Leopold Kronecker constructs in two articles published in 1869 and 1878, a theory which has its roots in Sturm’s work on the determination of the number of real solutions of an equation. The presentation of this theory of characteristics by Émile Picard will give rise to a controversy between the two mathematicians, who claimed the fame for a formula giving the number of solutions of certain systems of several equations. In this article, after an overview of the theory of (...) characteristics, we will show how, through this dispute, it is actually the way these two mathematicians approach mathematics that is at stake. (shrink)

It was Kronecker who sought to avoid the use in mathematics of all numbers other than the positive integers, and he outlined the means for carrying through this program. In the introductory sections of his memoir he briefly indicates the personal philosophy which made such a project appear desirable.

Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and (...) Brouwer. The book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene. (shrink)

La discusión sobre pureza del método suele enfatizar el estudio de demostraciones particulares de la práctica matemática. Una crítica a esta posición cuestiona el valor de la pureza al afirmar que el “progreso matemático” depende esencialmente del empleo de métodos impuros. Este artículo muestra que una perspectiva más holística, centrada en cómo las demandas de pureza pueden operar en la construcción de teorías autónomas, permite identificar un contexto donde la pureza del método adquiere valor en la práctica matemática. En particular, (...) se muestra la relevancia de tales demandas en el caso de las teorías algebraicas de números de Kronecker y Dedekind. (shrink)

In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can …nd a single marked object in a database of size N by using only O(pN ) queries of the oracle that identi…es the object. His result was generalized to the case of …nding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either M or K, M (...) < K, our task is to determine which is the case. We show how to solve this problem with a high probability of success using only iterations of Grover’s basic step (and no other algorithm). Let m be the required number of iterations; we prove that under certain restrictions on the sizes of M and K the estimation.. (shrink)

Today's number theory solves classical problems by structural tools that violate standard philosophical expectations in ontology. The far-reaching practical demands of this mathematics require on one hand that the tools be fully explicit and more rigorous than many philosophical theories are, and on the other hand that they relate as directly and as concisely as possible to guiding intuitions. The standard tools grew from a century-long trend of unifying algebra, topology, and arithmetic, notably in the Weil conjectures; and they rely (...) on devices that Kronecker produced for his idea of a pure arithmetic. Very large functors serve to organize individually simple kinds of data that can themselves even be depicted in simple pictures. Mathematicians and philosophers have debated issues of individuation and identity raised by these tools. (shrink)

Constructivism in mathematics is usually said to have its origin in the reaction of Leopold Kronecker, the great 19th-century German mathematician, to the rise of Georg Ferdinand Ludwig Philipp Cantor's set theory. This is in a sense misleading: Kronecker was not so much the first constructivist as the greatest representative at the time of a tradition of algebraists which includes Isaac Newton, Pierre de Fermat, Leonhard Euler, Carl Gustav Jacob Jacobi, Jacques Charles Francois Sturm – all responsible, as (...) Kronecker was, for the discovery of powerful algorithms – and which has continued since with names such as Adolf Hurwitz, Denes Konig, and Francis Macaulay. Kronecker obtained deep results in arithmetic of modular functions and the theory of elliptic functions with complex multiplication which turned out to be of fundamental importance for 20th-century mathematics. This fact is known to mathematicians. (shrink)

This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here (...) for the first time. (shrink)

In this article, I wish to discuss in an informal way the motivations and the motifs of the constructivist approach to logic and mathematics and by a natural extension to the general field of science, particularly theoretical physics. Foundational questions in those domains are not ruled by philosophical principles, but a critical philosophy of foundations could be the leitmotiv to the extent that it can be used as a criterion to decide between the theoretical options of scientific practices that are (...) often oblivious to their own doctrinal presuppositions. My objective is to provide the justificatory reasons for a constructivist option or posture in the field of scientific knowledge. (shrink)

