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)

On Numbers: On Prime Numbers (1990-1999)

"It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means.'' (Rob C Vaughan, 1990)

“Prime numbers. It was all so neat and elegant. Numbers that refuse to cooperate, that don’t change or divide, numbers that remain themselves for all eternity.” (Paul Auster, “The Music of Chance”, 1990)

"The zeta function is probably the most challenging and mysterious object of modern mathematics, in spite of its utter simplicity. [...] The main interest comes from trying to improve the Prime Number Theorem, i.e., getting better estimates for the distribution of the prime numbers. The secret to the success is assumed to lie in proving a conjecture which Riemann stated in 1859 without much fare, and whose proof has since then become the single most desirable achievement for a mathematician." (Martin C Gutzwiller, "Chaos in Classical and Quantum Mechanics", 1990)

"But natural selection does not explain how we came to understand the chemistry of stars, or subtle properties of prime numbers. Natural selection explains only that humans have acquired higher intellectual functions; it cannot explain why so much is understandable about the physical universe, or the abstract world of mathematics." (David Ruelle, "Chance and Chaos", 1991)

"The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers. This fact alone singles out the Riemann hypothesis as the main open question of prime number theory." (Enrico Bombieri,  "Prime Territory", The Sciences,  1992)

“To me, that the distribution of prime numbers can be so accurately represented in a harmonic analysis is absolutely amazing and incredibly beautiful. It tells of an arcane music and a secret harmony composed by the prime numbers.” (Enrico Bombieri, ”PrimeTerritory", The Sciences, 1992)

"Like the noble gases" (helium, neon, argon, krypton, xenon, and radon), primes exist in splendid isolation; conversely, any composite number is the product of a unique set of prime factors." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Why is it so important to find primes, or to show that a certain integer is one? A very practical application in cryptography rests on the fact that since it is extremely hard to factor very large numbers, a two-hundred-digit number that was the product of two primes could govern text encoding: It would be virtually impossible to guess what the two numbers were if you didn't know them in advance, and out of the question" (save perhaps on a state-of-the-art supercomputer) to go at it by trial and error." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

“If we imagine mathematics as a grand orchestra, the system of whole numbers could be likened to a bass drum: simple, direct, repetitive, providing the underlying rhythm for all the other instruments. There surely are more sophisticated concepts - the oboes and French horns and cellos of mathematics - and we examine some of these in later chapters. But whole numbers are always at the foundation.” (William Dunham, “The Mathematical Universe”, 1994)

"One reason nature pleases us is its endless use of a few simple principles: the cube-square law; fractals; spirals; the way that waves, wheels, trig functions, and harmonic oscillators are alike; the importance of ratios between small primes; bilateral symmetry; Fibonacci series, golden sections, quantization, strange attractors, path-dependency, all the things that show up in places where you don’t expect them [...] these rules work with and against each other ceaselessly at all levels, so that out of their intrinsic simplicity comes the rich complexity of the world around us. That tension - between the simple rules that describe the world and the complex world we see - is itself both simple in execution and immensely complex in effect. Thus exactly the levels, mixtures, and relations of complexity that seem to be hardwired into the pleasure centers of the human brain - or are they, perhaps, intrinsic to intelligence and perception, pleasant to anything that can see, think, create? - are the ones found in the world around us." (John Barnes, "Mother of Storms", 1994)

"Combinatorics is special. Most mathematical topics which can be covered in a lecture course build towards a single, well-defined goal, such as the Prime Number Theorem. Even if such a clear goal doesn’t exist, there is a sharp focus (e.g. finite groups). By contrast, combinatorics appears to be a collection of unrelated puzzles chosen at random. Two factors contribute to this. First, combinatorics is broad rather than deep. Second, it is about techniques rather than results. [...] Combinatorics could be described as the art of arranging objects according to specified rules. We want to know, first, whether a particular arrangement is possible at all, and if so, in how many different ways it can be done. If the rules are simple, the existence of an arrangement is clear, and we concentrate on the counting problem. But for more involved rules, it may not be clear whether the arrangement is possible at all." (Peter J Cameron, "Combinatorics: topics, techniques, algorithms", 1995)

"To be an engineer, and build a marvelous machine, and to see the beauty of its operation is as valid an experience of beauty as a mathematician's absorption in a wondrous theorem. One is not ‘more’ beautiful than the other. To see a space shuttle standing on the launch pad, the vented gases escaping, and witness the thunderous blast-off as it climbs heavenward on a pillar of flame - this is beauty. Yet it is a prime example of applied mathematics." (Calvin C Clawson,"Mathematical Mysteries", 1996)

