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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., Comment: 38 pages; to appear in Ghanita Bharati

We study the maximum ratio of the Euclidean norm to the comass norm of p-covectors in Euclidean n-space and improve the known upper bound found in the standard references by Whitney and Federer. We go on to prove stable systolic inequalities when the fundamental cohomology class of the manifold is a cup product of forms of lower degree., Comment: 11 pages. To appear in Differential Geometry and Its Applications

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., Comment: 9 pages. Published in London Mathematical Society Newsletter

The existence of nontrivial cup products or Massey products in the cohomology of a manifold leads to inequalities of systolic type, but in general such inequalities are not optimal (tight). Gromov proved an {optimal} systolic inequality for complex projective space. We provide a natural extension of Gromov's inequality to manifolds whose fundamental cohomology class is a cup product of 2-dimensional classes., Comment: 8 pages, published in Journal of Geometry

More than thirty years ago, Brooks and Buser-Sarnak constructed sequences of closed hyperbolic surfaces with logarithmic systolic growth in the genus. Recently, Liu and Petri showed that such logarithmic systolic lower bound holds for every genus (not merely for genera in some infinite sequence) using random surfaces. In this article, we show a similar result through a more direct approach relying on the original Brooks/Buser-Sarnak surfaces., Comment: 8 pages. To appear in Proceedings of the American Mathematical Society

We give a synthetic construction of a complete system of mutually unbiased bases in $\mathbb{C}^3$., Comment: 6 pages, to appear in Open Mathematics

We show that every closed nonpositively curved surface satisfies Loewner's systolic inequality. The proof relies on a combination of the Gauss-Bonnet formula with an averaging argument using the invariance of the Liouville measure under the geodesic flow. This enables us to find a disk with large total curvature around its center yielding a large area., Comment: 10 pages. To appear in Journal of Geometric Analysis

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., Comment: 32 pages

In his 1676 text De Quadratura Arithmetica, Leibniz distinguished infinita terminata from infinita interminata. The text also deals with the notion, originating with Desargues, of the perspective point of intersection at infinite distance for parallel lines. We examine contrasting interpretations of these notions in the context of Leibniz's analysis of asymptotes for logarithmic curves and hyperbolas. We point out difficulties that arise due to conflating these notions of infinity. As noted by Rodriguez Hurtado et al., a significant difference exists between the Cartesian model of magnitudes and Leibniz's search for a qualitative model for studying perspective, including ideal points at infinity. We show how respecting the distinction between these notions enables a consistent interpretation thereof., Comment: 19 pages

We provide choiceless proofs using infinitesimals of the global versions of Peano's existence theorem and Osgood's theorem on maximal solutions. We characterize all solutions in terms of infinitesimal perturbations. Our proofs are more effective than traditional non-infinitesimal proofs found in the literature. The background logical structure is the internal set theory SPOT, conservative over ZF., Comment: 19 pages, Journal of Logic and Analysis

In this paper we offer a reconstruction of the evolution of Leibniz's thought concerning the problem of the infinite divisibility of bodies, the tension between actuality, unassignability and syncategorematicity, and the closely related question of the possibility of infinitesimal quantities, both in physics and in mathematics. Some scholars have argued that syncategorematicity is a mature acquisition, to which Leibniz resorts to solve the question of his infinitesimals namely the idea that infinitesimals are just signs for Archimedean exhaustions, and their unassignability is a nominalist maneuver. On the contrary, we show that sycategorematicity, as a traditional idea of classical scholasticism, is a feature of young Leibniz's thinking, from which he moves away in order to solve the same problem, as he gains mathematical knowledge. We have divided Leibniz's path toward his mature view of infinitesimals into five phases, which are especially significant for reconstructing the entire evolution. In our reconstruction, an important role is played by Leibniz's text De Quadratura Arithmetica. Based on this and other texts we dispute the thesis that fictionality coincides with syncategorematicity, and that unassignability can be bypassed. On the contrary, we maintain that unassignability, as incompatible with the principle of harmony, is the ultimate reason for the fictionality of infinitesimals., Comment: 36 pages

We survey the effective foundations for analysis with infinitesimals developed by Hrbacek and Katz in 2021, and detail some applications. Theories SPOT and SCOT are conservative over respectively ZF and ZF+ADC. The range of applications of these theories illustrates the fact that analysis with infinitesimals requires no more choice than traditional analysis. The theory SCOT incorporates in particular all the axioms of Nelson's Radically Elementary Probability Theory, which is therefore conservative over ZF+ADC., Comment: 15 pages, to appear in Real Analysis Exchange

Currently the two popular ways to practice Robinson's nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach. It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets. We show that internal frameworks provide successful accounts of nonstandard hulls and Loeb measures. The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe., Comment: 34 pages, to appear in Bulletin of Symbolic Logic

Leibniz scholarship is currently an area of lively debate. We respond to some recent criticisms by Archibald et al., Comment: 2 pages, appeared in The Mathematical Intelligencer

In the history of infinitesimal calculus, we trace innovation from Leibniz to Cauchy and reaction from Berkeley to Mansion and beyond. We explore 19th century infinitesimal lores, including the approaches of Simeon-Denis Poisson, Gaspard-Gustave de Coriolis, and Jean-Nicolas Noel. We examine contrasting historiographic approaches to such lores, in the work of Laugwitz, Schubring, Spalt, and others, and address a recent critique by Archibald et al. We argue that the element of contingency in this history is more prominent than many modern historians seem willing to acknowledge., Comment: 60 pages

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., Comment: 19 pages, to appear in Antiquitates Mathematicae