arXiv subjects

Karl Kuhlemann

Publications and source records attributed to Karl Kuhlemann.

10 recordsLinked to original sources

arXiv2026

Marx versus Engels on infinitesimals: Chimera or triumph?

We document the evolution of Karl Marx's take on infinitesimals. We contrast his initial favorable stance with later criticisms, and examine the differing perspectives of Marx and Engels on the subject. Marx's favorable assessment was based on his study of Sauri's textbook. Later, influenced by Boucharlat's textbook, Marx reversed his position to an unfavorable stance, describing belief in infinitesimals as a `chimera'. Marxist scholar Guglielmo Carchedi claims that ``Marx differentiates with the eyes of the social scientist, of the dialectician'' but fails to note dialectician Engels' endorsement of infinitesimals. Struik linked Marx to Abraham Robinson, but missed the fact that the link passes via ... Fermat. Namely, there may be an affinity, as per Struik, between Marx's comments on the calculus and Robinson's nonstandard analysis, but the kernel of such an affinity resides in the techniques already found in the context of Fermat's adequality. To adapt Carchedi's metaphor, we could say that Marx may have differentiated with the eyes of adaequo of Pierre de Fermat. The first editor who worked on some of Marx's mathematical manuscripts in the mid-1920s was Emil J. Gumbel, though he is not mentioned in either the 1933 or the 1968 Soviet edition of Marx's mathematical manuscripts.

math.HO↗

arXiv2026

A philosophical history of infinitesimals

We explore the issue of providing a foundational framework for Leibnizian infinitesimals in the light of modern standard and nonstandard approaches. We outline a trichotomy of ordinals, cardinals and ringinals as a historiographic tool. A ringinal is a concept of infinite number, arithmetic in nature, different from Cantor's transfinite ordinals and cardinals. The continuum is not necessarily identifiable with R; even if one seeks such an identification, infinitesimals are not ruled out. Analysis with unlimited numbers (via the predicate standard) is possible in a conservative extension of Zermelo-Fraenkel set theory and in this sense is epistemologically 'safe'. We sketch a recent theory of infinitesimal analysis that formalizes Leibnizian definitions and heuristic principles while eschewing both the axiom of choice and ultrafilters, thus challenging received philosophical views on the nature of infinitesimals.

math.HO↗

arXiv2025

Formalism 25

Abraham Robinson's philosophical stance has been the subject of several recent studies. Erhardt following Gaifman claims that Robinson was a finitist, and that there is a tension between his philosophical position and his actual mathematical output. We present evidence in Robinson's writing that he is more accurately described as adhering to the philosophical approach of Formalism. Furthermore, we show that Robinson explicitly argued {against} certain finitist positions in his philosophical writings. There is no tension between Robinson's mathematical work and his philosophy because mathematics and metamathematics are distinct fields: Robinson advocates finitism for metamathematics but no such restriction for mathematics. We show that Erhardt's analysis is marred by historical errors, by routine conflation of the generic and the technical meaning of several key terms, and by a philosophical {parti pris}. Robinson's Formalism remains a viable alternative to mathematical Platonism.

math.HO↗

arXiv2025

Leibniz's contested infinitesimals: Further depictions

We contribute to the lively debate in current scholarship on the Leibnizian calculus. In a recent text, Arthur and Rabouin argue that non-Archimedean continua are incompatible with Leibniz's concepts of number, quantity and magnitude. They allege that Leibniz viewed infinitesimals as contradictory, and claim to deduce such a conclusion from an analysis of the Leibnizian definition of quantity. However, their argument is marred by numerous errors, deliberate omissions, and misrepresentations, stemming in a number of cases from flawed analyses in their earlier publications. We defend the thesis, traceable to the classic study by Henk Bos, that Leibniz used genuine infinitesimals, which he viewed as fictional mathematical entities (and not merely shorthand for talk about more ordinary quantities) on par with negatives and imaginaries.

math.HO↗

arXiv2024

A Leibniz/NSA comparison

We present some similarities between Leibnizian and Robinsonian calculi, and address some objections raised by historians. The comparison with NSA facilitates our appreciation of some Leibnizian procedures that may otherwise seem obscure. We argue that Leibniz used genuine infinitesimals and infinite quantities which are not merely stenography for Archimedean Exhaustion and that Leibniz's procedures therefore find better proxies in NSA than in modern Weierstrassian mathematics.

math.HO↗

arXiv2024

Exploring Felix Klein's contested modernism

An alleged opposition between David Hilbert and Felix Klein as modern vs countermodern has been pursued by marxist historian Herbert Mehrtens and others. Scholars such as Epple, Grattan-Guinness, Gray, Quinn, Rowe, and recently Siegmund-Schultze and Mazzotti have voiced a range of opinions concerning Mehrtens' dialectical methodology. We explore contrasting perspectives on Klein's contested modernism, as well as Hilbert's and Klein's views on intuition, logic, and physics. We analyze Jeremy Gray's comment on Klein's ethnographic speculations concerning Jewish mathematicians and find it to be untenable. We argue that Mehrtens was looking for countermoderns at the wrong address.

math.HO↗

arXiv2022

Three case studies in current Leibniz scholarship

We examine some recent scholarship on Leibniz's philosophy of the infinitesimal calculus. We indicate difficulties that arise in articles by Bassler, Knobloch, and Arthur, due to a denial to Leibniz's infinitesimals of the status of mathematical entities violating Euclid V Definition 4.

math.HO↗

arXiv2021

Leibniz on bodies and infinities: rerum natura and mathematical fictions

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part-whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals - as well as imaginary roots and other well-founded fictions - may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality.

math.HO↗

arXiv2021

Two-track depictions of Leibniz's fictions

Leibniz described imaginary roots, negatives, and infinitesimals as useful fictions. But did he view such 'impossible' numbers as mathematical entities? Alice and Bob take on the labyrinth of the current Leibniz scholarship.

math.HO↗

arXiv2020

Procedures of Leibnizian infinitesimal calculus: An account in three modern frameworks

Recent Leibniz scholarship has sought to gauge which foundational framework provides the most successful account of the procedures of the Leibnizian calculus (LC). While many scholars (e.g., Ishiguro, Levey) opt for a default Weierstrassian framework, Arthur compares LC to a non-Archimedean framework SIA (Smooth Infinitesimal Analysis) of Lawvere-Kock-Bell. We analyze Arthur's comparison and find it rife with equivocations and misunderstandings on issues including the non-punctiform nature of the continuum, infinite-sided polygons, and the fictionality of infinitesimals. Rabouin and Arthur claim that Leibniz considers infinities as contradictory, and that Leibniz' definition of incomparables should be understood as nominal rather than as semantic. However, such claims hinge upon a conflation of Leibnizian notions of bounded infinity and unbounded infinity, a distinction emphasized by early Knobloch. The most faithful account of LC is arguably provided by Robinson's framework. We exploit an axiomatic framework for infinitesimal analysis called SPOT (conservative over ZF) to provide a formalisation of LC, including the bounded/unbounded dichotomy, the assignable/inassignable dichotomy, the generalized relation of equality up to negligible terms, and the law of continuity.

math.HO↗