Discussions of the foundations of mathematics and their history are frequently restricted to logical issues in a narrow sense, or else to traditional problems of analytic philosophy. From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics illustrates the much greater variety of the actual developments in the foundations during the period covered. The viewpoints that serve this purpose included the foundational ideas of working mathematicians, such as Kronecker, Dedekind, Borel and the early Hilbert, and the (...) development of notions like model and modelling, arbitrary function, completeness, and non-Archimedean structures. The philosophers discussed include not only the household names in logic, but also Husserl, Wittgenstein and Ramsey. Needless to say, such logically-oriented thinkers as Frege, Russell and Gödel are not entirely neglected, either. Audience: Everybody interested in the philosophy and/or history of mathematics will find this book interesting, giving frequently novel insights. (shrink)

Figures of Thought looks at how mathematical works can be read as texts and examines their textual strategies. David Reed offers the first sustained and critical attempt to find a consistent argument or narrative thread in mathematical texts. Reed selects mathematicians from a range of historical periods and compares their approaches to organizing and arguing texts, using an extended commentary on Euclid's Elements as a central structuring framework. He develops fascinating interpretations of mathematicians' work throughout history, from Descartes to Hilbert, (...) Kronecker, Dedekind, Weil and Grothendieck. Reed traces the implications of this approach to the understanding of the history and development of mathematics. (shrink)

La philosophie de l’histoire des mathématiques entretient chez Cavaillès un rapport étroit et contrasté, avec celle que Brunschvicg expose dans La Modalité du jugement (1897). L’activité du jugement scientifique y est dite mixte, entre jugements (idéaux) d’intériorité et jugements (réalistes) d’extériorité. La forme mixte de l’activité historique de la connaissance est la modalité du possible. D’où une épistémologie historique qui revendique la filiation idéaliste kantienne et rejette l’idéalisme spéculatif. Cavaillès, penseur de la nécessité créatrice du devenir mathématique, réduisant le rôle (...) de la possibilité et de l’aventure intellectuelle dans son histoire, se détache par là d’un maître dont un Canguilhem serait demeuré plus proche. L’histoire mathématique récente, en préparant à un Kronecker sa revanche sur les conceptions ensemblistes abstraites qui inspiraient le nécessitarisme de Cavaillès, conduit à reconsidérer le radicalisme qui l’opposait à la philosophie brunschvicgienne de la modalité du jugement. (shrink)

La logique arithmétique est la logique interne de l'arithmétique, c'est la traduction ou l'interprétation de la logique formelle dans le langage de l'arithmétique. Cette arithmétique n'est pas l'arithmétique formelle de Frege et Peano, mais l'arithmétique classique de Fermat à Kronecker jusqu'à la théorie contemporaine des nombres. L'hypothèse proposée ici suppose qu'après l'arithmétisation de l'analyse, chez Cauchy et Weierstrass, et l'arithmétisation de l'algèbre, chez Kronecker, la logique formelle a amorcé son arithmétisation avec Hilbert pour atteindre son aboutissement avec l'informatique (...) théorique actuelle. Dans cette perspective, la méthode de la descente infinie de Fermat et l'arithmétique générale de Kronecker fournissent une critique constructiviste de l'induction transfinie en même temps qu'une preuve de consistance interne de l'arithmétique polynomiale. La position fondationnelle défendue dans l'ouvrage se réclame du constructivisme logicomathématique et constitue les assises d'un programme qu'on peut bien appeler " logique de la science " après Peirce et Carnap. Le motif recteur des travaux formels est d'ordre philosophique et c'est dans un esprit oecuménique que l'auteur a voulu mener ces recherches. (shrink)

Nies and Scholz formalized the notion of an infinite qubitstring and referred to it as a ‘state’. They defined ‘quantum Martin-Löf randomness’ for states. We give a notion of measurement of a state in a computable basis and introduce ‘quantum measurement randomness’, a randomness notion for states. A state is quantum measurement random if measuring it in any computable basis yields a Martin-Löf random bitstring with probability one. Our main result is that quantum Martin-Löf randomness strictly implies quantum measurement randomness. (...) This uses the construction of a quantum measurement random state which is not quantum Martin-Löf random. We prove two general results on which this construction relies: The first concerns Martin-Löf randomness relative to computable measures and extends a result of V. Vovk. The second is a combinatorial result about Kronecker products. (shrink)

This short paper investigates the prospects of designing semantically satisfactory fuzzy models for the formal language of precise truth. We start by showing that this language fails to admit fuzzy models based on Kronecker-Delta semantics for sharp truth-predications, and then we explore some alternative semantic possibilities.

