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Beppo Levi by Alessandro Terracini
We give below an English translation of the paper Commemorazione del Corrispondente Beppo Levi which appeared in the Proceedings of the Accademia Nazionale dei Lincei in 1963. The precise reference is:
Alessandro Terracini, Commemoration of Correspondent Beppo Levi, Proceedings of the Accademia Nazionale dei Lincei. Class of Physical, Mathematical and Natural Sciences (8) 34 (5) (1963), 590-606.
Alessandro Terracini, Commemoration of Correspondent Beppo Levi, Proceedings of the Accademia Nazionale dei Lincei. Class of Physical, Mathematical and Natural Sciences (8) 34 (5) (1963), 590-606.
Commemoration of Correspondent Beppo Levi by Alessandro Terracini.
In 1956, when the Accademia dei Lincei awarded Beppo Levi the Antonio Feltrinelli Prize for Mathematics reserved for Italian citizens, it was expressly stated that this was intended as a deserved recognition of the value of the results he had achieved, and to constitute the crowning achievement of a long and noble career. Now it can be said that words of this kind have rarely been used with greater justification. Beppo Levi's career was truly a long and noble one; he honoured mathematics throughout his long life, at all times, whether - in years now very distant - he carried out his activity as a professor in secondary schools, or in his university teaching, first as an assistant, and then as a professor in the universities of Turin, and then of Cagliari, and then of Parma, and then of Bologna, and then again - when he was removed from teaching in Italian universities - in the Universidad del Litoral, in Rosario, Argentina, where he remained in the last period of his life, continuing to honour mathematics in his teaching and in the work of spreading mathematical thought, which he carried out as director of the Mathematical Institute of that university, and in conversations with the people with whom he came into contact.
As for the first part of the motivation I mentioned, where the importance of the results achieved by Beppo Levi is alluded to, although it is a perfectly well-founded statement, I would be inclined to think that it was not the most welcome to Levi. Mathematics, according to him - and he has expressly stated this more than once, for example in his presentation of the journal "Mathematicae Notae" - is essentially a way of thinking; although it is a science in the ordinary sense of the positive sciences, that is, a system of knowledge, and although it interests those who apply it as a powerful instrument, it is essentially a way of thinking; it is a philosophy. On other occasions too he has maintained that the essence of mathematics is not constituted by numbers, but by thought; and in the sciences that apply mathematics what is characteristic is not, according to him, the possibility of introducing numbers or a certain formalism, but it is thought: the term "mathematics" does not correspond to the empirical or observational formulas that can be found in physics books. Even in statistical applications to physical theories, the dominant problem is not constituted by numbers, but by the search for hypotheses that suitably characterise the vague word "probability".
It is therefore clear that for him it was not generally the single result that was in view, but the fact that it was a product of mathematical thought. And, if in reviewing the scientific work of Beppo Levi, we must necessarily dwell, even briefly, on results obtained by him in some of the various fields that he studied, we wish to explicitly state that for him it was not these results that had value that were calibrated, but rather the circumstance of being the fruits of mathematical thought.
It goes without saying that the point of view outlined was also reflected in teaching: certainly to Beppo Levi apply the words that were written a few years ago for a completely different purpose: "the value of the teacher, even more than in the truth he discovers, lies in the training he offers to discovery".
Beppo Levi was born in Turin on 14 May 1875 (son of the lawyer Giulio Giacomo Levi, and of Mrs Mentina Levi Pugliese) fourth of ten brothers. Two of them died in the First World War: Decio, engineer and major in the army, fell on the field near Gorizia on 15 August 1916, and Eugenio Elia Levi, eminent mathematician, professor at the University of Genoa, war volunteer, captain of the engineers, fell in action, during the retreat of Caporetto, on 28 October of that same year.
Beppo Levi studied mathematics at the University of Turin, following the courses of professors to whom he remained attached for a long time: Corrado Segre, Giuseppe Peano, Vito Volterra, Enrico D'Ovidio, some of whom he felt the influence of for a long time.
