![]() In The Analyst, he supplied a critique of two central applications of Newton’s Method of Fluxions, namely, the derivative of a product and the derivative of a power. Thus, we can do the following:īerkeley, though an amateur Mathematician at best, was not satisfied with this reasoning. That is, o is an ‘infinitesimal’, so importantly not the same as 0. Then let h = o where o is an ‘infinitely small change in time’ (or infinitely small increment). Firstly, we need to find the ratio (or rate of change) of the formula between t and the increment h: To begin with, let me supply a rough approximation of how Newton found a derivative. ![]() For simplicity, I will mostly drop Newton’s terminology and use conventional descriptions henceforth. y = t 3 where the values of t are units of time, and a fluxion was the derivative of this function, e.g. A fluent was essentially a time varying function, e.g. The contradiction Berkeley identified centres around Newton’s notion of a fluxion of a fluent in his Method of Fluxions. This is, as I will suggest, what initially motivated the need for further developments in rigor for Calculus So, in this paper I want to argue for two things: (1) The Philosopher Berkeley, a mathematical outsider, successfully argued that there was an error in the foundations of Differential Calculus as constructed by Newton and (2) The identification of this error stimulated a fruitful debate amongst eighteenth-century British mathematicians over these foundations. Rather, in 1734 the Philosopher George Berkeley in The Analyst argued that Newton’s development of Calculus relied on a contradiction. It may be surprising to hear that Newton’s development of Calculus was not initially as ”pristine” as we may have thought. The development of Differential Calculus is a paradigm example of what I meant. “athematics is a human endeavour, sometimes stimulated by debate and controversy over the truth of theorems, despite its inhumanly pristine appearance in textbooks and the classroom.” ![]() ![]() In my previous article, “On the History of Proving False Theorems ”, I argued that: This isolation period of a year and a half was one of his most productive, in which he started to develop his theories of Newtonian Mechanics, Optics and Differential Calculus. It was here that the literal and metaphoric ‘apple fell from the tree’. Newton was forced to cease his studies at Cambridge and go into isolation at his home, Woolsthorpe. In 1665 Newton was in his twenties studying at Trinity College, Cambridge, when the Great Plague of London struck – the final resurgence of the bubonic plague in England. I want to focus on a contradiction in one of Newton’s discoveries while in isolation, in the field of Differential Calculus. Recently, you may have read or seen a few articles regarding various historical figures, such as Sir Isaac Newton, and the breakthroughs they made while in self-isolation during a pandemic. ![]()
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