Mathematicians From Italy

"Only the young men who sat under him could know his fascination as a teacher."

"Comprehensive books are very useful in the hands of those whose minds are already formed ; but experience has taught us that a judicious parsimony proves more successful in encouraging the mental efforts of young beginners, amid the many difficulties arising from the giddiness natural to their age, as well as from the number of their scholastic duties."

"Dylan teaches us that philosophical work is essentially work on our incompleteness, on our finitude, on the fact that we are conditioned by a whole host of things."

"Methodological atheism is precisely a way of living as if God did not exist, or regardless of God, or regardless of those who claim to impose their values in the name of God."

"Every culture has a need for the divine, more like a need for love."

"All evidence of the existence of God implies a kind of submission."

"For centuries, even millennia, we have been dragging along the dualism of mind and body, of soul and flesh. Even before the findings of biology and neurophysiology, this dualism was dismantled by two great figures of modern philosophy: the rationalist Baruch Spinoza and the empiricist David Hume. Starting from very different premises, the two converge in telling us that our ego, which claims to be immortal, is nothing more than a mode of nature."

"It seems clear that the current debate on the alleged dictatorship of relativism is a philosophical debate on the meaning and scope of science, critical reflection, political tolerance, and moral choice. (p. 20)"

"Fundamentalism arises, precisely with this word, in the context of Christianity, particularly Protestant Christianity, especially American. [...] There is also Catholic fundamentalism."

"Criticism is not only corrosive—if it dissolves old bonds, it is because it creates new solidarity. (p. 73)"

"Once upon a time, the heavens were populated by Deities whom we now consider to be more or less naive human projections. But for centuries, indeed for millennia, they appeared more powerful than men, if not their very creators. There is a Latin word that aptly describes this superiority: ‘'superstitio’'. (p. 18)"

"Against this fundamentalism, I believe we must return to the Enlightenment. [...] I particularly like what is called radical Enlightenment, that of Spinoza for example, that of Diderot, that of the Marquis de Sade. [...] How much violence... yes, he theorizes it, describes it in his novels, while some guardians of morality, such as various pedophile priests and those who have covered up for them and continue to do so, actually commit it."

"The libertine is constantly seeking new intellectual experiences. And so, in this sense, lust is the passion for knowledge, for moving forward, for never stopping, for never being satisfied with a point of view that has been reached."

"(About the death of Margherita Hack) She was a great researcher, a great fighter for the secular concept of freedom of research, and a great woman. Margherita was not afraid to be an atheist in the name of scientific freedom. :*‘’2013/06/29/morta-margherita-hack-trieste-astrofisica-aveva-91-anni/641222/ Margherita Hack dies at 91: astrophysicist between research and civil rights‘’, ‘'Il Fatto Quotidiano’', June 29, 2013."

"I fully identify with the anti-idolatrous component of Christianity."

"The core of Catholic interventionism is not, in principle, too different from so-called Islamic fundamentalism (although I personally dislike this term). In both cases, our response must be the same: ‘'They shall not pass!’'"

"Geymonat believes that scientific progress can be summed up in the slogan: “challenge and create”: the best service we can render it is therefore to challenge it."

"Yes, we have a soul. But it is made up of many small robots."

"Scientists are wrong when, faced with a part of reality that does not correspond to their theory, they try to erase reality instead of modifying the theory."

"The foundation of ethics is the practice of ethics; there is no need to base it on anything else."

"Too often we forget that the opposite of relativism is absolutism. (p. 15)"

"Fallibility is not a theory, but an attitude, a way of life, a philosophical option that Peirce did not see as dominating the practice of research, but rather as lurking in the folds of science. Absolute knowledge, that is, knowledge free from the constraints of criticism, would, in his view, have precluded any renewal of inquiry. (pp. 28-29)"

"Recently, I saw a photograph of Queen Elizabeth and the Pope meeting. They were the image of two remnants. The monarchy within the church and the monarchy within the state."

"The best thing is to discuss with confident, passionate people. And in this, Giussani was a true master."

"Any human behavior, even the most horrible in the light of certain values, is natural because it is permitted by the laws of nature."

"I believe that we can make sense of the world without necessarily bringing God into it."

"Darwin's ‘'Autobiography’' is not only a valuable testimony to the genesis of a concept and the personality of the man who produced it, but also a civic lesson in freedom."

"[al-Khwārizmī] not having taken algebra from the Greeks,. . . must have either invented it himself, or taken it from the Indians. Of the two, the second appears to me the most probable."