He graduated in 1896 with a thesis prepared under the guidance of Corrado Segre, which marked his initial orientation towards algebraic geometry, and among other things gave rise to the extensive Memoir 'On the variety of the chords of an algebraic curve' published two years later. In the same year 1896 he became, in Turin, assistant to the chair of Projective and Descriptive Geometry, then held by Luigi Berzolari. And later - after a brief period as a professor in secondary schools which I have already mentioned - he was professor of Projective and Descriptive Geometry at the University of Cagliari from 1906 to 1910; of Algebraic Analysis at the University of Parma from 1910, and then of Elements of the Theory of Functions at the University of Bologna from 1928; and of this same University - let us say immediately to conclude with his Italian university career - from 1951, now retired, he was professor emeritus. But from 1939 to his death, which occurred on 28 August 1961, the Argentine period of his life took place. Despite the pain he could not help but feel at being removed from his Italian chair - a pain that perhaps must have been more poignant in him, sensitive to a certain form of nationalism - he arrived in Argentina full of enthusiasm for his new duties, and threw himself into it with all his heart, organising the Mathematical Institute of the University of the Litoral, of which he assumed the position of director, and holding courses, separate from the official ones, on the most varied subjects, until later he also took on regular teaching, such as that of Analytical Geometry and Infinitesimal Calculus II, and then that of Rational Mechanics. In the meantime he was undertaking a series of publications for that institute, such as the "Publicaciones", the "monografias", and above all the journal "Mathematicae Notae", with which he found a way to extend his beneficial action even beyond those who were close to him, to Rosario, throughout Argentina, throughout the Hispanic American world, and indeed one might say everywhere. This journal, he wrote in introducing it, "is addressed first and foremost to the students of the Faculty to which the Institute is connected, but it hopes to find some sympathy even beyond, among young people who are approaching this very singular branch of knowledge for the first time": singular (he explains) because of the position of mathematics as a way of thinking, and here he continues with the words that I have already had occasion to recall before. And he continues by saying that the Journal aims to arouse interest in mathematical thought, and to publish simple articles, without claiming to be research in higher level fields, and often didactic articles.
The thought of this Journal, for many years, dominated him during many hours of his life. So in the period in which the existence of the Journal was a source of concern to him in the general situation of the country where he lived, and in the repercussions in the university field. So also, in another difficult period, when he interrupted the affectionate, very caring, continuous assistance that he gave to a person dear to him, seriously ill, only to return in thought from Torre Pellice - where he had come in those months - to Rosario, where his Journal had to continue, and continued.
As I have already mentioned, Beppo Levi presented himself for graduation with a dissertation discussed with Corrado Segre. It is to be assumed that it concerned several different topics, albeit connected to each other.
One certainly concerns the variety of chords of an algebraic curve, as is explicitly stated in the homonymous Memoir published a couple of years later.
On the other hand, it is to be remembered that at that time Corrado Segre was studying the decomposition of singular points of algebraic surfaces, extending to surfaces (not an easy extension) what Noether had done for the singularities of algebraic plane curves. On that topic Segre wrote a Memoir, published in the "Annali di Matematica", which soon became a classic, at the end of which he adds his thanks to Beppo Levi, and mentions his graduation thesis dedicated - he says - to the singularities of skew (hyperspatial) algebraic curves. The connection between the two topics is clear: the lines that are limit positions of chords whose two points of support have come to coincide in the same point are notoriously called improper chords of an algebraic line belonging, let us suppose, to ordinary space. If is simple, everything is obvious: not so when is multiple. This is precisely the case that Beppo Levi explores in depth.
But in the meantime, having fixed his attention on multiple points in space inevitably led Levi to tackle a problem considered to be fundamental.
The starting point for the formulation of this problem is provided by a theorem on algebraic plane curves, which is actually fundamental in the theory of these curves and which states the possibility of obtaining every algebraic plane curve without multiple components as a birational transform of another having only ordinary multiple points: one could also say that the given curve is birationally equivalent to a curve in space without singularities.
The fundamental role played by this theorem in the geometry on an algebraic curve, in that it allows one to reduce oneself to the study of only plane algebraic curves with ordinary singularities, could not fail to make an analogous theorem for surfaces appear as the substantial presupposition for the study of geometry on an algebraic surface. The question appeared in its fundamental character precisely in those years, in which Castelnuovo and Enriques carried out their work precisely to construct geometry on a surface with those geometric methods that were powerfully asserting themselves in the so-called Italian geometric school.
The problem is seen and stated with complete clarity by Corrado Segre in his Memoir of the "Annals": he begins by defining what is to be understood as an ordinary singularity of an algebraic surface of ordinary space.
