# Memories of Massera's academic and political life

We give an English translation of

*Memories of my academic and political life*, by José Luis Massera. The original was a lecture given at the National Museum of Anthropology in Mexico City on 6 March 1998. The Spanish text is in José Luis Massera, Recuerdos de mi vida academica y polltica, in*José Luis Massera. El científico y el hombre. Premio México de Ciencia y Tecnología 1997*(Faculty of Engineering, Montevideo, Uruguay, 1998), 45-58.**Memories of my academic and political life, by José Luis Massera**

I think that these memories, beyond a few episodes with a personal stamp, will contain generalisable experiences of interest to many of you.

When I was in the sixth year of the Primary system, I had a teacher who deeply marked my life. Nothing more, nothing less, he taught me to think. He was somewhat stern, he did not admit easy sympathies. One time that I have never forgotten he sanctioned me, and in the face of my father's protest he wrote down a single word in his notebook: "spoiled", which I was able to read, perhaps because he wanted me to read it. And I agreed with him. The essential thing was what I said before: beyond the knowledge of the programme, it was able to impress strongly on my mind that what was decisive was not this or that particular learning, but helping me to be able to understand myself.

Once, in the mathematics class of the first year of the Lyceum, the subject to be treated was the similarity of polygons. I had not studied the lesson and, precisely, they called me to give it. I could perhaps have invented some excuse, but I accepted the challenge and went to the blackboard where what I invented were proofs of the theorems. I honestly can't say they were correct, but I think they weren't essentially far off the mark. I don't know what conclusions my classmates drew; what I am sure of is that the teacher did not understand anything: he only knew what he had read in the text.

Later, now at the Liceo, in general I didn't like rote subjects - I never had a good memory and now it's much worse - particularly History. There was an exception with a teacher who was not very good but who tried a relatively rational explanation of the story: anyway, I liked that. In the third year, the mathematics teacher was a German who knew something and I liked it, perhaps as an incipient manifestation of my vocation.

More important was at that time - I was about fifteen years old - the revolution that I started in my house and that lasted several years. My father had a Hispanic American Encyclopaedic Dictionary in several volumes of a fairly good level. One day, on my way back from German teacher's class, it occurred to me to look up one of the words he had used in the dictionary, probably "equation". I came across a huge number of different equations, which I had not even suspected existed, nor how to approach them. My curiosity with algebra more than satisfied, I went to look for one of the others in the dictionary. Thus, day by day and word by word, I began a tour, extremely chaotic, without a doubt, which was giving me a harvest of mathematical terms and valuable information about them, which I was slowly accumulating and conceptualising.

Around that time, my parents travelled to Carlsbad for treatment and took me with my two sisters. On the way back we passed through Paris, and I accompanied my father to a large bookstore. There I bought two excellent little books, one on classical geometry and the other on trigonometry. On the way back I devoured them in no time: I had finally found reasonably organised mathematics texts. Nevertheless I continued to use the method, a silly thing to do, of the dictionary.

After the Liceo there were two years of Preparatory for different careers and I naturally chose Engineering, which was the one that used mathematics the most. But I was already further ahead of the syllabus. I read English and French well, but I knew the importance of German and booked a two-month intensive private course to learn the rudiments I needed to read mathematics. It was a complete success, I practiced - with the help of the dictionary - reading a thick volume of Projective Geometry.

The passage through Preparatory had other important consequences. There I met my colleague Rafael Laguardia, who had taken a higher education course with great French professors. We struck up a close friendship that lasted until his death in 1980. He knew some other young mathematics enthusiasts, with whom we formed a study group. Someone who had read an important text would then give a course to the others.

New horizons soon opened up. In Uruguay we had all read good textbooks by the Spanish mathematician then based in Argentina, Julio Rey Pastor, who made annual trips to Europe to update himself. We got in touch with him, and we agreed that he would make visits to Montevideo on Saturdays to pass on the news to our group. Thus, a fairly extensive course was formalised on what we then called "Abstract Spaces", today we would rather say "General Topology". Later, and as a consequence of the war in Spain and the fascist movements in Europe, Luís Santaló and other Spaniards, the great Italian mathematician Beppo Levi and others, immigrated to Argentina and became new teachers and friends of our group. On the other hand, our delegations attended the meetings of the Argentine Mathematical Union, which had existed for years. Around those same years, Misha Cotlar, the son of a modest Russian immigrant, who today is a world-renowned mathematician, joined our group. At that time, around the great movement of solidarity with the Spanish people, I began my political activity that I shared with mathematics for the rest of my life.

