☰On Number Theory (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)

"Beyond the theory of complex numbers, there is the much greater and grander theory of the functions of a complex variable, as when the complex plane is mapped to the complex plane, complex numbers linking themselves to other complex numbers. It is here that complex differentiation and integration are defined. Every mathematician in his education studies this theory and surrenders to it completely. The experience is like first love." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Imaginary numbers are not imaginary and the theory of complex numbers is no more complex than the theory of real numbers." (Mordechai Ben-Ari, "Just a Theory: Exploring the Nature of Science", 2005)

"We divide math up into separate areas (analysis, mechanics, algebra, geometry, electromagnetism, number theory, quantum mechanics, etc.) to clarify the study of each part; but the equally valuable activity of integrating the components into a working whole is all too often neglected. Without it, the stated aim of ‘taking something apart to see how it ticks’ degenerates imperceptibly into ‘taking it apart to ensure it never ticks again’." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Transcendental numbers then are numerous but exceedingly slippery. As a rule of thumb, a number that arises in mathematics is almost always transcendental unless it is obvious that it is not. However, showing that a particular number is transcendental can be exceedingly difficult. Number theory throws up endless problems of this kind where everyone feels sure what the answer must be but at the same time no-one has any real idea how it could ever by proved." (Peter M. Higgins, "Number Story: From Counting to Cryptography", 2008)

"There are highly symmetric tiling patterns in hyperbolic geometry. For each of them, we can construct complex functions that repeat the same values on every tile. These are known as modular functions, and they are natural generalisations of elliptic functions. Hyperbolic geometry is a very rich subject, and the range of tiling patterns is much more extensive than it is for the Euclidean plane. So complex analysts started thinking seriously about non-Euclidean geometry. A profound link between analysis and number theory then appeared. Modular functions do for elliptic curves what trigonometric functions do for the circle." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"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)

"While number theory looks for patterns in sequences of numbers, dynamical systems actually produce sequences of numbers [...]. The two merge when mathematicians look for number-theoretic patterns hidden in those sequences." (Kelsey Houston-Edwards, "Mathematicians Set Numbers in Motion to Unlock Their Secrets", Quanta Magazine, 2021)

☰On Number Theory (1975-1999)

"The theory of number is the epipoem of mathematics." (Scott Buchanan, "Poetry and Mathematics", 1975)

"[Number theory] produces, without effort, innumerable problems which have a sweet, innocent air about them, tempting flowers; and yet […] number theory swarms with bugs, waiting to bite the tempted flower-lovers who, once bitten, are inspired to excesses of effort!" (Barry Mazur, "Number Theory as Gadfly", The American Mathematical Monthly, Volume 98, 1991)

"[…] number theory […] is a field of almost pristine irrelevance to everything except the wondrous demonstration that pure numbers, no more substantial than Plato’s shadows, conceal magical laws and orders that the human mind can discover after all." (Sharon Begley, "New Answer for an Old Question", Newsweek, 5 July, 1993)

"Number theory [...] is a field of almost pristine irrelevance to everything except the wondrous demonstration that pure numbers, no more substantial than Plato's shadows, conceal magical laws and orders that the human mind can discover after all." (Sharon Begley, "New Answer for an Old Question", Newsweek, 1993)

"At first glance the theory of numbers is deprived of any geometricity. But this is actually not the case. At the contemporary stage of development of computers it has become possible to explain to a wide range of readers that visual geometry helps not only to illustrate some abstract situations from the number theory, but sometimes also to solve new problems." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Modem geometry and topology take a special place in mathematics because many of the objects they deal with are treated using visual methods. […] Each mathematician has his own system of concepts of the intrinsic geometry of his (specific) mathematical world and visual images which he associated with some or other abstract concepts of mathematics (including algebra, number theory, analysis, etc.). It is noteworthy that sometimes one and the same abstraction brings about the same visual picture in different mathematicians, but these pictures born by imagination are in most cases very difficult to represent graphically, so to say, to draw." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Number theory is so difficult, albeit so fascinating, because mathematicians try to examine additive creations under a multiplicative light." (William Dunham, 1994)

