What is Modern Mathematics

What Is Modern

Mathematics

G. Choquet

Mathematics Teaching Series

Educational Solutions Worldwide Inc.

First published in 1962. Reprinted 2009.

Copyright © 1962-2009 Educational Solutions Worldwide Inc. Author: G. Choquet Translator: Caleb Gattegno All rights reserved ISBN 0-00000-000-0

Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, NY 10003-4555 www.EducationalSolutions.com

Table of Contents

Presentation ................................................................ 1

Foreword.....................................................................3

Bourbaki And Analysis ........................................................... 3

Chapter 1: The Axiomatic Method ................................5

1 Structures ............................................................................ 6

2 Characteristics Of The Axiomatic Method ......................... 9

1 Economy Of Thought..................................................... 9

2 Multivalency A Guarantee Of Unity And Universality........10

3 Mutual Illumination Of Mathematical Entities: Dynamism................................................................. 11

4 Adaptability To The Physical Universe........................13

5 Validation Of Notions That Have Become Metaphysical. ............................................................13

3 Dangers Of The Axiomatic Method ...................................14

Chapter 2: Some Of The Tools Of The Axiomatic Method ................................. 17

1 Morphisms, Initial & Final Structures............................... 17

2 Sets And Universal Applications .......................................19

3 Categories And Functors ...................................................21

Chapter 3: Discovery Methods Linked With The Axiomatic Method .......................... 25

1 The Loosening Of Axioms.................................................. 25

2 The Tightening Of Axioms ................................................26

3 Study Of Structures Not Too Different ............................. 27

4 Generation Of Structures Answering Given Requirements......................................28

Chapter 4: Some Characteristics Of Bourbaki’s Contribution To Analysis......................... 29

1 Axiomatics And Multivalency............................................29

2 Bourbaki Is Essentially An Algebraist ..............................30

3 The Constant Renewal Of The Opus................................. 31

4 Choice Of Definitions ........................................................32

(1) Radon Measures. 34

(2) Invariant Measures On A Group. 35

(3) Measurable Functions. 35

5 Choice Of Contents And Theorems...................................36

Chapter 5: Modern Analysis In The World Today ...... 39

Chapter 6: The Impact Upon Modern Mathematics Education ............................... 43

Mathematical Activity As A Whole .......................................48

Index......................................................................... 49

Presentation

We publish as the first text in this series a short but illuminating study by one of the most versatile mathematicians of our time of what can be understood by modern mathematics. The author is an outstanding example of a thinker who has made a distinguished contribution to mathematics and who does not consider the problems of teaching mathematics unworthy of his most careful attention. President since 1952 of the International Commission for the Study and Improvement of the Teaching of Mathematics, he has given generously of his time and his substance to a number of groups involved in this field.

Quite convinced that mathematicians have a great deal to give to teachers of mathematics at all levels, he has been largely instrumental in the organization of courses for secondary school teachers. These courses have been in existence for some years now and have been warmly welcomed by his colleagues. The material for the courses was reprinted in the Bulletin de l’Association des Professeurs de Mathématiques and later in the form of books for the benefit of secondary school mathematics teachers in France and elsewhere who could not attend the

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courses. We hope that some of these books will appear in English in this series.

Monsieur Choquet is also aware that as a specialist he has had to concentrate on some aspects of the problems involved in the teaching of mathematics and that he may sometimes have insufficient knowledge of others. In courses which he attended as participant he has always been very keen to learn from his colleagues, whether they were teachers of infant and junior schools or administrators engaged in finding practical solutions to school problems. He considered it only right to sit as a pupil in Dr. T. J. Fletcher’s class in Madrid in 1957 to learn from him how to make a film about mathematics. He watched most attentively experiments in teaching young children with Cuisenaire rods and became both an adept at the approach and a skilled user of the rods. He travelled far and wide and studied local conditions in order to recast the proposals he made for his own country in a form to suit economic underdevelopment or manpower difficulties.

As the opening text of this series, the pithy presentation which follows will help readers to see the problem in a new light and to think more adequately of the articulation of the research of today into the teaching of today and of tomorrow. Its value for readers cannot but increase as they reach deeper and deeper understanding of the various sections so well arranged and so clearly presented. C. Gattegno, General Editor

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What Is Modern Mathematics

Foreword

Bourbaki And Analysis

In spite of the subtitle I do not intend in this short text to embark upon the rash project of trying to read the mind of that many-headed genius Bourbaki.

However, since I am concerned with the whole of analysis, and since Bourbaki has such clear-cut concepts and is so intimately associated with the development of mathematics in our time, we can hope that a study of ‘his’ philosophical and mathematical work may lead us to the essence of modern trends in analysis.

Such a study may serve to develop for all levels of education a teaching of mathematics better adapted to the needs of our time and to the level of awareness of our generation.

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Chapter 1:

The Axiomatic Method

The study of the history of mathematics shows clearly enough that after each period of research and extension there follows a period of review and synthesis during which more general methods are evolved and the foundation of mathematics consolidated. It is thus that Descartes’ contribution can be regarded as the culmination of a long period of apparently very diverse investigations which made it possible to relegate to the museums a large number of different procedures for the study of particular curves and functions and to replace them by one more nearly universal procedure. Today the number of research mathematicians is so great that the two processes, research and synthesis, can easily be carried out simultaneously. Still, the work of synthesis in the last fifty years, made possible by the theory of sets and the terminology proper thereto, has been particularly remarkable. One can find it clearly stated in Bourbaki, and it is there that I wish to study it.

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For Bourbaki there exists only one MATHEMATIC, and the main instrument for this evolution towards unity was the axiomatic method. In order to apply this method to the study of a theory, the mathematician ‘separates the main lines of reasoning that figure in it, then, taking each one in isolation, he considers it as an abstract principle and deduces from it its natural consequences; then, going back to the theory under study, he recombines the elements previously considered separately and sees how they react upon one another’ (Bourbaki). We find in this statement a more structural presentation of one of Descartes’ basic principles: to subdivide each difficulty into as many elements as are necessary to understand it.