I entered the Faculty of Engineering in 1935, the year in which Mr Eduardo García de Zúñiga, one of the first three engineers to graduate in Uruguay, was my professor of analysis; he was already very old. He chose to define the real numbers by the method of fundamental sequences, perhaps the most difficult of all. From my interventions in class, he knew what the level of my mathematical knowledge was, which explains the unusual fact that in 1937 I was appointed Assistant of Practical Classes; In the subsequent examinations, Zúñiga asked me, so that he could show me off, topics far in advance of those given in the courses. Zúñiga even published some of his own research, enriched the Faculty Library, which today bears his name, with entire collections of the world's leading mathematical journals and some very old scientific ones, Complete Works of the greatest mathematicians, etc., all of which has turned it into a bibliographical treasure probably unique in Latin America, worthy of being consulted by historians of science from other countries. Needless to say, this treasure gathered by our "great ancestor" as we affectionately called him, was an invaluable tool in our subsequent research.

Closing this long parenthesis, I must say that regulatory reasons forced me to finish the Engineering degree to receive the degree of Professor of Engineering, even away from mathematics. I did so, as an Industrial Engineer, in 1943. It was not a painful effort, the degree gave me a good training in Mechanics, Physics, Chemistry, very useful for my scientific culture.

On the initiative of Laguardia and with the intelligent support of Dean Vicente García, the Institute of Mathematics and Statistics was founded in 1942, which today bears his name. At last the group of fans of mathematics had a home, albeit a very modest one in its material reality: a spacious hall partitioned into smaller compartments. You could work, study, give courses, it was a qualitative change.

Meanwhile, in the world things continued to get worse. Once the Spanish Republic had been defeated, events turned towards the Nazi-Fascist offensive and the outbreak of World War II. My political activity consequently broadened and radicalised. I was appointed President of a fledgling Peace Movement and then General Secretary of a large allied solidarity movement: Anti-Nazi Action for Aid to Free Peoples, which collected money, warm clothes and shoes, and medicines and delivered them to representative diplomats of the Soviet Union, England and Free France "according to the will of the donor", expressed at the time of the donation. After the war, and facing the dangers of the "cold war", I returned to the struggle for peace, becoming a member of the World Peace Congress, which led the movement internationally. In 1942 I joined the Communist Party of Uruguay.

The end of the war opened new perspectives for academic life. The Rockefeller Foundation awarded Laguardia a scholarship to study mathematics in the United States. Later I also got a similar scholarship. Years before, I had solved many of the problems in the book "Aufgaben und Lehrsätze der Mathematik" by two famous Hungarian mathematicians Polya and Szegö, professors at Stanford, and I greatly admired the mathematical and didactic ability they showed in that work. So I wrote them a letter expressing my interest in taking some courses with them; attached a long list of the works I had studied so far. Szegö's answer was affirmative and betrayed a certain astonishment at the breadth of my knowledge.

So I decided to go to Stanford University in California to fulfil my scholarship. At that time it was not so easy to travel by plane to that place and I thought that the most practical thing was to go to New York and from there use other less expensive means of transport. Thus, I inaugurated a style of travel that I later applied on many occasions. I visited places and museums in New York; Quebec, Montreal, Ottawa and Toronto (the great museum of Chinese antiquities), the frozen Niagara Falls, the Colorado Canyon, beautiful San Francisco and many other places.

In Palo Alto, a small town that is located in front of the University, it was usual to house students in rooms in private houses; I did so in a huge mansion belonging to one of the owners of General Motors, where his very old wife lived, with whom I became friends and who allowed me to discover peculiar facets of North Americans. Once, she told me that she had received a visit from an FBI agent who had questioned her about me, "because you must know that he is a Communist." She replied that she had nothing to criticise me for, that my behaviour was correct and that I studied hard ... Being a person of that social class, the fact seemed significant to me.

