Mathematicians From Italy

"Matilde Marcolli describes how she came to mathematics influenced by her parents’ involvement in Italian contemporary art. The abstract art of her father and the conceptual art of her mother, together with atonal twentieth-century music, share with mathematics an appreciation of abstract structures. Her own art includes painting, where surrealism allows her to express contrasts inherent in the practice of mathematical research and to explore the inner world of patient, difficult, and painful hard work and also the bullying and “culture of cruelty” within the mathematics community. ... Besides painting, Marcolli is also a writer in many forms, including science fiction, short stories, poetry, and a theater play."

"The general discourse of scientists about science is marred by beliefs of the Ancient Greeks in the kalos kai agathos: that which is beautiful must also be good, and conversely. This leads inevitably to portraits of scientists as cartoonish heroes: the more profound and significant the science, ... In fact what is truly heroic about science is the fact that it does uncover beautiful truths about the universe despite the ugliness and brutality of the human beings involved."

"Violence, bullying, and intimidation exist and are practiced on a daily basis within the mathematical community, and there is a widespread "culture of cruelty" among its practitioners ..."

"It turns out that noncommutative geometry is a very good framework for theories of (modified) gravity coupled to matter. The main idea behind gravity and particle physics models based on noncommutative geometry is that "all forces become gravity" on an noncommutative space. In other words, it is only from the point of view of a slice of the geometry consisting of an ordinary spacetime manifold that we see a difference between gravity and the other forces, while from the point of view of the overall (noncommutative) geometry they are all seen together as gravity. As we will see, the main construction is not unlike the idea of "extra dimensions" many people are familiar with from string theory, except for the fact that the extra dimensions in these models are not only small, but also noncommutative, while the extended dimensions of spacetime maintain their commutative nature."

"The physicists' approach to the equivalence of Seiberg-Witten and Donaldson theory is based on Witten's interpretation of Donaldson's theory as a twisted supersymmetric Quantum Field Theory ... and on the concept of electro-magnetic duality."

"The study of the geometry of a Galois space Sr,q, i. e. of a projective r-dimensional space over a Galois field of order q = ph. where p, h are positive integers and p is a prime (the characteristic of the field), has recently been pursued and developed along new lines ... In it, both algebraic-geometric and arithmetical methods have been applied, including the use of electronic calculating machines; moreover, some of the problems dealt with are deeply connected with information theory, especially with the construction of q-ary error-correcting codes. It is actually a chapter of arithmetical geometry, which reduces to the investigation of certain questions on congruences mod p in the particular case when h = 1."

"He and Benedetto Croce actively attacked the regime from their seats in the Senate. After 1930 Volterra was dismissed from the University and stripped of his membership in all Italian scientific societies. The same thing later happened to Levi-Civita. To the honor of the Santa Sede, he and Volterra (both of whom were Jews) were soon thereafter appointed by Pope Pius XI to his Pontifical Academy."

"On my way home in May 1932, when I stopped in Rome to see Vito Volterra and explained my 92 Andre Weil formula to him, he jumped up out of his chair and ran to the back of the apartment, crying to his wife: "Virginia! Virginia! Il signor Weil ha dimostrato un gran bel teorema!" ("Mr. Weil has proved a very beautiful theorem!")"

"I did not hesitate at the Congress of Mathematicians at Paris to call the nineteenth century the century of the theory of functions, as the eighteenth century might have been called that of infinitesimal calculus."

"Si vous demandez à tout mathématicien si dans son esprit il fait une distinction les théories de l'élasticité et celles de l'électrodynamique, il vous dira qu'il n'en fait pas, car les types de équations différentielles qu'il rencontre, et les méthodes qu'il doit employer pour résoudre les problèmes qui se présentent, sont tout à les mêmes dans le deux cas. (If you ask any mathematician if in his mind he makes a distinction between the theories of elasticity and those of electrodynamics, he will tell you that he does not, because the types of differential equations he encounters, and the methods which he must employ to solve the problems which arise, are all the same in the two cases.)"

"C varieties in four-space were first investigated by Segre, in two memoirs ... which are still classic, and in which he gave a generation of those having more than six nodes, especially the one with ten nodes, while he also considered varieties containing a plane, and gave some of their properties."

