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Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic analysis. Some have doubted whether distinctively structural working methods have any impact in other fields such as differential equations. We show narrowly construed structuralism as offered by Benacerraf has no practical role in differential equations. But Dedekind’s approach to the continuum already did not fit that narrow sense, and little of mathematics today does. We draw on one calculus textbook, one celebrated analysis textbook, and a monograph on the Navier–Stokes equation to show structural methods like Dedekind’s have long been central to differential equations, and have philosophically respectable ontology and epistemology. [ABSTRACT FROM AUTHOR]

This paper defends a conceptualistic version of structuralism as the most convincing way of elaborating a philosophical understanding of structuralism in line with the classical tradition. The argument begins with a revision of the tradition of "conceptual mathematics", incarnated in key figures of the period 1850 to 1940 like Riemann, Dedekind, Hilbert or Noether, showing how it led to a structuralist methodology. Then the tension between the 'presuppositionless' approach of those authors, and the platonism of some recent versions of philosophical structuralism, is presented. In order to resolve this tension, we argue for the idea of 'logical objects' as a form of minimalist realism, again in the tradition of classical authors including Peirce and Cassirer, and we introduce the basic tenets of conceptual structuralism. The remainder of the paper is devoted to an open discussion of the assumptions behind conceptual structuralism, and—most importantly—an argument to show how the objectivity of mathematics can be explained from the adopted standpoint. This includes the idea that advanced mathematics builds on hypothetical assumptions (Riemann, Peirce, and others), which is presented and discussed in some detail. Finally, the ensuing notion of objectivity is interpreted as a form of particularly robust intersubjectivity, and it is distinguished from fictional or social ontology. [ABSTRACT FROM AUTHOR]

Let S (D) represent a set of proper nonzero ideals I (D) (respectively, t-ideals I t (D)) of an integral domain D ≠ q f (D) and let P be a valid property of ideals of D. We say S(D) meets P (denoted S (D) ◃ P) if each s ∈ S (D) is contained in an ideal satisfying P. If S(D) ◃ P , dim (D) cannot be controlled. When R = D [ X ] , I (D) ◃ P does not imply I(R) ◃ P while I t (D) ◃ P implies I t (R) ◃ P usually. We say S(D) meets P with a twist (written S (D) ◃ t P) if each s ∈ S (D) is such that, for some n ∈ N , s n is contained in an ideal satisfying P and study S (D) ◃ t P , as its predecessor. A modification of the above approach is used to give generalizations of almost bezout domains. [ABSTRACT FROM AUTHOR]

The paper discusses a relatively underexamined element of Wittgenstein’s philosophy of mathematics associated with his critique of set theory. I outline Wittgenstein’s objections to the theories of Dedekind and Cantor, including the confounding of extension and intension, the faulty definition of the infinite set as infinite extension and the critique of Cantor’s diagonal proof. One of Wittgenstein’s major objections to set theory was that the concept of the size of infinite sets, which Cantor expressed by means of symbols אₒ and c, had no application, i.e. that there was no grammatical technique that could show how such expressions were to be used. Notions of set theory are, so to speak, exterior – they find themselves outdoors, outside of what we usually do. They form a discourse that takes us beyond the horizon of everydayness and commonality. They are like an engine idling of the language of mathematics.

When it comes to the relation of modern mathematics and philosophy, most people tend to think of the three major schools of thought—i.e. logicism, formalism, and intuitionism—that emerged as profound researches on the foundations and nature of mathematics in the beginning of the 20th century and have shaped the dominant discourse of an autonomous discipline of analytic philosophy, generally known under the rubric of “philosophy of mathematics” since then. What has been completely disregarded by these philosophical attitudes, these foundational researches which seek to provide pure mathematics with a philosophically plausible justification by founding it on firm logico-philosophical bases, is that the genuine self-foundation of pure mathematics had been done before, namely during the 19th century, when it was developing into an entirely new and independent discipline as a concomitant of the continuous dissociation of mathematics from the physical world. This self-foundation of the 19th-century pure mathematics, however, was more akin to the German-idealist interpretations of Kant’s transcendental philosophy, than the post-factum, retrospective 20th-century researches on the foundations of mathematics. This article aims to demonstrate this neglected historical fact via delving into the philosophical inclinations of the three major founders of the 19th-century pure mathematics, Riemann, Dedekind and Cantor. Consequently, pure mathematics, with respect to its idealist origins, proves to be a formalization and idealization of certain activities specific to a self-conscious transcendental subjectivity.

