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Selected Lectures from the 7th International Congress on Mathematical Education

Québec, 17-23 August 1992, pp.147-155

## THE ORIGIN AND EVOLUTION OF MATHEMATICAL THEORIES

##### IMPLICATIONS FOR MATHEMATICAL EDUCATION

Miguel de Guzmán

Universidad Complutense, Spain

In this paper I am concerned to consider the right way to introduce young people to mathematics research. How should they be introduced to mathematical content and to mathematical theories? What is the attitude we should try to foster in them? What do those who are most successful in preparing young mathematicians actually do?

The questions need to be answered quite concretely for it must be admitted that the way to prepare researchers is not a matter of general agreement in the profession, it is not always carried out well, and too many details are often left to chance.

Of course there are many ways to involve students personally and actively in their learning of any mathematical topic, especially by motivating them with problems. For example:

*Here is a problem. Don’t read anything, just plunge in and try to solve it straight away.

*Read these several passages from these books carefully, then come and get a problem from me.

*Read this recent paper and then work on the problems it leaves open.

*I will be giving you suggestions for problems to solve throughout the course. Choose the ones that you think will be most productive, that most interest you, that you believe you have a chance of solving.

I strongly believe that the crucial insights in research in a particular field tend to come from a deep knowledge of the origins and evolution of the theory one is working with, and a familiarity with the style of thought in that area. This is acquired by learning its motivations, the circumstances of its origins (historical, social, personal), the right ways of asking questions, and so on.

I shall try to give substance to this claim by looking first of all at what a knowledge of the history of mathematics in general, and of the specific subject in particular, can offer us that is relevant to the context we are exploring here, and by briefly examining afterwards the lessons that can be derived from the knowledge of the evolution of a field in which I was personally involved some years ago.

**WHAT KNOWLEDGE ABOUT THE HISTORY OF MATHEMATICS AND ABOUT A PARTICULAR SUBJECT CAN OFFER THE STUDENT**

They offer a vision of science and mathematics as human activities.

We see that the truths, methods, and techniques of mathematics do not come out of the blue. They are not impersonal facts and skills without a history, but are the results of the efforts of passionate and deeply motivated people.

We see that, in spite of its many wonders, mathematics is not really a «godlike» or perfect science. Because it is an artefact of human beings it is also incomplete and fallible. Its history gives us many great discoveries and great discoverers to admire, but it also shows us that much of what we now take to be established and obvious truth was only arrived at after many errors and much controversy.

They offer a frame within which to organize the elements of our mathematical knowledge.

We see better how to relate events that took place centuries apart, how to appreciate the temporal contexts in which mathematical discoveries were made.

We see how people invested their efforts in the pursuit of certain questions, how «fashions» arose, and how the fashions of the past can alert us to those of the present.

We get a sense of how the various threads in the fabric of the subject we are working on were woven together over time,

They offer a dynamic vision of the evolution of mathematics.

We understand the driving forces at work developing the basic ideas and methods of mathematics. We get closer to the springs of creativity that generated particular subjects, consequently gaining a sense of their genesis and progress and a better appreciation of their true nature.

We get a flavor of the thrill and adventure of working in mathematics.

We are immersed creatively in the past and better able to understand our own problems.

There is the possibility of extrapolating towards the future.

We realize the tortuous paths of creativity, the ambiguities, obscurities, and partial illuminations that accompany the first attempts to shape the field.

We see how we can inject some dynamic, some life, into our educational tasks.

They offer an appreciation of the intertwining of mathematical thought and culture in human society: of the importance of mathematics as a part of human culture.

We see the influence of historical trends and developments on mathematics and, conversely, the impacts of mathematics on human culture, its sciences and philosophies, its arts and technologies.

They offer a more profound technical comprehension.

The more simple a theory is in the beginning the easier it is to understand and work with. Technical complications coming along later can begin to obscure the theory unless one grasps their motivations.

The lines of development of a theory point towards the future and provide guidelines for research.

They offer an awareness of the special life of any mathematical theory.

