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.

Larry J. Gerstein, Introduction to mathematical structures and proofs, New York, Sudbury MA, 1996, v.


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.”

Larry J. Gerstein, Introduction to mathematical structures and proofs, New York, Sudbury MA, 1996, v.


Like any science, mathematics is a thriving wonderland of research, with many mysteries to keep us humble despite the subject’s many remarkable achievements.

Larry J. Gerstein, Introduction to mathematical structures and proofs, New York, Sudbury MA, 1996, vi.


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.

Eric Gosset, Discrete Mathematics with Proof, Upper Saddle River, N. J. 2003, p. 29.


Paul R. Halmos

Set theory and algebra seem to stand to arithmetic and differential equations in the same ratio as words stand to numbers.

Paul. R. Halmos, I want to be a mathematician. An automatography, New York 1985, p. 5.


There is all the difference in the world between an exposition that cannot be misunderstood and one that is in fact understood.

Paul. R. Halmos, I want to be a mathematician. An automatography, New York 1985, p. 10.


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?

Paul. R. Halmos, I want to be a mathematician. An automatography, New York 1985, p. 69.


Books may be linearly ordered, but our minds are not.

Paul. R. Halmos, I want to be a mathematician. An automatography, New York 1985, p. 70.


Everything we learn changes everything we know and will help us later to learn more.

Paul. R. Halmos, I want to be a mathematician. An automatography, New York 1985, p. 71.


Serge Lang

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.

Serge Lang, Basic Mathematics, New York 1988, 102.


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.

George Pólya, Mathematics and plausible reasoning, vol. 1: induction and analogy in mathematics , Vol. 1, Princeton, New Jersey 1954, p. viii.