Then the fundamental problem that arises is that of birationally transforming any algebraic surface (without multiple parts) into another whose singularities are all ordinary. It is a question of demonstrating that this is possible.
Segre observes that the validity of this proposition had been asserted by Noether, but that in Noether there is no attempt at justification. After Noether, there was an attempt at proof by Pasquale Del Pezzo, but Segre shows that the proof attempted by Del Pezzo is not conclusive. After these critical considerations, conducted particularly in the light of the notion of composition of a superficial singularity, there is no systematic part in Segre relating to the proof in question.
This systematic part was carried out, in those years, by Beppo Levi, who in various publications that begin in 1897 and continue for some time, arrived at the theorem in question. It should also be added that no objection was raised against Levi's proof, although the ground on which it is developed is to be considered mined; so that for example the proof of a result, which was in the same order of ideas, due to Kobb, was recognised as invalid, as Levi himself contributed to prove.
For a long time Levi was considered the conqueror of the question. No objections were raised against his proof: it is cited as definitive by Picard, and also by those who sought proofs that proceeded in different ways, such as Severi or Albanese.
But, even in mathematics, there are things that are not definitive. And so, on the one hand, a few decades later, came the criticisms of Oscar Zariski who raised objections to the procedure used by Levi because he did not appear to be fully convinced that the alternating use of two certain types of transformations to gradually reduce singularities could not at some point generate difficulties.
On the other hand, with the passage of time, the very formulation of the question changed, as algebraic geometry prevailed with respect to any body, rather than with respect to the ordinary complex body, which was the only one considered during the first period.
And indeed I believe that just recently a result of notable generality was announced, established by the Japanese Hironaka.
Concerning Beppo Levi's work in this field, something must be added. First of all, many years later, in 1927, in a short piece contained in the second of the two volumes intended by the Kazan Physical-Mathematical Society to celebrate the centenary of Lobachevsky's discovery, he returned to the question, publishing what he intended to be the introduction to the generalisation to varieties with more dimensions. However, it must be said that - as far as I know - he never returned to the question again, except for the exception I am about to mention. To find a piece of Levi's work on the question, we must go back to one of his very last works, published in the "Mathematicae Notae" in 1957, a work written partly in Spanish and for a small part in Italian, which is read with emotion, as if it were a farewell, and ends with words of farewell to distant colleagues and friends, and precisely for this purpose he uses his mother tongue in this part. In his concluding words, he recognises the arduous and hidden nature of the whole question, and of the means he had used to resolve it, and he also recognises the obscurity that some explanatory notes he had drawn up around 1900 for Corrado Segre's personal use presented to him.
A separate mention, in the geometric production of that first period, is deserved by the works on the foundations of geometry, begun with the powerful Memoir on the Foundations of Projective Metrics.
But in the meantime, towards the beginning of the new century, Levi was moving away from algebraic geometry to take a position in the theory of functions and in that of sets. A Note of his, published in the "Rendiconti dell'Istituto Lombardo", dates back to 1902, On the Theory of Aggregates, which later writers on Set Theory have cited in relation to the so-called Zermelo Axiom. As early as 1890, Peano, in his Memoir on the Integrability of Differential Equations published in the "Mathematische Annalen", had explicitly stated the impossibility of applying an arbitrary law infinitely many times to extract a single element from a set; that is, to put it briefly, he had declared the inadmissibility of infinite arbitrary choices. As is well known, in 1904 Zermelo - with the aim of demonstrating that the set of real numbers, and indeed any set, can be ordered - adopted a point of view completely opposite to that of Peano, conferring validity on the infinite choices by means of an axiom constructed ad hoc; an axiom according to which, given an arbitrary set, there exists some correspondence law which associates its own element to each of its subsets. We will return to Zermelo's principle in a moment; here we only want to recall that, in his Note of 1902, Beppo Levi - very far, then and later, from the idea of being able to admit it, and therefore of applying it - mentioned it, avant la lettre, with words that those later used by Zermelo himself very closely resemble.
