On Literature: On God (From Fiction to Science-Fiction)

"We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus, we may quite well know that there is a God without knowing what He is." (Blaise Pascal, "Pensées", 1670)

"Sometimes the gods themselves forget the answers to their own riddles." (Edwin L. Arnold, "Lieut. Gullivar Jones: His Vacation", 1905)

"People believe in God because they’ve been conditioned to believe in God." (Aldous Huxley, "Brave New World", 1932)

"All perfection comes from within, and the perfection that is imposed from without is as frivolous and stupid as the trimmings on gingercake. The free man may be bad, but only the free man can be good. And all the kingdom and the power and the glory—call it of God, call it of Cosmos - must arise from the free will of man." (Anthony Boucher, "The Barrier", 1942)

"The gods do not speak the language of men, any more than men can speak the language of the gods." (Miriam Allen deFord, "The Apotheosis of Ki", 1956)

"Only the man who has had to face despair is really convinced that he needs mercy. Those who do not want mercy never seek it. It is better to find God on the threshold of despair than to risk our lives in a complacency that has never felt the need of forgiveness. A life that is without problems may literally be more hopeless than one that always verges on despair." (Thomas Merton, "No Man Is an Island", 1955)

"Man does not create gods, in spite of appearances. The times, the age, impose them on him." (Stanislaw Lem, "Solaris", 1961)

"If there are any gods whose chief concern is man, they cannot be very important gods." (Arthur C Clarke, "Rocket to the Renaissance", [revised version] 1962)

"To be an atheist is to maintain God." (Ursula K Le Guin, "The Left Hand of Darkness", 1969)

"I have seen God creating the cosmos, watching its growth, and finally destroying it." (Olaf Stapledon, "Nebula Maker", 1976)

"Man and the true God are identical—as the Logos and the true God are - but a lunatic blind creator and his screwed-up world separate man from God. That the blind creator sincerely imagines that he is the true God only reveals the extent of his occlusion." (Philip K Dick, "Valis", 1981)

"All the universe is just a dream in God's mind, and as long as he's asleep, he believes in it, and things stay real." (Orson Scott Card, "The Tales of Alvin Maker: Seventh Son", 1987)

"People needed to believe in gods, if only because it was so hard to believe in people." (Terry Pratchett, "Pyramids", 1989)

"God created two acts of folly. First, He created the Universe in a Big Bang. Second, He was negligent enough to leave behind evidence for this act, in the form of the microwave radiation." (Paul Erdős)

On Algebra: On Lie Algebra

"During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. [...] The invasion of algebra has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. The three subjects have been given independent but parallel developments." (Henri P Cartan & Samuel Eilenberg, "Homological Algebra", 1956)

"Algebraic deformation represents a method for the quantization of Lie groups and Lie algebras. Quantum groups are not groups, but Hopf algebras. [3]

"Lie groups describe finite symmetries or symmetries which smoothly depend on a finite number of real parameters. Lie algebras are the linearization of Lie groups at the unit element. The passage from Lie groups to Lie algebras simplifies considerably the approach. Lie algebras are frequently called infinitesimal symmetries." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"Representations of symmetries with the aid of linear operators (e.g., matrices) play a crucial role in modern physics. In particular, this concerns the linear representations of groups, Lie algebras, and quantum groups (Hopf algebras)" (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"Solvable Lie algebras are close to both upper triangular matrices and commutative Lie algebras. In contrast to this, semisimple Lie algebras are as far as possible from being commutative. By Levi’s decomposition theorem, any Lie algebra is built out of a solvable and a semisimple one. The nontrivial prototype of a solvable Lie algebra is the Heisenberg algebra." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"The fundamental Levi decomposition of a Lie algebra is the prototype of a semidirect product of Lie algebras." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"There is a fundamental relationship between Lie groupoids and Lie algebroids which is similar to the relationship between Lie groups and Lie algebras. There are, however, significant differences in the conventions involved, and it important to be aware of these." (Mike Crampin & David Saunders, "Cartan Geometries and their Symmetries: A Lie Algebroid Approach", 2016)

