Experience seems to show that the student usually finds a new theory difficult to grasp at a first reading. He needs to return to it several times before he becomes really familiar with it and can distinguish for himself which are the essential ideas and which results are of minor importance, and only then will he be able to apply it intelligently.
— quoted by Jean Dieudonne, in his preface to Foundations of Modern Analysis. (Academic Press, NY and London, 1969).
“I protest…against the use of infinite magnitude as if it were something finished; this use is not something admissible. The infinite is only a facon de parler, one has in mind limits approached by certain ratios as closely as desirable while other ratios may increase indefinitely.” Carl Friedrich Gauss, presumably the foremost mathematician of the 19th century, expressed this view in 1831 in reply to an idea of Schumacher and in the process uttered a horror infiniti which up to almost the end of the century was the common attitude of mathematicians and seemed unassailable considering the authority of Gauss. Mathematics should deal with finite magnitudes and finite numbers only while the treatment of the actual infinity, whether infinitely great or infinitely small, might be left to philosophy.
It was the mathematician Georg Cantor (1845-1918) who dared to fight this attitude and in the opinion of the majority of 20th century mathematicians, has succeeded in the task of bestowing legitimacy upon infinitely great magnitude. Besides the creative intuition and the artistic power of production, which guided Cantor in his work, an enormous amount of energy and perseverance was required to carry through the new ideas, which for two decades were rejected by his contemporaries with the arguments that they were meaningless or false or “brought into the world of mathematics a hundred years too early”. Not only Gauss and other outstanding mathematicians were quoted in evidence against the actual infinity but also leading philosophers such as Aristotle, Descartes, Spinoza and modern logicians. Set theory was even charged with violating the principles of religion, an accusation rejected by Cantor with particular vigour and minuteness. Only in the last years of the nineteenth century, when Cantor had ceased engaging in mathematical research, did set theory begin to infiltrate many branches of mathematics.
Cantor has shown how definite and distinctly great magnitudes can be handled — another evidence of the free creation which is characteristic of mathematics to a higher extent than that of the other sciences. It is no mere accident that at the birth of the set theory, the slogan was coined: the very essence of mathematics is its freedom.
Ref: Abstract Set Theory by Abraham A. Fraenkel.
Shared by Nalin Pithwa …(a piece that I enjoyed in the beginning of the book)
Reference: Chapter 1, Sets and Functions; Topology and Modern Analysis, G F Simmons, Tata McGraw Hill Pub.
It is sometimes said that math is the study of sets and functions. Naturally, this oversimplifies matters, but it does come as close to the truth as an aphorism can.
The study of sets and functions leads two ways. One path goes down, into the abysses of logic, philosophy, and the foundations of mathematics. The other goes up into the highlands of mathematics itself where these concepts are indispensable in almost all of pure mathematics as it is today. Needless to say, we follow the latter course. We regard sets and functions as tools of thought, and our purpose in this chapter is to develop these tools to the point where they are sufficiently powerful to serve our needs through the rest of the book.
As the reader proceeds, he will come to understand that the words set and function are not as simple as they may seem. In a sense, they are simple, but they are potent words, and the quality of simplicity they possess is that which lies on the far side of complexity. They are like seeds, which are primitive in appearance but have the capacity for vast and intricate developments.
Rudin is distilled to the essence.
A note to reader of Math books:
(I like this v much; I am reproducing it verbatim as the advice of G F Simmons, Topology and Modern Analysis, Tata McGraw Hill Publication, India)
Two matters call for special comment: the problems and proofs.
The majority of this problems are corollaries and extensions of theorems proved in the text, and are freely drawn upon at all later stages of the book. In general, they serve as a bridge between ideas just treated and development yet to come, and the reader is strongly urged to master them as he goes along.
In the earlier chapters, proofs are given in considerable detail, in an effort to smooth the way for the beginner. As our subject unfolds through the successive chapters and the reader acquires experience in following abstract mathematical arguments, the proofs become briefer and minor details are more and more left for the reader to fill in for himself. The serious student will train himself to look for gaps in proofs, and should regard them as tacit invitations to do a little thinking on his own. Phrases like “it is easy to see,” “one can easily show,” “evidently,” “clearly,” and so on are always to be taken as warning signals which indicate the presence of gaps, and they should put the reader on his guard.
It is a basic principle in the study of mathematics, and one too seldom emphasised, that a proof is not really understood until the stage is reached at which one can grasp it as a whole and see it as a single idea. In achieving this and, much more is necessary than merely following the individual steps in the reasoning. This is only the beginning. A proof should be chewed, swallowed, and digested, and this process of assimilation should not be abandoned until it yields a full comprehension of the overall pattern of thought.
Isn’t that beautiful advice for all budding math aspirants ?