Levi naturally returned to Zermelo's principle more than once. With regard to that principle, he highlighted the three parties into which mathematicians are divided: first of all, those who - following Peano - decisively reject any reasoning in which one resorts to infinite arbitrary choices: Levi, however, expressed the opinion that the followers of this tendency have gone, in the rigorous application of the prohibition, even further, arriving at denying the validity not only of infinite choices, but also of infinite operations, perfectly defined, whose results depend in a chain on each other, and therefore are not predictable a priori. The mathematicians of the second faction accept Zermelo's principle, and consider absolutely admissible a reasoning in which infinite choices intervene: not only in the case in which the infinite elements chosen arbitrarily correspond to the succession of natural numbers, but in a broader sense.
Then there is an intermediate party, in which Beppo Levi also placed himself, which from more to less is willing to adopt a pragmatic attitude, accepting or rejecting the infinite choices according to what common sense and the intuition of mathematical truth advise. In a more concrete way (and Beppo Levi expressed himself in these terms in a communication to the eighth Congress of Philosophy held in Rome in 1933) a criterion to characterise the reasonings in which the infinite choices are admissible can be found by accepting the reasonings whose conclusion remains unchanged when the conditions of time, place, and thinking subject are modified. The rather conciliatory position assumed by Levi in the words that I have somehow reproduced - and one could also look for examples in individual works of Levi - must not however make us forget his very firm attitude against an indiscriminate acceptance of the Zermelo principle.
Already when he wrote his Reflections on some points of the theory of sets and functions, published in 1918 in the volume of Works offered to Enrico D'Ovidio, Beppo Levi started from the premise that every mathematical reasoning presupposes, as the domain of its considerations, a set (or an infinite number of sets) within which it postulates the possibility of choosing an arbitrary element, as a first or irreducible act of thought: that set is what in later works (for example in the Memoir entitled by it published in the "Fundamenta mathematicae" of 1934) Levi would call the deductive domain. In Zermelo's principle, on the other hand, one goes far beyond the set in which one should operate, which takes away all legitimacy from that principle. Levi does not hesitate to affirm that the admission of Zermelo's principle contradicts the essential nature of mathematical analysis, and that therefore it must be rejected as devoid of any sense.
The notion of deductive domain has come to assume a fundamental importance in Levi, and it is worth dwelling on it for a moment, using some of the expositions that he subsequently dedicated to it, such as the Memoir of the "Fundamenta mathematicae", or the exposition in Spanish of 1942, one of the first fruits of his stay in Rosario. It is worthwhile, as I was saying, even if one does not always have the impression of being able to fully understand all of Beppo Levi's thought; and one confesses this all the more easily, since one thinks that even some reviewers of those works - and they are often well-known names - openly manifest a certain lack of certainty of having fully understood Levi's thought.
Let us therefore return to the words that I said when I spoke of the notion of deductive domain, and of the first set that is at its base. This first set is a primitive idea, and nothing more. Thus, arithmetic rests on the deductive domain of natural numbers. In it, it makes sense to choose one or more elements, only by designating them with a name. According to Levi, even real numbers constitute a primitive idea, which is formed in our mind at a certain moment of its development. The notion of a single real number can be explained as an example by means of the primitive set of natural numbers alone, but this explanation of an example must not be confused with the general idea of a real number, and of the class constituted by them. If we assume as primitive sets those of natural numbers and real numbers, we can construct Analysis, in its classical part. If we then assume three primitive sets, that is, in addition to the two previous ones - also that of functions, then we can also develop the modern part of Analysis.
When a deductive domain differs from another only because another is added to the primitive sets of the second, Levi calls the first an expansion of the second, and under certain conditions he speaks of a natural expansion. Indeed, Levi has enunciated a certain principle, which he calls the principle of approximation, which presides over such natural expansions.
Levi found himself starting his activity in the field of Analysis precisely in the years in which the notion of integral had been, one might say, revolutionised by the new notion of Lebesgue integral. Beppo Levi was among the first to enter the new order of ideas. In the meantime, in a series of Lincean Notes of 1906, he contributed to demonstrating some fundamental points of the new theory, even if Lebesgue did not show that he completely appreciated his intervention. And then Levi was too shrewd not to immediately appreciate the usefulness that the Lebesgue integral can offer in certain circumstances: he did not hesitate to use it - at least initially - in his memorable research, which will be discussed shortly, on the Dirichlet problem. Of considerable importance is the question, which preoccupied Beppo Levi for many years, of freeing the notion of the Lebesgue integral from the intervention of the concept of measure of a set; a concern that is well understood, if only from a didactic point of view. To answer the question, Beppo Levi intervened in an essential way with three large works: two of them in Italian, and bearing the same title: On the definition of the integral, the first published in the "Annals of Mathematics" of 1923, and the second in 1936 in the volume of Mathematical Works dedicated to Luigi Berzolari. The third work is an exposition in Spanish that dates back to the Argentine period.