At Stanford they welcomed me with a small family party. With Polya I had a cordial relationship, but not a deep one. Instead, I became very close friends with Szegó, his wife and his son Peter, whom I met again in New York as a fellow student, and who is today an outstanding mathematician, of whom I usually hear about now. All of them were excellent people, very progressive. The classes at the University were not of a high standard, except for a course in real number theory taught, as a visiting professor, by Rademacher. He once gave a very complicated proof of a fine approximation theorem of a real number, correct, but it annoyed me: in the next class I proposed a proof of a geometric nature, very brief, which delighted him.

Paradoxically, it was not at the University but outside of it that I found a source of great professional interest. I knew a person who had been a high-ranking officer in the Czarist Navy, married to a very nice Spanish woman. He was then working for the US Navy on ship stabilisation problems in rough seas, relying on theoretical studies by Russian mathematicians on stability issues, of which he had photocopies and knew that New York University had made a publication in English. I got hold of a copy of it and since I had acquired some notions of Russian in Uruguay, so with difficulty I could also have read the photocopies of my friend the sailor. All of which was of great interest to me; not so for the sailor who, although educated, was not capable of penetrating the mathematical meaning of things.

These topics fascinated me and although I was able to exchange ideas about them with some mathematicians who passed through the University, I was convinced that Stanford could not solve the problem for me. So I asked Rockefeller for a transfer to New York and Princeton, where they took care of it, which was granted. I therefore travelled to the East and stayed in a room in New York. The work plan was to enrol in the two Universities and travel by train two or three days a week; the very comfortable trains made it possible to study on the way.

In the first, directed by Courant - and which today bears his name - there were great mathematicians: Friedrichs, Artin and others. The first contact effectively overwhelmed me, they had worked on differential equations, but they had ceased to be an important centre in this regard. Courant had just finished a book on minimal surfaces and honoured me by asking me to proofread the text. It was the great "hobby" of his time, with a lot of fine mathematics but also with an arsenal of wires to define its contours and a recipe for soapy water; he was very proud of having achieved a minimal surface in the form of a Möbius strip. I worked a lot with him and Friedrichs on various topics and we became friends. As I was preparing to return home, he wrote a glowing letter to the Dean of the Faculty asking for his support in my mathematical career.

The main centre of my work in the East, however, was Princeton University, whose Mathematics department was headed by the great Russian mathematician Solomon Lefschetz. His original profession had actually been a Civil Engineer and he worked as such. Once, in France, he was directing a play and, in a fatal moment, he undid a steel sling that sliced both of his forearms. His life was saved and they placed two articulated prostheses with fingers supported by springs with which he could hold some elementary instruments for writing and other actions with incredible bodily efforts. Thus began another life for him, deepened the study of mathematics and, with amazing talent, became one of the founders of modern Algebraic Topology. By the time I arrived at Princeton he had been excited about the promising prospects for applications of topology to the study of differential equations, about which he had published a book. A deep friendship and mutual collaboration developed between us that lasted until his death. At last, he was the mathematician I needed.

In view of this new situation, I asked Rockefeller for an extension of the scholarship for another six months; The year was 1948. It was the time of greatest results in my research in the United States: reciprocal of the Lyapunov method in the stability of motion, description of the set of solutions of a system of differential equations in the plane, study of the solutions harmonic and subharmonic periodic. In the second and third of these subjects, I studied a paper by a well-known American mathematician and seemed to find a topological error that invalidated it. In some confusion, I consulted Lefschetz. The answer was blunt: "Well, if you think there is an error, say so!". From that came the work in which I corrected the error and exhaustively described the infinite set of solutions.

On the way back to Uruguay, I stopped at the Universities of New Orleans, Mexico and Rio de Janeiro, where I presented my results, as well as at the Montevideo Institute, where new young mathematicians were appearing: Villegas, Cabaña, Schäffer, Lumer Sebastiani, Lewowicz. There, Laguardia had taken new initiatives, one of which was important: the Colloquium, in which, at a frequent rhythm, each one presented the topics and research on which he was working; other Uruguayan mathematicians were doing their doctorates abroad. The Institute was already known and appreciated in other countries and high-ranking visiting professors came (Halmos, Laurent Schwartz, etc.), sometimes Latin American countries sent us their young people to study in Montevideo.

Meanwhile, I continued studying and publishing new results that completed my studies at Stanford. Quite frequently, Lefschetz invited me to attend the seminars he directed at the Science Tower of Mexico's National Autonomous University, which were very fruitful for me.