"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)

☰On Number Theory (1950-1974)

"On the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem-proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory." (Kurt Gödel, 1951)

"The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. ‘Perfect numbers’ certainly never did any good, but then they never did any particular harm." (John E Littlewood, "A Mathematician’s Miscellany", 1953)

"The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"The modern era has uncovered for combinatorics a wide range of fascinating new problems. These have arisen in abstract algebra, topology, the foundations of mathematics, graph theory, game theory, linear programming, and in many other areas. Combinatorics has always been diversified. During our day this diversification has increased manifold. Nor are its many and varied problems successfully attacked in terms of a unified theory. Much of what we have said up to now applies with equal force to the theory of numbers. In fact, combinatorics and number theory are sister disciplines. They share a certain intersection of common knowledge, and each genuinely enriches the other." (Herbert J Ryser, "Combinatorial Mathematics", 1963)

"No branch of number theory is more saturated with mystery than the study of prime numbers: those exasperating, unruly integerst hat refuse to be divided evenly by any integers except themselves and 1. Some problems concerning primes are so simple that a child can understand them and yet so deep and far from solved that many mathematicians now suspect they have no solution. Perhaps they are ‘undecidable’. Perhaps number theory, like quantum mechanics, has its own uncertainty principle that makes it necessary, in certain areas, to abandon exactness for probabilistic formulations." (Martin Gardner, "The remarkable lore of the prime numbers", Scientific American, 1964)

☰On Number Theory (1925-1949)

"Number theory is useful, since one can graduate with it." (Edmund Landau, "Vorlesungen über Zahlentheorie", ["Lectures on Number Theory"], 1927)

"Roughly it amounts to this: mathematical analysis as it works today must make use of irrational numbers (such as the square root of two); the sense if any in which such numbers exist is hazy. Their reputed mathematical existence implies the disputed theories of the infi nite. The paradoxes remain. Without a satisfactory theory of irrational numbers, among other things, Achilles does not catch up with the tortoise, and the earth cannot turn on its axis. But as Galileo remarked, it does. It would seem to follow that something is wrong with our attempts to compass the infinite." (Eric T Bell, "Debunking Science", 1930)

"The theory of numbers is the last great uncivilized continent of mathematics. It is split up into innumerable countries, fertile enough in themselves, but all the more or less indifferent to one another’s welfare and without a vestige of a central, intelligent government. If any young Alexander is weeping for a new world to conquer, it lies before him." (Eric T Bell, "The Queen of the Sciences", 1931)

"No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years." (Godfrey H Hardy, "A Mathematician's Apology", 1941)

"The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations." (Godfrey H Hardy, "A Mathematician's Apology", 1941)

☰On Number Theory (1900-1924)

"If one intends to base arithmetic on the theory of natural numbers as finite cardinals, one has to deal mainly with the definition of finite set; for the cardinal is, according to its nature, a property of a set, and any proposition about finite cardinals can always be expressed as a proposition about finite sets. In the following I will try to deduce the most important property of natural numbers, namely the principle of complete induction, from a definition of finite set which is as simple as possible, at the same time showing that the different definitions [of finite set] given so far are equivalent to the one given here." (Ernst Zermelo,  "Ueber die Grundlagen der Arithmetik", Atti del IV Congresso Internazionale dei Matematici, 1908)

[…] theory of numbers lies remote from those who are indifferent; they show little interest in its development, indeed they positively avoid it. [..] the pure theory of numbers is an extremely abstract thing, and one does not often find the gift of ability to understand with pleasure anything so abstract. […] I believe that the theory of numbers would be made more accessible, and would awaken more general interest, if it mere presented in connection with graphical elements and appropriate figures.”  (Felix Klein, “Elementary Mathematics from an Advanced Standpoint”, 1908)