Call a directed partially ordered cancellative divisibility monoid M a Riesz monoid if for all x , y 1 , y 2 ≥ 0 in M, x ≤ y 1 + y 2 ⇒ x = x 1 + x 2 where 0 ≤ x i ≤ y i . We explore the necessary and sufficient conditions under which a Riesz monoid M with M + = { x ≥ 0 ∣ x ∈ M } = M generates a Riesz group and indicate some applications. We call a directed p.o. monoid M Π -pre-Riesz if M + = M and for all x 1 , x 2 , ⋯ , x n ∈ M , glb (x 1 , x 2 , ⋯ , x n) = 0 or there is r ∈ Π such that 0 < r ≤ x 1 , x 2 , ⋯ , x n , for some subset Π of M. We explore examples of Π -pre-Riesz monoids of ∗ -ideals of different types. We show for instance that if M is the monoid of nonzero (integral) ideals of a Noetherian domain D and Π the set of invertible ideals, M is Π -pre-Riesz if and only D is a Dedekind domain. We also study factorization in pre-Riesz monoids of a certain type and link it with factorization theory of ideals in an integral domain. [ABSTRACT FROM AUTHOR]

'Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means.' Along this line, in The Open World, Hermann Weyl contrasted the desire to make the infinite accessible through finite processes, which underlies any theoretical investigation of reality, with the intuitive feeling for the infinite 'peculiar to the Orient,' which remains 'indifferent to the concrete manifold of reality.' But a critical analysis may acknowledge a valuable dialectical opposition. Struggling to spell out the infinity of real numbers mathematicians come to see the active role of emptiness. Pondering over the essence of self-awareness, the Japanese philosopher Nishida Kitarō comes to see the 'place' where it abides as absolute nothingness. Thus, the two ways of seeing coalesce into a perspective in which infinity and nothingness mirror each other. [ABSTRACT FROM AUTHOR]

Arithmetic is the study of numbers, their properties and relationships. Calculations related to the centralization of agriculture and counting had a practical impact on the formation of arithmetic. The first reliable information related to arithmetic was found in the historical monuments of Babylon and Ancient Egypt, dating back to the III-II millennia BC. In the Middle Ages, the sphere of trade and the conduct of approximate calculations were the main branches of the application of arithmetic. Initially, arithmetic developed in India and Muslim countries, and later began to spread in Western Europe. In modern times, marine astronomy, mechanics and complex commercial calculations posed new challenges and requirements, thereby contributing to new rounds in the development of astronomy. The article deals with the issues of comprehending the concept of number and its content in the framework of mathematical discourses of the 19th century. This issue is initially considered in a historical perspective, then the works of such mathematicians and philosophers as Richard Dedekind, Georg Cantor, Giuseppe Peano and the logic Gottlob Frege are analyzed. In the 1870s, the discourse on the foundations of arithmetic was updated. For the approval of mathematics as a consistent and correct science, the importance of determining the meaning of the concept of number, which is considered the basis of arithmetic, is noted. The article discusses questions about the essence and content of the concept of number. [ABSTRACT FROM AUTHOR]

Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism (advocated by Heine), game formalism (advocated by Thomae) and instrumental formalism (advocated by Hilbert). After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilbert's instrumental formalism. My primary aim there will be to develop its formalist elements more fully. These are, in the main, (i) its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, (ii) its departure from the traditional understanding of the basic nature of proof and (iii) its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert's formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert's observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all. [ABSTRACT FROM AUTHOR]

É comum considerar que a tendência estruturalista matemática começa no século XX, em algum momento depois do trabalho sobre los fundamentos da teoria dos conjuntos e torna-se a sua propagação através do grupo Bourbaki. Neste artigo argumentamos que essa tendência estilística estava presente em Richard Dedekind (1831-1916) desde 1854 na sua dissertação de habilitação como professor na Universidade de Göttingen. O objetivo principal deste artigo é mostrar como se desenvolve o estilo estructuralista nos trabalhos de Dedekind e argumentar por qué consideramos que é importante para comprendeer a arquitetura do continuum aritmético.