Hilbert's programme is shown to have been inspired in part by what we can call Kronecker's programme in the foundations of an arithmetic theory of algebraic quantities.While finitism stays within the bounds of intuitive finite arithmetic, metamathematics goes beyond in the hope of recovering classical logic. The leap into the transfinite proved to be hazardous, not only from the perspective of Gödel's results, but also from a Kroneckerian point of view.

Vector logic is a mathematical model of the propositional calculus in which the logical variables are represented by vectors and the logical operations by matrices. In this framework, many tautologies of classical logic are intrinsic identities between operators and, consequently, they are valid beyond the bivalued domain. The operators can be expressed as Kronecker polynomials. These polynomials allow us to show that many important tautologies of classical logic are generated from basic operators via the operations called Type I and (...) Type II products. Finally, it is described a matrix version of the Fredkin gate that extends its properties to the many-valued domain, and it is proved that the filtered Fredkin operators are second degree Kronecker polynomials that cannot be generated by Type I or Type II products. (shrink)

Edmund Husserl (1859–1938) is the founder of the phenomenological movement which has profoundly influenced twentieth‐century Continental philosophy. The historical setting in which his thought took shape was marked by the emergence of a new psychology (Herbart, von Helmholtz, James, Brentano, Stumpf, Lipps), by research into the foundation of mathematics (Gauss, Rieman, Cantor, Kronecker, Weierstrass), by a revival of logic and theory of knowledge (Bolzano, Mill, Boole, Lotze, Mach, Frege, Sigwart, Meinong, Erdmann, Schröder), as well as by the appearance of (...) a new theory of language (Peirce, Marty). This context is thus very like that which gave birth to the Vienna Circle. Though Husserl's study of classical thinkers began with the British empiricists (Locke, Berkeley, and in particular Hume), he later turned almost exclusively to the writings of Kant, Descartes, and Leibniz. As for his contemporaries outside the phenomenological movement, Husserl's closest – though always critical – engagement was with the neo‐Kantians (Rickert and, above all, Natorp). (shrink)

I show on the basis of unpublished sources how Hilbert’s conviction of the solvability of all mathematical problems originated from an engagement with Kant’s philosophy of mathematics. Furthermore, I consider other sense of the “solvability” or “decidability” of mathematical problems which Hilbert thought about later: decidability in finitely many steps, which is an issue Hilbert inherited from Kronecker, “finitistic decidability” which Hilbert develops by reflecting on Kronecker’s methodological strictures, and finally the decision-problem as raised by Behmann in the (...) 1920s. I argue that these different preoccupations have different historical and biographical roots, and should also be kept conceptually distinct. (shrink)

The period 1873–1947 was productive in fostering ideas about observing, measuring, and quantifying repetitive human movements, prior to the rise of occupational health and ergonomics within industrial psychology. Starting with physiological experimentation in the lab, instruments of graphic inscription were then applied in the industrial workplace, initially as a benevolent measurement for monitoring worker health, but elsewhere as a more invasive measurement for the surveillance of worker efficiency. Herman Helmholtz’s invention of the myograph, and an adaptation called the ergograph, would (...) help form what Kronecker (1873) and later Mosso (1891) termed the ‘curve of fatigue’, and were used in extensive research on factory workers for Jules Amar’s Le Moteur humain in 1914. Meanwhile, in Britain in 1915 the physiologist Sherrington was observing workers in munitions factories, feeding into the formation of the Industrial Fatigue Research Board in 1919, which produced official reports. In the United States, similar but more high-profile research was conducted by Frederick Winslow Taylor and Lillian and Frank Gilbreth, who studied movement efficiency to maximize industrial productivity by innovating upon photographic and chronophotographic techniques. Further physiological research was taken up in Lawrence J. Henderson’s Harvard Fatigue Laboratory between 1927 and 1947 on subjects situated in environmental extremes. (shrink)