"I have decided first to consider the majority of the authors who up to now have written about [algebra], so that I can fill in what they have missed out. They are very many, and among them Mohammed ibn Musa [Al-Khwarizmi], an Arab, is believed to be the first [...] I believe that the word “algebra” came from him, because some years ago, Brother Luca [Pacioli] of Borgo San Sepolcro of the Minorite order, having set himself the task of writing on this science, as much in Latin as in Italian, said that the word “algebra” was Arabic [...] and that the science came from the Arabs. Many who have written after him have believed and said likewise, but in recent years, a Greek work on this discipline has been discovered in the Library of our Lord in the Vatican, composed by a certain Diophantus of Alexandria, a Greek author [...] Antonio Maria Pazzi and I have translated five books (of the seven) [...] In this work we have found that he cites the Indian authors many times, and thus I have been made aware that this discipline belonged to the Indians before the Arabs."

"The spectacular results in the fluctuation theory of sums of independent random variables, obtained in the last 15 years by , , , , , , and others, have gradually led to the realization that the nature of the problem, as well as that of the methods of solution, is algebraic and combinatorial. After Baxter showed that the crux of the problem lay in simplifying a certain operator identity, several algebraic proofs (, , Wendel) followed. It is the present purpose to carry this algebraization to the limit: the result we present amounts to a solution of the for s. The solution is not presented as an algorithm, but by showing that every identity in a Baxter algebra is effectively equivalent to an identity of symmetric functions independent of the number of variables. Remarkably, the identities used so far in the combinatorics of fluctuation theory "translate" by the present method into classical identities of easy verification. The present method is nevertheless also useful for guessing and proving new combinatorial identities: by way of example, it will be shown in the second part of this note how it leads to a generalization of the Bohnenblust-Spitzer formula for the action of arbitrary ."

"It has been observed that whereas s and s are likely to be embarrassed by references to the beauty in their work, mathematicians instead like to engage in discussions of the beauty of mathematics. Professional artists are more likely to stress the technical rather than the aesthetic aspects of their work. Mathematicians, instead, are fond of passing judgment on the beauty of their favored pieces of mathematics. Even a cursory observation shows that the characteristics of mathematical beauty are at variance with those of artistic beauty. For example, courses in “art appreciation” are fairly common; it is however unthinkable to find any “mathematical beauty appreciation” courses taught anywhere. The purpose of the present paper is to try to uncover the sense of the term “beauty” as it is currently used by mathematicians."

"It cannot be a complete coincidence that several outstanding logicians of the twentieth century found shelter in s at some time in their lives: Cantor, , Gödel, Peano, and are some. was one of the saner among them, though in some ways his behavior must be classified as strange, even by mathematicians' standards. He looked like a cross between a and a large owl. He spoke softly in complete paragraphs which seemed to have been read out of a book, evenly and slowly enunciated, as by a . When interrupted, he would pause for an uncomfortably long period to recover the thread of the argument. He never made casual remarks: they did not belong in the baggage of ."

"The more experimental scientists and s are, the more common sense they have, and so on until you get to the mathematicians, who are totally devoid of common sense."

"To avoid this it is best, indeed necessary, that female minds – as they say – having been elevated enough on their own and filled, be kept away entirely from the liberal disciplines and be content with the management of domestic affairs, busying themselves with the needle and the spindle; these things, and others of this kind, are proper to women, unlike pen and paper, since nothing is really more irritating than a learned woman in a debate. Most equitable judges, these things are the fortifications and foundations that must be overturned and destroyed. I dare to promise that the opinion of all your adversaries will be rejected by me as a thing of no importance”."

"Man always acts to achieve goals; the goal of the Christian is the glory of God, “I hope my studies have brought glory to God, as there were useful to others, and derived from obedience, because that was my father’s will. Now I have found better ways and means to serve God, and to be useful to others."

"Is it a surprise that into the vessel, in which the mercury has no inclination and no repugnance, not even the slightest, to being there, it should enter and should rise in a column high enough to make equilibrium with the weight of the external air which forces it up?"

"Many have argued that a vacuum does not exist, others claim it exists only with difficulty in spite of the repugnance of nature; I know of no one who claims it easily exists without any resistance from nature."

"Sola enim Geometria inter liberales disciplinar acriter exacuit ingenium, idoncumque reddit ad civitates exornandas in pace et in bello defendendas: caeteris enim paribus, ingenium quod exercitatum sit in Geometrica palestra, peculiare quoddam, et virile robur habere solet: praestabitque semper, et antecellet, circa studia architecturae, rei bellicae, nauticaeque, etc."