"Arithmetic does not present to us that feeling of continuity which is such a precious guide; each whole number is separate from the next of its kind and has in a sense individuality; each in a manner is an exception and that is why general theorems are rare in the theory of numbers; and that is why those theorems which may exist are more hidden and longer escape those who are searching for them." (Henri Poincaré, "Annual Report of the Board of Regents of the Smithsonian Institution", 1909)

"The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics. The accusation is one against which there is no valid defence; and it is never more just than when directed against the parts of the theory which are more particularly concerned with primes. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found it in it a mysterious attraction impossible to resist." (Godfrey H Hardy, 1915)

"The theory of numbers is unrivalled for the number and variety of its results and for the beauty and wealth of its demonstrations. The Higher Arithmetic seems to include most of the romance of mathematics." (Louis Mordell, 1917)

"The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. […] Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers." (Godfrey H. Hardy, "The Theory of Numbers", Nature 1922)

📝Misquoted: Gauss on the Queen of Mathematics

The following two quotes from Carl Friedrich Gauss are frequently met in books on Mathematics and Science (sometimes with slight differences in formulation as they were translated from German):

"Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics." 
"Mathematics is the Queen of the Sciences, and Number Theory the Queen of Mathematics."

The earliest occurrence of the quotes I met first in Eric T Bell's "Mathematics: Queen & Servant of Science" (1951), which is probably one of the most quoted sources for the quotes:

"Mathematics is Queen of the Sciences and Arithmetic the Queen of Mathematics. She often condescends to render service to astronomy and other natural sciences, but under all  circumstances the first place is her due."

However, doing a little research I found an earlier reference in John T Merz' "European Thought in the Nineteenth Century" Vol II (1903), who cites Sartorius von Waltershausen's "Gauss zum Gedächtniss" (1856) as source:

"Mathematics is the Queen of the Sciences, and arithmetic the Queen of Mathematics. She frequently condescends to do service for astronomy and other natural sciences, but to her belongs, under all circumstances, the foremost place."

Arithmetic, at least the way introduced in schools, is the study of numbers and their properties in report with the basic operations (addition, subtraction, multiplication, division, exponentiation and extraction of roots). In exchange, Number Theory appeared while studying the intrinsic properties of integers and integer-valued functions, what is known also as Higher Arithmetic. Even if the earliest discoveries date from early antiquity, the basis were put by Fermat, Euler, Gauss and others mathematicians, incorporating over time domains like Complex Analysis, Group theory or Galois theory.

According to Merriam-Webster, "Number Theory" was used for the first time around 1864 and it took some time until it was incorporated in mathematical texts. Arithmetic and Number Theory become synonymous in the 20th century (which might be strange for some). Thus, in some occurrence of the quote "Arithmetic" was replaced by "Number Theory" to reflect nowadays’ interpretation of Gauss' words. The two quotes seem to mean the same thing, though they probably need to be interpreted upon case through the historical context.  

'Queen' is used here metaphorically as an analogy between the meanings associated with a queen (ruler, fertility, authority), respectively the roles taken by Mathematics and Arithmetic in science. Therefore, a science considered as a queen could be seen as having a high fecundity of ideas and authority in another field. As "Queen of Science" were considered also Metaphysics (Immanuel Kant), Philosophy and even Theology. Also Physics and Chemistry have a statute of royalty.

The importance of Higher Arithmetic in respect to Mathematics, is stressed by Gauss in a letter to Dirichlet from 1838: 

"I place this part of Mathematics above all others (and have always done so)."

The idea is reiterated by David Hilbert half of century later:

"We thus see how arithmetic, the queen of mathematical science, has conquered large domains and has assumed the leadership. That this was not done earlier and more completely, seems to me to depend on the fact that the theory of numbers has only in quite recent times arrived at maturity." (David Hilbert, "Theorie der Algebraischen Zahlkörper", Bericht der Mathematiker-Vereinigung,' vol. IV, 1897)

Paraphrasing Godfrey H Hardy, even if Number Theory began as an experimental science, its role in the further development of Mathematics is important, worthily of the status of a 'queen'.