This paper proposes a reading of the history of equivalence in mathematics. The paper has two main parts. The first part focuses on a relatively short historical period when the notion of equivalence is about to be decontextualized, but yet, has no commonly agreed-upon name. The method for this part is rather straightforward: following the clues left by the others for the 'first' modern use of equivalence. The second part focuses on a relatively long historical period when equivalence is experienced in context. The method for this part is to strip the ideas from their set-theoretic formulations and methodically examine the variations in the ways equivalence appears in some prominent historical texts. The paper reveals several critical differences in the conceptions of equivalence at different points in history that are at variance with the standard account of the mathematical notion of equivalence encompassing the concepts of equivalence relation and equivalence class. [ABSTRACT FROM AUTHOR]

A central claim of Deleuze's reading of Bergson is that Bergson's distinction between space as an extensive multiplicity and duration as an intensive multiplicity is inspired by the distinction between discrete and continuous manifolds found in Bernhard Riemann's 1854 thesis on the foundations of geometry. Yet there is no evidence from Bergson that Riemann influences his division, and the distinction between the discrete and continuous is hardly a Riemannian invention. Claiming Riemann's influence, however, allows Deleuze to argue that quantity, in the form of 'virtual number', still pertains to continuous multiplicities. This not only supports Deleuze's attempt to redeem Bergson's argument against Einstein in Duration and Simultaneity, but also allows Deleuze to position Bergson against Hegelian dialectics. The use of Riemann is thereby an important element of the incorporation of Bergson into Deleuze's larger early project of developing an anti-Hegelian philosophy of difference. This article first reviews the role of discrete and continuous multiplicities or manifolds in Riemann's Habilitationsschrift, and how Riemann uses them to establish the foundations of an intrinsic geometry. It then outlines how Deleuze reinterprets Riemann's thesis to make it a credible resource for Deleuze's Bergsonism. Finally, it explores the limits of this move, and how Deleuze's later move away from Bergson turns on the rejection of an assumption of Riemann's thesis, that of 'flatness in smallest parts', which Deleuze challenges with the idea, taken from Riemann's contemporary, Richard Dedekind, of the irrational cut. [ABSTRACT FROM AUTHOR]

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We discuss the differences between the ancient and the modern notion of mathematical continuity. We focus on three ancient approaches to the continuum, namely the monist, the atomist and the Aristotelian approach. Afterwards, we analyze the construction of real numbers by Dedekind, Weierstrass and Cantor. The modern continuum is characterized by these constructions, but is a more general notion. We compare the ancient conception of continuity and the modern approach in order to show that the modern concept of mathematical continuity cannot be interpreted as part of the ancient theoretical framework, or as some kind of extension of this framework. [ABSTRACT FROM AUTHOR]

In this paper, I study Richard Dedekind and Heinrich Weber's 1882 Theorie der algebraischen Funktionen einer Veränderlichen , with a focus on the inherently arithmetical aspects of their work. I show that their paper provides an arithmetical rewriting of Riemannian function theory, i.e. a rewriting built on elementary arithmetical notions such as divisibility. I start with contextual elements concerning what is “arithmetical”, to put Dedekind and Weber's works into perspective from that viewpoint. Then, through a detailed analysis of the 1882 paper and using elements of their correspondence, I suggest that Dedekind and Weber deploy a strategy of rewriting parts of mathematics using arithmetic, and that this strategy is essentially related to Dedekind's specific conception of numbers and arithmetic as intrinsically linked to the human mind. [ABSTRACT FROM AUTHOR]

Aim of this paper is to confute two views, the first about Schröder's presumptive foundationalism, according to he founded mathematics on the calculus of relatives; the second one maintaining that Schröder only in his last years (from 1890 onwards) focused on an universal and symbolic language (by him called pasigraphy). We will argue that, on the one hand Schröder considered the problem of founding mathematics already solved by Dedekind, limiting himself in a mere translation of the Chain Theory in the language of the relatives. On the other hand, we will show that Schröder's pasigraphy was connaturate to himself and that it roots in his very childhood and in his love for foreign languages. [ABSTRACT FROM AUTHOR]

