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)
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.
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 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)
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)
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)
Kant notably holds that arithmetic is synthetic a priori and has to do with the pure intuition of time. This seems to run against our conception of arithmetic as universal and topic neutral. Moreover, trained in the tradition constituting the aftermath of W.V. Quine's attack on the the a priori and on the analytic/synthetic distinction, the modern philosopher of arithmetic is likely to consider Kant's position a nonstarter, and leave settling the question of what Kant's philosophy (...) of arithmetic is exclusively to the Kant scholar and the historian of the philosophy of arithmetic. I argue that this conclusion is misguided because it rests on the unfounded supposition that the pure intuition of time is the basis for Kant's syntheticity and a priority theses. I recover Kant's grounds for holding those theses and their significance to contemporary philosophy of arithmetic. ;I consider and reject Friedman's eliminativist attempt at making Kant palatable to the contemporary philosopher. I argue that Kant's ideas about the mathematical method in 1763, before he explicitly draws the analytic/synthetic distinction, inform the appreciation of Kant's mature view. The idea of construction in intuition is a key to Kant's Critical position that explains the relation between the intellectual and the sensible aspects in Kant's thought. I show that Kant employs a distinct notion of pure formal intuition that is associated with arithmetical necessity construed as peculiarly mathematical; irreducible to logical or sensible modality. Kant's claim is not that the intuition of time serves to justify arithmetical judgments, I argue, but that we cannot represent time as we do unless we think of it arithmetically. According to Kant, arithmetic is not reducible to logic but it is nonetheless just as fundamental to thought in general. The singularity numerical judgments in relation to the category of quantity is shown to involve a notion of a form of an object that is primary with respect to the concept of an object in general. Finally, I reconstruct Kant's notion of symbolic construction and explicates Kant's conception of a constructive procedure. I argue that a Kantian view of the ontology of arithmetic takes the numbers to be nominalizations of construction procedures for intuitable symbolic types. (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 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)
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. 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)
Frege's life and character -- The project -- Frege's new logic -- Defining the numbers -- The reconception of the logic, I-"Function and concept" -- The reconception of the logic, II- "On sense and meaning" and "on concept and object" -- Basic laws, the great contradiction, and its aftermath -- On the foundations of geometry -- Logical investigations -- Frege's influence on recent philosophy.
This book is an expanded version of Joan Weiner's introduction to Frege's work in the Oxford University Press ‘Past Masters’ series published in 1999. The earlier book had chapters on Frege's life and character, his basic project, his new logic, his definitions of the numbers, his 1891 essay ‘Function and concept’, his 1892 essays ‘On Sinn and Bedeutung’ and ‘On concept and object’, the Grundgesetze der Arithmetik and the havoc wreaked by Russell's paradox, and a final brief chapter on Frege's (...) influence. To this, Weiner has added two further chapters on Frege's dispute with Hilbert on the foundations of geometry and on the three late essays of his ‘Logical investigations’. There is little change to the content of the earlier chapters, but they have been divided into sections, each with its own heading, which makes it easier to find one's way around.With the two additional chapters, the book provides an excellent introduction to Frege's work from his earliest Begriffsschrift , which gives the first presentation of his new logic, to his three late essays , which expound his views on logic, truth, and thought. As in the earlier version, the focus is on the logicist project that dominated Frege's career: the attempt to demonstrate that arithmetic is reducible to logic. In all her writings on Frege, Weiner has been particularly sensitive to the philosophical …. (shrink)
The paper considers Fregean and neo-Fregean strategies for securing the apriority of arithmetic. The Fregean strategy recovers the apriority of arithmetic from that of logic and a family of explicit definitions. The neo-Fregean strategy relies on a principle which, though not an explicit definition, is given the status of a stipulation; unlike the Fregean strategy it relies on an extension of second order logic which is not merely a definitional extension. The paper argues that this methodological difference is (...) important in assessing the success of the neo-Fregean strategy. (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)
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)