Rigorization of Calculus. 18 th Century Approaches,Cauchy, Weirstrass,

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1 Rigorization of Calculus 18 th Century Approaches,Cauchy, Weirstrass,

2 Basic Problem In finding derivatives, pretty much everyone did something equivalent to finding the difference ratio and letting. Of course, if it makes no sense to divide by it. Various explanations were given to try to make sense of this.

3 Proposed Solutions Patch Proponents Problems Early Limit Ideas, Version 1: A limit is a quantity which the ratio can never surpass, but can approach to within any given difference. Then 2x becomes the limit of 2x + h as h goes to zero. Early Limit Ideas, Version 2: The limit is the value of the ratio at the last instant of time before h an evanescent quantity has vanished. At that moment, 2x+ h becomes 2x. Newton, d Alembert, Lacroix, MacLaurin Berkeley, Lagrange: First, by this very definition of limit, 2x + h can never really become 2x, and the secant line can never become the tangent line. Moreover, the whole process is unintelligible: Either h is zero or it isn t. If it is zero, the difference ratio is undefined. Summing up, h cannot be got rid of.

4 Proposed Solutions Patch Proponents Problems Compensation of Errors: The methods of the calculus get correct results even though the foundations didn t make sense, because the error made in replacing 2x + h by 2x is offset by the error made in treating the curve at the given point as though it coincided with the tangent at two different points, which it does not. Berkeley, Carnot Lagrange: Although the offsetting errors could actually be verified for some functions, it was not at all clear that it could be verified in general.

5 Proposed Solutions Patch Proponents Problems Greek Exhaustion: The process could be made rigorous by proving that, if the rate of change of y is R, R cannot be less than 2x and cannot be more than 2x, so R must be exactly 2x. This method is similar to that used by Archimedes prove that the area of the circle couldn t be less than 2π, and could not be greater than 2π. Maclaurin This might work as a proof method, but still the only method of finding the rates of change involved the suspect methods of infinitesimals, fluxions, etc. In addition, the proofs in general cases were quite difficult.

6 Proposed Solutions Patch Proponents Problems Orders of Zeros: The quantity h can Euler; Laplace Lagrange: This is essentially be made less than any given quantity and so when added to a finite quantity like 2x it is actually zero. But when considered in equivalent to the limit argument above, so the critiques of Berkeley and Lagrange still hold. ratios, it is not zero, so the difference ratio is not meaningless.

7 Proposed Solutions Patch Proponents Problems Algebraic Methods: The equation 2 is just a special case of the more general series representation of a function: We then define the rate of change of y with respect to x as p(x). There is no need to describe what happened to h. In the case of, the derivative is exactly 2x. Lagrange; John Landen, L.F.A. Abrogast, J. P. Gruson It is not clear how the series is related to other approaches to tangents or rates of change. How do we know there is a series for each function? Two functions can have the same Taylor series.

8 Proposed Solutions These were the attempts, such as they were, to address the problems with the foundations of the calculus in the 18 th century. Of all the 18 th century mathematicians, Lagrange probably moved farthest in the direction of recognizing the need for a more rigorous footing for calculus, and he helped lay the foundation for that work. The real solution to these issues didn t really surface until the work of Cauchy and his colleagues in the 19 th century. We look at that next.

9 The Rigorization of Calculus The big idea of what happened was to base calculus on the bedrock of algebra, which everyone was comfortable with. There are several aspects of algebra that made it a natural candidate for providing a rigorous foundation for calculus.

10 The Rigorization of Calculus 1. Algebra had generality and certainty. It was seen as generalized arithmetic, where variables stood for numbers and algebraic manipulations were operatons on those generalized numbers. Because numbers themselves seemed certain, algebra seemed a good candidate for a foundation for calculus.

11 The Rigorization of Calculus 2. Algebra, using as it did the heuristic power of symbols, brought that power to the foundational problems of calculus. 3. Infinite series, infinite products, and continued fractions were all considered to be part of algebra. It was this idea that allowed Lagrange to attempt to define derivatives in terms of the coefficients of a Taylor s series expansion of a function. In this way, Lagrange helped to reduce calculus to algebra, and to rid the derivative of any dependence on the infinitely small.