On Edward Beltrami - Historical Perspectives

"Early mathematicians had no difficulty accepting the first four of Euclid’s axioms. But the fifth was thought not to be so obvious as the other four, and mathematicians tried into the 19th century to derive the fifth postulate from the first four. All such attempts were doomed to failure, but in making the effort, investigators developed a great deal of modern mathematics. It was finally discovered (in the 19th century by Bolyai, Gauss, Lobachevsky, Beltrami, and others) that the fifth postulate is in fact independent of the first four and that one gets a perfectly legitimate geometry by discarding Euclid’s parallel axiom and replacing it with a different one: 'The hyperbolic parallel axiom: Given a line l and a point P not on l, there are infinitely many lines through P that are parallel to l.'" (Bruce Crauder et al, "Functions and Change: A Modeling Approach to College Algebra and Trigonometry", 2008)

"At any rate, long before the curvature of space was first detected, Beltrami’s construction of the hyperbolic plane showed that more than one kind of geometry is possible. Beltrami assumed that Euclidean space exists, and constructed a non-Euclidean plane inside it, with nonstandard definitions of 'line' and 'distance' (namely, line segments in the unit disk and pseudodistance). This shows that the geometry of Bolyai and Lobachevsky is logically as valid as the geometry of Euclid: if there is a space in which 'lines' and 'distance' behave as Euclid thought they do, then there is also a surface in which 'lines' and 'distance' behave as Bolyai and Lobachevsky thought they might." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"Beltrami started this train of thought in 1865, by asking which surfaces can be mapped onto the plane in such a way that their geodesics go to straight lines. He found that the answer is precisely the surfaces of constant Gaussian curvature. For example, great circles on the sphere can be mapped to lines on the plane, and the map that does the trick (to be precise, for the hemisphere) is central projection [...]. Rays from the center O to any great circle form a plane, which of course meets any other plane in a line. Thus projection from O to a plane sends great circles to lines, though only half of the sphere is mapped." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"When real numbers are used as coordinates, the number of coordinates is the dimension of the geometry. This is why we call the plane two-dimensional and space three-dimensional. However, one can also expect complex numbers to be useful [...]. What is remarkable is that complex numbers are if anything more appropriate for spherical and hyperbolic geometry than for Euclidean geometry. With hindsight, it is even possible to see hyperbolic geometry in properties of complex numbers that were studied as early as 1800, long before hyperbolic geometry was discussed by anyone. This was noticed by the third great contributor to non-Euclidean geometry after Beltrami and Klein - the French mathematician Henri Poincaré [...]" (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"[...] quaternions give a nice approach to symmetric objects in three-dimensional space: the regular polyhedra. But this leads in turn to the regular polytopes, a family of four-dimensional symmetrical objects as remarkable as the regular polyhedra. One then becomes convinced that four-dimensional space is not just a set of quadruples; it is a world of genuine geometry." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

On Literature: On Probability (From Fiction to Science-Fiction)

"That is probable which for the most part usually comes to pass, or which is a part of the ordinary beliefs of mankind, or which contains in itself some resemblance to these qualities, whether such resemblance be true or false." (Marcus T Cicero, "De Inventione", cca. 86–84 BC)

"Take away probability, and you can no longer please the world; give probability, and you can no longer displease it." (Blaise Pascal, "Thoughts", 1670)

"Ignorance gives one a large range of probabilities." (George Eliot, "Daniel Deronda", 1876)

"When you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth." (Arthur C Doyle, "The Sign of Four", 1890

"Every probability - and most of our common, working beliefs are probabilities - is provided with buffers at both ends, which break the force of opposite opinions clashing against it […]" (Oliver W Holmes, "The Autocrat of the Breakfast-Table", 1891) 

"It is more than possible; it is probable." (Arthur C Doyle, "The Memoirs of Sherlock Holmes", 1893)

"If everything, everything were known, statistical estimates would be unnecessary. The science of probability gives mathematical expression to our ignorance, not to our wisdom." (Samuel R Delany, "Time Considered as a Helix of Semi-Precious Stones", 1969)

"People are entirely too disbelieving of coincidence. They are far too ready to dismiss it and to build arcane structures of extremely rickety substance in order to avoid it. I, on the other hand, see coincidence everywhere as an inevitable consequence of the laws of probability, according to which having no unusual coincidence is far more unusual than any coincidence could possibly be." (Isaac Asimov, "The Planet That Wasn't", 1976)

