Preface

Preface for the Student.

1. Please do not read the following Preface for the Expert!

2. I presuppose only logical thinking and the English language as known; nothing from school mathematics, let alone from higher mathematics.

To forestall objections: One number, no number, two cases, all things from a given totality, and the like, are clear verbal constructs of the English language. Theorem 1, Theorem 2, . . ., Theorem 301 (likewise with axioms, definitions, chapters, sections) or 1), 2) and the like in case distinctions are labels which distinguish the theorems, axioms, . . ., cases, and which are more convenient for looking things up than if I were to speak, say, of Theorem Light Blue, Theorem Dark Blue, and the like. Up to "301" the introduction of the so-called positive integers would in fact present no difficulty; the first difficulty — overcome in Chapter 1 — lies in the totality of the positive integers

with the mysterious row of dots after the comma (called natural numbers in Chapter 1), in the definition of the arithmetical operations to be performed with them, and in the proofs of the associated theorems.

I develop everything corresponding in turn: in Chapter 1 for the natural numbers, in Chapter 2 for the positive fractions and positive rational numbers, in Chapter 3 for the positive (rational and irrational) numbers, in Chapter 4 for the real numbers (positive, negative, and zero), in Chapter 5 for the complex numbers; thus I speak only of such numbers as you have already dealt with at school.

In this sense:

3. Please forget everything you learned at school; for you did not learn it.

Please keep in mind throughout the corresponding parts of the school curriculum; for you have not forgotten them after all.

4. The multiplication table, even the theorem

does not occur; but I recommend to you, as an exercise for Chapter 1, § 4, to define

and to prove that theorem.

5. Forgive me for addressing you familiarly; this happens not only because one is accustomed to address the reader with "read" and "see", but because this book is written in part in usum delphinarum, inasmuch as my daughters, as is well known (see E. Landau, Vorlesungen über Zahlentheorie, Vol. 1, p. V), have already been studying for several semesters (chemistry), believe they learned differential and integral calculus at school, and to this day do not know why

holds.

Berlin, December 28, 1929.

Edmund Landau.

Preface for the Expert.

This little book is a concession to those colleagues (unfortunately in the majority) who do not share my standpoint on the following question.

While at school one must naturally forgo a rigorous and gapless construction of elementary mathematics, university instruction in mathematics should acquaint the student not only with the material and the results, but also with the methods of proof. Even one who learns mathematics chiefly for its applications to physics and other sciences, and who therefore must often work out further mathematical lemmas for himself, can proceed securely along the beaten path only if he has learned to walk, i.e., if he can distinguish between false and true, between conjectures and proofs (or, as some so nicely say, between non-rigorous and rigorous proofs).

Therefore I consider it right — following some of my teachers and colleagues, some authors from whose writings I have drawn, and most of my students — that the student should learn already in the first semester on which basic facts, assumed as axioms, analysis is built up without gaps, and how this construction can be begun. In the choice of the axioms one may, as is well known, proceed in various ways; hence I do not declare it to be wrong, but rather almost diametrically opposed to my personal standpoint, when for real numbers numerous of the usual laws of arithmetic and Dedekind's fundamental theorem 205 of the following work are postulated as axioms. To be sure, I do not prove the consistency of the five Peano axioms (for the very reason that it cannot be done); but each of them is obviously independent of the preceding ones. On the other hand, with that enlarged number of axioms the question at once forces itself upon the student whether not quite a few of them can be proved (a clever fellow would add: or refuted) from what precedes; and since the provability of all these things has been known for many decades, the student really deserves to learn the (throughout quite easy) proofs at the beginning of his studies.

I do not even want to speak at length about the fact that in many cases not even Dedekind's fundamental theorem (or its equivalent surrogate when the real numbers are grounded by fundamental sequences) is taken as a basis; so that then things like the mean value theorem of the differential calculus, the theorem resting on it that a function with a derivative vanishing in an interval is constant there, or, e.g., the theorem that a steadily decreasing, bounded sequence of numbers tends to a limit, appear without any proof or, what is still worse, with a supposed proof which is not one. The number of representatives of this extreme variety of the other standpoint seems to me not only to decrease monotonically; but the limit toward which this number tends, according to the theorem just mentioned, is perhaps even zero.

But only rarely does one begin with a grounding of the natural numbers. I too confess that I never failed to go through the theory of the real numbers after Dedekind, but formerly presupposed the properties of the integers and the rational numbers. The last three times, however, I preferred to begin with the integers. Once, and also for the coming summer semester, as a concession to those hearers who want to differentiate right away, or who do not even want to learn the whole elucidation of the number concept in the first semester (or possibly not at all), I did divide my lectures into two simultaneous ones, one of which was called "Grundlagen der Analysis". In this one I proceed, starting from the Peano axioms for the natural numbers, up to the theory of the real numbers and of the complex numbers; incidentally, the hearer does not yet need the complex numbers in the first semester; but their introduction is quite simple and can be inserted at once without effort.

Now in the entire literature there is no textbook that sets itself the modest goal of grounding only the arithmetic of numbers in the above sense. And even the comprehensive works in which this is undertaken in the introductory chapters leave (consciously or unconsciously) quite a bit to the reader.