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George B Mathews - Collected Quotes

"A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined." (George B Mathews, "Theory of Numbers", 1892)

"As Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious." (George B Mathews, "Theory of Numbers", 1892)

"It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory." (George B Mathews, "Theory of Numbers", 1892)

"Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories."  (George B Mathews, "Theory of Numbers", 1892)

"The invention of the symbol ≡ by Gauss affords a striking example of the advantage which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic." (George B Mathews, "Theory of Numbers", 1892)

"That a formal science like algebra, the creation of our abstract thought, should thus, in a sense, dictate the laws of its own being, is very remarkable. It has required the experience of centuries for us to realize the full force of this appeal." (George B Mathews)

☰On Number Theory (1850-1899)

"The properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations." (Leonhard Euler, cca. 1856/57)

"The Mathematics, like language, (of which indeed they may be considered a species,) comprehending under that designation the whole science of number, space, form, time, and motion, as far as it can be expressed in abstract formulas, are evidently not only one of the most useful, but one of the grandest of studies." (Edward Everett, [address] 1857)

"In the Theory of Numbers it happens rather frequently that, by some unexpected luck, the most elegant new truths spring up by induction." (Carl Friedrich Gauss, Werke, 1876)

"The concept of power, which includes as a special case the concept of whole number, that foundation of the theory of number, and which ought to be considered as the most general genuine origin of sets [Moment bei Mannigfaltigkeiten], is by no means restricted to linear point sets, but can be regarded as an attribute of any well-defined collection, whatever may be the character of its elements. [...] Set theory in the conception used here, if we only consider mathematics for now and forget other applications, includes the areas of arithmetic, function theory and geometry. It contains them in terms of the concept of power and brings them all together in a higher unity. Discontinuity and continuity are similarly considered from the same point of view and are thus measured with the same measure." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)

"In science nothing capable of proof ought to be accepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in […] that part of logic which deals with the theory of numbers. In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number concept entirely independent of the notions of intuition of space and time, that I consider it an immediate result from the laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

"As Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious." (George B Mathews, "Theory of Numbers", 1892)

"Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories." (George B Mathews, "Theory of Numbers", 1892)

"The combinatory analysis in my opinion holds the ground between the theory of numbers and algebra, and is the proper passage between the realms of discontinuous and continuous quantity. It would appear advisable [...] to consider the theory of partitions an important part of combinatory analysis." (Percy A MacMahon, "Combinatory Analysis: A Review of the Present State of Knowledge", Proceedings of the London Mathematical Society Vol. s1-28 No. 1, 1896)

On Numbers: Perfect Numbers II

"It is the greatest that will ever be discovered, for, as they [perfect numbers] are merely curious without being useful, it is not likely that any person will attempt to find one beyond it." (Peter Barlow, "Theory of Numbers", 1811)

“The Perfect numbers are also like the virtues, few in number; whilst the other two classes are like the vices - numerous, inordinate and indefinite.” (W Wynn Westcott, “Numbers: Their Occult Power and Mystic Virtues”, 1911)

“The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. ‘Perfect numbers’ certainly never did any good, but then they never did any particular harm.” (John E Littlewood, “A Mathematician’s Miscellany”, 1953)

”Maybe some simple combination of a dozen or so primes in fact yield an odd perfect number!” (Stan Wagon, “The evidence: perfect numbers”, Mathematical Intelligencer 7(2), 1985)

”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)

“We should not leave unmentioned the principal numbers […] those which are called ‘perfect numbers’. These have parts which are neither larger nor smaller than the number itself, such as the number six, whose parts, three, two, and one, add up to exactly the same sum as the number itself. For the same reason twenty-eight, four hundred ninety-six, and eight thousand one hundred twenty-eight are called perfect numbers.” (Stanley J Bezuszka & Margaret J Kenney, ”Even perfect numbers”, Math. Teacher 90, 1997)