"When we think of π, let’s not always think of circles. It is related to all the odd whole numbers. It also is connected to all the whole numbers that are not divisible by the square of a prime. And it is part of an important formula in statistics. These are just a few of the many places where it appears, as if by magic. It is through such astonishing connections that mathematics reveals its unique and beguiling charm." (Sherman K Stein, "Strength in Numbers", 1996)

"Yet, I believe the problem stands like a unconquerable fortress. For all that is known, it would be almost by luck that an odd perfect number would be found. On the other hand, nothing that has been proved is promising to show that odd perfect numbers do not exist. New ideas are required." (Paulo Ribenboim, "The New Book of Prime Number Records", 1996)

"Maybe so, but something is going on with the primes." (Carl B Pomerance, [lecture] 1997) [response to Albert Einstein's "God doesn't play dice"]

"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, 1997)

"The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. […] A beautiful theorem may not be blessed with an equally beautiful proof; beautiful theorems with ugly proofs frequently occur. When a beautiful theorem is missing a beautiful proof, attempts are made by mathematicians to provide new proofs that will match the beauty of the theorem, with varying success. It is, however, impossible to find beautiful proofs of theorems that are not beautiful." (Gian-Carlo Rota,"The Phenomenology of Mathematical Beauty", 1997)

"Distributed control means that the outcomes of a complex adaptive system emerge from a process of self-organization rather than being designed and controlled externally or by a centralized body." (Brenda Zimmerman et al, "A complexity science primer", 1998)

"To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people." (Andreas Knauf, "Number Theory, Dynamical Systems and Statistical Mechanics", 1998)

“Rather mathematicians like to look for patterns, and the primes probably offer the ultimate challenge. When you look at a list of them stretching off to infinity, they look chaotic, like weeds growing through an expanse of grass representing all numbers. For centuries mathematicians have striven to find rhyme and reason amongst this jumble. Is there any music that we can hear in this random noise? Is there a fast way to spot that a particular number is prime? Once you have one prime, how much further must you count before you find the next one on the list? These are the sort of questions that have tantalized generations.” (Marcus du Sautoy, “The Music of the Primes”, 1998)

On History of Mathematics (2000-)

"I can attest to the benefits brought by the use of history of mathematics through my personal experience. The study of history of mathematics, though it does not make me a better mathematician, does make me a happier man who is ready to appreciate the multi-dimensional splendour of the discipline and its relationship to other cultural endeavours. It does enhance the joy derived from my job as a mathematics teacher when I try to share this kind of feeling with my class." (Siu Man-Keung, "The ABCD of Using History of Mathematics in the (Undergraduate) Classroom", 2000)

"The history of algebra shows us that nothing is more unsound than the rejection of any method which naturally arises, merely because of one or more apparently valid cases in which such a method leads to erroneous results. Such cases should indeed teach caution, but not rejection. For if the latter had been preferred to the former, negative quantities, and still more, their square roots, would have been an effectual bar to the progress of algebra. And think of those immense fields over which even the rejecters of divergent series now roam without fear! Those fields would not even have been discovered, much less cultivated and settled." (Gavin Hitchcock, "A Window on the World of Mathematics, 1870 Reminiscences of Augustus De Morgan - a dramatic presentation", 2000)

"The history of mathematics contains a wealth of material that can be used to inform and instruct in today's classrooms. Among these materials are historical problems and problem solving situations. While for some teachers, historical problem solving can be a focus of a lesson, it is probably a better pedagogical practice to disperse such problems throughout the instructional process. Teachers who like to assign a "problem of the week" will find that historical problems nicely suit the task. Ample supplies of historical problems can be found in old mathematics books and in many survey books on the history of mathematics. These problems let us touch the past but they also enhance the present. Their contents reveal the mathematical traditions that we all share. Questions originating hundreds or even thousands of years ago can be understood, appreciated, and answered in today's classrooms. What a dramatic realization that is!" (Frank J Swetz, "Problem Solving from the History of Mathematics", 2000)

"There are quite divergent opinions about the role the history of mathematics could play in the presentation of mathematics itself. A very common attitude is simply to ignore it, arguing that a deductive approach is better suited for this purpose, since in this way all concepts, theorems and proofs can be introduced in a clearcut way. On the other extreme, a rather naive attitude is to follow the historical development of a mathematical discipline as closely as possible, presumably using original books, papers, and so on. It is clear that both methods have serious defects." (Constantinos Tzanakis, "Presenting the Relation between Mathematics and Physics on the Basis of their History: a Genetic Approachl", 2000)