This work — if it is found suitable by him — is meant to give every colleague of the other pedagogical persuasion, who therefore does not go through the foundations, at least the possibility of referring to a source where what is missing, and only what is missing, is presented in gapless coherence. The reading is quite easy if one — as is indeed the case — has already learned the results at school, and if one has gotten past the abstract first four or five pages.

I come before the public with this work with hesitation, because in doing so I publish in a field in which I have (apart from an oral communication of Mr. Kalmar) nothing new to say; but no one else has taken upon himself my labor, in part tedious.

The definitive impulse for this "flight into the public", however, was given by a concrete incident.

The other persuasion always thinks that in the later course of his studies the student will learn the matter anyway, by means of a lecture course or the literature. And none of those honored friends and enemies of mine would have doubted that, e.g., in my lectures everything necessary is to be found. I too believed that. And then the following terrible adventure happened to me. On the basis of my lecture notebook, my then assistant and dear colleague Privatdozent Dr. Grandjot (now professor at the University of Santiago) lectured on the foundations of analysis and returned my manuscript to me with the remark that he had found it necessary to add further axioms to the Peano axioms in the later course of the development, since the usual path I had followed exhibited a certain gap. Before I go into the details, I will at once mention by way of anticipation:

1. Grandjot's objection was justified.
2. Axioms which cannot be enumerated at the beginning of the whole (because they tie in with later concepts) are very regrettable.
3. Grandjot's axioms are all provable (as we could already have learned from Dedekind), and matters remain (see the entire following work) with the Peano axioms.

There are three places at which the objection takes hold:

I. At the definition of $x + y$ for natural numbers.

II. At the definition of $x \cdot y$ for natural numbers.

III. At the definition of $\sum_{n=1}^{m} x_n$ and $\prod_{n=1}^{m} x_n$, after one already has $x + y$ and $x \cdot y$ for some domain of numbers.

Since the matter is analogous in all three cases, I speak here only of $x + y$ for natural numbers $x, y$. If, say, in a lecture course on number theory I prove some theorem about natural numbers by first proving its correctness for $1$ and then deriving its correctness for $x + 1$ from its correctness for $x$, a hearer will occasionally raise the objection that I had not previously proved the assertion for $x$ at all. The objection is unjustified, but pardonable; the student had simply never heard of the induction axiom. Grandjot's objection sounds similar; with the difference that it was justified, so that I also had to pardon it. On the basis of his five axioms, Peano defines $x + y$ for fixed $x$ and all $y$ as follows:

and he and his successors mean thereby: $x + y$ is defined generally; for the set of $y$ for which it is defined contains $1$, and with $y$ also $y'$.

But one has not defined $x + y$ at all.

It would be in order if one had the concept "numbers $\leqq y$" (which is not the case on Peano's path, since order is introduced only after addition) and spoke of the set of $y$ for which there exists an $f(z)$, defined for $z \leqq y$, with the properties:

This is how Dedekind's grounding proceeds. With the kind help of my colleague von Neumann in Princeton, I had worked out such a path for this little book, after a prior introduction of order (which would not have been convenient for the reader). At the last hour, however, I learned of a very much simpler proof from Dr. Kalmar in Szeged; now the matter looks so simple, and the proof so similar to the other proofs of the first chapter, that even the expert would not have noticed this point, had I not put my above confession of guilt and atonement on record at such length. With $x \cdot y$ it goes exactly the same way; $\sum_{n=1}^{m} x_n$ and $\prod_{n=1}^{m} x_n$, to be sure, is possible only by Dedekind's path; but from Chapter 1, § 3 on, one does have the set of the $z \leqq y$.

To make things as easy as possible for the reader, I have repeated some (not very extensive) sets of words in several or all chapters. For the expert it would of course suffice, e.g., to say once and for all at the proofs of Theorems 16 and 17: This reasoning holds for every class of numbers for which the signs $<$ and $=$ are defined and have certain properties established earlier. Such repeated arguments concern theorems that had to occur in all the chapters in question, because the theorems were applied in what follows. But $\sum_{n=1}^{m} a_n$ and $\prod_{n=1}^{m} a_n$ need only be introduced in the last chapter in order to have them thereby also for the lower kinds of numbers. Hence I wait with them until the complex numbers, and likewise with the theorems on subtraction and division; the former hold, of course, e.g. for natural numbers, only when every minuend is greater than the subtrahend, the latter, e.g. for natural numbers, only when all the divisions come out even.

My book is written, forgoing side remarks, in the merciless telegraphic style ("Axiom", "Definition", "Theorem", "Proof", only occasionally "Preliminary Remark"; rarely words belonging to none of these five rubrics) which is in place for so easy a subject matter.

I hope, after decades of preparation, to have composed this work in such a way that a normal student can read it in two days. And then he may even forget the entire contents (since he already knows the formal rules from school) except for the induction axiom and Dedekind's fundamental theorem.

But if even one or another colleague of the other persuasion should find the matter so easy that he presents it in his lectures for beginners (along the following or any other path), I would have attained a goal for which on a larger scale I do not dare to hope.

Berlin, December 28, 1929.

Edmund Landau.