In the 1923 Memoir, Levi had given a new direct definition of the integral, which then in fact coincided with the Lebesgue integral in the case of measurable limited functions: Levi had however expressed doubts about the possibility of imagining the extension of his definition also to the case of non-measurable functions. Indeed, shortly after that first Memoir by Beppo Levi, Vitali had intervened, demonstrating that the condition of measurability is superfluous. But Levi did not consider himself completely satisfied since Vitali was still using ideas that were within the scope of Lebesgue's theory, and in particular he made use of the concept of measure. Thus arose the second Memoir by Levi, intended to arrive at the main properties of the integral directly, so that one could instead arrive at the theory of measure as an application of the theory of the integral. This is how in the new Memoir Levi obtains both the theorem on the integral of the limit function of an increasing succession of functions, and the so-called Fubini theorem.
A special mention must be made of Beppo Levi's studies on the Dirichlet problem, that is, the so-called first boundary value problem. It is worth remembering that it concerns the search for a harmonic function at the points of a region, which assumes assigned values on its boundary. The uniqueness of the solution is easily proven. Things are different for the proof of the existence of a solution. It is well known that Riemann had thought he could affirm this existence, on the basis of the so-called Dirichlet principle, according to which the solution is necessarily that function which - among all the functions assuming the assigned values on the boundary - minimises the so-called Dirichlet integral, that is, the integral , extended to the domain, of the sum of the squares of the first partial derivatives of the function u, with respect to each of the two independent variables . But, as is also well known, Weierstrass had removed any illusion about the validity of the alleged existential demonstration based on Dirichlet's principle, since - while the existence of a lower limit for the values assumed by the integral considered is evident, - it is not permissible to affirm without further ado that the lower limit is a minimum, that is, to conclude the existence of a function in correspondence with which that integral actually assumes the value . Around 1900 Hilbert had drawn the attention of mathematicians to Dirichlet's principle, in view of the possibility of transforming it into a variational method suitable for existential demonstration; and indeed Hilbert had achieved this goal in particular cases. The basis of the demonstration is the consideration of a minimising sequence such that
.
If the inversion of the limit sign with the integral sign were permissible, it would be
,
that is, it would be concluded that
Approximately the same period also includes two notable contributions by Beppo Levi to number theory. One is a group of four Notes, published between 1906 and 1908 in the "Proceedings of the Academy of Sciences of Turin" and summarised in a communication to the International Congress of Mathematicians held in Rome in 1908.
They concern the third degree determined equation (let's say of genus one, with reference to an obvious geometric interpretation) pursuing, independently, an aim also proposed by Hurwitz. Several authors, starting with Fermat, had indicated geometrically spontaneous ways to deduce a rational solution from one, or two known solutions, let's say briefly to deduce a rational point of the proposed cubic, from one or two known rational points. Levi has made a systematic treatment of the subject, based on the concept he introduced of a finite configuration of rational points, obtained when the procedures mentioned are applied both to some starting rational points and to those subsequently deduced by them through the same procedures. Levi's other contribution to the Theory of Numbers is represented by a Memoir of the "Circolo matematico di Palermo" which completes in an essential point a work by Minkowski relating to a system of n linear forms in as many variables: Levi has also demonstrated a result that Minkowski had indicated as presumable, without possessing the proof.
Although I must now give up dwelling further on particular aspects of Beppo Levi's mathematical activity, I would just like to mention two of his expositions, one very short, the other short on Mathematical Logic. The first is the homonymous entry in the Italian Encyclopedia. The second is the text of a short series of lectures held by Levi at the Argentine University of Tucumàn. The title of this last publication would translate into Italian as "Scorrieria nella logica". A little less common is perhaps the Spanish term "correría" corresponding to the Italian "sconceria", so much so that Levi felt obliged to begin with some explanatory words in that term, taking them from the Dictionary of the Academy, which records two meanings: the first of "hostility that people of war practice by extorting and pillaging the country" and the second of "generally short journey, made from various points, with return to the starting point". And Levi clarified that for his lectures the second meaning could be accepted in full, and the first as a somewhat exaggerated hyperbole.