"Non vi sarebbe quindi da stupirsi se le geometrie di Galois venissero and avere in futuro applicazioni anche al campo della fisica, da cui attualmente sembrano molto lontane esce anzi tali spazi finiti portassero alla costruzione di schemi a modelli dove i fenomeni fisici trovassero interpretazioni matematiche più semplici di quelle consuete. (It would not be much of a surprise if Galois geometry in the future came to have applications in the field of the physics, from which these finite geometries are currently far removed. These finite geometries might lead to the construction of models in which physical phenomena have simpler mathematical interpretations than the models now used. — modified from the original translation by Tallini)"

"A nonsingular cubic surface F can be rationally represented upon a plane α if and only if F contains a rational point."

"Bombieri is one of the guardians of the Riemann Hypothesis and is based at the prestigious Institute for Advanced Study in Princeton, once home to Einstein and Gödel. He is very softly spoken, but mathematicians always listen carefully to anything he has to say."

"When things get too complicated, it sometimes makes sense to stop and wonder: Have I asked the right question?"

"Probabilistic reasoning—always to be understood as subjective—merely stems from our being uncertain about something."

"My thesis, paradoxically, and a little provocatively, but nonetheless genuinely, is simply this : PROBABILITY DOES NOT EXIST.The abandonment of superstitious beliefs about the existence of Phlogiston, the Cosmic Ether, Absolute Space and Time, ... , or Fairies and Witches, was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs."

"The most notable difference (of the American character) lies in the psychology of work. In the Orient one works to live; in Europe one works to consume; in America one works to work. These are the three stages of a progressive evolution."

"A fairly large part, if not, indeed, the very nucleus, of the Fascist movement has been built up of ex-Socialists who abandoned their party because of, or in consequence of, the war. This observation is particularly true of the younger element in the Socialistic party, including young men of a practical turn, often restless in temperament, who had rallied to the Socialist party not so much because of its positive economic program, as because of its negative program of protest against the aimless individualism of the Liberal regime, and who found in Fascism the means for effectuating their desire to take a part and to reconstruct."

"Euclid … manages to obtain a rigorous proof without ever dealing with infinity, by reducing the problem [of the infinitude of primes] to the study of finite numbers. This is exactly what contemporary mathematical analysis does."

"The age-long history of thinking on gravitation, too, was erased from the collective consciousness, and that force somehow became the serendipitous child of Newton's genius. The new attitude is well illustrated by the anecdote of the apple, a legend spread by Voltaire, one of the most active and vehement erasers of the past. … The need to build the myth of an ex nihilo creation of modern science gave rise to much impassioned rhetoric."

"The oft-heard comment that Leonardo [da Vinci]'s genius managed to transcend the culture of his time is amply justified. But his was not a science-fiction voyage into the future as much as a plunge into the past."

"From semantics to shipbuilding, from dream theory to propositional logic, any specialist … is invariably astonished to discover that modern knowledge was foreshadowed at the time. … Should we not replace these foreshadowings by the study of the influences of Hellenistic thought on modern thought?"

"Many scholars have felt that the Heronian passage [on a pipe-organ moved by an anemourion-like wheel] can be disregarded because it is not confirmed by other writings. Heron presumably mentioned the anemourion in a moment of distraction, forgetting that it had not been invented yet. We know that he was given to such lapses."

"Today Eratosthenes' method [of calculating the circumference of the earth] seems almost banal … yet it is inaccessible to prescientific civilizations, and in all of Antiquity not a single Latin author succeeded in stating it coherently."

"Unfortunately, the optimistic view that "classical civilization" handed down certain fundamental works that managed to include the knowledge contained in the lost writings has proved groundless. In fact, in the face of a general regression in the level of civilization, it's never the best works that will be saved through an automatic process of natural selection."

"About Archimedes one remembers that he did strange things: he ran around naked shouting Heureka!, plunged crowns into water, drew geometric figures as he was about to be killed, and so on. … One ends up forgetting he was a scientist of whom we still have many writings."

"Since UFO stands for "unidentified flying object", the word ufology means approximately "knowledge about unknown flying objects", and is therefore a "science" whose content is void by definition. Similar considerations hold for parapsychology."

"The ancient Greek philosopher, Democritus, propounded an hypothesis of the constitution of matter, and gave the name of atoms to the ultimate unalterable parts of which he imagined all bodies to be constructed. In the 17th century, Gassendi revived this hypothesis, and attempted to develope it, while Newton used it with marked success in his reasonings on physical phenomena; but the first who formed a body of doctrine which would embrace all known facts in the constitution of matter, was Roger Joseph Boscovich, of Italy, who published at Vienna, in 1759, a most important and ingenious work, styled Theoria Philosophiæ Naturalis ad unicam legem virium, in Natura existentium redacta. This is one of the most profound contributions ever made to science; filled with curious and important information, and is well worthy of the attentive perusal of the modern student. In more recent days, the theory of Boscovich has received further confirmation and extension in the researches of Dalton, Joule, Thomson, Faraday, Tyndall, and others."

