arXiv subjects
Publications and source records attributed to Stefan Neuwirth.
We study the relationship between the growth rate of an integer sequence and harmonic and functional properties of the corresponding sequence of characters. In particular we show that every polynomial sequence contains a set that is Lamba(p) for all p but is not a Rosenthal set. This holds also for the sequence of primes.
This constant is the maximum of the sum $|c_0|+|c_1|+|c_2|+|c_3|$ of the moduli of the coefficients of a trigonometric polynomial $c_0+c_1e^{it}+c_2e^{2it}+c_3e^{3it}$ bounded by 1. Its value is still unknown, but I will present some ideas on how to compute it and describe a distinguished torus of extremal functions.
We compare two henselisations of a residually discrete valuation domain. Our constructive proof that a certain natural morphism is an isomorphism is also a proof in classical mathematics. Although this isomorphism is implicitly accepted as obvious in the literature, it seems that no proof was previously available.
Let M be a finitely generated submodule of a free module over a multivariate polynomial ring with coefficients in a discrete coherent ring. We prove that its module MLT(M ) of leading terms is countably generated and provide an algorithm for computing explicitly a generating set. This result is also useful when MLT(M ) is not finitely generated. Suppose that the base ring is strongly discrete coherent. We provide a Buchberger-like algorithm and prove that it converges if, and only if, the module of leading terms is finitely generated. We also provide a constructive version of Hilbert's syzygy theorem by following Schreyer's method.
In 1955, Paul Lorenzen is a mathematician who devotes all his research to foundations of mathematics, on a par with Hans Hermes, but his academic background is algebra in the tradition of Helmut Hasse and Wolfgang Krull. This shift from algebra to logic goes along with his discovery that his ``algebraic works [...] have been concerned with a problem that has formally the same structure as the problem of consistency of the classical calculus of logic'' (letter to Carl Friedrich Gethmann dated 14 January 1988). After having provided a proof of consistency for arithmetic in 1944 and published it in 1951, Lorenzen inquires still further into the foundations of mathematics and arrives at the conviction that analysis can also be given a predicative foundation. Wilhelm Ackermann as well as Paul Bernays have pointed out to him in 1947 that his views are very close to those proposed by Hermann Weyl in Das Kontinuum (1918): sets are not postulated to exist beforehand; they are being generated in an ongoing process of comprehension. This seems to be the reason for Lorenzen to get into contact with Weyl, who develops a genuine interest into Lorenzen's operative mathematics and welcomes with great enthusiasm his Einf{\"u}hrung in die operative Logik und Mathematik (1955), which he studies line by line. This book's aim is to grasp the objects of analysis by means of inductive definitions; the most famous achievement of this enterprise is a generalised inductive formulation of the Cantor-Bendixson theorem that makes it constructive. This mathematical kinship is brutally interrupted by Weyl's death in 1955; a planned visit by Lorenzen at the Institute for Advanced Study in Princeton takes place only in 1957--1958. As told by Kuno Lorenz, Lorenzen's first Ph.D. student, a discussion with Alfred Tarski during this visit provokes a turmoil in Lorenzen's operative research program that leads to his abandonment of language levels and to a great simplification of his presentation of analysis by distinguishing only between ``definite'' and ``indefinite'' quantifiers: the former govern domains for which a proof of consistency is available and secures the use of the law of excluded middle; the latter govern those for which there isn't, e.g. the real numbers. Lorenzen states in his foreword to Differential und Integral (1965) that he is faithful to Weyl's approach of Das Kontinuum in this simplification. This history motivates a number of mathematical and philosophical issues about predicative mathematics: how does Weyl's interest into Lorenzen's operative mathematics fit with his turn to Brouwer's intuitionism as expressed in ``{\"U}ber die neue Grundlagenkrise der Mathematik'' (1921)? Why does Lorenzen turn away from his language levels and how does this turn relate to Weyl's conception of predicative mathematics? What do Lorenzen's conceptions of mathematics reveal about Weyl's conceptions?