”Throughout both ancient and modern history the feverish hunt for perfect numbers became a religion.” (Clifford A Pickover, “Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning”, 2001)

"One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery - and more totally useless - than the perfect numbers." (Martin Gardner)

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Resources:Wikipedia (2018) List of perfect numbers [Online] Available from: https://en.wikipedia.org/wiki/List_of_perfect_numbers

Carl B Pomerance - Collected Quotes

"Factoring big numbers is a strange kind of mathematics that closely resembles the experimental sciences, where nature has the last and definitive word. […] as with the experimental sciences, both rigorous and heuristic analyses can be valuable in understanding the subject and moving it forward. And, as with the experimental sciences, there is sometimes a tension between pure and applied practitioners." (Carl B Pomerance, "A Tale of Two Sieves", The Notices of the American Mathematical Society 43, 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"]

"With randomness it is very unlikely to be embarrassed, but even if you get embarrassed, you can't replicate it'' (Carl B Pomerance, [lecture] 1997)

"Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems." (Carl B Pomerance, 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 & Carl B Pomerance, "Prime Numbers: A Computational Perspective", 2001)

"You can ask the question about these ancient topics, such as perfect numbers and amicable numbers [...] and ask, are these good problems [...] studying them helped us develop all of elementary number theory and from elementary number theory we developed the rest of number theory, and also you can argue that from elementary number theory came algebra." (Carl B Pomerance, "Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory", [lecture] 2013)

Barry Mazur - Collected Quotes

"[Number theory] produces, without effort, innumerable problems which have a sweet, innocent air about them, tempting flowers; and yet…number theory swarms with bugs, waiting to bite the tempted flower-lovers who, once bitten, are inspired to excesses of effort!" (Barry Mazur, "Number Theory as Gadfly", The American Mathematical Monthly, Volume 98, 1991)

"Architectural conjectures are mathematically precise assertions, as well milled as minted coins, provisionally usable in the commerce of logical arguments; less than ‘coins’ and more aptly, promissory notes to be paid in full by some future demonstration, or to be contradicted. These conjectures are expected to turn out to be true, as, of course, are all conjectures; their formulation is often away of "formally" packaging, or at least acknowledging, an otherwise shapeless body of mathematical experience that points to their truth." (Barry Mazur, "Conjecture", Synthese 111, 1997)

"A seemingly modest change of notation may suggest a radical shift in viewpoint. Any new notation may ask new questions." (Barry Mazur, "Imagining Numbers", 2003)

"If we think of square roots in the geometric manner, as we have just done, to ask for the square root of a negative quantity is like asking: ‘What is the length of the side of a square whose area is less than zero?’ This has more the ring of a Zen koan than of a question amenable to a quantitative answer." (Barry Mazur, "Imagining Numbers", 2003)

“[…] history of mathematics. It might be described as a moment of restless anticipation in the face of a slowly emerging act of imagining." (Barry Mazur, "Imagining Numbers", 2003)

 "The wonderful thing about mathematics is that, in the end as well as in the beginning, it can depend upon no authority other than one’s own (your own) mind; its verification comes from thinking alone, an activity open to anyone. If we have no theoretical equipment, we use the mathematical eyes and ears with which we were born and just experiment with our guesses to see whether we have faith in them." (Barry Mazur, "Imagining Numbers", 2003)

"At some point in his or her life every working mathematician has to explain to someone, usually a relative, that mathematics is hardly a finished project. Mathematicians know, of course, that it is far too soon to put the glorious achievements of their trade into a big museum and just become happy curators. In many respects, the study of mathematics has hardly begun." (Barry Mazur, [foreword] 2006)

"It is a hard balancing act: to explain important and beautiful mathematical ideas - to truly explain them - to people with a general cultural background but no technical training in math, and yet not to slip away from the full seriousness and ambitious goals of the subject being explained." (Barry Mazur, [foreword] 2006)