1 Structures

The ‘main lines of reasoning’ are the structures. For example the set R of the real numbers possesses various structures: those of a group, of a field, of a vector space, of order and of a topological space. Conversely, one and the same structure can be found in a number of distinct theories. For example, the structure of a group found in the study of R is also found in that of the integers modulo-p and in that of spatial displacements.

For the study of a structure to be applicable to different theories, the sets considered must necessarily be very general; in particular, the nature of their elements must not be a factor, but only the relationships between them. These relationships are clearly stated in the axioms defining the structure.

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Thus the order structure of an arbitrary set S is a binary-relation on S, denoted , which satisfies the following axioms or postulates.

For any x, y,z belonging to S we have

1) x x

2) (x y and y x) ⇒ (x= y)

3) (x y and y Z) ⇒ (x z)

Some of these structures have a more fundamental significance because they are met in all theories. They are called the mother-structures and include the structures associated to an equivalence relation; the order structures; the algebraic structures, the topologic structures etc.

When we compare structures to each other, we can see that some are “richer” than others. Thus the structures called finite abelian groups or fields, are richer than any structure that is merely a group structure.

Some structures are more complex because they display several mother-structures bound together by conditions of compatibility. These are called multiple-structures. For example a topological group is a set which displays at the same time a group structure and a topological structure made compatible by stipulating that the operations (x, y) x. y and x x–1 are to be continuous.

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Chapter 1: The Axiomatic Method

Topological algebra and algebraic topology are concerned with multiple-structures; differential geometry and differential algebra consider structures which are still richer. At the apex of the building we find “Crossroads-structures” which involve very many structures. Potential theory is a particularly good example of such a structure. It is the multiplicity of the mother-structures found in such theories that explain why mathematicians so different from each other find in them such an interest: every step forward in the study of the constituent structures has repercussions on the whole theory. It is easy to establish that the progress in potential theory corresponds to progress in other theories such as Lebesgue’s integration, topological spaces, topological vector spaces, Radon measure, Abelian locally compact groups, distributions, etc., etc.

Such structures are the actual field of study of the analysts and we shall define analysis as the complete set of crossroads structures. But as these have not been defined rigidly the boundary regions are numerous; our definition of analysis only establishes a hierarchy.

A theory A will be classified as belonging more to analysis than a theory B if the structures studied in A are richer than those studied in B.

Analysis appears as a universe whose complexity recalls that of life itself. While algebra is a world of minerals whose beauties are those of crystals with their pure forms, analysis is inhabited by beings whose shapes are as uncertain as algae, hydrae or spongae; it is an exuberant world full of opportunities where

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explorations can follow anyone of a great many courses and where everyone can put the imprint of his personality on the area he explores.

2 Characteristics Of The Axiomatic Method

1 Economy Of Thought Recent developments in mathematics and in industry show interesting analogies; the axiomatic method is analogous to an automatic production line; the mother-structures to the machine-tools.

The axiomatic method effects an economy in thinking and in notation, for the important theorems needed in various forms, in different contexts, are established once and for all in a sufficiently general system of axioms so as to make them include all useful applications. In that general system of reference, terminology and notation are chosen to fit in the various special cases, and preference is always given to the more suggestive words which evoke resonance and appeal to intuition. This carefulness in the choice of terms goes hand in hand with a concern for clarity in presentation; modern mathematicians have developed a precise and austere style, and are only satisfied when their papers only consist of the bare bones of definitions, lemmas, theorems, corollaries, remarks and warnings

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2 Multivalency A Guarantee Of Unity And Universality. The first axiomatic systems were categorical or univalent, such as the axiomatisation of elementary geometry by Euclid and by Hilbert; the definition of integers by Peano. By contrast fundamental structures are multivalent, that is to say that the axioms which define them can be applied to vast classes of sets carrying non-isomorphic structures.

It is this multivalency which is a guarantee of their adaptability to most varied situations. It follows that it is sometimes difficult to tell whether a statement belongs to algebra, to geometry or to analysis.

Thus we can say that the elementary geometry of space is nothing else than linear algebra in a three dimensional vector-space on which a scalar product is defined, and that the study of the quadratic forms in that space is equivalent to the study of the conics of the plane.

In a similar way to study Hilbert space is, of course, to be doing geometry (since one there talks of spheres, angles, perpendiculars) but it is equally to be doing algebra and analysis.

For example for Henri Cartan, to sweep (balayage) in potential theory, is the same as to project orthogonally on a convex cone of a Hilbert space. More generally, though convex sets belong to geometry, they become one of the basic tools of the analyst who is studying topological vector spaces.

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This multivalency of the large structures is therefore a unifying factor which allows of a mutual widening of the various mathematical theories. Such a phenomenon is not new. We already had examples in the geometrical representation of complex numbers; the synthesis of algebra and geometry effected by Descartes; the use Monge made of geometry in his research in analysis. But it was left to the algebra of sets and its universal language, to amplify this phenomenon. Here are only a few typical examples:

• Zariski’s topology in algebraic geometry.

• topological interpretation and proofs of a number of important theorems in Logic.

• Leray’s theory of fiber bundles first studied in algebraic topology but now invading algebra and analysis.

3 Mutual Illumination Of Mathematical Entities: Dynamism. It follows from this multivalency that isolated entities are no longer studied; what is studied is families of entities being in some mutual relationships. Not only do theorems acquire much greater generality thereby, but at the same time each entity is individually better known, since its relationships with other entities, bring out its own various aspects. Here too, what is new is not this use of a “context” but the awareness of the phenomenon and of its generality:

It is well known that a tangent to a curve at a point has been defined by a family of secants; that analytic functions of a real

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variable became better known when they were studied in the complex plane, and that for some time, “normal families” of analytical functions have been a powerful tool. Modern mathematics are “relational” and this gives them an inner dynamism reflected in the special vocabulary and special typographic signs: mappings, injections, jets, springs, arrows, and arrowed schemata.