"If this same balance, even though corporeal, were considered to be not on the earth's surface but in the highest regions beyond the sun's sphere, then the threads, while still drawn to the centre of the earth, would be very much less convergent to each other, would be quasi-parallel. Let us imagine a mechanical balance transported beyond the starry balance [i.e., the constellation of that name] in the firmament, to an infinite distance. It will be understood by everybody that the suspension threads would no longer be convergent, but would be exactly parallel. ... The Geometer has the special privilege to carry out, by abstraction, all constructions [operationes] by means of the intellect. Who, then, would wish to prevent me from freely considering figures hanging on a balance imagined to be at an infinite distance beyond the confines of the world?"

"I have absolutely no proof, but I believe in the existence of extraterrestrial life somewhere in the cosmos. I think it would be an unjustifiable waste to create an endless and varied universe like ours that had terrestrial life as its sole final result."

"As far as we know, entropy increases throughout the portion of the universe observable from Earth. It does not seem probable to us, but in any case nothing excludes, that beyond the particle horizon which marks the maximum limit of observations there exist regions in which the arrow of time is reversed compared to ours and in which entropy decreases. I dare not think of the theoretical and observational complications that would arise if the matter contained in one of these anomalous regions began to interact with ours."

"Fermi was somewhat skeptical about the possibility of an extraterrestrial civilization so technologically advanced that it could freely move between the stars. One of his favorite arguments was that such a civilization would be able to colonize the entire Galaxy in less than 300 million years, a short time compared to its age of about 15 billion years. That civilization should have already reached Earth, leaving appreciable traces. But since these traces were not found, Fermi came to the conclusion that the hypothetical civilization does not exist."

"Not even mathematics can be considered as a closed and complete system of axioms and theorems. The mathematical world is inexhaustible, no finite set of postulates and deductions will ever be able to give us the answer to all questions. Gödel's theorem, whose statement dates back to about half a century ago, brutally put an end to all attempts to condense mathematics into a list of axioms from which the truth or falsity of each of its assertions should follow. If the same mathematical language that physics uses to describe the world remains intrinsically incomplete, it is not reasonable to expect that the universe can be describable starting from a finite set of natural laws. The incompleteness of mathematics and consequently that of physics is repugnant to many, but it must be said that for the exact sciences, Gödel's theorem is by no means a defeat: on the contrary, it provides us with an intellectual push towards ever broader and more fruitful developments."

"The Bible and Eastern religions have always attracted people of various backgrounds who have wanted to recognize extraordinary anticipations of cosmology and atomic physics in sacred texts. I remain deeply skeptical on this point."

"Man is a product of Darwinian evolution. His mental structure and logical categories have been profoundly influenced by the struggle for survival in the nature in which he evolved. In particular, we know in greater detail and intuitively those natural laws and mathematical rules that allow us to survive. This basic approach imposes on us a substantially anthropomorphic vision of the world around us."

"I am convinced that the race backwards in space and time, to the origins of the Big Bang and the elementary structure of matter, is infinite; logically and spatially inaccessible. I therefore believe that the quarks that make up electrons and protons are not the most elementary particles at all; similarly, there is no single chain of events that led from the Big Bang to us, but an infinity of equally plausible possibilities. Mine is still a faith, because I have no proof of anything."

"The notion of Algebraic Complete Integrability (ACI) of certain mechanical systems, introduced the early 1980s, has given great impetus to the study of moduli spaces of holomorphic vector bundles over an algebraic curve (or a higher-dimensional variety, still at a much less developed stage). Several notions of 'duality' have been the object of much interest in both theories. There is one example, however, that appears to be a beautiful isolated feature of genus-2 curves. In this note such example, which belongs to a 'universal' class of ACIs, namely (generalized) s, is interpreted in the setting of the classical geometry of Klein's quadratic complex, following the Newstead and Narasimhan-Ramanan programme of studying moduli spaces through projective models."

"... As a mathematician coming of age in the early years when women were underrepresented, namely the 1970s, I received informal mentoring in the form of experiential advice from rare encounters with females who had achieved professional recognition; their words of wisdom were substantive resources that allowed me to persevere. I am inspired by the book Every Other Thursday (Daniell 2006) which tells the story of a group of professional women, including scientists, university professors, and administrators, who met twice a month for twenty-five years, establishing specific practices such as goal setting, networking, and checking on each other's progress. I am inspired to see more intentional examples of mentoring taking panel for women and girls interested in the ."

"The nature of s comes down to a differential equation and a duality. The interplay between the two variables is still something of a mystery (to this writer). By virtue of the lattice of periods, the theta function is at the same time one of the most powerful objects of algebraic geometry. Much classical mathematics of curve theory (Riemann surfaces) is derived using this algebraic aspect. The key idea is to interpret the moduli space of line bundles over the curve as a principally polarized abelian variety. Exploitng its self-dual property provides two variables whose duality establishes a linearization of the class of non-linear s that have as a prototype."

"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."