"In the real world irrational things happened, impossible coincidences happened, because probability required that coincidences rarely, but not never, occur." (Orson Scott Card, "Ender’s Game", 1985)

"One of the elementary rules of nature is that, in the absence of a law prohibiting an event or phenomenon, it is bound to occur with some degree of probability. To put it simply and crudely: Anything that can happen does happen." (Kenneth W Ford)

On Systems: On Paths

"The state of a system at a given moment depends on two things - its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets? We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow recti-linear paths and obey the laws of impact and elastic bodies; yet as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible, that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance, and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known, and we want to find the law; or that, instead of deducing effects from causes, we wish to deduce the causes." (Henri Poincaré, "Science and Hypothesis", 1902)

"It is not enough to know the critical stress, that is, the quantitative breaking point of a complex design; one should also know as much as possible of the qualitative geometry of its failure modes, because what will happen beyond the critical stress level can be very different from one case to the next, depending on just which path the buckling takes. And here catastrophe theory, joined with bifurcation theory, can be very helpful by indicating how new failure modes appear." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"The pinball machine is one of those rare dynamical systems whose chaotic nature we can deduce by pure qualitative reasoning, with fair confidence that we have not wandered astray. Nevertheless, the angles in the paths of the balls that are introduced whenever a ball strikes a pin and rebounds […] render the system some what inconvenient for detailed quantitative study." (Edward N Lorenz, "The Essence of Chaos", 1993)

"In a free-market economy, then, uncertainty is a necessary element. Only when the economy is in a state of uncertainty can the participants efficiently search for solutions to problems and find creative answers. In addition, only a system that depends on uncertainty can survive unexpected shocks. A complex process can take multiple paths to an optimal solution. It does not require 'ideal' conditions; in fact, shocks often force it to find a better solution, a higher hill in the fitness landscape. The 'creative destruction' identified by the Austrian school suggests that a free-market economy is not only resilient to shocks, but is also creative and capable of generating innovation. It can only do so while in a high state of uncertainty." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999)

"It is, however, fair to say that very few applications of swarm intelligence have been developed. One of the main reasons for this relative lack of success resides in the fact that swarm-intelligent systems are hard to 'program', because the paths to problem solving are not predefined but emergent in these systems and result from interactions among individuals and between individuals and their environment as much as from the behaviors of the individuals themselves. Therefore, using a swarm-intelligent system to solve a problem requires a thorough knowledge not only of what individual behaviors must be implemented but also of what interactions are needed to produce such or such global behavior." (Eric Bonabeau et al, "Swarm Intelligence: From Natural to Artificial Systems", 1999)

"Complexity theory shows that great changes can emerge from small actions. Change involves a belief in the possible, even the 'impossible'. Moreover, social innovators don’t follow a linear pathway of change; there are ups and downs, roller-coaster rides along cascades of dynamic interactions, unexpected and unanticipated divergences, tipping points and critical mass momentum shifts. Indeed, things often get worse before they get better as systems change creates resistance to and pushback against the new. Traditional evaluation approaches are not well suited for such turbulence. Traditional evaluation aims to control and predict, to bring order to chaos. Developmental evaluation accepts such turbulence as the way the world of social innovation unfolds in the face of complexity. Developmental evaluation adapts to the realities of complex nonlinear dynamics rather than trying to impose order and certainty on a disorderly and uncertain world." (Michael Q Patton, "Developmental Evaluation", 2010)

"In the 'computation' that is the economy, large and small probabilistic events at particular non-repeatable moments determine the attractors fallen into, the temporal structures that form and die away, the technologies that are brought to life, the economic structures and institutions that result from these, the technologies and structures that in turn build upon these; indeed the future shape of the economy - the future path taken. The economy at all levels and at all times is path dependent. History again becomes important. And time reappears." (W Brian Arthur, "Complexity and the Economy", 2015)

"Feedback systems are closed loop systems, and the inputs are changed on the basis of output. A feedback system has a closed loop structure that brings back the results of the past action to control the future action. In a closed system, the problem is perceived, action is taken and the result influences the further action. Thus, the distinguishing feature of a closed loop system is a feedback path of information, decision and action connecting the output to input." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

On Paths: On Shortest Path

"The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain." (Paul Painlevé, "Analyse des travaux scientifiques", 1900)