A very suggestive and convenient notation was produced to indicate relations and transformations.

x f (x); A Ā; x~y; x< y; A × B; ∏ Ai; E/R etc.

In the hands of mathematicians the entities are fashioned like precious stones in the hands of a jeweler and each of the transformations which are brought to it, reveals a new facet, an unexpected aspect.

This relational aspect of mathematics is in agreement with the well known principle that in order to know a notion well one has to study its forms and its opposites. It is also in agreement with a principle that seems to dominate all modern scientific investigations, i.e. that we cannot reach the “essence” of the entities studied, only the relationships of one to another. An experiment in physics only reveals a relation between the Universe and the experimental set-up. What is essential in a telephone network is not the nature of the wires or their shape, but its circuitry. For the mathematician two isomorphically structured sets are equivalent.

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The virtuosity with which the young generation of mathematicians nurtured in the new methods, makes use of the dynamism of relationships and the enjoyment they get out of it, seem to prove that this dynamism is well adapted to the structure of man’s brain.

4 Adaptability To The Physical Universe. This multivalency of theories is a guarantee of a greater possibility of their being used in physics. Thus, Hilbert space has served to interpret better the quantum field theories; the geometry of Riemannian spaces, and the exterior differential calculus have formed the framework for General Relativity. Modern theoretical physics itself is now developing its own axiomatic structures: some fundamental facts are taken as postulates and from them are deduced consequences the experimental verification of which is sought afterwards. Naturally it is understood that the axioms selected correspond to only one aspect of the physical universe.

5 Validation Of Notions That Have Become Metaphysical. A good axiomatic system is quite often the only way of getting out of metaphysical difficulties. Thus complex numbers only lost their mystery and their “absurdity” when their set was identified with R2 to which two well-defined operations were associated. In more recent times the foundations of probability theory were very hazy when the latter was based on the theory of games, the theory of errors and the stochastic theory. Probability theory only found its unity and firm foundations when Kolmogorov gave it an axiomatic presentation. From these axioms it appears

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as a branch of the theory of measure but a special branch which shows much vigor, has its own language and problems and could enrich itself with the results of the theory of measure while fertilizing classical analysis. A brilliant illustration of this latter fact is to be found in the close relation that was recently shown to exist between Markov’s processes and potential theory.

3 Dangers Of The Axiomatic Method

Although axiomatic systems are the machine-tools of mathematics it will easily be understood that they are of interest only if their output is good. It is relatively easy to construct axiomatic systems, by slightly modifying known systems; one finds unfortunately that too many are being produced and published as theses or papers. Their authors may have got a lot of pleasure out of formulating them and that leads them to exaggerate their importance. A number of these vast theories have a few applications or none at all. An urgent question is thus raised: which are the useful axiomatic systems?

There is probably no absolute criterion that would permit one to decide on this matter. Nevertheless one could agree that there is no need for a tank if one only wants to kill a fly. A general theory will be justified if it reveals unexpected and fruitful bonds between theories up till then apparently unrelated, or if it solves a challenge so far unanswered. The fact that a theory is general does not imply that it is going to be useful, particularly if the light that it throws is too weak. We shall see further on with what stringency Bourbaki has selected the theories to which he was to give right of place. But it is interesting to look for the

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safeguards which will protect Bourbaki from falling into the temptation to develop axiomatic systems as an aim in itself.

For André Weil, “if logic represents the rules of hygiene for a mathematician, the staple food from which he lives is formed of the great problems”. This is another way of saying what Hilbert used to say, “A branch of science is full of life so long as it presents problems in abundance; absence or lack of problems is a sign of death”.

Hilbert is for Bourbaki a model and almost a father figure. The son looks up to the father for the elegance and the simplicity of his papers “due to the fact that he drew out of the soil, where no one had been able to see them yet, the fundamental principles which made it possible to trace up to the solution the royal road which had been until then sought in vain”. He is the Master of the axiomatic method, whether one considers univalent structures (as in elementary geometry) or multivalent ones and he has taught mathematicians to think axiomatically. “He never falls in the pitfall of some of his disciples, of creating an elaborate theory for a few meager results and he never generalizes for the pleasure of generalizing” (Dieudonné). He is a lover of the special problem, precise and concrete. It was in order to solve such problems that he has created the tools whose importance has not diminished yet: direct method in the calculus of variations based on semi-continuity in order to solve Dirichlet’s problem; definition and use of “Hilbert spaces” for the solution of integral equations etc., etc.

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The great problems to which he called the attention of mathematicians at the 1900 Congress have continued to stimulate fruitful research. Today, for example, Riemann’s problem of the zeros of the ζ(s)-function is still provoking a number of attempts at solution, though the true nature of the problem still seems to evade everybody.

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Chapter 2:

Some Of The Tools Of The

Axiomatic Method

A modern mathematician studying a structure is bound to use auxiliary structures. In order to construct them he needs a guide that will lead him to good definitions. We shall now examine some of the procedures that have proved themselves valuable and have been found to be good guides.

1 Morphisms, Initial & Final Structures

A structure on a set S is defined by several axioms expressed in terms of the elements of S and possibly of auxiliary sets. The form of these axioms defines what is called a species of structure of which we shall only give a few examples.

The postulates for the group define a species, commutative groups form a sub-species. Other examples: the species of the

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vector spaces on R, that of compact topological spaces, and that of differentiable varieties.

