Making the calculus rigorous
Before you read: Should the calculus be rigorous? Why? Do you know who made it rigorous?
I. Scan the article to say:
- what Augustin-Louis Cauchy did during the 1820s
- when the graph of a parabola is continuous and discontinuous judging by the picture
- wben a function f(x) is differentiable
- how Cauchy defined the integral of a function f(x) between the values a and b
Monge's educational ideas were opposed by Lagrange, who favoured a more traditional and theoretical diet of advanced calculus and rational mechanics (the application of the calculus to the study of the motion of solids and liquids). Eventually Lagrange won, and the vision of mathematics that was presented to the world was that of an autonomous subject that was also applicable to a broad range of phenomena by virtue of its great generality, a view that has persisted to the present day.
During the 1820s Augustin-Louis Cauchy lectured at the École Polytechnique on the foundations of the calculus. Since its invention it had been generally agreed that the calculus gave correct answers, but no one had been able to give a satisfactory explanation of why this was so. Cauchy rejected Lagrange's algebraic approach and proved that Lagrange's basic assumption that every function had a power series expansion was in fact false. Newton had suggested a geometric or dynamic basis for calculus, but this ran the risk of introducing a vicious circle when the calculus was applied to mechanical or geometric problems. Cauchy proposed basing the calculus on a sophisticated and difficult interpretation of the idea of two points or numbers being arbitrarily close together. Although his students disliked the new approach, and Cauchy was ordered to teach material that the students could actually understand and use, his methods gradually became established and refined to form the core of the modern rigorous calculus, a subject now called mathematical analysis.
Traditionally, the calculus had been concerned with the two processes of differentiation and integration and the reciprocal relation that exists between them. Cauchy provided a novel underpinning by stressing the importance of the concept of continuity, which is more basic than either. He showed that, once the concepts of a continuous function and limit are defined, the concepts of a differentiable function and an integrable function can be defined in terms of them. Unfortunately, neither of these concepts is easy to grasp, and the much-needed degree of precision they bring to mathematics has proved difficult to appreciate. Roughly speaking, a function is continuous at a point in its domain if small changes in the input around the specified value only produce small changes in the output.
Figure 4: Continuous and discontinuous functions.
Date: 2015-01-29; view: 723