"Some mathematical models have been blindly used - their presuppositions as little understood as any legal fine print one ‘agrees to’ but never reads - with faith in their trustworthiness. The very arcane nature of some of the formulations of these models might have contributed to their being given so much credence. If so, we mathematicians have an important mission to perform: to help people who wish to think through the fundamental assumptions underlying models that are couched in mathematical language, making these models intelligible, rather than (merely) formidable Delphic oracles." (Barry Mazur, "The Authority of the Incomprehensible", 2014)

"In mathematics, we often depend on the proof of a statement to offer not only a justification of its truth, but also a way of understanding its implications, its connections to other established truths - a way, in short of explaining the statement. But sometimes even though a proof does its job of showing the truth of a result it still leaves us with the nagging question of why.’ It may be elusive - given a specific proof - to describe in useful terms the type of explanation the proof actually offers. It would be good to have an adequate vocabulary to help us think about the explanatory features of mathematics (and, more generally, of science)." (Barry Mazur, "On the word ‘because’ in mathematics, and elsewhere", 2017)

♾️On Complex Numbers VI

"We do not perceive any quantity such as that its square is negative!" (Bhāskara II, "Bijaganita", 12th century)

"These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen,) are reputed to imply that the Case proposed is Impossible. And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible that any Number (Negative or Affirmative) Multiplied into it- self can produce (for instance) -4. Since that Like Signs (whether + or -) will produce +; and there- fore not -4. But it is also Impossible that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing or any Number Fewer than None. Yet is not that Supposition(of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing. Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense." (John Wallis, in "Treatise of Algebra", 1685)

“Even zero and complex numbers are not excluded from the signification of a variable quantity.” (Leonhard Euler, “Introductio in Analysin Infinitorum” Vol. I, 1748)

“Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers” (Carl F Gauss, “Disquisitiones arithmeticae” [“Arithmetical Researches”], 1801)

"Whether or not I have found a logic, by the role of which operations with imaginary quantities are conducted, is not now the question. but surely this is evident that since they lead to right conclusions they must have a logic! […] Till the doctrines of negative and imaginary quantities are better taught than they are at present taught in the University of Cambridge, I agree with you that they had better not be taught [...]" (Robert Woodhouse, [letter to Baron Meseres] 1801)

“At the beginning I would ask anyone who wants to introduce a new function in analysis to clarify whether he intends to confine it to real magnitudes (real values of the argument) and regard the imaginary values as just vestigial –or whether he subscribes to my fundamental proposition that in the realm of magnitudes the imaginary ones a+b√−1 = a+bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analysis is, to my mind, a self-sufficient science. It would lose immeasurably in beauty and symmetry from the rejection of any fictive magnitudes. At each stage truths, which otherwise are quite generally valid, would have to be encumbered with all sorts of qualifications. “ (Carl F Gauss, [letter to Bessel,] 1811)

"Our general arithmetic, so far surpassing in extent the geometry of the ancients, is entirely the creation of modern times. Starting originally from the notion of absolute integers, it has gradually enlarged its domain. To integers have been added fractions, to rational quantities the irrational, to positive the negative .and to the real the imaginary. This advance, however, has always been made at first with timorous and hesitating step. The early algebraists called the negative roots of equations false roots, and these are indeed so when the problem to which they relate has been stated in such a form that the character of the quantity sought allows of no opposite. But just as in general arithmetic no one would hesitate to admit fractions, although there are so many countable things where a fraction has no meaning, so we ought not to deny to, negative numbers the rights accorded to positive simply because innumerable things allow no opposite. The reality of negative numbers is sufficiently justified since in innumerable other cases they find an adequate substratum. This has long been admitted, but the imaginary quantities - formerly and occasionally now, though improperly, called impossible-as opposed to real quantities are still rather tolerated than fully naturalized, and appear more like an empty play upon symbols to which a thinkable substratum is denied unhesitatingly by those who would not depreciate the rich contribution which this play upon symbols has made to the treasure of the relations of real quantities." (Carl F Gauss, "Theoria residuorum biquadraticorum, Commentatio secunda", Göttingische gelehrte Anzeigen, 1831)