"Imagine the forehead of a bull, with the protuberances from which the horns and ears start, and with the collars hollowed out between these protuberances; but elongate these horns and ears without limit so that they extend to infinity; then you will have one of the surfaces we wish to study. On such a surface geodesics may show many different aspects. There are, first of all, geodesics which close on themselves. There are some also which are never infinitely distant from their starting point even though they never exactly pass through it again; some turn continually around the right horn, others around the left horn, or right ear, or left ear; others, more complicated, alternate, in accordance with certain rules, the turns they describe around one horn with the turns they describe around the other horn, or around one of the ears. Finally, on the forehead of our bull with his unlimited horns and ears there will be geodesics going to infinity, some mounting the right horn, others mounting the left horn, and still others following the right or left ear. [...] If, therefore, a material point is thrown on the surface studied starting from a geometrically given position with a geometrically given velocity, mathematical deduction can determine the trajectory of this point and tell whether this path goes to infinity or not. But, for the physicist, this deduction is forever useless. When, indeed, the data are no longer known geometrically, but are determined by physical procedures as precise as we may suppose, the question put remains and will always remain unanswered." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"A variety of natural phenomena exhibit what is called the minimum principle. The principle is displayed where the amount of energy expended in performing a given action is the least required for its execution, where the path of a particle or wave in moving from one point to another is the shortest possible, where a motion is completed in the shortest possible time, and so on." (James R Newman, "The World of Mathematics" Vol. II, 1956)

"We frequently find that nature acts in such a way as to minimize certain magnitudes. The soap film will take the shape of a surface of smallest area. Light always follows the shortest path, that is, the straight line, and, even when reflected or broken, follows a path which takes a minimum of time. In mechanical systems we find that the movements actually take place in a form which requires less effort in a certain sense than any other possible movement would use. There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies - so to say, economical tendencies of nature. Nature seems to follow the tendency of economizing certain magnitudes, of obtaining maximum effects with given means, or to spend minimal means for given effects." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

On Søren Kierkegaard - Collected Quotes

"How close men, despite all their knowledge, usually live to madness? What is truth but to live for an idea? When all is said and done, everything is based on a postulate; but not until it no longer stands on the outside, not until one lives in it, does it cease to be a postulate." (Søren Kierkegaard, "The Journals of Søren Kierkegaard" 1A75, 1835)

"Language has time as its element; all other media have space as their element." (Søren Kierkegaard, "Either/Or: A Fragment of Life", 1843)

"One should not think slightly of the paradoxical; for the paradox is the source of the thinker's passion, and the thinker without a paradox is like a lover without feeling: a paltry mediocrity." (Søren Kierkegaard, "Philosophical Fragments: Or, A Fragment of Philosophy" 1844)

"So it happens at times that a person believes that he has a world-view, but that there is yet one particular phenomenon that is of such a nature that it baffles the understanding, and that he explains differently and attempts to ignore in order not to harbor the thought that this phenomenon might overthrow the whole view, or that his reflection does not possess enough courage and resolution to penetrate the phenomenon with his world-view." (Søren Kierkegaard, 1844)

"Take a book, the poorest one written, but read it with the passion that it is the only book you will read-ultimately you will read everything out of it, that is, as much as there was in yourself, and you could never get more out of reading, even if you read the best of books." (Søren Kierkegaard, "Stages on Life's Way", 1845)

"All essential knowledge relates to existence, or only such knowledge as has an essential relationship to existence is essential knowledge." (Søren Kierkegaard, "Concluding Unscientific Postscript", 1846)

"It is the duty of the human understanding to understand that there are things which it cannot understand, and what those things are. Human understanding has vulgarly occupied itself with nothing but understanding, but if it would only take the trouble to understand itself at the same time it would simply have to posit the paradox." (Søren Kierkegaard, [The Journals of Søren Kierkegaard] 1847)

"Take away the paradox from a thinker and you have a professor." (Søren Kierkegaard, 

[journal entry] 1849)

"Once a man acts in a decisive sense and comes out into reality, existence can get a grip on him and providence educate him." (Søren Kierkegaard, [4journal entry] 1850)

On Numbers: On Prime Numbers (Unsourced)

