See Landau's Contents for a list of the chapter and section headings. The work, aimed at university students, contains two Prefaces, one for the student and another for the teacher. Both were written in Berlin on 28 December 1929. We give a version of Steinhardt's translation of the two Prefaces below. They tell us much about Landau and his attitude towards teaching mathematics:
## FOUNDATIONS OF ANALYSIS## THE ARITHMETIC OF |

- Please don't read the preface for the teacher.
- I will ask of you only the ability to read English and to think logically - no high school mathematics, and certainly no higher mathematics.
To prevent arguments: a number, no number, two cases, all objects of a given totality, and so on, are completely unambiguous phrases. Theorem 1, Theorem 2, ..., Theorem 301, or 1), 2), etc. for distinguishing the various cases, are labels which distinguish the theorems and the cases; similarly for axioms, definitions, chapters and sections. These are more convenient to refer to than if we were to speak, say, of Theorem Light-blue, Theorem Dark-blue, and so on. As a matter of fact, the introduction of the so-called positive integers up to "301" would not offer any difficulty whatsoever. The first difficulty - overcome in Chapter I - lies in the totality of the positive integers

1, ...

with the mysterious series of dots after the comma (called natural numbers in Chapter I), in the definition of the arithmetical operations with these numbers, and in the proofs of the associated theorems.

I develop analogous material, first for the natural numbers in Chap. I; then for the positive fractions and positive rational numbers, in Chap. II; next for the positive (rational and irrational) numbers, in Chap. III; next for the real numbers (positive, negative, and zero), in Chap. IV; and finally for the complex numbers, in Chap. V. Thus I speak only of such numbers as you have already met with in high school.

Apropos:

- Please forget everything you have learned in school; for you haven't learned it.
Please keep in mind at all times the corresponding portions of your school curriculum; for you haven't actually forgotten them.

- The multiplication table will not occur in this book, not even the theorem
2 × 2 = 4,

but I would recommend, as an exercise for Chap. I, § 4, that you define

2 = 1 + 1,

4 = (1 + 1) + 1) + 1), and then prove the theorem.

- Forgive me for "theeing" and "thouing" you.[ In the German edition Professor Landau uses the familiar "du" (thou) throughout this preface. [Trans.]] One reason for my doing so is that this book is written partly
*in usum delphinarum,*[For Delphine use. The Delphin Classics were prepared by French scholars for the use of the Dauphin of France, son of King Louis XIV. [Trans.]] since, as is well known (cf. E Landau,*Vorlesungen über Zahlentheorie,*Vol. 1, p. V), my daughters have been studying (chemistry) for several semesters, think they have learned differential and integral calculus in school, and yet even today don't know why*x*×*y*=*y*×*x*is true.

Berlin, December 28, 1929

EDMUND LANDAU

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

While a rigorous and complete exposition of elementary mathematics can not, of course, be expected in the high schools, the mathematical courses in colleges and universities should acquaint the student not only with the subject matter and results of mathematics, but also with its methods of proof. Even one who studies mathematics mainly for its applications to physics and to other sciences, and who must therefore often discover auxiliary mathematical theorems for himself, can not continue to take steps securely along the path he has chosen unless he has learned how to walk - that is, unless he is able to distinguish between true and false, between supposition and proof (or, as some say so nicely, between non-rigorous and rigorous proof).

I therefore think it right - as do some of my teachers and colleagues, some authors whose writings I have found of help, and most of my students - that even in his first semester the student should learn what the basic facts are, accepted as axioms, from which mathematical analysis is developed, and how one can proceed with this development. As is well, known, these axioms can be selected in various ways; so that I do not declare it to be incorrect, but only to be almost diametrically opposite to my point of view, if one postulates as axioms for real numbers many of the usual rules of arithmetic and the main theorem of this book (Theorem 205, Dedekind's Theorem). I do not, to be sure, prove, the consistency of the five Peano axioms (because that can not be done), but each of them is obviously independent of the preceding ones. On the other hand, were we to adopt a large number of axioms, as mentioned above, the question would immediately occur to the student whether some of them could not be proved (a shrewd one would add: or disproved) by means of the rest of them. Since it has been known for many decades that all these additional axioms can be proved, the student should really be allowed to acquaint himself with the proofs at the beginning of his course of study - especially since they are all quite easy.

I will refrain from speaking at length about the fact that often even Dedekind's fundamental theorem (or the equivalent theorem in the development of the real numbers by means of fundamental sequences) is not included in the basic material; so that such matters as the mean-value theorem of the differential calculus, the corollary of the mean-value theorem to the effect that a function having a zero derivative in some interval is constant in that interval, or, say, the theorem that a monotonically decreasing bounded sequence of numbers converges to a limit, are given without any proof or, worse yet, with a supposed proof which in reality is no proof at all. Not only does the number of proponents of this extreme variant of the opposite point of view seem to me to be decreasing monotonically, but the limit to which, in conformity with the above-mentioned theorem, this number converges, may even be zero.

Only rarely, however, is the foundation of the natural numbers taken as the starting point. I confess that while I myself have never failed to cover the (Dedekind) theory of real numbers, in my earlier courses I assumed the properties of the integers and of the rational numbers. But the last three times I preferred to begin with the integers. For the next Spring term (as once before) I have divided my course into two simultaneous courses one of which has the title "Grundlagen der Analysis" (Foundations of Analysis). This is a concession to those hearers who want, after all, to do differentiation right away, or who do not want to learn the whole explanation of the number concept in the first semester (or perhaps not at all). In the Foundations of Analysis course I begin with the Peano axioms for the natural numbers and get through the theory of the real and of the complex numbers. The complex numbers, incidentally, are not needed by the student in his first semester, but their introduction, being quite simple, can be made without difficulty.