It is beginning to be rather well known that Edmund Husserl, the founder of phenomenological philosophy, was originally a mathematician; he studied with Weierstrass and Kronecker in Berlin, wrote his doctoral dissertation on the calculus of variations, and was then a colleague of Cantor in Halle until he moved to the Göttingen of Hilbert and Klein in 1901. Much of Husserl’s writing prior to 1901 was about mathematics, and arguably the origin of phenomenology was in Husserl’s attempts to give (...) philosophical foundations first for analysis and later for the formal sciences in general. However, what exactly Husserl’s thoughts about mathematics were is relatively little known. Stefania Centrone’s book Logic and Philosophy of Mathematics in the Early Husserl fills this lacuna. Centrone deciphers Husserl’s early texts about mathematics and logic somewhat selectively, but also extremely accurately and carefully into lucid English and in terms we know, without however reading contemporary views into Husserl’s views. Her analysis reveals for example that Husserl’s view of logic is close to what is today taught in the standard textbooks and that he viewed abstract mathematics as a theory of structures. Her work also uncovers Husserl’s relationship to the algebraists of logic as well as to Bolzano, Frege, and Hilbert.The book is composed of three rather independent chapters. In contrast to most of the other commentaries on early Husserl, the organization of the book is thematic rather than an attempt to document the various stages in the development of Husserl’s views. The first chapter discusses Husserl’s major work on arithmetic, the second the idea of pure logic, and the last the imaginary in mathematics. (shrink)

A partir del concepto themata de Gerald Holton, sugiero la noción de “tensiones temáticas” en un intento por abordar asuntos relacionados con la necesidad de establecer criterios de identidad en la evolución de controversias científicas. Por “tensiones temáticas” entiendo una variedad de presiones de fondo que moldean el desarrollo de ciertas controversias. Aplico la noción a dos disputas distantes en el tiempo para esclarecer su parentesco: la controversia que sostuvieron platónicos y aristotélicos entre los siglos iii a. c. y iii (...) d. c. sobre el modo de existencia del infinito, y la controversia que sostuvieron Georg Cantor y Leopold Kronecker a finales del siglo xix sobre la legitimidad de los números transfinitos. (shrink)

After discussing Husserl’s development from his years as student up to the publication of the _Philosophie der Arithmetik_ in 1891 in Part 1, we will now examine the various authors and theories that influenced Husserl at the time, starting with his mathematical background. Husserl began his studies in 1876 in Leipzig, attending “lectures in mathematics, physics, astronomy and philosophy” and continued with mathematics and philosophy in Berlin, where he studied under Karl Weierstrass and Leopold Kronecker. Subsequently, Husserl obtained his (...) Ph.D in mathematics in Vienna with former Weierstrass-student Leo Königsberger in January 1883, with the technical work _Beiträge zur Variationsrechnung_ (“Contributions to the Calculus of Variations”). Through both his mathematical and philosophical background Husserl learns about and is influenced by Bernard Bolzano. In the years of ÜBZ and PA Husserl was mainly acquainted with Bolzano’s work as a mathematician and logician, especially his _Paradoxien des Unendlichen_, while later on in Husserl’s development Bolzano’s _Wissenschaftslehre_ will become an important influence. One of the most pervasive influences throughout Husserl’s career will remain his Brentanian background. Husserl was converted to philosophy by Franz Brentano himself and continued his studies with Carl Stumpf, a relatively orthodox Brentanist. In Husserl’s _Philosophie der Arithmetik_ we see much attention going out to the endeavours of other members of the School of Brentano, such as Alexius Meinong and Benno Kerry, and in later works also Kazimierz Twardowski and Anton Marty. After having examined these various significant forces in Husserl’s early philosophy, a general conclusion to the discoveries in Part 1 and Part 2 will follow. (shrink)

Kronecker’s views on the foundations of mathematics are often reduced to jokes and regarded as an outdated set of ill-assorted ideas. A closer look however shows that they constitute an original and coherent doctrine justified by epistemological convictions. This doctrine appears in the article On the concept of number, published in the Journal of Crelle (1887) and, especially, in the last course taught by Kronecker, in Berlin in the summer semester of 1891. This article would attempt to state (...) the principles and insights of Kro-necker’s doctrine by comparing it to other foundation schools of thought so as to underline its originality and show the influence this doctrine exerted on Kronecker’s own mathematical work. (shrink)

