I've just noticed that a very nice book about number theory by Melvin Fitting and Greer Fitting is freely available online here. Here's a bit from the introduction.
A preface is supposed to explain why you should read the book. Like most prefaces, this one will make more sense after the fact. Nevertheless, here goes.
Most mathematicians believe (rather strongly) that numbers behave in certain well-defined ways. This belief can not be justified by personal experience. No mathematician has `seen' more than a finite, probably small, collection of numbers. Instead mathematicians justify their beliefs by giving proofs. In practice, this means that certain facts about numbers are accepted as `obvious', and used in carefully reasoned arguments for the correctness of other facts that are less obvious, or possibly not obvious at all. Since mathematicians generally are concerned to establish the nonobvious, little thought is customarily given as to why the `obvious' facts are correct.
Now, it is an observation as old as Aristotle that one can not provesomething from nothing. One must always begin with some body of `obvious' facts and proceed from there. In practice, most mathematicians contentedly place hundreds of facts in this `obvious' category in order to get on with their proper business of discovery and verification of the non-obvious.
But at least once in a mathematician's career, it is good to take a sharp look at the status of the `obvious' facts; and it is probably best to do it early, and get it over with. As we remarked above, it is not possible to do away with all assumptions, even in mathematics. But, one of the great achievements of 19th and early 20th century mathematics was the careful and precise limitation of exactly what a working mathematician must `accept on faith'. That is, it was discovered what can constitute an irreducible minimum of `obvious' facts.
It is the purpose of this book to present such an irreducible minimum, and show how most commonly assumed facts about numbers follow directly. Nonetheless, this book is a bit of a fraud, because like all mathematicians we still assume that some obvious facts are more obvious than others. This is a book about the number systems, so for our purposes we assume as `sufficiently obvious' a variety of pre-numerical facts. Specifically, we assume, without being too explicit about the matter, several principles about the behaviour of sets or collections. Now this material too has been subjected to a similar treatment, also around the turn of the century. Today one can find basic set theory developed from a small number of axioms in many books on elementary set theory. But our book is long enough already, so we elected to omit this material here. For us the issue is: given a variety of `obvious facts' from set theory, what elementary properties of numbers must one accept in order to logically derive the entire basic framework of mathematics.
Comments