Let two sets S and S′ be endowed with structures of the same species, then a bijection (that is one-one correspondence) f of S on S′ is called an isomorphism if it exchanges the structures of S and S′ in a way easy to state in each case.

In a more general way a morphism of A in B is an application of A in B which has some properties that are connected with the structure. The definition of the morphisms is made in such a way that it is ascertained that the product of two of them is also a morphism and that if a bijection f of A upon B is a morphism, as well as f–1, then f is an isomorphism. For example for the species of structure formed of the topological spaces, the class of the continuous applications forms a class of morphisms; the open applications (i.e. those that change any open set into an open set) also form another class of morphisms which is nevertheless less useful than the first class.

Let A be a given set, (Bi) a family of sets endowed with a structure of a given species, and for each i let fi be an application of A upon Bi. A question that arises is: can we endow A with a structure from the same species in such a way that each fi be a morphism? Under certain conditions this is possible and among all the possible solutions there is one that is privileged and which is called the initial structure associated to the (Bi, fi). This is the way that is used, for example, for the species of topological spaces in order to define the reciprocal image of a topology: the

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topology induced, on a sub-set of a given space; the product of a family of topological spaces.

When fi is an application of Bi in A, the solution of the problem, if it exists, is called the final structure associated to the (Bi, fi); this is the way in which one defines a topology on the quotient-set A of a topological space B by an equivalence relation R.

2 Sets And Universal Applications

Let S and Τ be two species of structures, let A be a set of species S; let us give ourselves a family of applications called (SΤ)-applications of A in the sets of species Τ and a family of applications called Τ-applications of the sets of species Τ into the sets of same species; let us also assume that these families are transitive in the sense that the product of an (SΤ)-application by a Τ-application is again one (SΤ)-application and that the product of two Τ-applications is still a Τ-application. The question then is to find whether there exists a set B of species Τ and an (SΤ)-application Φ of A into B such that any (SΤ)-application (of A into a B) can be written as φ=f o Φ where f is a Τ-application of B into B. Under very general sufficient conditions, this problem has a solution and even a infinity of non-isomorphic solutions. In order to determine a unique solution, the following condition must be added: the image Φ (A) of A into B is such that two Τ-applications of B into a B which coincide in Φ (A) also coincide in B. This space B thus obtained is called the universal space associated with A and Φ is called the universal application associated with A.

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Chapter 2: Some Of The Tools Of The Axiomatic Method

Examples.

1) Compact groups associated with a topological group.

A is a topological group, Τ is the species of compact topological groups, the (SΤ)-applications and the Τ-applications are formed by the arbitrary continuous homomorphisms. It can be shown that there is identity between the almost-periodic functions defined on A and the functions φ o Φ where g is any continuous function on B.

This example shows what interest the universal sets can present for analysis.

2) Tensor-product of two vector spaces.

A here is the (Cartesian) product of two vector spaces S1 and S2 (on the field R1), Τ is the species of the vector spaces on R; the (SΤ)-applications are the bilinear applications defined in S1 × S2; the Τ-applications are the linear applications. The universal vector space B is called the tensor product of the spaces S1, S2; through it the study of the bilinear applications defined in S1 × S2 is reduced to the linear applications defined in B.

Here are a few other universal sets; free algebraic structures

ring and field of fractions

completion of a uniform space

compactification of Stone—Cech

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free topological groups

Albanese variety (in algebraic geometry).

3 Categories And Functors

Of the large tools of mathematics the theory of “categories” is the most recently developed. It is one more plunge into abstraction, for the relations that it considers are no longer relations between elements of one set, but relations among the entities of one “category” or even of different categories.

It is almost miraculous that such generality be not synonymous with emptiness and facility. But in fact this theory has become an indispensable guide of the young generation of mathematicians in several fields.

In this text we shall be content with a few examples and some definitions to give an idea of what it is all about.

All groups form a category.

So do all vector spaces,

all topological spaces,

all ordered sets,

and more generally the category of all sets endowed with a species of structure upon which morphisms exist.

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A category therefore is not a set; it is convenient to think of it as a class of objects which is vaster than a set. Now let be a class of objects. To any X,ϒ ∈ we associate a set denoted by Hom (X,ϒ) whose elements are called homo-morphisms or morphisms of X upon ϒ, and for all X,ϒ,Z ∈ we assume an application (f,g) g o f (called composition) of Hom (X,ϒ) × Hom (ϒ,Z) into Hom (X,Z). We shall say that endowed with its homomorphisms is a category if the following axioms are fulfilled:

K1: composition is associative: h o (g o f) = (h o g) o f.

K2: to each X ∈ there exists an element ex of Hom (X,X) called unit of X and such that ex o f = f and f o ex=f for all homomorphisms f (when these expressions have a meaning).

We shall then call isomorphism of X into ϒ (X,ϒ∈ ) any u ∈ Hom (X,ϒ) such that there exists v ∈ Hom (ϒ,X) for which u o v = eх and v o u = ey.

The relations between different categories are established via functors. Let , be two categories and let F be a law which associates to each X∈ an element X′∈ , denoted by F(X) and let us assume that, to any X,ϒ ∈ and to any u∈Hom (X,ϒ), F associates u′ ∈ Hom (X′,ϒ ′) (u′ is denoted F(u)).

F is said to be a functor if:

1 when u is a unit so is F (u)

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2 for all u,v such that u o v has a meaning, we have F(u o v) = F(u) o F(v).

From these two notions of categories and functors it is possible to build an algebra which becomes richer as the categories are specialized.

Let us show on an easy example how categories can serve as a guide.

From the study of various “concrete” classical categories in which the notion of a product (ordered sets, groups, topological spaces) exists, a schema is abstracted expressible in terms of general categories, hence the notion of a category with product. If we now meet a new concrete category not yet endowed with a product, the general schema not only indicates whether one can define that product but also formulates its definition.