"A prime number, which exceeds a multiple of four by unity, is only once the hypotenuse of a right triangle." (Pierre de Fermat)

"God may not play dice with the universe, but something strange is going on with the prime numbers." (Paul Erdős)

"If [the Riemann Hypothesis is] not true, then the world is a very different place. The whole structure of integers and prime numbers would be very different to what we could imagine. In a way, it would be more interesting if it were false, but it would be a disaster because we've built so much round assuming its truth." (P  Sarnak)

"[...] in one of those unexpected connections that make theoretical physics so delightful, the quantum chorology of spectra turns out to be deeply connected to the arithmetic of prime numbers, through the celebrated zeros of the  Riemann zeta function: the zeros mimic quantum energy levels of a classically chaotic system. The connection is not only deep but also tantalizing, since its basis is still obscure - though it has been fruitful for both mathematics and physics." (Michael V Berry)

"[Looking at the distribution of the primes is like the] the feeling of being in the presence of one of the inexplicable secrets of creation." (Don Zagier) 

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate." (Leonhard Euler)

"Observation more than books and experience more than persons, are the prime educators." (Amos B Alcott)

"Prime numbers are the most basic objects in mathematics. They also are among the most mysterious, for after centuries of study, the structure of the set of prime numbers is still not well understood […]" (Andrew Granville) 

"The consequences [of the Riemann Hypothesis] are fantastic: the distribution of primes, these elementary objects of arithmetic. And to have tools to study the distribution of these of objects." (H Iwaniec)

"We found a beautiful and most general proposition, namely, that every integer is either a square, or the sum of two, three or at most four squares. This theorem depends on some of the most recondite mysteries of numbers, and it is not possible to present its proof on the margin of this page." (Pierre de Fermat)

On Numbers: On Prime Numbers (1970-1989)

"But the laws of addition and multiplication (the associative laws, for example) are not a human invention. They are unintended consequences of human invention, and they were discovered. And the existence of prime numbers - indivisible numbers that are the product only of themselves and unity - is also a discovery, no doubt quite a late one. The prime numbers were discovered in the series of natural numbers, not by everyone but by people who studied these numbers and their special peculiarities - by real mathematicians." (Karl R Popper, "Notes of a Realist on the Body-Mind Problem", [in "All Life is Problem Solving", 1999] 1972)

"The communication of modern science to the ordinary citizen, necessary, important, desirable as it is, cannot be considered an easy task. The prime obstacle is lack of education. [...] There is also the difficulty of making scientific discoveries interesting and exciting without completely degrading them intellectually. [...] It is a weakness of modern science that the scientist shrinks from this sort of publicity, and thus gives an impression of arrogant mystagoguery." (John M Ziman,"The Force of Knowledge: The Scientific Dimension of Society", 1976)

"I hope that ... I have communicated a certain impression of the im- mense beauty of the prime numbers and the endless surprises which they have in store for us." (Don Zagier, 1977) 

"There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers belong to the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision.” (Don Zagier, “The First 50 Million Prime Numbers”, The Mathematical Intelligencer Vol. 0, 1977)

“[…] there is no apparent reason why one number is prime and another not. To the contrary, upon looking at these numbers one has the feeling of being in the presence of one of the inexplicable secrets of creation.” (Don Zagier, “The First 50 Million Prime Numbers”, The Mathematical Intelligencer, Volume 0, 1977)

"Prime numbers have always fascinated mathematicians, professional and amateur alike. They appear among the integers, seemingly at random, and yet not quite: there seems to be some order or pattern, just a little below the surface, just a little out of reach." (Underwood Dudley, “Elementary Number Theory”, 1978)

"Some order begins to emerge from this chaos when the primes are considered not in their individuality but in the aggregate; one considers the social statistics of the primes and not the eccentricities of the individuals." (Philip J Davis & Reuben Hersh, “The Mathematical Experience”, 1981)

"Meaning does not reside in the mathematical symbols. It resides in the cloud of thought enveloping these symbols. It is conveyed in words; these assign meaning to the symbols." (Marvin Chester, "Primer of Quantum Mechanics", 1987)

"A tendency to drastically underestimate the frequency of coincidences is a prime characteristic of innumerates, who generally accord great significance to correspondences of all sorts while attributing too little significance to quite conclusive but less flashy statistical evidence." (John A Paulos, "Innumeracy: Mathematical Illiteracy and its Consequences", 1988)