La critica che Ludwig Wittgenstein muove ai presupposti filosofici del dibattito sui fondamenti della matematica si estende oltre le tematiche specifiche di tale dibattito ed investe un’intera tradizione di pensiero. Dietro alle argomentazioni di Frege e Russell, alle dimostrazioni di Cantor e di Dedekind, al programma di Hilbert e al teorema di Gödel, c’è uno sfondo filosofico che le acute osservazioni del filosofo ci permette di smascherare. Questa ricerca, articolata in quattro studi, intende presentare alcuni momenti fondamentali del dibattito sui fondamenti e la relativa critica di Wittgenstein. Si prenderà in considerazione la definizione di limite di Cantor, la sezione di Dedekind, la definizione logicista di numero naturale ed infine il problema della scoperta in matematica. In conclusione si cercherà di interpretare il cosiddetto anti-platonismo di Wittgenstein.

O objetivo deste trabalho é traçar alguns paralelos históricos que existem entre a oposição Darstellung/Beschreibung tal como ela é usada por Wittgenstein no período intermediário de seu pensamento e a mesma oposição em alguns matemáticos do século XIX, pertencentes a duas escolas distintas que divergiam quanto ao método de produção matemática. Em seguida, retornaremos aos escritos de Wittgenstein, buscando mostrar em que medida esta investigação histórica pode ser útil para iluminar alguns pontos de sua filosofia e levantar alguns problemas que surgiram neste contexto.

If numbers were identified with any of their standard set-theoretic realizations, then they would have various non-arithmetical properties that mathematicians are reluctant to ascribe to them. Dedekind and later structuralists conclude that we should refrain from ascribing to numbers such ‘foreign’ properties. We first rehearse why it is hard to provide an acceptable formulation of this conclusion. Then we investigate some forms of abstraction meant to purge mathematical objects of all ‘foreign’ properties. One form is inspired by Frege; the other by Dedekind. We argue that both face problems. [ABSTRACT FROM PUBLISHER]

Deleuze's theory of time set out in Difference and Repetition is a complex structure of three different syntheses of time - the passive synthesis of the living present, the passive synthesis of the pure past and the static synthesis of the future. This article focuses on Deleuze's third synthesis of time, which seems to be the most obscure part of his tripartite theory, as Deleuze mixes different theoretical concepts drawn from philosophy, Greek drama theory and mathematics. Of central importance is the notion of the cut, which is constitutive of the third synthesis of time defined as an a priori ordered temporal series separated unequally into a before and an after. This article argues that Deleuze develops his ordinal definition of time with recourse to Kant's definition of time as pure and empty form, Hölderlin's notion of 'caesura' drawn from his 'Remarks on Oedipus' (1803) and Dedekind's method of cuts as developed in his pioneering essay 'Continuity and Irrational Numbers' (1872). Deleuze then ties together the conceptions of the Kantian empty form of time and the Nietzschean eternal return, both of which are essentially related to a fractured I or dissolved self. This article aims to assemble the different heterogeneous elements that Deleuze picks up on and to show how the third synthesis of time emerges from this differential multiplicity. [ABSTRACT FROM AUTHOR]

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research. [ABSTRACT FROM AUTHOR]

A Dedekind algebra is an ordered pair (B, h), where B is a non-empty set and h is a similarity transformation on B. Among the Dedekind algebras is the sequence of the positive integers. From a contemporary perspective, Dedekind established that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. The purpose here is to show that this seemingly isolated result is a consequence of more general results in the model theory of second-order languages. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ℵ0 isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on ω called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type that occurs in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. The second-order theory of any countably infinite Dedekind algebra is categorical, and there are countably infinite Dedekind algebras whose second-order theories are not finitely axiomatizable. It is shown that there is a condition on configuration signatures necessary and sufficient for the second-order theory of a Dedekind algebra to be finitely axiomatizable. It follows that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the rings of Gorenstein homological dimensions less than or equal to 1. We call these Gorenstein (semi)hereditary rings and call a particular subclass of these strongly Gorenstein (semi)hereditary rings. [ABSTRACT FROM AUTHOR]

