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.

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 small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The (...) casual mathematical reader may be satisfied to read the text of the five act play, whereas the others may wish to delve into the 130 footnotes, some of which contain elucidation of the mathematics or comments on the history. (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)

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.

In this article, we analyse how Ernst Cassirer’s approach of a phenomenology of knowledge deals with the general question of disunity in science and society. By elaborating on the concept of functional unity, which presupposes difference, Cassirer’s work helps to revise foundational concepts of modern science and society, such as pluralism and truth. Relating Cassirer’s approach to the current interest in political epistemology, we show the implications of Cassirer’s theory of knowledge and analyses of modern science, particularly physics. In these (...) analyses, Cassirer carves out the relational logic of scientific knowledge and its consequences on epistemological and ethico-political levels. While this logic of relations relativises absolute claims on either level, it entails its distinct normative criteria as elements of the scientific ideal. Cassirer’s project defends this ideal and thus can simultaneously defend plurality _in_ science (including the humanities) in terms of methods, perspectives and aims, and preserve the unity _of_ science as a normative ideal and symbolic form in constant interaction with other forms. Thus, Cassirer’s pluralism must be distinguished from mere historicist or relativist conceptions. (shrink)

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)

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.

In this paper we compare the interpretation of the differential calculus that Hegel proposes in the Science of Logic with that which Cohen develops in The Principle of the Infinitesimal Method and its History and in the second edition of the Kantian Theory of Experience. Although both philosophers find in the calculus the founding qualitative principle of the quantitative, Cohen argues that Hegel does not solve the problem of the relation between mathematics and natural science.

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)