In the paper, a detailed analysis of some new logical aspects of Cantor’s diagonal proof of the uncountability of continuum is presented. For the first time, strict formal, axiomatic, and algorithmic definitions of the notions of potential and actual infinities are presented. It is shown that the actualization of infinite sets and sequences used in Cantor’s proof is a necessary, but hidden, condition of the proof. The explication of the necessary condition and its factual usage within the framework of Cantor’s (...) proof makes, Cantor’s proof invalid. It’s shown that traditional Cantor’s proof has a second necessary, but hidden as well, condition which is teleological by its nature, i.e., is not mathematical. The explication of the second necessary condition makes Cantor’s statement on the uncountability of continuum unprovable from the point of view of classical logic.One of the most dramatic facts in history of science is connected just with the notion of actual infinity and consists in the following. On the one hand, Aristotle, Berkeley, Locke, Descartes, Spinoza, Gauss, Kant, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and a lot of other outstanding creators of classical logic and classical mathematics, during millenniums, stated categorically and insistently warned about that the actual infinity is a self-contradictory notion and its usage in mathematics is inadmissible (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered”). On the other hand, a lot of outstanding scientists of the XX c.c., such as Hilbert, Church, Turing, Gödel, etc., and modern axiomatic set theory as a whole ignored this (intuitive though) opinion of genii and accepted Cantor’s “transfinite paradise”, based on the actual infinity. In the paper some logical, philosophical, and psychological aspects of this historical “favorable opposition” are analyzed, and the fact that the scientific intuition of genii has won the historical battle against the mytho-’logic’ of Cantor’s transfinite ‘paradise’ is explained and logically justified. (shrink)

The narrative about the nineteenth century favored by many philosophers of mathematics strongly influenced by either logic or algebra, is that geometric intuition led real and complex analysis astray until Cauchy and Kronecker in one sense and Dedekind in another guided mathematicians out of the labyrinth through the arithmetization of analysis. Yet the use of geometry in most cases in nineteenth century mathematics was not misleading and was often key to important developments. Thus the geometrization of complex numbers was (...) essential to their acceptance and to the development of complex analysis; geometry provided the canonical examples that led to the formulation of group theory; and geometry, transformed by Riemann, lay at the heart of topology, which in turn transformed much of modern mathematics. Using complex numbers as my case study, I argue that the best way to teach students mathematics is through a repertoire of modes of representation, which is also the best way to make mathematical discoveries. (shrink)

In earlier work, representations ofr nucleons were constructed by taking therth Kronecker product of self-representations of the complete homogeneous Lorentz groupL 0, where these were in the form of a four-component Dirac spinor with components corresponding to the internal symmetries of spin, parity, and charge. When permutations that include every possible exchange of spin, charge, and coordinate, are factored out, the4 F coordinates of flat Minskowski space are contracted by an isometry φ such that energy levels correspond to troughs (...) or saddle points in the new nuclear manifoldM. This is a symmetric space, and using the critical point theory of Morse for the neighborhood of an energy level, it is found that the symmetry σ associated with φ separatesM into just the sets of rotational and vibrational levels. Furthermore, by employing only one parameter, corresponding to the range of energies encountered, good agreement is found with the experimental levels and state labels of 10 B, 10 Be, 10 C, 10 C, and 10 O. The supermultiplet theory of Wigner is a necessary condition for the existence of the states. (shrink)

Dedekind used to refer to Riemann as his main model concerning mathematical methodology, particularly regarding the use of abstract notions as a basis for mathematical theories. So, in passages written in 1876 and 1895 he compared his approach to ideal theory with Riemann’s theory of complex functions. In this paper, I try to make sense of those declarations, showing the role of abstract notions in Riemann’s function theory, its influence on Dedekind, and the importance of the methodological principle of avoiding (...) ‘forms of representation’ in shaping ideal theory. In order to emphasize the abstract viewpoint of Riemann and Dedekind, I compare their work with that of their great german contemporaries, Weierstrass and Kronecker; so, an influential ‘Göttingen group’ is confronted with the ‘Berlin school’ of mathematics. Some light is also thrown on the relation between Riemann and Dedekind, particularly with respect to Dedekind’s interest on Riemann’s ideas --in function theory, geometry and topology--, beginning in the 1850s and through later works. The abstract approach of the ‘Göttingen group’ was determinant for the turn to abstract mathematics in our century, and the direct influence of Riemann on Dedekind shows how this approach developed. (shrink)