On a trip to Moscow for my political-social activities, a large group of Russian mathematicians invited me to the house of one of them where they told me that, reading the Annals of Mathematics, one day they were surprised to see the inverse problem of Lyapunov's method, which several of them had tried unsuccessfully to solve for years, and which was written by a mathematician from Montevideo (where would that country be?). The old Italian mathematician Sansone, who also dealt with differential equations, invited me several times to the Centro Matematico Estivo, which the Unione Matematica Italiana organised in the summers in a villa he owned, on the shores of Lake Como (a wonderful landscape!), where they held talks by invited mathematicians from Italy and from other countries, including Halmos, where I presented my results for several years and learned about other ongoing research.

Another activity that I developed for years was that of reviewing for Mathematical Reviews and its German counterpart: they sent me copies of publications so that I could prepare their reviews. In particular, as I understood mathematical Russian reasonably well, it was not at all difficult for me to review articles in journals in that language for which reviewers were not often found, while also receiving fresh information on new research. I was thus up to date on cutting-edge problems that fed my own new discoveries.

When Schäffer returned from Switzerland, where he had received his Ph.D. which showed that if the second term of a linear differential equation was bounded, there was also a bounded solution; and he told me that surely there was a vast "generalisation of that result." I agreed, and we resolved to "get our teeth into it" by working collaboratively. My "nose" had not deceived me, and relatively soon we published several papers with successive generalisations in the same journal. We then resolved to "grab the bull by the horns" and undertake a sweeping generalisation of Perron's theorem. The independent variable of the equations was still the real variable t, but the dependent variable became itself a measurable function in a Banach space $X$, that is, the dependent variable "moved" in a function space.

In order to make the following concepts clear, for at least for some of you, I am forced to introduce some mathematical technicalities, although not totally precise. We will consider two types of linear differential equations, the homogeneous equation

$\large\frac{dx}{dt}\normalsize + Ay = 0$ (1)

and non-homogeneous one
$\large\frac{dx}{dt}\normalsize + Ax = f(x)$ (2)

where $A$ is a linear operator acting on $X$ and $f$ a measurable function on $X$. We will say that the subspace $Y$ of $X$ induces a "dichotomy" of the solutions of (1) if when the initial conditions of a solution belong to $Y$, the solution is stable, and unstable otherwise. We will say that the pair $(B, D)$ - where $B$ and $D$ are two function spaces in $X$ - is "admissible" for (2) if for each $f$ belonging to $B$ there exists a solution of (2) belonging to $D$ (recall the theorem of Perron). The main results of the investigation can then be summarised in the following theorem: "If there is an admissible pair for the non-homogeneous equation (2), then the homogeneous equation (1) has a dichotomy, and vice versa." The results of this research were published in a 400-page book published in the United States that was translated into Russian. Later, I was invited to present them at an international conference on function spaces that was held in Israel.
In 1955, the Communist Party of Uruguay went through a serious crisis: its General Secretary was expelled by the Central Committee (C.C.) that I was part of, and replaced by Rodney Arismendi, an old friend of mine, whose report was stirring. The idea was making its way that the Uruguayan revolution could not yet be socialist. A first step on this path of "Advanced Democracy" was in 1962 by the Left Liberation Front (FIDEL) chaired by Luis Pedro Bonavita, a small rural producer, former member of one of the bourgeois parties.

At that meeting of the Central Committee to which I alluded, I became a member of the Executive Committee of the Communist Party. In 1963-1972 I was elected national deputy on FIDEL lists. The parliamentary task, very absorbing, prevented me from any academic activity, except the simple dictation of classes in the Faculty. The political situation in the country began to become tense, there were violent repressions, students murdered in the streets; the University, of which my colleague and friend Oscar Maggiolo was Rector, was severely harassed.

At the same time, in 1971 the creation of a new political alliance matured, the Broad Front (FA), today the third national political force, and with the possibility of contesting the presidency of the Republic in the 1999 elections to the two traditional parties. It is made up of socialists, communists, fidelistas, Christian Democrats, anarchists and other leftist political groups, and some new offshoots of the traditional "white" and "colorado" bourgeois parties. To preside over it, General Líber Seregni was elected and with him dozens of democratic military arrived at the Broad Front.