There are two general questions which many views in the philosophy of mathematics can be seen as addressing: what are mathematical objects, and how do we have knowledge of them? Naturally, the answers given to these questions are linked, since whatever account we give of how we have knowledge of mathematical objects surely has to take into account what sorts of things we claim they are; conversely, whatever account we give of the nature of mathematical objects must be accompanied by a corresponding account of how it is that we acquire knowledge of those objects. The connection between these problems results in what is often called “Benacerraf’s Problem”, which is a dilemma that many philosophical views about mathematical objects face. It will be my goal here to present a view, attributed to Richard Dedekind, which approaches the initial questions in a different way than many other philosophical views do, and in doing so, avoids the dilemma given by Benacerraf’s problem. [ABSTRACT FROM AUTHOR]

In this article we give a modern interpretation of Kummer’s ideal numbers and show how they developed from Jacobi’s work on cyclotomy, in particular the methods for studying “Jacobi sums” which he presented in his lectures on number theory and cyclotomy in the winter semester 1836/37. [ABSTRACT FROM AUTHOR]

We examine when summands of completely decomposable modules over a domain R are again completely decomposable. We show that this is the case if R is an h-local Prüfer domain. If R is 1-dimensional Noetherian, then the problem reduces locally if almost all localizations are integrally closed. If R is 1-dimensional Noetherian and local, then the integral closure of R must have at most two maximal ideals. [ABSTRACT FROM AUTHOR]

Let R be an integral domain with quotient field K and integral closure R'. Anderson and Zafrullah called R an "almost valuation domain" if for every nonzero x ∈ K, there is a positive integer n such that either xn ∈ R or x-n ∈ R. In this article, we introduce a new closely related class of integral domains. We define a prime ideal P of R to be a "pseudo-strongly prime ideal" if, whenever x, y ∈ K and xyP ⊆ P, then there is a positive integer m ≥ 1 such that either xm ∈ R or ymP ⊆ P. If each prime ideal of R is a pseudo-strongly prime ideal, then R is called a "pseudo-almost valuation domain" (PAVD). We show that the class of valuation domains, the class of pseudo-valuation domains, the class of almost valuation domains, and the class of almost pseudo-valuation domains are properly contained in the class of pseudo-almost valuation domains; also we show that the class of pseudo-almost valuation domains is properly contained in the class of quasilocal domains with linearly ordered prime ideals. Among the properties of PAVDs, we show that an integral domain R is a PAVD if and only if for every nonzero x ∈ K, there is a positive integer n ≥ 1 such that either xn ∈ R or ax-n ∈ R for every nonunit a ∈ R. We show that pseudo-almost valuation domains are precisely the pullbacks of almost valuation domains, we characterize pseudo-almost valuation domains of the form D + M, and we use this characterization to construct PAVDs that are not almost valuation domains. We show that if R is a Noetherian PAVD, then R has Krull dimension at most one and R' is a valuation domain; we show that every overring of a PAVD R is a PAVD iff R' is a valuation domain and every integral overring of R is a PAVD. [ABSTRACT FROM AUTHOR]

The commutative algebra of invariants of a Lie super-algebra need not be affine, but does have a common ideal with an affine algebra, in all the known examples. This leads us to extend a class of algebras C to a class which we call “nearly C”, by admitting those algebras C having a common ideal A with an algebra (containing C) in C such that C/A∈C. We generalize this notion slightly, study the prime ideals of such algebras, and extend some of the standard theorems about affine algebras, Noetherian rings, and Dedekind domains. Our main theorem is that nearly affine domains are catenary, and the Krull dimension equals the transcendence degree of the quotient field. Nevertheless, it is known that nearly affine domains need not be Mori. [Copyright &y& Elsevier]

The structure of Ext is fundamental to understanding rings and their modules. For example, relative injectivity can be interpreted as a structural property of Ext as in when Ext[SUP1,SUBR](A, B) is torsion-free or zero. The purpose of this work is to examine Ext[SUP1,SUBR] (A,B under the assumption that A and B are torsion-free modules of finite rank, and R is a Dedekind domain. It is the abelian group theorist's mantra that their results extend to modules over Dedekind domains. This is almost always the case, however the structure of Ext is strongly influenced by the rank of the completion of R; unlike the case for the integers, the completion of a domain may have finite rank over the domain. Below, R will represent an integral domain with quotient field Q, and we will assume that R ≠ Q. In most case R will be assume to be Dedekind. All unadorned Ext, Hom, and ⊗ symbols are with respect to R. [ABSTRACT FROM AUTHOR]