We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.
There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics.
We provide a constructive treatment of basic results in the theory of central simple algebras. One main issue is the fact that one starting result, Wedderburn's Theorem stating that a simple algebra is a matrix algebra over a skew field, is not constructively valid. We solve this problem by proving instead a dynamical version of this theorem. One can use this to give constructive proofs of basic results of the theory of central simple algebras, such as Skolem-Noether Theorem. We illustrate this development by giving an elementary constructive proof of a theorem of Becher (which is itself a consequence of a celebrated theorem of Merkurjev).
We study etale topology and the notion of Azumaya algebra over a commutative ring constructively. As an application of the syntactic version of Barr's Theorem, we show the equivalence between two definitions of Azumaya algebra.
Paul Lorenzen, mathematician and philosopher of the 20th century, mentions October 1947 as the date of a crisis in his mathematical and philosophical investigations. An autograph dated 15 October 1947 documents this crisis. This article proposes a translation and a commentary of it and sketches the circumstances of its writing on the base of his correspondence with Paul Bernays. A letter from Lorenzen to Carl Friedrich Gethmann dated 14 January 1988 carves out the story of this crisis by showing how he soaks up the indications of his correspondents and transmutes them into an absolutely original research. Paul Lorenzen, math{é}maticien et philosophe du 20e si{è}cle, {é}voque le mois d'octobre 1947 comme la date d'une c{é}sure dans ses investigations tant math{é}matiques que philosophiques. Un {é}crit autographe dat{é} du 15 octobre 1947 marque cette c{é}sure. Cet article en propose une traduction et un commentaire ligne {à} ligne, puis esquisse les circonstances de son {é}criture sur la base de la correspondance avec Paul Bernays, figure tut{é}laire de la logique math{é}matique et l'un de ses principaux interlocuteurs {à} cette {é}poque, {à} l'instar de Heinrich Scholz et Oskar Becker. Une lettre de Lorenzen {à} Carl Friedrich Gethmann dat{é}e du 14 janvier 1988 {é}toffe le r{é}cit de cette c{é}sure en montrant comment il s'impr{è}gne des indications que lui fournissent ses correspondants pour les transmuter en une recherche absolument originale.
This article proposes a reading of Book I of Euclid's Elements with an emphasis of the experiences of infinity supplied by it, as a preparatory study for a research on the embodied cognition of infinity and its material anchors. -- Cet article propose une lecture du premier livre des Éléments d'Euclide. Il soutient qu'une telle lecture procure des expériences de l'infini. Il s'agit selon son auteur d'une étude préliminaire à une recherche sur la cognition incarnée de l'infini et de ses ancrages matériels.
This article proposes a fresh and direct reading of foundational texts of philosophy and aims at bringing back the inflamed debates that are contemporaneous with the birth of Greek axiomatics, and indeed at understanding what is timeless in the questions addressed by Zeno. -- Cet article propose une lecture fraiche et directe de textes fondateurs de la philosophie et veut faire revivre les d{é}bats enflamm{é}s qui ont vu naitrel'axiomatique l{é}gu{é}e par les Grecs, voire de comprendre ce qu'il y a d'intemporel dans les questions pos{é}es par Z{é}non.]
This article gives an account of a teaching experience carried out from 2008 to 2021 at the university of Franche-Comt{é} as an answer to the ministerial command of proposing cross-disciplinary courses in the curricula. The goal of the experience was to develop simultaneously a discourse on mathematics and a discourse on literature, two independent discourses, but each filled with the gap between one domain and the other and mindful of the flashes of thought that are lightening from one to the other, witnesses of unity in their reasoning. -- Cet article rend compte d'une exp{é}rience d'enseignement men{é}e de 2008 {à} 2012 {à} l'universit{é} de Franche-Comt{é} en r{é}ponse {à} l'injonction minist{é}rielle de proposer des unit{é}s transversales dans les maquettes de dipl{ô}me. Le but de cette exp{é}rience a {é}t{é} de d{é}velopper {à} la fois un discours sur les math{é}matiques et un discours sur la litt{é}rature, deux discours ind{é}pendants, mais chacun rempli de l'{é}cart entre l'un des domaines et l'autre, et attentif aux {é}clairs de la pens{é}e qui jaillissent de l'un vers l'autre, t{é}moignages d'unit{é} dans la d{é}marche intellectuelle.