The right-wing military coup of June 1973 was being prepared. In the Faculty of Engineering, where another great friend, engineer Julio Ricaldoni, was Dean, there was an explosive accident that was attributed to left-wing students, the University was involved, almost all the Deans were imprisoned, including Ricaldoni, for several years.

Many prominent intellectuals and militants were killed; others were able to go into exile in countries of America and Europe. The Mexican embassy was home to many refugees who later lived in the country for many years and profess a deep love for Mexico.

Most of the mathematicians went into exile. I went into hiding for a few years. Lewowicz - before he went into exile - took me in to shelter in his house for several nights. For a time I was the secretary of the Communist Party of Uruguay in the national territory. My son, who was very sick, died at the beginning of 1973; my daughter went into exile with her family in Brazil, where they lived until the end of the dictatorship. Martha's daughter, whom I love as if she were mine, went into exile in Mexico with her two little children.

I was imprisoned in October 1975. On the first day, while I was standing and with my hands and ankles tied, a soldier grabbed me by the shoulders and moved me roughly, I fell and fractured the neck of my right femur, despite which the standing continued until they were convinced that it was impossible for me to stand up and they put me to bed on a wire mattress where I spent a month without assistance until a doctor ordered me to be taken to the military hospital for an X-ray. From there, with the help of a rudimentary cane, the organism itself was in charge of welding the fracture, until today.

I was imprisoned for nine and a half years in two barracks and a prison known as Penal de Libertad; the sarcastic name is due to the fact that it is located in the vicinity of a town with that name. The prison cells housed two prisoners, with minimal possibilities of communication with the rest, except in some collective tasks of daily life, in the so-called "recesses" of an hour in which those who could did sports and those who could not, as it was in my case, we walked talking with another prisoner; and family visits every fortnight, also lasting one hour. As can be seen, human relations were almost exclusively limited to the cell.

In it one could read books from the good library that had been formed with the donations made by the relatives of the prisoners when they allowed it. Of course, excluding political books ... and mathematics. Who knows what mysterious messages those strange and incomprehensible signs could contain! There was some flaw in that provision and it happened to me, precisely. Once, I asked the library for a book whose title was confusing; the book arrived, it did not have covers but a "homemade" cover made with thick paper. I began to read it and its style was somehow familiar to me; indeed, a few pages later the mystery was clarified in a footnote: it was a Soviet edition translated into Spanish of a text by the famous English mathematician Turing in which he explained the "mechanism" of the so-called machine that bears his name and which has such theoretical importance for branches of modern mathematics. I studied it conscientiously, with a mischievous smile that never left my lips. Years later, an excellent theatrical version of Turing's life was performed in Uruguay for several years. The library of the prison contained a great wealth of literary works, among them Mexican, that I read during those long years. I especially remember "The Death of Artemio Cruz" by Fuentes.

There were several cellmates, communists and tupamaros [members of a Marxist-Leninist urban guerrilla group], with whom we freely conversed on the most varied topics. A communist paper factory worker accompanied me for years and we became great friends; very intelligent and restless, we talked about the most diverse topics. Even remembering the teachings of the Faculty, I was able to give him short courses in physics, chemistry, etc., which he absorbed with passion. In other cells, young mathematicians such as Markarian and Accinelli were imprisoned, whom we only saw at breaks and in some collective task; taking some risks, we did some mathematical homework like the one entitled "Is it true that two plus two always equals four?", which could interest and intrigue non-mathematical prisoners.

To all this, Martha, my wife, had also fallen prisoner, was tortured and confined in a women's prison, installed in what had been a monastery. She was there for three years until late in 1979; she was able to recover our apartment, which had been occupied and looted by the military. She managed to live giving private classes in philosophy and came to visit me at the Prison, overcoming the endless inconveniences and demands placed on her. While she was in prison we corresponded sporadically because our letters were not always delivered. So, she would visit me and take care of my younger sister.

Exile in other American and European countries was the destiny, also difficult, for many other compatriots. In the case of mathematicians and other professional scientists, it was easier for them to be recognised for their abilities. For many others, getting jobs and work cost a lot. One of my main tasks was to develop the movement of solidarity and struggle against the Uruguayan military dictatorship. In particular, I must express my profound gratitude for the wide spread that the demand for my freedom had in America and Europe; here in Mexico the activity and strength of solidarity was notable, particularly among mathematicians. This solidarity had a peculiar form of expression in that more than ten universities - among them that of Puebla - awarded me the title of Doctor Honoris Causa.

