ABSTRACTA probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson’s transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields may have advantages over hyperreals in probabilistic modeling. (...) We show that probabilities developed over such fields are less expressive than hyperreal probabilities. (shrink)

The aim of this paper is to analyze Dimitry Gawronsky’s doctrine of actual infinitesimals. I examine the peculiar connection that his critical idealism establishes between transcendental philosophy and mathematics. In particular, I reconstruct the relationship between Gawronsky’s differentials, Cantor’s transfinite numbers, Veronese’s trans-Archimedean numbers and Robinson’s hyperreal numbers. I argue that by means of his doctrine of actual infinitesimals, Gawronsky aims to provide an interpretation of calculus that eliminates any alleged given element in knowledge.

ABSTRACT The aim of this paper is to discuss how Bruno Bauch deals with the problem of the coordination between empirical concepts and spatiotemporal objects. We shall argue that Bauch reformulates the Kantian distinction between concepts and intuitions by means of a philosophical consideration of differential calculus and that he thereby explains the possibility of such coordination, avoiding certain difficulties of the Kantian doctrine. RESUMO O propósito deste trabalho é discutir como o neokantismo de Bruno Bauch trabalha o problema da (...) coordenação de conceitos e objetos espaço-temporais. Sustentaremos que Bauch reformula a distinção kantiana de conceitos e intuições mediante uma consideração filosófica do cálculo diferencial e que, assim, explica a possibilidade de tal coordenação, evitando certas dificuldades da doutrina de Kant. (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 offers an introduction to Hermann Cohen’s Das Princip der Infinitesimal-Methode, 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, as well as those of several minor figures, 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. (shrink)

This paper offers an introduction to Hermann Cohen’s Das Princip der Infinitesimal-Methode, 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, as well as those of several minor figures, 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. (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...) calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

This thesis argues for the significance of G.W Leibniz’s concepts of ‘expression’, ‘force’ and ‘perspective’ to the writings of Walter Benjamin and Gilles Deleuze. By triangulating the philosophical projects of Benjamin, Deleuze and Leibniz, as has not yet been done, the thesis opens up new perspectives and provides new readings of all three. Designating a structure of relations in which every simple substance or monad serves as a ‘living mirror’ of the universe, Leibniz’s concept of ‘expression’ denotes virtual inclusion or (...) immanence. His concept of ‘force’ denotes the self-incurred drive that motivates the monad to action, while his ‘perspectivism’ defines the monads individuality through their infinite points of view on the world. Deleuze and Benjamin, I suggest, appropriate Leibniz’s concepts as part of their respective critiques of epistemology, which target Kant’s conception of experience as a hierarchic relation of representation, allowing them to redefine experience as non-hierarchal, de-centred and embodied. At the same time, for both, Leibniz’s philosophy serves to criticize historicist views of chronological time. Leibniz’s perspectivism is reformulated by Deleuze and Benjamin as part of their respective critical theories of the image, culminating in their later formulations of the ‘dialectical’ and ‘crystal’ image, respectively. The conclusion however, highlights the diverging paths formed by their returns to Leibniz. Benjamin develops a politically effective ‘historical perspectivism’ in which the discontinuity of history enables ‘true historical time’ to replace chronological time. Deleuze, on the other hand, opts for an ahistorical pure form of temporality, his ‘mannerist perspectivism’ describing a continuous, perpetually repeated ‘becoming’. (shrink)

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.