Each theory has its own peculiar character, molded by the special circumstances that gave rise to it. It was born at a particular moment, the result of particular concerns. It was motivated by curiosity about some phenomenon, the wish to apply some known results, to expand some collection of techniques, to complete some existing theory, and so on.

Each theory developed according to its particular style, its expectations and disappointments, its correct intuitions and its false starts.

Each theory inhabits its own «local» atmosphere generated by the personal and social forces that surrounded it.

It seems to me that one can conclude that: Familiarity with the origin and evolution of a mathematical theory has profound lessons to offer to anyone trying to be inducted into the field.

**A CASE STUDY: THE DIFFERENTIATION OF INTEGRAL**S

By following the lines of development of a specific theme in mathematical analysis from this century -Lebesgue differentiation and its extensions- I shall now try to show some of the lessons which can be extracted from this study that could be of use to those seeking to do research in this area. If I am fortunate, the example will have some lessons for people working in other fields too.

Some of the ideas and methods stimulated by this theory during this century have proved very useful in other areas of mathematical analysis, particularly in Fourier analysis and in some aspects of geometrical measure theory. I will present a non-technical description of the main highlights of the theory, taking into account that we are not interested in its technicalities here but rather in the educational implications for those wanting to be introduced to the subject. For the sake of brevity I will trace the main points of the theory from its origins in 1904 to the time its progress was interrupted by the Second World War.

**The beginning: the Lebesgue differentiation theorem**

Towards the completion of an interesting theory.

The Lebesgue differentiation theorem, the equivalent of the fundamental theorem of calculus, was the culminating point of his measure theory. He first proved it for R1 (1904): If f is a function in L(R1) then at almost every point x

Essentially this meant that the means of an integrable function over intervals containing a point x converge, at almost every point, to the value of the function at that point when the intervals contract to the point. The idea followed by Lebesgue in the proof was ingenious but not translatable to R2. Since the order structure of the real line is so crucial for the proof in R1, what might be the corresponding tool for R2?

As in so many other cases, the first impulse to develop new techniques came from the need to extend a theory to more general situations.

**Vitali’s covering theorem**

Sharpening tools that have proved to be interesting, deep, and useful; pushing their scope further.

At the end of the 19th century a number of covering theorems were discovered that helped substantially to clear up the structure of Euclidean space from an analytic point of view. The so-called Heine-Borel covering theorem, the Lindelöf theorem, and others, became important tools in this respect. Vitali’s covering theorem was an important advance: Let M be a measurable set in the plane with a Vitali cover V for M (i.e. for every point of M a sequence of square intervals centered on the corresponding point and contracting to that point is given). Then one can extract from V a sequence {Qk} of disjoint squares such that

Vitali’s theorem was not invented for the purpose of obtaining a proof of the Lebesgue differentiation theorem in R2, but this was the use Lebesgue made of it in 1910, showing that his theorem for the line could be generalized to the plane if one takes the means of an integrable function over squares or circles containing the corresponding point.

*The Lebesgue differentiation theorem in R2*

The solution of an interesting problem often leads to deeper questions; a good problem is never exhausted

The result Lebesgue obtained was quite satisfactory, but it led immediately to a natural question: Can one replace the squares by more general intervals (e.g. by rectangles in the direction of the coordinate axes, or perhaps by rectangles in arbitrary position)? These natural questions turned out to be quite challenging and these problems remained open for a long time, as we shall see.

**The value of paradox**

A paradoxical situation can be the beginning of a new development.

From 1908 until 1924 there was in the air a belief that Vitali’s theorem would also hold if intervals were substituted for squares. The theorem of Lebesgue would then admit a nice and direct generalization. The fact, first proved by H. Bohr (1918) and first published by Banach (1924), that intervals in the plane do not satisfy Vitali’s lemma seemed counterintuitive. This sort of paradox made the study of the covering properties of different systems of sets in the plane more challenging, and at the same time started to throw some new light on the subject.

