Foundation for fluxions

“When am I ever going to use this?” As a math teacher, this is the number one question that I hear from students. It is also a wrong question; it isn’t the question the student truly intended to ask. The question they are really asking is “Why should I value this?” and they expect a response in terms of how math will solve their problems. But should we study math only because it is useful? Or should we study math because it is true? It is my contention that valuing mathematical inquiry as a pursuit of truth is a better mindset in which to approach the practice of mathematics, rather than exalting practicality. This paper will demonstrate one unexpected reason to support such a philosophical view: it actually leads to more practical applications of mathematical endeavors than would otherwise be discovered. Support for this theory may be found in the life of George Berkeley. This paper will examine the historic mathematical implications of Berkeley’s philosophical convictions: the refinement of real analysis and the development of nonstandard analysis. Berkeley not only answers the question of why we need philosophical integration in mathematics, but also how we approach such integration: through Christian faith. This paper will close by examining the latter.

BOOK OF ABSTRACTS, 2010

Wherein it is examined whether the objects, principles, and inferences of the modern analysis really are more distinctly conceived, or more evidently deduced than the humble infinitesimal calculus.

Resumo: Neste artigo analisamos as críticas apresentadas por George Berkeley, em The analyst (1734), ao método das fluxões e à inconsistência intrínseca à noção de infinitésimo do cálculo diferencial e integral, introduzido por Isaac Newton. Procuramos mostrar que as críticas de Berkeley não eram de todo infundadas, uma vez que foram necessários quase duzentos anos para que viesse a ser introduzida por Karl Weierstrass a definição rigorosa de limite, que propiciou uma solução para o problema dos infinitésimos. São mencionadas ainda duas outras teorias contemporâneas, com abordagens distintas para a solução da questão do infinitésimo: a análise não-standard de Abraham Robinson e o cálculo diferencial paraconsistente proposto por Newton da Costa. Apesar de serem citados alguns autores importantes para o desenvolvimento do cálculo, este artigo não se propõe a analisar suas obras e não pretende apresentar uma história do cálculo diferencial e integral. Palavras-chave: George Berkeley, método das fluxões, infinitésimo, cálculo diferencial paraconsistente.

Our spinning school is in a thriving way. The children begin to find a pleasure in being paid in hard money. 1 In the early 1730s George Berkeley began to explore the conceptual field between ideas and spirits that he previously claimed to be empty. In this field he found a rich set of concepts including "notions," "principles," "beliefs," "opinions," and even "prejudices." Elsewhere I have referred to this phase in Berkeley's thought as his "second conceptual revolution." 2 I believe that it was motivated by his increasing need to develop a language to discuss the social, moral and theological concerns vital to him and his circle. This second conceptual revolution made possible two of his most important contributions to 18 th century thought: The Analyst (1734) and The Querist (1735-37). Even though they were written almost simultaneously, these texts are rarely discussed together, since the former is categorized as a ...

The infinitely small and the infinitely large are essential in calculus. They have appeared throughout its history in various guises: infinitesimals, indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small magnitudes, infinite sums, power series, limits, and hyperreal numbers. And they have been fundamental at both the technical and conceptual levels -as underlying tools of the subject and as its foundational underpinnings. We will consider examples of these aspects of the infinitely small and large as they unfolded in the history of calculus from the 17 th through the 20 th centuries. We will also present 'didactic observations' at relevant places in the historical account.

Continental Philosophy Review, 2010

The aim of this paper is to explore the uses made of the calculus by Gilles Deleuze and G.W.F. Hegel. I show how both Deleuze and Hegel see the calculus as providing a way of thinking outside of finite representation. For Hegel, this involves attempting to show that the foundations of the calculus cannot be thought by the finite understanding, and necessitate a move to the standpoint of infinite reason. I analyse Hegel’s justification for this introduction of dialectical reason by looking at his responses to Berkeley’s criticisms of the calculus. For Deleuze, instead, I show that the differential must be understood as escaping from both finite and infinite representation. By highlighting the sub-representational character of the differential in his system I show how the differential is a key moment in Deleuze’s formulation of a transcendental empiricism. I conclude by dealing with some of the common misunderstandings which occur when Deleuze is read as endorsing a modern mathematical interpretation of the calculus.

2005

Gödel's incompleteness results are two of the most fundamental and important contributions to logic and the foundations of mathematics. He showed that no axiomatizable formal system strong enough to capture elementary number theory can prove every true sentence in its language. This theorem is an important limiting result regarding the power of formal axiomatics, but has also been of immense importance in other areas, such as the theory of computability.

K. Lee. Lerner. "The Elaboration of the Calculus." (Preprint) Originally published in Schlager, N. Science and Its Times: Understanding the Social Significance of Scientific Discovery. Thomson Gale, 2001

Many of the most influential advances in mathematics during the 18th century involved the elaboration of the calculus, a branch of mathematical analysis which describes properties of functions (curves) associated with a limit process. Although the evolution of the techniques included in the calculus spanned the history of mathematics, calculus was formally developed during the last decades of the 17th century by English mathematician and physicist Sir Isaac Newton (1643-1727) and, independently, by German mathematician Gottfried Wilhelm von Leibniz (1646-1716). Although the logical underpinnings of calculus were hotly debated, the techniques of calculus were immediately applied to a variety of problems in physics, astronomy, and engineering. By the end of the 18th century, calculus had proved a powerful tool that allowed mathematicians and scientists to construct accurate mathematical models of physical phenomena ranging from orbital mechanics to particle dynamics. Although it is clear that Newton made his discoveries regarding calculus years before Leibniz, most historians of mathematics assert that Leibniz independently developed the techniques, symbolism, and nomenclature reflected in his preemptory publications of the calculus in 1684 and 1686. The controversy regarding credit for the origin of calculus quickly became more than a simple dispute between mathematicians. Supporters of Newton and Leibniz often arguing along bitter and blatantly nationalistic lines and the feud itself had a profound influence on the subsequent development of calculus and other branches of mathematical analysis in England and in Continental Europe.

This essay investigates the rhetoric surrounding the appearance of the concept of the infinitesimal in the seventeenth-century Calculus of Sir Isaac Newton and Gottfried Wilhelm Leibniz. Although historians often have positioned rhetoric as a supplemental discipline, this essay shows that rhetoric is the “material” out of which a new and powerful mathematical system emerges. At the height of empiricism, the infinitesimal, thought by Newton and Leibniz to be evanescent or nascent, made available no recourse to empirical or geometric verification. Instead, the infinitesimal found its “substance” in the rhetorical arguments surrounding it, which ultimately precipitated an epistemic shift in scientific and mathematical practice.