To sum up: We have just considered some tools of very general characters; others exist such as, for example, exact sequences and diagrams, which are in constant use in algebra and in algebraic topology. The use of these tools is inseparable from a very precise set of notations whose field of application is constantly widening. It is a new language, closed to the layman, but clear and evocative to the initiated. Of course, these tools are no magic wand and are only worth what their user is worth.

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Chapter 3:

Discovery Methods Linked With

The Axiomatic Method

Though no tool and no method can generate creative gifts if these are lacking, they can considerably increase efficiency where they do exist. We have studied some of the tools of the axiomatic methods, we shall now consider some of the discovery methods which only gain their full meaning within the study of multivalent structures. Any serious investigator discovers them for himself, but it is not without interest to make them explicit.

1 The Loosening Of Axioms

A certain analyst believes that a statement s concerning a crossroad-structure S defined by a number of axioms, is correct. The statement s has being formulated in simple terms that would still have a meaning in another axiomatic system S′ less rich in axioms than S (this would not mean that the statement

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would be true in S′). The analyst can then use the following method which reduces itself to a “loosening” of some axioms. He will try to prove s in S′; there are fewer combinations of axioms in S′ and this may assist in finding the proof; if he were lucky, either he would have proved s in S′, hence also in S, or he would find in S′ a counter-example C disproving s. A careful study of C may lead him to formulate a supplementary property P which added to the axioms of S′ would make him prove s. All that will be left is to go back to S′ to see whether P could be proved from it. The proof of s will follow.

2 The Tightening Of Axioms

The method I, consisted in temporarily suppressing some of the axioms of the system S; another method of research consists in adding new ones, i.e. to study particular cases.

The additional axioms will make it possible to use tools which were not available in S; in that way we get unexpected statements and proofs; going back to S one has to attempt to adapt to S the results obtained.

A well known particular case of this method is found in the use of discrete or finite models. In probability theory for example, the Markov processes owe much to the study of the processes on finite or discrete sets. In potential theory the study of the kernels on a finite set reveals phenomena unsuspected in the general case.

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3 Study Of Structures Not Too Different

If it is not known how to prove a theorem Τ concerning a crossroad-structure S, but it is possible to prove it for a structure S′ with axioms differing little from those of S, a large part of the lemmas, in which the proof of Τ in S′ can be broken down, may also be valid in S. The examination of the others may lead to their reformulation so as to obtain statements also valid in S.

Thus, for example, as it is not yet possible to prove Riemann’s hypothesis, one studies the problems relating to finite fields, hoping either to be able to transpose the results thus obtained to the classic question, or even to make such cases appear as particular cases of one and the same arithmetic-algebraic problem. Such a more general problem may be more easily solved. The history of mathematics is full of examples showing that by moving to an adequate level of generality one often gains in flexibility and that in it the secret springs of the proofs are made more evident.

It is however important when we cannot solve a problem, not to fall into the trap of solving easier ones and of believing that progress in the original question has been made. Such attempts may be excellent ways of coming closer to the challenge, but it is often preferable not to publish them.

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Chapter 3: Discovery Methods Linked With The Axiomatic Method

4 Generation Of Structures Answering Given Requirements

Present day technology can produce on demand machine-tools answering complex requirements. We are not too far from the day when chemists will be able to produce synthetically, fibers that will satisfy all requirements of the public. In mathematics, the theory of categories makes us see how now it is possible to produce the structures which will have the properties needed for such and such a question.

The state of mind of a young mathematician is no longer that of a builder in contact with matter; he no longer builds stage by stage, and brick by brick, the complex entities he needs; he only demands of these entities that they should have some given (and non-contradictory) mutual relations; they hence form a category that one can study by ordinary methods. The concretization of the elements of the category as sets endowed with a certain structure is one of the last stages in research.

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Chapter 4:

Some Characteristics Of

Bourbaki’s Contribution To

Analysis

We examined above the tools and the principles; let us now consider how the Bourbaki group and the members individually have made use of them in the realization which is their work.

1 Axiomatics And Multivalency

In agreement with his principles Bourbaki shows a predilection for multivalent structures. He likes general statements. He says “when it does not cost more, any theory will be produced in the most general framework.” This, though it proves to save much thought, requires a greater effort on the part of the reader.

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Thus, not only vector spaces are studied in relation to an arbitrary field, but whenever possible, their study is replaced by that of modules on a ring endowed with a unit (this, of course, forces one to adopt definitions which are valid in the general case, for example that of the tensor product). Similarly differential equations such as x′=f(x,t) are studied not in spaces of finite dimension but in normed spaces and f (x,t) is supposed to be only Lipschitzian in x and ruled in t.*

2 Bourbaki Is Essentially An Algebraist

The promoters of Bourbaki had discovered algebra while working with the great German algebraists at a time when modern algebra was not known in France. Because of that their analysis is impregnated with algebra and algebraic notations: algebra of sets, of course, but also groups, linear and multi-linear algebras, duality. They like transformations and properties that are expressed as algebraic relations. When a theory which is classically classified as belonging to analysis can be made into an algebraic one in part or in toto, Bourbaki does not forgo the pleasure of doing it.

In the past, analysis was essentially the study of functions defined on R or Rn whose values were on R or Rn and of operators of differentation and integration. At present, for the Bourbakist, R is mainly a commutative field of zero characteristic and this seems to be quite often what he needs. When he is working in

* A function is said to be ruled in t if it is uniform limit of step functions.

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the field of real numbers, he knows that all he must add is that it is ordered and locally compact.

In the hands of Claude Chevalley, the study of Lie groups or Lie algebras is free of all foreign elements: analysis plays in them a very restricted role; it only serves to prove the existence of entities endowed with such or such property or in the definition of such or such operation. For example, in differentation, one only retains its character of being a linear application of an algebra on itself, satisfying identically the relation D (x,y) = xD(y)+D(x) y.

