Quotes
Table of content:
Larry J. Gerstein
Students who are new to higher mathematics are often startled to discover that mathematics is a subject of ideas, and not just formulaic rituals, and that they are now expected to understand and create mathematical proofs. Mastery of an assortment of technical tricks may have carried the students through calculus, but it is no longer a guarantee of academic success.
Students need experience in working with abstract ideas at a nontrivial level if they are to achieve the sophisticated blend of knowledge, discipline, and creativity that we call “mathematical maturity.”
Like any science, mathematics is a thriving wonderland of research, with many mysteries to keep us humble despite the subject’s many remarkable achievements.
Eric Gosset
Although logic is an essential tool for the mathematician (to prove theorems), it may appear to be of only minor interest to other people. However, this is not the case. We all frequently encounter logical (or seemingly logical) presentations of facts and conclusions. We need to be able to identify which claims do have logical support and which do not.
Paul R. Halmos
Set theory and algebra seem to stand to arithmetic and differential equations in the same ratio as words stand to numbers.
There is all the difference in the world between an exposition that cannot be misunderstood and one that is in fact understood.
Study actively. Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
My advice is to read till you come to a definition new to you, and then stop and try to think of examples and non-examples, or till you come to a theorem new to you, and then stop and try to understand it and prove it for youself - and, most important, when you come to an obstacle, a mysterious passage, an unsolvable problem, just skip it. Jump ahead, try the next problem, turn the page, go to the next chapter, or even abandon the book and start another one. Books may be linearly ordered, but our minds are not.
Everything we learn changes everything we know and will help us later to learn more.
Steven G. Krantz
We all know that one way to work out our thoughts is to engage in an animated discussion with someone whom we respect. But you can instead, à la Descartes, have that discussion with yourself. And a useful way to do so is by writing.
Serge Lang
Very few books are meant to be read from beginning to end, and there are many ways of reading a book.
In writing the book, the whole subject has to be organized in a totally ordered way, along lines and pages, which is not the way our brain works naturally. But it is unavoidable that some topics have to be placed before others, even though our brain would like to perceive them simultaneously. This simultaneity cannot be achieved in writing, which thus gives a distortion of the subject.
In any book, it is impossible to avoid some mistakes, some confusion, some incorrectness of language, and some misuse of notation. If you find any such things in the present book, then correct them or improve them for yourself, or write your own book. This is still the best way to learn a subject, aside from teaching it.
George Pólya
Now, observe that inductive reasoning is a particular case of plausible reasoning. Observe also (what modern writers almost forgot, but some older writers, such as Euler and Laplace, clearly perceived) that the role of inductive evidence in mathematical investigation is similar to its role in physical research. Then you may notice the possibility of obtaining some information about inductive reasoning by observing and comparing examples of plausible reasoning in mathematical matters. And so the door opens to investigating induction inductively.