In Chapter 3 of his Notes on constructive mathematics, Martin-L{\"o}f describes recursively constructed ordinals. He gives a constructively acceptable version of Kleene's computable ordinals. In fact, the Turing definition of computable functions is not needed from a constructive point of view. We give in this paper a constructive theory of ordinals that is similar to Martin-L{\"o}f's theory, but based only on the two relations "$x \leq y$" and "$x < y$", i.e., without considering sequents whose intuitive meaning is a classical disjunction. In our setting, the operation "supremum of ordinals" plays an important r\^ole through its interactions with the relations "$x \leq y$" and "$x < y$". This allows us to approach as much as we may the notion of linear order when the property "$\alpha \leq \beta$ or $\beta \leq \alpha$" is provable only within classical logic. Our aim is to give a formal definition corresponding to intuition, and to prove that our constructive ordinals satisfy constructively all desirable properties. Note that by adding classical logic, we would recover the ordinals of usual classical mathematics, at the cost of a loss of computability for most statements given in the usual form.
Let R be a local domain, v a valuation of its quotient field centred in R at its maximal ideal. We investigate the relationship between R^h, the henselisation of R as local ring, and {v}, the henselisation of the valuation v, by focussing on the recent result by de Felipe and Teissier referred to in the title. We give a new proof that simplifies the original one by using purely algebraic arguments. This proof is moreover constructive in the sense of Bishop and previous work of the authors, and allows us to obtain as a by-product a (slight) generalisation of the theorem by de Felipe and Teissier.
This essay inquires how mathematical beings could be inserted into the architecture of modes of existence proposed by Bruno Latour in the framework of his pluralist and renewed ontology of the modern world. After a description of the problem, the work of Reviel Netz on the emergence of Greek mathematics, and of Charles Sanders Peirce on the diagrammatic dimension of mathematical practice are presented, as well as their impact on our essay. Its central part is the development of an empirical conception of mathematics that plays a central rôle in the sequel. Our analysis is based on the notion of experience according to William James; it is also inspired by certain aspects of Per Martin-Löf's philosophy. It provides a way of thinking the firm certainty with which proofs endow theorems, while invalidating the interpretation of this certainty as the mark of a direct access to an absolute and transcendental truth. The sequel of our essay builds on this analysis for defining a sort of quasi-mode of existence appropriate for mathematical beings that respects the principal features of modes of existence according to the latourian ontology. In the conclusion, the way this quasi-mode might be integrated into this ontology is discussed, in particular with respect to the mode of reference that prevails in many other sciences.
Inspired by the work of Lorenzen on the theory of preordered groups in the forties and fifties, we define regular entailment relations and show a crucial theorem for this structure. We also describe equivariant systems of ideals {à} la Lorenzen and show that the remarkable regularisation process invented by him yields a regular entailment relation. By providing constructive objects and arguments, we pursue Lorenzen's aim of "bringing to light the basic, pure concepts in their simple and transparent clarity"
Krull's Fundamentalsatz, the generalisation of the main theorem of elementary number theory to integral domains, is the starting point of Lorenzen's career in mathematics. This article traces a conceptual history of Lorenzen's successive reformulations of the Fundamentalsatz on the basis of excerpts of his articles. An edition of the extant correspondence of Lorenzen with Hasse, Krull, and Aubert provides a better understanding of the context of these investigations.