On Numbers: On Prime Numbers (2000-)

 “As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to such an extent that it can legitimately be asked whether the subject of study of arithmetic is not the human mind itself. From this a strange fascination arises: how can it be that these numbers, which lie so deeply within ourselves, also give rise to such formidable enigmas? Among all these mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant." (Gerald Tenenbaum & Michael M France, “The Prime Numbers and Their Distribution”, 2000)

“One of the remarkable aspects of the distribution of prime numbers is their tendency to exhibit global regularity and local irregularity. The prime numbers behave like the ‘ideal gases”’which physicists are so fond of. Considered from an external point of view, the distribution is - in broad terms - deterministic, but as soon as we try to describe the situation at a given point, statistical fluctuations occur as in a game of chance where it is known that on average the heads will match the tail but where, at any one moment, the next throw cannot be predicted.” (Gerald Tenenbaum & Michael M France, “The Prime Numbers and Their Distribution”, 2000)

"The seeming absence of any ascertained organizing principle in the distribution or the succession of the primes had bedeviled mathematicians for centuries and given Number Theory much of its fascination. Here was a great mystery indeed, worthy of the most exalted intelligence: since the primes are the building blocks of the integers and the integers the basis of our logical understanding of the cosmos, how is it possible that their form is not determined by law? Why isn't 'divine geometry' apparent in their case?" (Apostolos Doxiadis, “Uncle Petros and Goldbach's Conjecture”, 2000)

“Prime numbers belong to an exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoy simple, elegant description, yet lead to extreme - one might say unthinkable - complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In modern times, while theoreticians continue to grapple with the profundity of the prime numbers, vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes in other domains." (Richard Crandall and Carl Pomerance, “PrimeNumbers: A Computational Perspective”, 2001) 

"Although the prime numbers are rigidly determined, they somehow feel like experimental data." Timothy Gowers,"Mathematics: A Very Short Introduction", 2002)

“[Primes] are full of surprises and very mysterious […] They are like things you can touch. […][ In mathematics most things are abstract, but I have some feeling that I can touch the primes, as if they are made of a really physical material. To me, the integers as a whole are like physical particles.” (Yoichi Motohashi, “The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics”, 2002)

“The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so. […] Although the prime numbers are rigidly determined, they somehow feel like experimental data." (Timothy Gowers, “Mathematics: A Very Short Introduction”, 2002)

“[…] despite their apparent simplicity and fundamental character, prime numbers remain the most mysterious objects studied by mathematicians. In a subject dedicated to finding patterns and order, the primes offer the ultimate challenge.” (Marcus du Sautoy, “The Music of the Primes”, 2003)

“Our world resonates with patterns. The waxing and waning of the moon. The changing of the seasons. The microscopic cell structure of all living things have patterns. Perhaps that explains our fascination with prime numbers which are uniquely without pattern. Prime numbers are among the most mysterious phenomena in mathematics.” (Manindra Agrawal, 2003)

“The primes have been a constant companion in our exploration of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts of the greatest mathematical minds to explain the modulation and transformation of this mystical music, the primes remain an unanswered riddle.” (Marcus du Sautoy, “The Music of the Primes”, 2003)

"Until [the RH is proved], we shall listen enthralled by this unpredictable mathematical music, unable to master its twists and turns. The primes have been a constant companion in our exploration of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts of the greatest mathematical minds to explain the modulation and transformation of this mystical music, the primes remain an unanswered riddle. We still await the person whose name will live for ever as the mathematician who made the primes sing." (Marcus du Sautoy, "The Music of the Primes", 2003)

"The concept of proof perhaps marks the true beginning of mathematics as the art of deduction rather than just numerological observation, the point at which mathematical alchemy gave way to mathematical chemistry." (Marcus du Sautoy,"The Music of the Primes", 2004)

"The concept of proof perhaps marks the true beginning of mathematics as the art of deduction rather than just numerological observation, the point at which mathematical alchemy gave way to mathematical chemistry." (Marcus du Sautoy, "The Music of the Primes", 2004)

