The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes

Ehrlich, Philip

doi:10.1007/s00407-005-0102-4

Cited by 94 publications

(83 citation statements)

References 100 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Cauchy's definition in terms of null sequences is a reasonable definition, and one that connects well with the sequential approach of the ultrapower construction. 52 Mathematicians viewed infinitesimals with deep suspicion due in part to a conflation of two separate criticisms, the logical one and the metaphysical one, by Berkeley, see Sherry [143]. Thus, the emphasis on the elimination of infinitesimals in the traditional account of the history of analysis is misplaced.…”

Section: A Critique Of the Weierstrassian Scientific Methodsmentioning

confidence: 99%

“…When Stolz meekly objected that Cantor's "proof" does not apply to his system, Cantor responded by the "abomination" remark (see Section 6 for more details). Now Cantor's proof contains an error that was exhastively analyzed by Ehrlich [52]. As it stands, it would "prove" the non-existence of the surreals!…”

Section: A Question Sessionmentioning

confidence: 99%

“…No contradiction here. The otherwise excellent study by Ehrlich [52] contains a curious slip with regard to Poisson. Poisson describes infinitesimals as being "less than any given magnitude of the same nature" [123, p. 13-14] Answer.…”

Section: A Question Sessionmentioning

confidence: 99%

See 2 more Smart Citations

A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

Katz

2011

Found Sci

View full text Add to dashboard Cite

Abstract. 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 an ontologically limitative disposition on historiography, teaching, and research.

show abstract

Section: A Critique Of the Weierstrassian Scientific Methodsmentioning

confidence: 99%

Section: A Question Sessionmentioning

confidence: 99%

See 1 more Smart Citation

A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

Katz

2011

Found Sci

View full text Add to dashboard Cite

show abstract

“…Via Veronese he introduced the Marburg community to a tradition going back to the work of Otto Stolz (1883Stolz ( , 1891 and Paul Du BoisReymond (1877), which Hans Hahn (1907) had recently shown was still fruitful (cf. Ehrlich, 2006 for more details). Unfortunately, Gawronsky's book is burdened by an apologetic agenda that probably rendered it unreadable outside Cohen's restricted group.…”

Section: The Marburg Divide: Gawronsky Cassirer and Natorp On The Inmentioning

confidence: 99%

Hermann Cohen's Das Princip der Infinitesimal-Methode: The history of an unsuccessful book

Giovanelli¹

2016

Studies in History and Philosophy of Science Part A

View full text Add to dashboard Cite

This paper offers an introduction to Hermann Cohen's Das Princip der Infinitesimal-Methode (1883), and recounts the history of its controversial reception by Cohen's early sympathizers, who would become the so-called 'Marburg school' of Neo-Kantianism, as well as the reactions it provoked outside this group. By dissecting the ambiguous attitudes of the best-known representatives of the school (Paul Natorp and Ernst Cassirer), as well as those of several minor figures (August Stadler, Kurd Lasswitz, Dimitry Gawronsky, etc.), this paper shows that Das Princip der Infinitesimal-Methode is a unicum in the history of philosophy: it represents a strange case of an unsuccessful book's enduring influence. The "puzzle of Cohen's Infinitesimalmethode," as we will call it, can be solved by looking beyond the scholarly results of the book, and instead focusing on the style of philosophy it exemplified. Moreover, the paper shows that Cohen never supported, but instead explicitly opposed, the doctrine of the centrality of the 'concept of function', with which Marburg Neo-Kantianism is usually associated.

show abstract

“…The latter is used to date in large parts of physics and engineering, and was used historically in mathematics by e.g. Archimedes, Leibniz, Euler, and Newton until the advent of the Weierstraß 'epsilon-delta' framework, as detailed in [36].…”

mentioning

confidence: 99%

To be or not to be constructive, that is not the question

Sanders

2018

Indagationes Mathematicae

View full text Add to dashboard Cite

Abstract. In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of mathematics, called intuitionism. Intuitionism was revolutionary in many respects but stands out -mathematically speaking-for its challenge of Hilbert's formalist philosophy of mathematics and rejection of the law of excluded middle from the 'classical' logic used in mainstream mathematics. Out of intuitionism grew intuitionistic logic and the associated BrouwerHeyting-Kolmogorov interpretation by which 'there exists x' intuitively means 'an algorithm to compute x is given'. A number of schools of constructive mathematics were developed, inspired by Brouwer's intuitionism and invariably based on intuitionistic logic, but with varying interpretations of what constitutes an algorithm. This paper deals with the dichotomy between constructive and non-constructive mathematics, or rather the absence of such an 'excluded middle'. In particular, we challenge the 'binary' view that mathematics is either constructive or not. To this end, we identify a part of classical mathematics, namely classical Nonstandard Analysis, and show it inhabits the twilight-zone between the constructive and non-constructive. Intuitively, the predicate 'x is standard' typical of Nonstandard Analysis can be interpreted as 'x is computable', giving rise to computable (and sometimes constructive) mathematics obtained directly from classical Nonstandard Analysis. Our results formalise Osswald's longstanding conjecture that classical Nonstandard Analysis is locally constructive. Finally, an alternative explanation of our results is provided by Brouwer's thesis that logic depends upon mathematics.

show abstract

The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes

Cited by 94 publications

References 100 publications

A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

Hermann Cohen's Das Princip der Infinitesimal-Methode: The history of an unsuccessful book

To be or not to be constructive, that is not the question

Contact Info

Product

Resources

About