If I may insert a personal memory, I would like to add that those lectures of Levi remind me of some pleasant days in the autumn of 1942 spent in Tucumàn as in a dive - at that time and in that place - into the Italian intellectual world. Beppo Levi had come to Tucumàn to give those lectures, and at the same time my friend Leone Lattes, professor of forensic medicine at the University of Pavia, then resident in Buenos Aires, whom I had brought on behalf of the Argentine Scientific Society to speak about blood groups. My brother Benvenuto and my other friend Renato Treves, now a professor at the University of Milan, were also in Tucumàn as professors. It was truly a small Italy that we had created in those days.
I would not like to pass over in silence four volumes by Beppo Levi, which date back to different moments of his life.
Of the two written in Spanish, one is an exposition of fundamental theorems of existence for systems of partial differential equations where a systematic procedure is assigned, which leads either to ascertain their incompatibility, or to calculate the solutions. This book is a reworking of a course in Higher Analysis held by him in Bologna, and also of a course of lectures held in Rosario in 1941: it bears an affectionate dedication to his brother Eugenio Elia "who was his dearest friend and companion in studies of all kinds". The other, Leyendo a Euclides, is a book, pleasant to read, written with the intention of presenting without pretensions to a non-mathematical public the thoughts of a mathematician occasioned by the reading of Euclid. In Beppo Levi, Euclid's Elements appear to be the fruit of the conceptions of the circle of mathematicians and philosophers who surrounded Socrates rather than the work of an Alexandrian from 300 BC. The confusion that once existed between Euclid and Euclid of Megara, not feared by Beppo Levi, has given rise to more than one criticism of his book, which nevertheless remains highly interesting. The two volumes in Italian, published twenty years apart, together constitute a short course in Analysis. Particularly notable is the first, which dates back to the times of Parma, and develops the algebraic part with great simplicity and originality, operating in any field, and thus anticipating the treatises - which came much later - of Modern Algebra, or Abstract Algebra.
Alongside these four, one could perhaps include a fifth little book, dedicated to children, the Abacus from 1 to 20, today unfortunately almost unobtainable, in which - based on the recognition that numbers constitute the means of the counting operation, and not its result - one is taught, in a first arithmetic initiation, to conceive numbers as the elements of an ordered succession formed by the words that designate them.
I have already spoken about the great activity of Beppo Levi during the Argentine period: in some moments of this period, his protection was also very considerable, even if very often he gave it his own characteristics.
Sometimes he has taken up, in order to spread it among a new public, some of his expositions from previous periods. Sometimes he has availed himself of collaborators. Sometimes he also faced new problems, not without leaving here and there the impression that his voice was no longer the same, as happened for example with the work, in collaboration with Santalò and Celestino De Maria, on the determination of the order of the cone which, in differential projective geometry, generalises the well-known cone of Del Pezzo.
Beppo Levi had an eminently critical spirit, even with regard to himself and what he was writing, so that it also happened that, at the moment in which he exposed a certain point of view, reservations and limitations arose that induced him to superimpose on it perhaps completely opposite points of view that immediately surfaced in his speech. This must be attributed to the circumstance already mentioned that the reader - or the listener - sometimes found himself in difficulty about the conviction of having fully grasped his thought, even if it had presented itself to him in the clearest way.
In whatever period one examines Beppo Levi's productive activity, a common characteristic is his great versatility. He was capable of taking a deep interest in problems of a very varied nature. In this sense he still belongs to the generation of great mathematicians of the past, who had the virtue of dominating many fields of mathematics, without enclosing themselves in that one-sidedness of interest, which can be explained, but which nevertheless does not cease to be a characteristic of modern mathematics.
Against modern mathematics, and not for this reason alone, Beppo Levi was rather severe. In a period of time that is approaching the century - he wrote - the form of mathematical thought has changed significantly in a direction less suited to my own thought.
As for severity, we would be completely wrong to consider it in general a trait of his character, which was instead indulgent and affectionate. In particular, he felt the most affectionate feelings towards his family, even if life's circumstances had led him to live far away from some of them: he felt deeply attached to them, and he constantly turned his thoughts to them - both near and far.
Last Updated March 2025