“The origin and the immediate purpose for the introduction of complex number into mathematics is the theory of creating simpler dependency laws (slope laws) between complex magnitudes by expressing these laws through numerical operations. And, if we give these dependency laws an expanded range by assigning complex values to the variable magnitudes, on which the dependency laws are based, then what makes its appearance is a harmony and regularity which is especially indirect and lasting.” (Bernhard Riemann, “Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse”, 1851)

“The difficulties which so many have felt in the doctrine of Negative and Imaginary Quantities in Algebra forced themselves long ago on my attention […] And while agreeing with those who had contended that negatives and imaginaries were not properly quantities at all, I still felt dissatisfied with any view which should not give to them, from the outset, a clear interpretation and meaning [...] It early appeared to me that these ends might be attained by our consenting to regard Algebra as being no mere Art, nor Language, nor primarily a Science of Quantity; but rather as the Science of Order in Progression.” (William R Hamilton, “Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method… “, 1853) 

“The reason and the immediate purpose for the introduction of complex quantities into mathematics lie in the theory of uniform relations between variable quantities which are expressed by simple mathematical formulas. Using these relations in an extended sense, by giving complex values to the variable quantities involved, we discover in them a hidden harmony and regularity that would otherwise remain hidden.” (Bernhard Riemann, “Gesammelte Mathematische Werke”)

On Numbers: Perfect Numbers I

“A perfect number is that which is equal to the sum of its own parts.” (Euclid, “Elements”, cca. 300 BC)

“If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.” (Euclid, “Elements”, cca 300 BC)

“Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect. And those which are said to be opposite to each other, the superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too little.” (Nicomachus of Gerasa, “Introductio Arithmetica”, cca. 100 AD)

"There exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way. First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096; and then they must be totalled each time there is a new term, and at each totaling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect. If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term. If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity." (Nicomachus of Gerasa, “Introductio Arithmetica”, cca. 100 AD)

"Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect." (Saint Augustine, "The City of God", 426 AD)

“We should not leave unmentioned the principal numbers […] those which are called ‘perfect numbers’. These have parts which are neither larger nor smaller than the number itself, such as the number six, whose parts, three, two, and one, add up to exactly the same sum as the number itself. For the same reason twenty-eight, four hundred ninety-six, and eight thousand one hundred twenty-eight are called perfect numbers.” (Hrotsvit of Gandersheim, “Sapientia”, 10th century)

"[…] I think I am able to prove that there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number. But I can see nothing which would prevent one from finding numbers of this sort. For example, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3, 7, 11, 13, one would have 198585576189, which would be a perfect number. But, whatever method one might use, it would require a great deal of time to look for these numbers […]" (René Descartes, [a letter to Marin Mersenne] 1638)

“The existence of an odd perfect number – its escape, so to say, from the complex web of conditions which hem it in on all sides – would be little short of a miracle.” (James J Sylvester)

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Resources:
Wikipedia (2018) List of perfect numbers [Online] Available from: https://en.wikipedia.org/wiki/List_of_perfect_numbers

☰On Number Theory (-1849)

"[…] in the science of numbers ought to be preferred as an acquisition before all others, because of its necessity and because of the great secrets and other mysteries which there are in the properties of numbers. All sciences partake of it, and it has need of none." (Boethius, cca. 6th century)

"The properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations." (Leonhard Euler, "Specimen de usu observationum in mathesi pura" ["Example of the use of observation in pure mathematics"], cca. 1753)

"There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square." (Pierre de Fermat, [Letter to Frénicle] 1657)

"In the Theory of Numbers it happens rather frequently that, by some unexpected luck, the most elegant new truths spring up by induction.” (Carl F Gauss, Disquisitiones Arithmeticae, published in 1801)

"Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers" (Carl F Gauss, "Disquisitiones arithmeticae" ["Arithmetical Researches"], 1801)