"Using history of mathematics in the classroom does not necessarily make students obtain higher scores in the subject overnight, but it can make learning mathematics a meaningful and lively experience, so that (hopefully) learning will come easier and will go deeper. The awareness of this evolutionary aspect of mathematics can make a teacher more patient, less dogmatic, more humane, less pedantic. It will urge a teacher to become more reflective, more eager to learn and to teach with an intellectual commitment." (Siu Man-Keung, "The ABCD of Using History of Mathematics in the (Undergraduate) Classroom", 2000)

"Using the history of mathematics as an introduction to a critical and cultural study of mathematics is one of the most important challenges for mathematics teachers and for students. There are many possibilities in mathematics education for the use of history [...]" (Lucia Grugnetti, "The History of Mathematics and its Influence on Pedagogical Problems", 2000)

"The most amazing event in the history of Greek mathematics has to have been the discovery of irrational numbers. This was not merely a fact about real numbers, which didn’t exist yet. It was a blow to Pythagorean philosophy, one of the main tenets of which was that all was number and all relations were thus ratios. And it was a genuine foundational crisis: the discovery of irrational numbers invalidated mathematical proofs. More than that, it left open the question of what one even meant by proportion and similarity." (Craig Smoryński, "History of Mathematics: A Supplement", 2008)

"The history of mathematics can be studied chronologically, thematically, topically, and biographically. I have used in this course elements of each approach." (Israel Kleiner, Excursions in the History of Mathematics", 2012)

"The issue of rigorous foundations for calculus began with gropings in the early seventeenth century and concluded with a 'final' resolution in the 1870s. This rather slow evolution toward a logical grounding is not atypical in the history of mathematics. Rigor, formalism, and the logical development of a concept, result, or theory usually come at the end of a process of mathematical evolution. In the case of calculus, mathematicians achieved very impressive results during the seventeenth and eighteenth centuries by intuitive, heuristic reasoning, and therefore had no compelling reasons to put their subject on firm foundations. This does not mean that there was no concern during these two centuries for the logic behind the algorithms of calculus; and there were attempts, albeit unsuccessful, to supply it." (Israel Kleiner, Excursions in the History of Mathematics", 2012)

"The history of mathematical ideas is often very difficult to untangle, and because ideas evolve gradually over a long period of time it is impossible to draw an exact boundary between a given theory and its offspring. It is consequently a painful task to credit some developments to a small number of authors." (Barnaby Sheppard, "The Logic of Infinity", 2014)

On Girard Desargues - Historical Perspectives

"We shall also demonstrate the following property, of which the original inventor is M. Desargues, of Lyon, one of the great minds of our time, and most versed in mathematics, amongst other topics, in conics, and whose writings on this subject, although small in number, have given ample testimony to those who have wished to receive of its knowledge. I am willing to confess that I owe the little I have found on this subject to his writings, and that I have endeavored, as far as possible, to imitate his method [...]" (Blaise Pascal, "Essais pour les coniques", 1640

"The famous geometer Desargues worked on the lines of Kepler and is now commonly credited with the authorship of some of the ideas of his predecessor. [...] the oneness of opposite infinities followed simply and logically from a first principle of Desargues, that every two straight lines, including parallels, have or are to be regarded as having one common point and one only. A writer of his insight must have come to this conclusion, even if the paradox had not been held by Kepler, Briggs, and we know not how many others, before Desargues wrote. [...] Desargues must have learned directly or indirectly from the work in which Kepler propounded his new theory of these points, first called by him the Foci (foyers), including the modern doctrine of real points at infinity." (Charles Taylor, "The Geometry of Kepler and Newton", 1899)

"In general this geometry instead of dealing with definite triangles, polygons, circles, etc., in the Euclidean manner, is based on a consideration of all points of a straight line, of all lines through a common point and of the possible effects of setting up an orderly one-to-one correspondence between them. In particular, Desargues makes a comparative study of the different plane sections of a given cone, deducing from known properties of the circle analogous results for the other conic sections." (William T Sedgwick, "A Short History of Science", 1917)

"We owe to Desargues the theory of involution and of transversals; also the beautiful conception that the two extremities of a straight line may be considered as meeting at infinity, and that parallels differ from other pairs of lines only in having their points of intersection at infinity. He re-invented the epicycloid and showed its application to the construction of gear teeth, a subject elaborated more fully later by La Hire." (Florian Cajori, "A History of Mathematics", 1919)