This is in many cases the effect of perceived paradoxes. It has been reported that in the midst of working on a difficult problem the physicist Niels Bohr was overheard to say: «How wonderful! We have met a paradox. Now we have some hope of making progress.»

*An impasse concerning the strong density problem (1924-1934)*

There are periods of impasse when progress can come from many directions; one must remain open to all possibilities.

Since Vitali’s lemma fails for intervals, what will happen to the differentiation theorem of Lebesgue? Even in the case when the function f is the characteristic function of a measurable set, the problem of generalizing the theorem to the plane seemed to be quite difficult. This so-called strong density problem (the local density of a measurable set with respect to intervals in the plane) remained open for many years -until 1933, when Saks was able to prove the strong density theorem. In the meantime, many mathematicians were looking in other directions to find some light that could illuminate this challenging question.

*The role of a good game*

A good mathematical game can be the beginning of a deep theory.

In 1917 S. Kakeya proposed a problem that looked like a puzzle: What is the infimum of the areas of those plane figures within which a needle of length one can be inverted by continuous motions? The problem has a very long and interesting history in which some important mathematicians show up: e.g., Besicovitch, Perron, Rademacher, Schoenberg. Those interested in the ramifications of it are invited to consult the bibliography proposed at the end of the paper. (By the way, the surprising solution, given by Besicovitch in 1928, is that the infimum mentioned in the statement of the problem is zero.) Here it should suffice to mention that the problem has had very profound implications for the subject of differentiation of integrals and for Fourier analysis. By means of the tools developed in order to solve it C. Fefferman in 1971 was able to solve an important problem which had remained open for many years (the multiplier problem for the ball).

**Different lines of thought concerning a theory**

At the point where different subjects intersect can often be found many deep questions and much light on those subjects.

At the beginning of the century the theory of Lebesgue measure was recognized as an important tool in many connections in mathematical analysis. It generated a strong interest in the geometric structure of measurable sets. Some of the questions proposed at the time later proved to have deep implications for differentiation theory. In 1926 Banach proposed the question: How large in measure can a linearly accessible set in the unit square Q be? («Linearly accessible» means that each point of the set can be reached by a straight line originating outside Q.) In 1927 Nikodym solved the problem in a long and complicated paper by constructing a set N contained in Q and having measure 1 (a set of full measure in Q) such that through each of its points there is a straight line not intersecting the set N again. N is a strange set that, in spite of «filling» Q, seems to leave many more points of Q in its complement Q – N. At the end of Nikodym’s paper appears an observation of Zygmund that shows that the collection of all rectangles in the plane is an unsatisfactory system for proving the Lebesgue differentiation theorem, and, further, that the density theorem with respect to the system of plane rectangles does not hold.

Later on, R.O. Davies, working in this same direction, constructed still more paradoxical and spectacular sets than that of Nikodym.

**The versatility of mathematical tools**

When you find a good tool, try to make use of it in some connected problems.

The strong density theorem was proved by Saks in 1933. By then F. Riesz was already in possession of a powerful tool concerning continuous functions in R1, the so-called rising sun lemma (also called the water flowing lemma). He was able to apply it to solving several interesting problems of the moment with ease, presenting another simple and easy proof of the strong density theorem in 1934.

In this same year, Jessen, Marcinkiewicz, and Zygmund were able to give the definitive theorem in the direction of differentiation of functions by the system of intervals in R» : If f is a function in L(logL)n-1(Rn) then the intervals differentiate the integral of f and this space of functions is in some sense the best one.

**After climbing the peak**

When the evolution of a theory along a particular path seems to be close to the summit, one may need to start looking for different lines of thought.

After the Jessen-Marcinkiewicz-Zygmund theorem, the attention of the mathematicians concerned with the differentiation of integrals turned in a natural way in other directions. Busemann and Feller took a new path in 1934 and R. de Possel yet another in 1936.

**The abstract and the concrete**

Examine the concrete and try to discover a general pattern.

