of implicit commitment which reconciles epistemic stability with the latter two notions, and argue that all three ideas are in fact compatible. (shrink)

In this paper I present a formalist philosophy mathematics and apply it directly to Arithmetic. I propose that formalists concentrate on presenting compositional truth theories for mathematical languages that ultimately depend on formal methods. I argue that this proposal occupies a lush middle ground between traditional formalism, fictionalism, logicism and realism.

Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system--Axiom V--which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does (...) prove each of the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be "The Basic Laws of Cardinal Number", as Frege understood them. Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, "Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?". (shrink)

The goal of the research programme I describe in this article is a realist epistemology for arithmetic which respects arithmetic's special epistemic status (the status usually described as a prioricity) yet accommodates naturalistic concerns by remaining fundamentally empiricist. I argue that the central claims which would allow us to develop such an epistemology are (i) that arithmetical truths are known through an examination of our arithmetical concepts; (ii) that (at least our basic) arithmetical concepts are accurate mental representations (...) of elements of the arithmetical structure of the independent world; (iii) that (ii) obtains in virtue of the normal functioning of our sensory apparatus. The first of these claims protects arithmetic's special epistemic status relative, for example, to the laws of physics, the second preserves the independence of arithmetical truth, and the third ensures that we remain empiricists. Preliminaries Justifying and grounding concepts Cameras and filters An epistemology for arithmetic Concluding remarks. (shrink)

The philosophy of arithmetic of Wittgenstein's Tractatus is outlined and the central role played in it by the general notion of operation is pointed out. Following which, the language, the axioms and the rules of a formal theory of operations, extracted from the Tractatus, are presented and a theorem of interpretability of the equational fragment of Peano's Arithmetic into such a formal theory is proven.