"But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes. We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line."

"For a long time the celebrated theory of Boscovich was the ideal of physicists. According to his theory, bodies as well as the ether are aggregates of material points, acting together with forces, which are simple functions of their distances. ...When Lord Salisbury says that nature is a mystery, he means... that this simple conception of Boscovich is refuted almost in every branch of science, the Theory of Gases not excepted. The assumption that the gas molecules are aggregates of material points, in the sense of Boscovich, does not agree with the facts."

"The phrase "ahead of his time" is overused. I'm going to use it anyway. I'm not referring to Galileo or Newton. Both were definitely right on time, neither late or early. Gravity, experimentation, measurement, mathematical proofs … all these things were in the air. Galileo, Kepler, Brahe, and Newton were accepted - heralded! - in their own time, because they came up with ideas that scientific community was ready to accept. Not everyone is so fortunate. Roger Joseph Boscovich … speculated that this classical law must break down altogether at the atomic scale, where the forces of attraction are replaced by an oscillation between attractive and repulsive forces. An amazing thought for a scientist in the eighteenth century. Boscovich also struggled with the old action-at-a-distance problem. Being a geometer more than anything else, he came up with the idea of "fields of force" to explain how forces exert control over objects at a distance. But wait, there's more! Boscovich had this other idea, one that was real crazy for the eighteenth century (or perhaps any century). Matter is composed of invisible, indivisible a-toms, he said. Nothing particularly new there. Leucippus, Democritus, Galileo, Newton, and other would have agreed with him. Here's the good part: Boscovich said these particles had no size; that is, they were geometrical points … a point is just a place; it has no dimensions. And here's Boscovich putting forth the proposition that matter is composed of particles that have no dimensions! We found a particle just a couple of decades ago that fits a description. It's called a quark."

"In 1763 a Croatian Jesuit named Roger Joseph Boscovich (1711 - 1787) identified the ultimate implication of this mechanical atomic theory. One of the crucial aspects of Isaac Newton's laws of motion is their predictive capability. If we know how an object is moving at any instant - how fast, and in which direction - and if, furthermore, we know the forces acting on it, we can calculate its future trajectory exactly. This predictability made it possible for astronomers to use Newton's laws of motion and gravity to calculate, for example, when future solar eclipses would happen. Boscovich realized that if all the world is just atoms in motion and collision, then an all-seeing mind "could, from a continuous arc described in an interval of time, no matter how small, by all points of matter, derive the law [that is, a universal map] of forces itself … Now, if the law of forces were known, and the position, velocity and direction of all the points at any given instant, it would be possible for a mind of this type to foresee all the necessary subsequent motions and states, and to predict all the phenomena that necessarily followed from them.""

"Boscovich's ideas exerted a deep influence on the work on the next following generation of physicist ... Our esteem for the purposefulness of Boscovich's great scientific work, and the inspiration behind it, increases the more as we realize the extent to which it served to pave the way for the later developments."

"After spending a year or so in France, Scotland, and England, he returned to Milan as professor of science, and shortly afterward was elected to a chair at Pavia. Here he divided his time between debauchery, astrology, and mechanics. His two sons were as wicked and passionate as himself: the elder was in 1560 executed for poisoning his wife, and about the same time Cardan in a fit of rage cut off the ears of the younger who committed some offence; for this scandelous outrage he suffered no punishment, as Pope Gregory XIII granted him protection."

"A great number of writers on the history of medicine have indicated important observations and suggestions which made their intitial appearance with Cardano."

"When Cardano's Consolation or Comforte was translated into English in 1573... one of the readers is known to have been William Shakespeare. ...Hamlet's thoughts on death and slumber are believed to have been inspired by... passages in Comforte..."

"Cardano was a man of universal interests, and much of his ability must have been inherited from his father, Fazio Cardano... a lawyer... but also deeply steeped in the medical sciences, mathematics, and all kinds of occult lore... he had... a high reputation as a scholar in his native town; even Leonardo da Vinci notes several times that he consulted Messer Fazio on geometric questions... he was appointed as a public lecturer in geometry."

"Most important for the history of science is the fact that Liber de Ludo Aleae, "The Book of Games of Chance," contains the first study of the principles of probability. ...it would seem much more just to date the beginnings of probability theory from Cardano's treatise rather than the customary reckoning from Pascal's discussions with his friend de Méré and the ensuing correspondence with Fermat... at least a century after Cardano..."