Now in the entire literature there is no textbook which has the sole and modest aim of laying the foundation, in the above sense, for operations with numbers. The larger books which attempt that task in their introductory chapters leave (consciously or not) quite a bit for the reader to complete.

The present book should give to any of my colleagues of the other pedagogical faction (who therefore does not go through the foundations) at least the opportunity, provided he considers this book suitable, of referring his students to a source where the material he leaves out-and that material only-is treated in full. After the first four or five rather abstract pages the reading is quite easy if - as is actually the case - one is acquainted with the results from high school.

It is not without hesitation that I publish this little book, because in so doing I publish in a field where (aside from an oral communication of Mr Kalmár) I have nothing new to say; but nobody else has undertaken this labour which in part is rather tedious.

But the immediate cause for venturing into print was furnished by a concrete incident.

The opposition party likes to believe that the student would eventually learn these things anyway during the course of his study from some lecture or from the literature. And of these honoured friends and enemies, none would have doubted that everything needed could be found in, say, my lectures. I, too, believed that. And then the following gruesome adventure happened to me. My then assistant and dear colleague Privatdozent Dr Grandjot (now Professor at the University of Santiago) was lecturing on the foundations of analysis and using my notebook as a basis for the lectures. He returned my manuscript to me with the remark that he had found it necessary to add further axioms to Peano's in the course of the development, because the standard procedure, which I had followed, had proved to be incomplete at a certain point. Before going into details I want to mention at once that

- Grandjot's objection was justified.
- Axioms which, because they depend on later concepts, cannot be listed at the very beginning, are very regrettable.
- Grandjot's axioms can all be proved (as we could have learned from Dedekind), so that everything remains based on Peano's axioms (cf. the entire following book).

There were three places where the objection came in:

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

II. At the definition of *x* × *y* for the natural numbers.

III. At the definition of ∑ *x*_{n} and of ∏ *x*_{n} , after one already has *x* + *y* and *x* × *y*, for some domain of numbers.

Since the situations in all three cases are analogous, I will speak here only about the case of *x* + *y* for natural numbers *x*, *y*. When I prove some theorem on natural numbers, say in a lecture on number theory, by first establishing it as true for 1 and then deducing its validity for *x* + 1 from its validity for *x*, then occasionally some student will raise the objection that I have not first proved the assertion for *x*. The objection is not justified but it is excusable; the student just had never heard of the axiom of induction. Grandjot's objection sounds similar, with the difference that it was justified; so I had to excuse it also. On the basis of his five axioms, Peano defines *x* + *y* for fixed *x* and all *y* as follows:

x+ 1 =x'

x+y'= (x+y)"

and he and his successors then think that *x* + *y* is defined generally; for, the set of *y*'s for which it is defined contains 1, and contains *y*' if it contains *y*.

But *x* + *y* has *not* been defined.

All would be well if - and this is not done in Peano's method because order is introduced only after addition - one had the concept "numbers ≤ y" and could speak of the set of y's for which there is an *f*(*z*), defined for *z* ≤ *y*, with the properties

f(1) =x,

f(z') = (f(z))'forz<y.

Dedekind's reasoning does follow these lines. With the kind help of my colleague von Neumann in Princeton I had worked out such a procedure, based on a previous introduction of ordering, for this book. This would have been somewhat inconvenient for the reader. At the last minute, however, I was informed of a much simpler proof by Dr Kalmár in Szeged. The matter now looks so simple and the proof so similar to the other proofs in the first chapter, that not even the expert might have noticed this point had I not given above a detailed confession of crime and punishment. For x.y the same simple type of proof applies; however, ∑ *x*_{n} and ∏ *x*_{n} are possible only with the Dedekind procedure. But from Chap. I, § 3 on, one has the set of the *x* ≤ *y* anyway.

To make it as easy as possible for the reader I have repeated in several chapters, or sometimes in all, certain (not very lengthy) phrases. For the expert it would of course be sufficient to say once and for all, for instance in the proof of Theorems 16 and 17: This reasoning holds for every class of numbers for which the symbols < and = are defined and have certain properties mentioned earlier. Such repeated deductive reasonings occurred in connection with theorems which had to be given in all the chapters concerned because the theorems are used later on. But it suffices to introduce ∑ *a*_{n} and ∏ *a*_{n} since they will then apply to the preceding types of numbers. I therefore defer their introduction to the chapter on complex numbers, and do the same for the theorems on subtraction and division; the former hold for the natural numbers, say, only if the minuend is larger than the subtrahend, the latter for the natural numbers, say, only if the division leaves no remainder.

My book is written, as befits such easy material, in merciless telegram style ("Axiom," "Definition," "Theorem," "Proof," occasionally "Preliminary Remark," rarely words which do not belong to one of these five categories).

I hope that I have written this book, after a preparation stretching over decades, in such a way that a normal student can read it in two days. And then (since he already knows the formal rules from school) he may forget its contents, with the exception of the axiom of induction and of Dedekind's fundamental theorem.

Should, however, any of my colleagues who holds the other point of view find the matter so easy that he presents it in his lectures for beginners (in the following or in any other way), I would have achieved a success which I do not even dare hope will be realized on any large scale.

Berlin, December 28, 1929

EDMUND LANDAU

JOC/EFR August 2007

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