Anderson has always had wide ranging interests in mathematics.
Logic was an early fascination, geometry was one of the courses he enjoyed designing and teaching, and the history of mathematics became a strong interest. Victor J. Katz, born in Philadelphia, received his Ph. He has long been interested in the history of mathematics and, in particular, in its use in teaching.
Its first edition received the Watson Davis Prize of the History of Science Society, a prize awarded annually by the Society for a book in any field of the history of science suitable for undergraduates.
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He has written and edited about thirty books, mainly on graph theory and the history of mathematics. Victor J. Katz, born in Philadelphia, received his Ph. He has long been interested in the history of mathematics and, in particular, in its use in teaching.
Its first edition received the Watson Davis Prize of the History of Science Society, a prize awarded annually by the Society for a book in any field of the history of science suitable for undergraduates. He has written and edited about thirty books, mainly on graph theory and the history of mathematics. His research interests focus mainly on British mathematics, especially in the 19th and early 20th centuries, and on the history of graph theory and combinatorics. Who Gave You the Epsilon? Moreover, since based his argument on the physics of the problem and this analytic expression satisfies the wave equation, it the known facts about musical vibrations discovered must be twice differentiable.
It was generally recognized In , Euler wrote a paper on the same problem at the time that musical sounds and, in particular, in which he agreed completely with dAlembert con- vibrations of a musical string are composed of fun- cerning the solution but differed from him on its inter- damental frequencies and their harmonic overtones. Euler contended that dAlemberts solution This physical evidence, and some loose mathemati- was not the most general, as the latter had claimed.
By an arbitrary function he meant an arbitrary shape of b by a curve drawn free-hand. Relying on the eighteenth-century article of others. But then one arrived at the man- called its functions. Since Bernoullis initial that century by work in partial differential equations shape of the string was given by an analytic ex- , p.
Bernoulli retorted , The work of Monge in the s, giving a ge- p. Ravetz , p. Eulers no-mans land between the two. The debate did, however, have important consequences for the Fourier and Fourier series evolution of the function concept. Its major effect was to extend that concept to include: Fouriers work on heat conduction submitted to the Paris Academy of Sciences in , but published a Functions defined piecewise by analytic expres- only in in his classic Analytic Theory of Heat sions in different intervals.
It upset several tenets of eighteenth-century of function he gave in his Introductio with the mathematics. The result was known to Euler and La- following definition given in , in which the term grange among others , but only for certain functions. This is a very comprehensive notion arbitrary. An infinity of values being given to and comprises in itself all the modes through the abscissa x, there are an equal number of or- which one quantity can be determined by dinates f x.
We do not now clear that two functions given by different an- suppose these ordinates to be subject to a com- alytic expressions can agree on an interval without mon law; they succeed each other in any manner necessarily agreeing outside the interval. Some of Eulers discontinuous functions Fouriers proof of his theorem was loose even by were shown to be representable by a Fourier the standards of the early nineteenth century.
In seriesan analytic expressionand were thus fact, it was formalism in the spirit of the eighteenth continuous in Eulers sense. To convince the As we shall see, all this forced a re-evaluation of the reluctant mathematical community of the reasonable- function concept. See , , , , ,  for ness of his claim, Fourier needed to show that: details.
A new subjectanalysisbegan to take made sense for arbitrary functions f x , not nec- form, in which the concept of function was central. The concept that interval between the initial segments of the of function was in a state of fluxan analytic expres- resulting Fourier series and the functional values sion an arbitrary formula , then a curve drawn of f x.
Fourier accomplished all this using mathematical Both the subject of analysis certainly its basic no- reasoning that would be clearly unacceptable to us tions and the concept of function were ripe for a today. However, re-evaluation and a reformulation. This is the next It was, no doubt, partially because of his very stage in our development. Dirichlets concept of function Fouriers work raised the analytic algebraic ex- Dirichlet was one of the early exponents of the criti- pression of a function to at least an equal footing cal spirit in mathematics ushered in by the nineteenth with its geometric representation as a curve.
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His century others were Gauss, Abel, Cauchy. He un- work had a fundamental and far-reaching impact on dertook a careful analysis of Fouriers work to make it subsequent developments in mathematics. For ex- mathematically respectable. The task was not simple: ample, it forced mathematicians to re-examine the notion of integral, and was the starting point of the To make sense out of what he [Fourier] did took researches that led Cantor to his creation of the the- a century of effort by men of more critical ory of sets.
As for its impact on the evolution of the genius, and the end is not yet in sight , function concept, Fouriers work: p. Thus, it was by its Fourier series was, of course, incorrect. The Fourier series con- p. He classifies functions as simple and mixed. See , , , , , definite integral, and  for Cauchys contribution. Now let us consider Dirichlets definition of func- b clear understanding of the function concept.
Also, it is irrelevant in usher in a new spirit of rigor in analysis see the what way this correspondence is established. In his famed Cours dAnalyse of and subsequent The novelty in Dirichlets conception of function works, he rigorously defined the concepts of conti- as an arbitrary correspondence lies not so much in the nuity, differentiability, and integrability of a function definition as in its application. Mathematicians from in terms of limits.
Bolzano had done much of this Euler through Fourier to Cauchy had paid lip service earlier, but his work went unnoticed for fifty years. Dirichlet was the first to take seriously the no- tinuous. This paradoxical situation, Cauchy claims, see , p. The Dirichlet function: cannot happen when his definition of continuity is used. Thus, a function can have infinitely many dis- range over all real numbers.
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See , , , , continuities which can be dense in any interval and , ,  for details about Dirichlets work. In connection with the representability of a function in a contrast to Dirichlets function D x , this one is given Fourier series. Certain classes of functions were in- by an analytic expression and is Riemann-integrable. Dirichlets works. It planted the discontinuous firmly The character of analysis began to change.
Since upon the mathematical scene. The importance of this the seventeenth century, the processes of analysis development can be inferred from the following state- were assumed to be applicable to all functions, but ment of Hawkins , p.
In fact, the investigation is essentially a history of attempts to extend the of various classes of functionssuch as continuous integral concept to as many discontinuous func- functions, semi-continuous functions, differentiable tions as possible; such attempts could become functions, functions with non-integrable derivatives, meaningful only after existence of highly dis- integrable functions, monotonic functions, continu- continuous functions was recognized and taken ous functions that are not piecewise monotonic seriously.
One example is Dinis study of continuous non-differentiable func- In , Weierstrass startled the mathematical tions, for which he defined the so-called Dini deriva- community with his famous example of a continu- tives. Bolzano had Bois Reymond and Darboux.
Who Gave You the Epsilon?: and Other Tales of Mathematical History
The first major step in these developments was This example was contrary to all geometric intuition. As we recall, the coefficients of a except possibly at a finite number of points! See , Fourier series are given by integrals. Cauchy had de- p.