"One of the first important steps to be taken in modern times... was due to Desargues. In a work published in 1639 Desargues set forth the foundation of the theory of four harmonic points, not as done today but based on the fact that the product of the distances of two conjugate points from the center is constant. He also treated the theory of poles and polars, although not using these terms." (David E Smith, "History of Mathematics" Vol. 1, 1923)

"Desargues the architect was doubtless influenced by what in his day was surrealism. In any event, he composed more like an artist than a geometer, inventing the most outrageous technical jargon in mathematics for the enlightenment of himself and the mystification of his disciples. Fortunately Desarguesian has long been a dead language." (Eric T Bell, "The Development of Mathematics", 1940

"[...] all the laws of algebra correspond to projective coincidences, and von Staudt showed that all the required coincidences follow from the theorems of Pappus and Desargues. Then in 1899 David Hilbert showed that all laws of algebra except the commutative law for multiplication follow from the Desargues theorem. And in 1932 Ruth Moufang showed that all except the commutative and associative laws follow from the little Desargues theorem. Thus the Pappus, Desargues, and little Desargues theorems are mysteriously aligned with the laws of multiplication!" (John Stillwell, "The Four Pillars of Geometry", 2000)

"Calculation with numbers is the obvious model for calculation with letters, but a geometric model is also conceivable, since numbers can be interpreted as lengths. Indeed, the coordinate geometry of Fermat and Descartes was based on algebra. They found that the curves studied by the Greeks can be represented by equations, and that algebra unlocks their secrets more easily and systematically than classical geometry. But to apply algebra in the first place, Fermat and Descartes assumed classical geometry. In particular, they used Euclid’s parallel axiom and the concept of length to derive the equation of a straight line,"(John Stillwell, "The Four Pillars of Geometry", 2000)

"The Pappus and Desargues theorems show that certain coincidences - three points lying on the same line - are in fact inevitable. In fact, all such coincidences can be explained as consequences of these two theorems [...] the Pappus and Desargues theorems do more than explain projective coincidences - they also explain where basic  algebra comes from!" (John Stillwell, "The Four Pillars of Geometry", 2000)

On Geometrical Figures: On Parabola

"To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle. According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient." (Proclus Lycaeus, cca 5th century)

"Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. The straight line is simple and unique; the classes of curves are indefinitely many, - circle, parabola, hyperbola, ellipse, etc." (Pierre de Fermat, "Introduction aux Lieux Plans et Solides", 1679)

"The Ellipse is the most simple of the Conic Sections, most known, and nearest of Kin to a Circle, and easiest describ'd by the Hand in plano. Though many prefer the Parabola before it, for the Simplicity of the Æquation by which it is express'd. But by this Reason the Parabola ought to be preferr'd before the Circle it self, which it never is. Therefore the reasoning from the Simplicity of the Æquation will not hold. The modern Geometers are too fond of the Speculation of Æquations." (Isaac Newton, "Arithmetica Universalis", 1707)

"Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members. On the other hand, the mathematics of variable magnitudes - function theory or analysis - considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the student by a point moving in accordance to this law, is the parabola. [...] If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates, - reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology […] bears to anatomy." (Christian H Dillmann," Die Mathematik die Fackelträgerin einer neuen Zeit", 1889)

"A surface which can be regarded as the set of successive position of a curve moving in space is said to be generated by the curve. The utility of this notion in constructing a surface geometrically, in a picture or as a model is increased as the complexity of the generator and its motion is decreased. When the generator is a straight line, it is called a ruled surface. Since you can exchange X and Y in the above analysis, the hyperbolic paraboloid is generated by a line in two ways. It is a doubly ruled surface." (George K Francis, "A Topological Picturebook", 1987)

On Probability (1825-1849)

"The calculus of probability is equally applicable to things of all kinds, moral and physical and, if only in each case observations provide the necessary numerical data, it does not at all depend on their nature." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The probability of an event is our reason to believe that it will occur or occurred. [...] Probability depends on our knowledge about an event; for the same event it can differ for different persons. Thus, if a person only knows that an urn contains white and black balls, whereas another person alsoknows that there are more white balls than black ones, the latter has more grounds to believe in the extraction of a white ball. In other words, for him, that event has a higher probability than for the former." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible' (equally like to happen)." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"I consider the world probability as meaning the state of mind with respect to an assertion, a coming event, or any other matter on which absolute knowledge does not exist." (Augustus De Morgan, "Essay on Probability", 1838)