In the hands of Henri Cartan the theory of functions of several complex variables is purified: integration remains a fundamental tool but he studies it locally, drawing out of it the algebraic properties which from now on are the only ones to be utilized. In his “Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes”, he prefers Weierstrass’ viewpoint to that of Cauchy and in his Chapter I he extracts from the algebraic study of formal series the maximum of information about their composition, their inversion and their differentiability. When he studies the Potential Theory he prefers algebraic tools: formula of composition of kernels; the interpretation of the “sweeping operation” (balayage) as an orthogonal projection in a Hilbert space.

3 The Constant Renewal Of The Opus

Bourbaki’s monument is not a balance sheet of the past but a living construction in constant evolution and oriented towards

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Chapter 4: Some Characteristics Of Bourbaki’s Contribution To Analysis

the future. Bourbaki integrates into his work recent developments that have proved themselves, and because of the new trend he is ready to recast whole branches even if they are of major importance (often discovering, by so doing, unexpected and fascinating new results). For instance, the older books of the treatise are going to be recast in terms of categories implicitly or explicitly: non-separated topological spaces have acquired the freedom of the city since their importance in various theories (and in particular in algebraic geometry) was discovered; linear partial differential equations are dealt with in terms of distributions, convolutions and of Fourier and Laplace transforms.

On the other hand, Bourbaki on some questions experiences irrational phobias. For example he has an interesting conception of the measure theory but it is too rigid in terms of spaces locally compact and of vague convergence; he relegates abstract measures, to the Chamber of Horrors, closing to his disciples thereby the gate to the theory of probability which even though it has not yet found its optimal tools is certainly at present showing an astonishing vitality.

4 Choice Of Definitions

The finding of good definitions is one of the essential parts of Bourbaki’s efforts. People object to the excessively deductive and formal character of this work: Bourbaki states the postulates and draws the consequences but does not explain his choice of the axioms nor the theorems he proves. The reason is that the history of these choices would be too long. Anyone who

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has attempted to provide the axiomatics of a hitherto confused theory, knows that the good definitions are only found after a number of unsuccessful attempts and that these attempts should be discarded, lest one weaken one’s mind by retaining in it a number of similar axiomatic theories. The real justification for a good axiomatic theory is its success.

Let us observe Bourbaki at work on a choice of definitions. While the classical analyst would start from “natural” definitions in a historic context and deduce key-theories from them, keeping the definitions as they were in the beginning while forging ahead in the theory, Bourbaki would change the definitions under the influence of the key theorems. He would use the key theorems as definitions if I may use an inaccurate but expressive phrase. This is one of the most important aspects of the Bourbakization of theories.

In a more precise manner, when a theorem establishes that entities E defined by a definition D have a property P which reveals itself in the course of the development as more adaptable than D, or that remains valid over a larger range than D and thus permits further generalizations, Bourbaki gives to P the role held earlier by D thus obtaining either a definition of E equivalent to the first, but more manageable, or a widening of the class of E to which the theory is applicable.

Here are a few illustrations of this fruitful method.

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1) Radon Measures. A theorem due to F. Riesz proved that, upon R there is identity between the Stieltjes integrals (defined upon a function locally of bounded variation) and the continuous linear forms on the space H (R) of continuous numerical functions vanishing outside a compactum.

Taken now as a definition this provides several advantages over the ordinary one of the Radon measure: immediate extension not only to R but to any locally compact space; greater flexibility in the study of the operations on measures (product of measures, images of measures, etc.); perfect adaptations to the definition of the weak topology on the space of measures, which proved itself to be the best suited of all topologies for that space.

Hence the definition of Radon’s measures now well known.

The developments that followed showed that this definition was apt not only in the case of integration. A Radon measure is nothing else than a continuous linear form on a certain topological vector space; new entities can now easily be defined by the following process. Let V be a topological space; the continuous linear forms on V are new entities that form a vector space V′ dual of V; the theory of duality, now well established, will provide various topologies on V′ which would ease the study of V. This gives a world of possibilities. Let us mention for example L. Schwartz’s distributions, de Rham’s currents, L. C. Young’s generalized surfaces.

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2) Invariant Measures On A Group. Lebesgue’s integral on R, can be defined by a well known process of continuation of the integral of continuous functions with compact support; on the space H (R) of these functions it is a linear form I which is positive in the sense that. I(f)>o for all f>o and is invariant by translation, in the sense that I (f)=I(g) when g is obtained from f by translation.

It can be shown that any function which has these properties only differs from Lebesgue’s integral by a constant coefficient. Hence the axiomatic definition of Lebesgue’s integral on R (up to a constant factor): it is a positive linear form on H (R) which is invariant with respect to the translations on R. This new definition is not only more manageable because it brings out the properties of the integral that are directly useful but also because it can be immediately adapted to the case of locally compact arbitrary groups.

3) Measurable Functions. In classical analysis measurable applications of Rn upon R are defined as follows:

f is said to be measurable if for any number λ the set of x such that f(x)<λ is measurable (with respect to Lebesgue’s measure).

Lusin’s theorem proves the equivalence of that definition with the following:

f is measurable if, for every compactum K of R, and for every number ∈ > o there exists a sub-compactum K′ of K such that (I)

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the measure of (K—K′) is smaller than ∈ ;(2) the restriction of f to K′ is continuous. The property involved in this second definition is both suggestive and more convenient in a number of applications. On the other hand it still has an interesting meaning when one substitutes for measure a more general set-function, as for example capacity in potential theory. Finally it is immediately usable in the definition of measurable applications of a locally compact space (endowed with a positive Radon measure) into an arbitrary topological space.

This second definition is therefore preferable to the classical definition and should be adopted.