Busemann and Feller introduced into the field the consideration of what has been called the halo of a measurable set with respect to a differentiation basis (a generalization of the system of all spheres or of all intervals used in the differentiation theorem). From the concrete ideas introduced by Saks and Riesz in their treatment of the strong density theorem, the idea of the halo was a natural development. By means of it Busemann and Feller were able to present the characterizations of systems of sets that would have good differentiation properties. The time had come to try to leap from the concrete cases to some more general formulations which could be used in other cases. They managed it by giving a quantitative characterization by means of something which was later perceived to be a (1,1) weak-type inequality for the maximal Hardy-Littlewood operator -of the systems of sets used for differentiating the integral.

For his part, R. de Possel proceeded in a similar vein, from the concrete to the abstract. He observed what happens in the plane with respect to the differentiation and covering properties of the different systems:

a) Squares satisfy Vitali’s lemma; squares allow the differentiation of L1(R2).

b) Rectangles in arbitrary directions do not satisfy Vitali’s lemma; rectangles do not have the strong density property.

c) Intervals allow the differentiation of LlogL(R2); but not of L1 (R2).

In a natural way, he decided to try to explore what are the covering properties, if any, of the system of intervals. He was able in this way to initiate an interesting line of research, looking for the quantitative connections between the differentiation properties and the covering properties of a differentiation basis.

So we can see here in action another interesting principle, which should be kept in mind:

When you notice a qualitative connection, try to find the quantitative reasons for it.

The progress of the theory was interrupted by the Second World War. After it came many other interesting developments: in particular, from the work of Besicovitch in connection with Geometric Measure Theory, and from the intervention of many analysts working in Fourier Analysis. Those interested in following up this subject in detail are invited to consult some of the references below.

Banach, S. (1924). Sur un théoréme de M. Vitali. Fundamenta Mathematicae, 5,130-136.

Bruckner, A.M. (1971). Differentiation of integrals. American Mathematical Monthly, 78 (Slaught Memorial Paper No. 12).

Busemann H., & Feller, W. (1934). Zur Differentiation der Lebesgueschen Integrale. Fundamenta Mathematicae, 22, 226-256.

Fefferman, C. (1971). The multiplier problem for the ball. Annals of Mathematics, 94, 330-336.

Guzmán, M. de (1975). Differentiation of integrals in Rn. Lecture Notes in Mathematics, 481. Berlin: Springer.

Guzmán, M. de (1981). Real variable methods in Fourier analysis. North Holland Mathematical Studies, 46 .

Guzmán, M. de, & Welland, G. (1971). On the differentiation of integrals. Revista de la Unión Matemática Argentina, 25, 253-276.

Jessen, B., Marcinkiewicz, J., & Zygmund, A. (1935). Note on the differentiability of multiple integrals. Fundamenta Mathematicae, 25, 217-234.

Kakeya, S. (1917). Some problems on maxima and minima regarding ovals. Tohoku Science Reports, 6, 71-88.

Lebesgue, H. (1904). Leçons sur l’intégration et la recherche des fonctions primitives. Paris: Gauthier-Villars.

Lebesgue, H. (1910). Sur l’intégration des fonctions discontinues. Annales scientifiques de l’École normale supérieure, 27, 361-450.

Nikodym, O. (1927). Sur la mesure des ensembles plans dont tous les points sont rectilinéairement accessibles. Fundamenta Mathematicae, 10, 116-168.

Possel, R. de, (1936). Sur la dérivation abstraite des fonctions d’ensemble. Journal de mathématiques pures et appliquées, 15, 391-409.

Riesz, F. (1934). Sur les points de densité au sens fort. Fundamenta Mathematicae, 22, 221-265.

Saks, S. (1933). Théorie de l’intégrale. Varsovie.

Vitali, G. (1908). Sui gruppi di punti e sulle funzioni di variabili reali. Atti Accademia di Scienze di Torino, 43, 75-92.

Zygmund, A. (1934). On the differentiability of multiple integrals. Fundamenta Mathematicae, 23, 134-149.