The theme « Truth and Certainty » is reminiscent of Hegel’s dialectic of prominent in the Phänomenologie des Geistes, but I want to treat it from a different angle in the perspective of the constructivist stance in the foundations of logic and mathematics. Although constructivism stands in opposition to mathematical realism, it is not to be considered as an idealist alternative in the philosophy of mathematics. It is true that Brouwer’s intuitionism, as a variety of constructivism, (...) has idealistic overtones, but my main concern in this paper is located in the mathematical tradition of constructive mathematics from the Greeks to Fermat, Gauss and Kronecker, and from the logical side, in the finitist doctrine of Hilbert and his followers. (shrink)

My discussion centers around Ulrich Kohlenbach's Applied Proof Theory: Proof Interpretations and their Use in Mathematics [21 ] which appears as a major work in the "proof mining project" of recent proof theory. The emphasis is on the tradition of proof theory originating with Hilbert and his motivations. I examine also Hilbert's Kroneckerian inspiration and I maintain that Hilbert's programme is the continuation of Kronecker's programme by logical and metamathematical means. I conclude that despite Kreisel's proposal of a shift (...) of emphasis in traditional proof theory, Hilbert's original proof-theoretical ideal is still alive and proof mining is in the purview of Hilbert programme. © 2011 Elsevier B.V., All rights reserved. (shrink)

From the known coordinate representation of these operators, a unified treatment of the abstract operators for curvilinear coordinates and their canonically conjugate momenta is given for systems in three dimensions. A configuration representation, corresponding to classical configuration space, exists in which description is simplified; the three-dimensional ket space factors into a direct product of one-dimensional spaces. Four cases are examined, according to the range of the continuous curvilinear coordinate. In addition to normalization of momentum eigenstates to the Kronecker delta (...) for finite range of coordinate and normalization to the Dirac delta for range on the entire real axis, normalization occurs to δ (or δ+) for range on the positive (or negative) part of the real axis. The domain of Hermiticity of curvilinear coordinate and conjugate momentum operators in each case is the entire three-dimension ket space. The fundamental commutation relations hold in all representations. Statements to the contrary for the radial and azimuth coordinates in spherical polar coordinates are seen to be erroneous. (shrink)

In this article we describe the logical performances displayed by a context-dependent associative memory model. This model requires the existence of a network able to construct the Kroneker product of two vectors, and then to send the composed vector to a correlation distributed memory. This system of nets is capable to sustain all the operations of the classical propositional calculus. This fact implies the existence of vector logics where the logical functions are displayed by matrix operators constructed using the properties (...) of the Kronecker product. When the basic binary matrix operators act on fuzzy inputs, a probabilistic many-valued logic emerges. The present approach implies the potential existence of alternative vector logics. We describe a vector Shefferian “Iogic”, and we comment the potentialities of multidimensional vector logics. (shrink)