5 Choice Of Contents And Theorems

In the writing of his treatise Bourbaki is forced to make choices at every moment. We just saw how he chooses definitions. When he chooses the content of his chapters he is equally careful.

His main interest is in tools and only in those which have proved their usefulness especially. Elegant results or even profound results do not hold his attention if they are tail ends of theories or lead to impasses. He leaves out, unconcerned with completeness, notions that are close to those he judged to be the most fundamental ones. If he thinks that a theory is not sufficiently mature for a choice to be made among its various possible axiomatic foundations, he prefers to wait to include it until the theory has matured. He has little taste for hors-d’oeuvres, for embellishment, or incidental developments without much connection with the rest of mathematics.

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He builds like the Romans, solidly. If the construction is by chance elegant, it is due to the beauty of its own inner structure; above all, he looks for simplicity, strength, usefulness, effectiveness.

In general topology, following Haussdorf, he made a sober choice in a maze of notions. A choice of convenient postulates for general topological spaces; a choice of a good notation for compactness; the introduction of filters (H. Cartan) simplified the notion of convergence; that of uniform spaces (A. Weil) brought together a number of notions which until then were considered as unrelated. This introduction of uniform spaces was further justified when the relations between compact and uniform spaces were discovered.

In functional analysis he was able to put into the right perspective the notions and the techniques consecrated by their use: local convexity of topological vector spaces; duality; theorem of the closed graph; theorem on the separation of convex sets; the theorems of Krein and Milman and of Stone-Weierstrass.

We have already mentioned his exclusive choice, in the theory of integration, of Radon measures on locally compact spaces, which in his hands have become a remarkable tool. In the “elementary volumes” the classical questions are treated with an unusual economy of means and a great generality. The theorem of finite increments is given for functions whose values are in a normed space; convex functions are treated elementarily but in a sufficiently complete fashion to meet most needs of analysis;

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primitives are defined in reference to ruled functions; finally, we already noted the generality of his “elementary” study of differential equations.

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Chapter 5:

Modern Analysis In The World

Today

I stated at the beginning of this text that the study of the works of Bourbaki and his disciples would give a fairly good and faithful idea of the modern tendencies in analysis.

After the brief examination of the salient features of these works we can attempt to verify this statement by having a look at what is being done in Analysis in the world as a whole. For that purpose let us open the “Mathematical Reviews”. About two thirds of what is being written could still be done with the tools available thirty or so years ago; a good number of these papers are valuable, some contain very profound and ingenious reasonings; important notions are being introduced in them and tools created and tested in a specialized field. But one can lament that too many writers do not seem to be aware of the existence of basic tools which have been thoroughly tested and which they rediscover, with ingenuity but painstakingly, and in

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the restricted domain of special cases of already known theorems.

In the remaining third, the writers use modern tools. There, too, one finds the inevitable waste that goes with all scientific production; too many papers are shallow or hollow and do not add anything to the construction of the “mathematic temple”. But in the best papers the modern techniques show a terrific yield. Each year brings with it the solution of one or more of the problems considered as beyond reach and sees bridges created between theories that seemed to have nothing in common.

Here is a list of the most nourishing branches of Analysis:

Topological groups and Lie’s theory.

Topological algebra.

Measure and integration.

Functions of several complex variables and analytic varieties (where there are many algebraic techniques, fiber bundles, filtrated spaces).

Partial differential equations (in which one uses distributions and other generalized functions; study of the nonlinear case).

Potential theory (general kernels, study of the principles and of the relations with probability theory).

Harmonic analysis on general groups, functions of positive type.

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Functional analysis (locally convex topological vector spaces; convexity; spectral theory of operators.)

General topology.

Differential geometry.

Differential topology.

Probability theory.

These branches have developed to their full vigor following the same principles as Bourbaki; the language used is the same. In specialized colloquia devoted to them, the best among the specialists use the same methods, the same language, have the same preoccupations. In its most active parts, therefore modern analysis is manifesting a great unity.

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Chapter 5: Modern Analysis In The World Today

Chapter 6:

The Impact Upon Modern

Mathematics Education

From time immemorial teaching has been adapted to the evolution of knowledge. But this adaptation has sometimes lagged behind, to the great detriment of both science and teaching. During the last fifty years or so scientific progress has been so rapid that delay in adaptation was inevitable. In mathematics, the “new look” resulting from the use of the theory of sets and the axiomatic method has been a revolution which makes a renovation of teaching urgent at all levels: primary, secondary, and university.

This renovation is required.

First, for mathematics’ sake itself. Indeed it is not the old people or even those of middle age who are producing the best work; it is imperative that we clear the way for the younger generation. In order that they may assimilate mathematics more easily, we

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must show them clearly the great simplifying ideas, teach them how to handle complex situations by teaching them the unifying theories, which throw bridges between the various fields. This will require sacrifice and agreeing to abandon such and such elegant theories which, though polished by centuries of work, are now seen to be as isolated branches.

Second, for the users of mathematics (who are everyday more and more numerous). On the one hand a number of mathematical techniques have become indispensable or useful in physics and engineering: matrices; Fourier and Laplace transforms; partial differential equations; distributions; Hilbert spaces; etc. On the other hand the new mathematics have brought simplifications and economy of thought to all fields from which the physicist and the engineer can benefit as well as the future mathematician.

It is obvious that the text-books for such a renewal are still wanting. Entirely absorbed by their investigations, the professional mathematicians have let a deep rift form between research and teaching. But in the last ten years, frightened by the sight of the growing gulf, they have reacted. They first changed their own way of teaching, then they turned round to their colleagues of the secondary schools and started a profitable dialogue with them. Still they need to gather their courage for an urgent and essential task. The time for mere criticism and vague indications is over. They must now sit down and write the needed textbooks or help their colleagues, technicians, or secondary school teachers to write them. The purpose will not be to copy the Bourbaki production conceived for advanced

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students, but to adapt to each level the language, methods and techniques of the mathematics of today.