"By degree of probability we really mean, or ought to mean, degree of belief [...] Probability then, refers to and implies belief, more or less, and belief is but another name for imperfect knowledge, or it may be, expresses the mind in a state of imperfect knowledge. (Augustus De Morgan, "Formal Logic: Or, The Calculus of Inference, Necessary and Probable", 1847)

"Without doubt, matter is unlimited in extent, and, in this sense, infinite; and the forces of Nature mould it into an innumerable number of worlds. Would it be at all astonishing if, from the universal dice-box, out of an innumberable number of throws, there should be thrown out one world infinitely perfect? Nay, does not the calculus of probabilities prove to us that one such world out of an infinite number, must be produced of necessity? (Philippe Buchez & William B Greene, "Remarks on the Science of History: Followed by an a priori autobiography", 1849)

On Geometrical Figures: On Ellipses

"To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle. According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient."  (Proclus Lycaeus, cca 5th century)

"Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. The straight line is simple and unique; the classes of curves are indefinitely many, - circle, parabola, hyperbola, ellipse, etc." (Pierre de Fermat, "Introduction aux Lieux Plans et Solides", 1679)

"The Ellipse is the most simple of the Conic Sections, most known, and nearest of Kin to a Circle, and easiest describ'd by the Hand in plano. Though many prefer the Parabola before it, for the Simplicity of the Æquation by which it is express'd. But by this Reason the Parabola ought to be preferr'd before the Circle it self, which it never is. Therefore the reasoning from the Simplicity of the Æquation will not hold. The modern Geometers are too fond of the Speculation of Æquations." (Isaac Newton, "Arithmetica Universalis", 1707)

"The perturbations which the motions of planets suffer from the influence other planets, are so small and so slow that they only become sensible after a long interval of time; within a shorter time, or even within one or several revolutions, according to circumstances, the motion would differ so little from motion exactly described, according to the laws of Kepler, in a perfect ellipse, that observations cannot show the difference. As long as this is true, it not be worth while to undertake prematurely the computation of the perturbations, but it will be sufficient to adapt to the observations what we may call an osculating conic section: but, afterwards, when the planet has been observed for a longer time, the effect of the perturbations will show itself in such a manner, that it will no longer be possible to satisfy exactly all the observations by a purely elliptic motion; then, accordingly, a complete and permanent agreement cannot be obtained, unless the perturbations are properly connected with the elliptic motion." (Carl F Gauss, "Theoria motus corporum coelestium in sectionibus conicis solem ambientium", 1809)

"Beauty cannot be defined by abscissas and ordinates; neither are circles and ellipses created by their geometrical formulas." (Carl von Clausewitz, "On War", 1832)

"But it is a third geometry from which quantity is completely excluded and which is purely qualitative; this is analysis situs. In this discipline, two figures are equivalent whenever one can pass from one to the other by a continuous deformation; whatever else the law of this deformation may be, it must be continuous. Thus, a circle is equivalent to an ellipse or even to an arbitrary closed curve, but it is not equivalent to a straight-line segment since this segment is not closed. A sphere is equivalent to any convex surface; it is not equivalent to a torus since there is a hole in a torus and in a sphere there is not. Imagine an arbitrary design and a copy of this same design executed by an unskilled draftsman; the properties are altered, the straight lines drawn by an inexperienced hand have suffered unfortunate deviations and contain awkward bends. From the point of view of metric geometry, and even of projective geometry, the two figures are not equivalent; on the contrary, from the point of view of analysis situs, they are.” (Henri Poincaré, “Dernières pensées”, 1913)

"A circle no doubt has a certain appealing simplicity at the first glance, but one look at a healthy ellipse should have convinced even the most mystical of astronomers that that the perfect simplicity of the circle is akin to the vacant smile of complete idiocy. Compared to what an ellipse can tell us, a circle has nothing to say." (Eric T Bell, "The Handmaiden of the Sciences", 1937)

"Since the ellipse is a closed curve it has a total length, λ say, and therefore f(l + λ) = f(l). The elliptic function f is periodic, with 'period' λ, just as the sine function is periodic with period 2π. However, as Gauss discovered in 1797, elliptic functions are even more interesting than this: they have a second, complex period. This discovery completely changed the face of calculus, by showing that some functions should be viewed as functions on the plane of complex numbers. And just as periodic functions on the line can be regarded as functions on a periodic line - that is, on the circle - elliptic functions can be regarded as functions on a doubly periodic plane - that is, on a 2-torus." (John Stillwell, "Yearning for the impossible: the surpnsing truths of mathematics", 2006)