12 The Rigorization of Calculus 4. The focus of 18 th century algebra was on solving equations, and when they could not be solved exactly, approximations in various forms were used. The most desirable were those expressed as infinite series. Other recursive methods were sometimes used, but when possible the results of these methods were expressed as infinite series.

13 The Rigorization of Calculus Toward the later part of the 18 th century, issues of convergence and approximation of error terms for series came to the forefront. With these concerns came increased use of inequalities to express bounds on error terms, etc. These tools (e.g. inequalities) soon became part of the standard algebraic tool kit, especially for dealing with errors (i.e. small deviations).

14 The Rigorization of Calculus Enter Augustin Louis Cauchy ( ). He succeeded in providing an algebraic foundation for derivative, integral, and other fundamental ideas of calculus. His two most important insights were: Limits could be given a rigorous definition based on inequalities, and The derivative, integral, continuity, and infinite series could all be given a rigorous foundation based on limits.

15 The Definition of Limit Limit was defined as: When the successively attributed values of the same variable indefinitely approach a fixed value, so that finally they differ from it by as little as desired, the last is called the limit of all the others.

16 The Definition of Limit This was essentially the same as a definition of Lacroix in There were no deltas or epsilons in it, but he still resorted to ideas of inequalities in proofs. Weierstrass provided the full blown deltaeplison definition in the 1870 s. By the way, the use of epsilon is probably related to the French term for error.

17 A Little About Cauchy Studied under Lagrange and Laplace, who convinced him to leave civil engineering in favor of mathematics. Worked in analysis, complex analysis, differential equations, probability, and mathematical physics (wave propagation on the surface of a liquid).

18 A Little About Cauchy Cauchy comes second to Euler in terms of productivity, filling 27 volumes with his discoveries (8 full length books, 789 papers). He founded his own journal to publish his work, and another journal had to limit submissions to four pages in order to cope with his output (that rule is still in effect).

19 A Little About Cauchy For all his genius, Cauchy was strongly disliked by most of his contemporaries, and has even been described as a narrowminded bigot, especially in religious matters. He was an ardent supporter of the House of Bourbon in France. Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done." Niels Abel

20 A Little About Cauchy... I managed to approach my too rigid judge at his residence... just as he was leaving... During this very short and very rapid walk, I quickly perceived that I had in no way earned his regards or his respect as a scientist... without allowing me to say anything else, he abruptly walked off, referring me to the forthcoming publication of his Leçons à 'École Polytechnique where, according to him, 'the question would be very properly explored'. Jean Victor Poncelet

21 A Little About Cauchy [His lectures] were very confused, skipping suddenly from one idea to another, from one formula to the next, with no attempt to give a connection between them. His presentations were obscure clouds, illuminated from time to time by flashes of pure genius.... of the thirty who enrolled with me, I was the only one to see it through. Luigi Menabrea

22 A Little About Cauchy Numerous terms in mathematics bear Cauchy's name: the Cauchy integral theorem in the theory of complex functions, the Cauchy Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy Riemann equations and Cauchy sequences.

23 Karl Weierstrass October 1815 (Westphalia) February 1897 (Berlin) Father wished him to study finance, but guess what? He like math.

24 Karl Weierstrass He reacted to the conflict inside him by pretending that he did not care about his studies, and he spent four years of intensive fencing and drinking.... when I became aware of [a letter from Abel to Legendre] in Crelle's Journal during my student years, [it] was of the utmost importance. The immediate derivation of the form of the representation of the function given by Abel..., from the differential equation defining this function, was the first mathematical task I set myself; and its fortunate solution made me determined to devote myself wholly to mathematics; I made this decision in my seventh semester...

25 Karl Weierstrass The standards of rigour that Weierstrass set, defining, for example, irrational numbers as limits of convergent series, strongly affected the future of mathematics. He also studied entire functions, the notion of uniform convergence and functions defined by infinite products. His effort are summed up in the Encyclopaedia Britannica as follows: Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations. He also advanced the theory of bilinear and quadratic forms.

26 Karl Weierstrass Health problems plagued him in his later years. During his last three years he was confined to a wheelchair, immobile and dependent. He died of pneumonia.

27 QUIZ

28 Reincarnation? Kevin Karl