The first half of the twentieth century was marked by the simultaneous development of logic and mathematics. Logic offered the necessary means to justify the foundations of mathematics and to solve the crisis that arose in mathematics in the early twentieth century. In European science in the late nineteenth century, the ideas of symbolic logic, based on the works of J. Bull, S. Jevons and continued by C. Pierce in the United States and E. Schroeder in Germany were getting popular. (...) The works by G. Frege and B. Russell should be considered more progressive towards the development of mathematical logic. The perspective of mathematical logic in solving the crisis of mathematics in Ukraine was noticed by Professor of Mathematics of Novorossiysk University Ivan Vladislavovich Sleshynsky. Sleshynsky is a Doctor of Mathematical Sciences, Professor of Novorossiysk University. After studying at the University for two years he was a Fellow at the Department of Mathematics of Novorossiysk University, defended his master’s thesis and was sent to a scientific internship in Berlin, where he listened to the lectures by K. Weierstrass, L. Kronecker, E. Kummer, G. Bruns. Under the direction of K. Weierstrass he prepared a doctoral dissertation for defense. He returned to his native university in 1882, and at the same time he was a teacher of mathematics in the seminary, Odesa high schools, and taught mathematics at the Odesa Higher Women’s Courses. Having considerable achievements in the field of mathematics, in particular, Pringsheim’s Theorem proved by Sleshinsky on the conditions of convergence of continuous fractions, I. Sleshynsky drew attention to a new direction of logical science. The most significant work for the development of national mathematical logic is the translation by I. Sleshynsky from the French language “Algebra of Logic” by L. Couturat. Among the most famous students of I. Sleshynsky, who studied and worked at Novorossiysk University and influenced the development of mathematical logic, one should mention E. Bunitsky and S. Shatunovsky. The second period of scientific work of I. Sleshynsky is connected with Poland. In 1911 he was invited to teach mathematical disciplines at Jagiellonian University and focused on mathematical logic. I. Sleshynsky’s report “On Traditional Logic”, delivered at the meeting of the Philosophical Society in Krakow. He developed the common belief among mathematicians that logic was not necessary for mathematics. His own experience of teaching one of the most difficult topics in higher mathematics – differential calculus, pushed him to explore logic, since the requirement of perfect mathematical proof required this. In one of his further works of this period, he noted the promising development of mathematical logic and its importance for mathematics. He claimed that for the mathematics of future he needed a new logic, which he saw in the “Principles of Mathematics” by A. Whitehead and B. Russell. Works on mathematical logic by I. Sleszynski prompted many of his students in Poland to undertake in-depth studies in this field, including T. Kotarbiński, S. Jaśkowski, V. Boreyko, and S. Zaremba. Thanks to S. Zaremba, I. Sleshynsky managed to complete the long-planned concept, a two-volume work “Theory of Proof”, the basis of which were lectures of Professor. The crisis period in mathematics of the early twentieth century, marked by the search for greater clarity in the very foundations of mathematical reasoning, led to the transition from the study of mathematical objects to the study of structures. The most successful means of doing this were proposed by mathematical logic. Thanks to Professor I. Sleshynsky, who succeeded in making Novorossiysk University a center of popularization of mathematical logic in the beginning of the twentieth century the ideas of mathematical logic in scientific environment became more popular. However, historical events prevented the ideas of mathematical logic in the domestic scientific space from the further development. (shrink)

Ce recueil comporte les traductions de textes classiques de Bolzano à Kronecker sur ce que Felix Klein a baptisé l’arithmétisation de l’analyse. L’auteure y a ajouté des commentaires introductifs d’ordre biographique et des notes explicatives à contenu technique et biblio-graphique. Ces commentaires et ces notes sont souvent utiles, mais on pourra sou-ligner leur manque de pertinence à l’occasion.

Mathematical objects are divided into (1) those which are autonomous, i.e., not dependent for their existence upon mathematicians’ conscious acts, and (2) intentional objects, which are so dependent. Platonist philosophy of mathematics argues that all objects belong to group (1), Brouwer’s intuitionism argues that all belong to group (2). Here we attempt to develop a dualist ontology of mathematics (implicit in the work of, e.g., Hilbert), exploiting the theories of Meinong, Husserl and Ingarden on the relations between autonomous and intentional (...) objects. In particular we develop a phenomenology of mathematical works, which has the stratified intentional structure discovered by Ingarden in his study of the literary work. (shrink)

We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches to the extreme value theorem. We argue that a given pre-mathematical phenomenon may have several aspects that are not necessarily captured by a single formalisation, pointing to a complementarity rather than a rivalry of the approaches.

This paper sets out to adduce visual evidence for Kroneckerian finitism by making perspicuous some of the insights that buttress Kronecker’s conception of arithmetization as a process aiming at disclosing the arithmetical essence enshrined in analytical formulas, by spotting discrete patterns through algorithmic fine-tuning. In the light of a fairly tractable case study, it is argued that Kronecker’s main tenet in philosophy of mathematics is not so much an ontological as a methodological one, inasmuch as highly demanding requirements (...) regarding mathematical understanding prevail over mere preemptive reductionism to whole numbers. (shrink)

This chapter relates the debate analyzed in Chapter 5 to the only instance in which Wittgenstein attempted (successfully) the constructivization of a classical proof, viz. Euler’s proof for the infinitude of primes. This analysis is used to claim, against much of the literature, that during the “transitional period” Wittgenstein had a constructivist, more precisely Kroneckerian, position on mathematics.