Thirdly, for those who will become neither mathematicians nor users of mathematics. It is universally agreed that from the study of this discipline they can gain an intellectual flexibility which cannot be acquired otherwise. Modern mathematics will perhaps give more to them than to the others. Because there are not many technicalities involved, they can learn the theory of sets as related to logic and find that attractive and useful. The simplicity of the multivalent axiomatic systems makes them accessible to all and as they have a number of varied applications they will not appear as a mere game.

It is out of the question to attempt here to outline a syllabus. All that can be done is to indicate a few principles that follow from the examination we have undertaken in this text:

To make our pupils used to thinking as soon as possible in terms of sets and operations. At a very early age they should be taught to use the language and the algebra of sets, as its symbolism is simple and precise. Experiments in teaching have shown that pupils like using it.

Concurrently with the algebra of sets, the elements of logic can be taught them in connection with the grammatical analysis of their own language. It has been observed that senior students of 19 years of age are unable to reason, cannot give the negation of a proposition, nor state correctly a definition or a theorem; this,

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Chapter 6: The Impact Upon Modern Mathematics Education

we think is due to starting so late to train them in this sort of exercise.

Very early, too, our pupils must grasp clearly the notion of a function. To that end, they must have studied and constructed various examples drawn from life, from algebra, from arithmetic, geometry, physics etc. They should know how to compose functions, take the reciprocal function of a one valued function, recognize a transformation and a group of transformations. Progressively they will be introduced to the larger structures of equivalence, and order, and topological, and algebraic structures. These structures can be studied, at various levels, from the beginning of the secondary school (around 12 years of age).

The aim is to give our pupils some tools and teach them how to use them. We must avoid losing ourselves in generalities; on the contrary we must go straight to the key theorems which include large numbers of special theorems with immediate applications.

For example, very early in elementary geometry, the affine structure of the plane or of space, will be found and the algebra of vectors will be used. Afterwards in one or other way the scalar-product may be introduced and this will reduce to a few simple calculations the essential parts of ordinary metric geometry.

Similarly, at university level, the more powerful tools will be brought to light: theorems on compact spaces; metric of uniform convergence; Stone-Weierstrass theorem; method of successive

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approximations etc.: the students should be trained to recognize which structures are involved in the statements met; this presupposes that the selected definitions and statements always stress structures. For example, Lebesgue’s integral on R must at a certain stage, appear to them as a positive linear form on H (Rn), invariant with respect to translations; the Laplacian must appear as the only 2nd order differential operator invariant with the respect to displacements etc.

In this text, much has been said about mathematics in general and little about analysis in particular. The reason is that it is no longer possible to divide the teaching of mathematics into its classical divisions of algebra, geometry and analysis.

The proper bases for the teaching of analysis even at school levels, are algebra (algebra of sets, study of the field R, linear algebra, groups) and topology. The same algebraic bases are needed for the study of geometry (which in secondary schools today means the study of a vector space of two or three dimensions endowed with a scalar-product).

It becomes therefore essential to think of a teaching whose main fabric will be the fundamental structures. Algebra and geometry will support each other, algebra bringing its symbolism and its operations, geometry its language loaded with intuitions. Geometry will provide analysis with its topological framework, the tool of convexity and a convenient interpretation of integration and differentiation; analysis in its turn will produce for algebra a rich collection of groups and of vector spaces.

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Chapter 6: The Impact Upon Modern Mathematics Education

Mathematical Activity As A Whole.

A little has been said here about the methods of research and a little more has been said about the already existing mathematical theory. To conclude this examination of mathematics undertaken in order to assist in meeting the problems of mathematical education, let us add one word about one aspect of the mathematical activity which has not been dealt with at all.

Every mathematical activity is formed of cycles, smaller or larger, in which one can roughly recognize the four stages that follow: observation, mathematization, deduction, applications.

These four stages are essential, in particular a purely deductive teaching would be traumatic and sterile.

Each of the large cycles corresponds to the conquest of a new notion; its four stages are the stages necessary to allow the brain to restructure itself and to shift from one level of thought to another. This is as valid for the research worker as for the pupil whose creative activity cannot function unless we allow him, and perhaps help him, to follow the path which leads to knowledge.

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Index

Abstract measures, 32 Affine structure, 46 Albanese, 21 Algebra, 7, 8, 10, 11, 20, 21, 23, 27, 30, 31, 32, 40, 45, 46, 47 of sets, 11, 30, 45, 47 of vectors, 46 Algebraic geometry, 11, 21, 32 structures, 7, 20, 46 topology, 7, 11, 23 Analysis, 3, 8, 10, 11, 13, 20, 29, 30, 31, 35, 37, 39, 40, 41, 45, 47 Applications, 9, 14, 18, 19, 20, 35, 36, 45, 46, 48 Arithmetic, 27, 46 Auxiliary structures, 17 Axiomatic method, 5, 9, 14, 15, 17, 25, 43 systems, 9, 14, 45 theories, 33 Axiomatics, 29, 32

Axioms, 6, 9, 10, 13, 17, 22, 25, 26, 27, 32 Balayage, 10, 31 Bijection, 18 Bourbaki, 3, 5, 6, 14, 15, 29, 30, 31, 32, 33, 36, 39, 41, 44 Bourbakization, 33 Capacity, 36 Cartan, Henri, 10, 31 Categories, 21, 22, 23, 28, 32 Category with product, 23 Cauchy, 31 Chevalley, Claude, 31 Commutative groups, 17 Compact groups, 8, 20 topological spaces, 18 Compactum, 34, 35 Continuous applications, 18 Homo-morphisms, 22 Convexity, 37, 40, 47 Convolutions, 32 Crossroads-structures, 7

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What is Modern Mathematics

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