epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Introduction and Basic
Terminology
Have you ever felt that the words mathematics dnid frustration have a lot
in common? There are many people who do, including, at times, some very
good mathematicians. At the beginner's level, the level for readers of this
book, this feeling is often the result of the use of an unproductive and often
unsystematic (and panicky) approach that leads to hours of unfruitful work.
When anxiety sets in, memorization may look like the way to "survival," but
memorization without a thorough understanding is usually a poor and risky
approach, both in the short and in the long run. It is difficult to recall
successfully a large amount of memorized material under the pressure of an
exam or a deadhne. It is very easy for most of this material to quickly fall
into oblivion. The combination of these two aspects will render most of the
work done completely useless, and it will make future use of the material
very difficult. Moreover, no ownership of the subject is gained.
The construction of airtight logical constructions ("proofs") represents
one of the major obstacles that mathematical neophytes face when making
the transition to more advanced and abstract material. It might be easy to
believe that all results already proved are true and that there is no need to
check them or understand why they are true, but there is much to be learned
from understanding the proofs behind the results. Such an understanding
gives us new techniques that we can use to gain an insider's view of the
subject, obtain other results, remember the results more easily, and be able
to derive them again if we want to.