"The worst aspect of the term 'complex' - one that condemns it to eventual extinction in my opinion - is that it is also applied to structures called 'simple'. Mathematics uses the word 'simple' as a technical term for objects that cannot be 'simplified'. Prime numbers are the kind of thing that might be called 'simple'" (though in their case it is not usually done) because they cannot be written as products of smaller numbers. At any rate, some of the 'simple' structures are built on the complex numbers, so mathematicians are obliged to speak of such things as 'complex simple Lie groups'. This is an embarrassment in a subject that prides itself on consistency, and surely either the word 'simple' or the word 'complex' has to go." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"There are many ways to use unique prime factorization, and it is rightly regarded as a powerful idea in number theory. In fact, it is more powerful than Euclid could have imagined. There are complex numbers that behave like 'integers' and 'primes', and unique prime factorization holds for them as well. Complex integers were first used around 1770 by Euler, who found they have almost magical powers to unlock secrets of ordinary integers. For example, by using numbers of the form a + b√ -2. where a and b are integers, he was able to prove a claim of Fermat that 27 is the only cube that exceeds a square by 2. Euler's results were correct, but partly by good luck. He did not really understand complex 'primes' and their behavior." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Number-theoretic equivalences of the Riemann hypothesis provide a natural method of explaining the hypothesis to nonmathematicians without appealing to complex analysis. While it is unlikely that any of these equivalences will lead directly to a solution, they provide a sense of how intricately the Riemann zeta function is tied to the primes"  (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007

"So the prime number theorem is a relatively weak statement of the fact that an integer has equal probability of having an odd number or an even number of distinct prime factors." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

“The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists! We just saw a clever argument to prove that there are infinitely many primes, but we don't know any argument to prove that there are infinitely many pairs of twin primes.” (David Ruelle, “The Mathematician's Brain”, 2007)

"The first and easies proofs [of the prime number theorem] are analytic and exploit the rich connections between number theory and complex analysis. It has resisted trivialization, and no really easy proof is known. This is especially true for the so-called elementary proofs, which use little or no complex analysis, just considerable ingenuity and dexterity. The primes arise sporadically and, apparently, relatively randomly, at least in thes ense that there is no easy way to find a large prime number with no obvious congruences. So even the amount of structure implied by the prime number theorem is initially surprising." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"For both primes and symmetries, zeta functions act as black boxes. They are built from a formula which binds together the numbers you are trying to understand. The hope is that the zeta function will reveal new insights into the numbers of symmetries. It provides a way of getting from part of the mathematical world where chaos seems to reign to a completely different region where one can start to pick out patterns." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Mathematicians call them twin primes: pairs of prime numbers that are close to each other, almost neighbors, but between them there is always an even number that prevents them from truly touching. […] If you go on counting, you discover that these pairs gradually become rarer, lost in that silent, measured space made only of ciphers. You develop a distressing presentiment that the pairs encountered up until that point were accidental, that solitude is the true destiny. Then, just when you’re about to surrender, you come across another pair of twins, clutching each other tightly.” (Paolo Giordano, “The Solitude of prime numbers”, 2008)

"Riemann had found a passageway from the familiar world of numbers into a mathematics which would have seemed utterly alien to the Greeks who had studied prime numbers two thousand years before. He had innocently mixed imaginary numbers with his zeta function and discovered, like some mathematical alchemist, the mathematical treasure emerging from this admixture of elements that generations had been searching for. He had crammed his ideas into a ten-page paper, but was fully aware that his ideas would open up radically new vistas on the primes." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Since they represent so natural a sequence, it is almost irresistible to search for patterns among the primes. There are however no genuinely useful formulas for prime numbers. That is to say there is no rule that allows you to generate all prime numbers or even to calculate a sequence that consists entirely of different primes." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008) 

“[…] if all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime.” (Martin Gardner, “When You Were a Tadpole and I Was a Fish”, 2009)

"The significance of Fourier’s theorem to music cannot be overstated: since every periodic vibration produces a musical sound" (provided, of course, that it lies within the audible frequency range), it can be broken down into its harmonic components, and this decomposition is unique; that is, every tone has one, and only one, acoustic spectrum, its harmonic fingerprint. The overtones comprising a musical tone thus play a role somewhat similar to that of the prime numbers in number theory: they are the elementary building blocks from which all sound is made." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)