"Cardano's entertaining books on science and curiosities were among the best read and most pirated works in the sixteenth century. ...his work on the "Great Art" has been characterized as the first that goes decisively beyond the attainments of classical Greek mathematics."

"The application of the theory [of probability] to mortality tables in any large way may be said to have started with John Graunt... The first tables of great importance, however, were those of Edmund Halley... however... Cardan seems to have been the first to have been the first to consider the problem in a printed work, although his treatment is very fanciful. He gives a brief table in his proposition "Spatium vitae naturalis per spatium vitae fortuitum declarare," this appearing in the De Proportionibus Libri V..."

"The law which asserts that the equation X = 0, complete or incomplete, can have no more real positive roots than it has changes of sign, and no more real negative roots than it has permanences of sign, was apparently known to Cardan; but a satisfactory statement is possibly due to Harriot (died 1621) and certainly to Descartes."

"The problem of the biquadratic equation was laid prominently before Italian mathematicians by Zuanne de Tonini da Coi, who in 1540 proposed the problem, "Divide 10 parts into three parts such that they shall be continued in proportion and that the product of the first two shall be 6." He gave this to Cardan with the statement that it could not be solved, but Cardan denied the assertion, although himself unable to solve it. He gave it to Ferrari, his pupil, and the latter, although then a mere youth, succeeded where the master had failed. ...This method soon became known to algebraists through Cardan's Ars Magna, and in 1567 we find it used by Nicolas Petri [of Deventer]."

"He... gave thirteen forms of the cubic which have positive roots, these having already been given by Omar Kayyam."

"He states that the root of x^3 + 6x = 20 is{{center|1=x = \sqrt[3]{\sqrt{108} + 10} - \sqrt[3]{\sqrt{108} - 10}.}}"

"Cardan's originality in the matter seems to have been shown chiefly in four respects. First, he reduced the general equation to the type x^3 + bx = c; second, in a letter written August 4, 1539, he discussed the question of the irreducible case; third, he had the idea of the number of roots to be expected in the cubic; and, fourth, he made a beginning in the theory of symmetric functions. ...With respect to the irreducible case... we have the cube root of a complex number, thus reaching an expression that is irreducible even though all three values of x turn out to be real. With respect to the number of roots to be expected in the cubic... before this time only two roots were ever found, negative roots being rejected. As to the question of symmetric functions, he stated that the sum of the roots is minus the coefficient of x2"

"[Zuanne de Tonini] da Coi... impuned Tartaglia to publish his method, but the latter declined to do so. In 1539 Cardan wrote to Tartaglia, and a meeting was arranged at which, Tartaglia says, having pledged Cardan to secrecy, he revealed the method in cryptic verse and later with a full explanation. Cardan admits that he received the solution from Tartaglia, but... without any explanation. At any rate, the two cubics x^3 + ax^2 = c and x^3 + bx = c could now be solved. The reduction of the general cubic x^3 + ax^2 + bx = c to the second of these forms does not seem to have been considered by Tartaglia at the time of the controversy. When Cardan published his Ars Magna however, he transformed the types x^3 = ax^2 + c and x^3 + ax^2 = c by substituting x = y + \frac{1}{3}a and x = y - \frac{1}{3}a respectively, and transformed the type x^3 + c = ax^2 by the substitution x = \sqrt[3]{c^2/y}, thus freeing the equations of the term x^2. This completed the general solution, and he applied the method to the complete cubic in his later problems."

"The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). This was devoted primarily to the solution of algebraic equations. It contained the solution of the cubic and biquadratic equations, made use of complex numbers, and in general may be said to have been the first step toward modern algebra."

"Most of his analysis of cubic equations seems to have been original; he shewed that if the three roots were real, Tartaglia's solution gave them in a form which involved imaginary quantities. Except for the somewhat similar researches of Bombelli a few years later, the theory of imaginary quantities received little further attention from mathematicians until John Bernoulli and Euler took up the matter after the lapse of nearly two centuries. Gauss first put the subject on a systematic and scientific basis, introduced the notation of complex variables, and used the symbol i, which had been introduced by Euler in 1777, to denote the square root of (-1): the modern theory is chiefly based on his researches."

"The Ars Magna is a great advance on any algebra previously published. Hitherto algebraists had confined their attention to those roots of equations which were positive. Cardan discussed negative and even complex roots, and proved that the latter would always occur in pairs, though he declined to commit himself to any explanation as to the meaning of these "sophistic" quantities which he said were ingenious though useless."