International solidarity and the democratic struggle of the Uruguayan people themselves were corroding the foundations of the dictatorship. At the end of 1980, the military themselves called a plebiscite to reform the Constitution to their liking, and they lost. At the end of 1983, all the democratic parties and the social forces called for a vast demonstration against the regime presided over by a podium with well-known figures from all of them, including my wife. The audience was probably the largest ever seen in Montevideo: the place was a large park and the photograph taken from above shows a compact crowd unfolding along the winding curve of a wide avenue. Those photos were published by the conservative newspaper "El País" which was received in the prison library; the companions who worked there spread the event to all the prisoners, including the red colour of Martha's dress.

On 3 March 1984, I was released. From then on and for months, my house was invaded by hundreds of friends who came to greet me. Also in March, Seregni was released, whom we met at his home, in the midst of a large spontaneous street demonstration. That day an Italian delegation in solidarity with me had arrived in Montevideo, one of its members commented that it reminded him of the fall of fascism in his country.

I participated with Martha in many public demonstrations that demanded the release of those who were still imprisoned, with exciting reunions with old friends, some who had been imprisoned and other active opponents of the dictatorship.

The members of the Frente Amplio participated in the so-called Programmatic Agreement, in which representatives of the democratic political parties prepared an advanced proposal for the new government. Unfortunately, this form of democracy, almost direct, was minimally applied by the democratically elected government in 1985.

In that same year, we attended with the ex-Dean Ricaldoni - who had been released before me - and with other professors, a student assembly in the Faculty of Architecture: as they had not lived in a democratic University, they it was not easy for them to understand the criteria of democratic participation in its leadership.

In the summer of 1984-85 I was also touched by a meeting with Uruguayan mathematicians and other scientists, under the banner of UNESCO that sponsored it, who came to Montevideo from exile for a few days to discuss the Program for the Development of Basic Sciences (PEDECIBA). The objective was to rebuild science in Uruguay and promote the return of those who were outside. Personally, it was the reunion with colleagues, disciples and friends and the resumption of academic work. The world-renowned Uruguayan biologist Dr Caldeyro-Barcia was elected as its president; I was part of the first Board of Directors. PEDECIBA is currently financed by the State and by the United Nations (UNDP) and is a very important piece of the Uruguayan scientific research team.

With the democratic return to the University in 1985, Luis Abete was elected as Dean of the Faculty, his excellent colleague Agustín Cisa and I as members of the Council of our area of studies. The rectification of the university conduct imposed by the dictatorship was not easy at all and encountered many obstacles from other members from the previous period; the prevailing climate was not yet the most suitable. I then proposed to hold interviews with each of these "professors" to agree with them on the relatively interesting books that each one should study in order to be able to teach classes, and so it was done. The result was brilliant: almost all of them resigned and were replaced by better ones. Simultaneously, we worked with gifted young students who are now mathematicians and researchers and received their doctorates in Brazil: Markarian, the Paternain brothers, Catsigeras, Enrich, Accinelli and others, not counting returnees from exile.

After such tough tests, today the Uruguayan mathematical school is strong. Dynamical Systems and Chaos are studied at the Institute of Mathematics and Statistics Rafael Laguardia (IMERL). In another important venue, the Mathematics Centre of the Faculty of Sciences, other branches are studied.

At the same time, I have integrated (first with Ricaldoni and Rafael Guarga, current Dean, and then with María Simon and Arturo Lezama) the Scientific Research Commission of the Faculty of Engineering.

Already in 1973, I had foreseen that it would be impossible to resume my mathematical research after the dictatorship, having spent almost thirty years without doing so. I really liked the history of mathematics and science in general and I was very interested in building a small library on these subjects. Only much later was I able to undertake this project and the results were not numerous, although they were of some interest. A well-known French philosopher invited me to collaborate on a book that he edits, on science and dialectics, in which distinguished scientists from various branches write. I accepted the proposal; the book is expected to appear this year. The contact with Séve has enriched me a lot in philosophical matters.

Finally. I apologise for the length of this talk and I hope that it has been useful to you as an experience of the formation of a school, in this case the Mathematical School of Uruguay.

Montevideo, February 1998

Last Updated February 2023