the most auspicious tetrahedron &/or two pairs of charming mathematicians of celestial extraction

Abel, Galois, Riemann, Clifford; 

Some very basic equations and notions of analytic geometry (which goes back to Fermat and Descartes); 

Discovery/Invention of Imaginary Numbers (by Gödelian Neoplatonists); 

Charles Fourier, Le reveur sublime: une vie une oeuvre (avec René Scherer et Simone Debout/France Culture, 2002); 

Jean-Pierre Serre & Alain Connes: Alexandre Grothendieck (Fondation Hugot du Collège de France/Youtube);

The Quadratic Formula: Why do we Complete the Square? (Presh Talwalkar/Youtube);

Transforming Algebraic Function (Professor Dave Explains/Youtube); 

Shifting, Stretching, and Reflecting Parent Function Graphs (Mario's Math Tutoring/Youtube);

Graphing Polynomials (Eddie Woo/Youtube); 

Graphing Cubic Functions (Eddie Woo/Youtube); 

500 Years of NOT Teaching the Cubic Formula (Burkard Polster, Mathologer/Youtube); 

The Insolvability of the Quintic (Aleph 0/Youtube); 

The Subtle Reason Taylor Series Work (Morphocular/Youtube 2024); 

Group and Abelian Group [this is an useful and clear explanation with examples, not a proof] (Neso Academy/Youtube); 

Group Definition: Abstract Algebra (Socratica/Youtube); 

Algebraic Structures: Groups, Rings, and Fields (James Hamblin/Youtube); 

What is a Tensor? (Dan Fleisch/Youtube); 

Intro to the Fundamental Group: Algebraic Topology (Trefor Bazett & Tom Crawford/ Youtube); 

Necessity of Complex Numbers (Barton Zwiebach, MIT/Youtube, Spring 2016) [an open minded, honest & serious nutshell explanation of a fundamental question in physics & mathematics many scientists would rather not talk about];

Imaginary Numbers Are Real: Riemann Surfaces (Welch Labs/Youtube); 

Secret Kinks of Elementary Functions (Imaginary Angle/Youtube 2024); 

Colon Nancarrow's Study for Player Piano n. 41a (Youtube); 

Algebraic Operations & Geometrical Constructions on a Plane (Gödelian Neoplatonists, A/Z 2022); 

Understanding Exponentiation and Multiplication with Negative Numbers (A/Z 2022);

en exergue: accounts to reconcile, anecdotes to pick up, inscriptions to make out with the truth of the story or—fracas (!):

"Ce qui m'a donné le plus de peine a été de toucher le fond au sujet de la conception de l'espace apportée par le cubisme: problème qui touche profondément non seulement à la métaphysique mais aux dernières conceptions de la physique: de Riemann, Einstein jusqu'à Jeans et Edington. J'ai fait là très attention et j'espère que le cubisme est bien montré en ce sens: qu'il fut une véritable révolution de la conception de l'espace dans la peinture, chose aussi importante sur le plan esthétique que le furent la Réforme et le Jansénisme sur le plan théologique."

André Masson (Lettre à Daniel-Henry Kahnweiler, 1939/Écrits, anthologie établie par Françoise Levaillant)

"Les notes musicales deviennent des nombre, et si votre esprit est doué de quelque aptitude mathématique, la mélodie, l'harmonie écoutée, tout en gardant son caractère voluptueux et sensuel, se transforme en une vaste opération arithmétique, où les nombres engendrent les nombres, et dont vous suivez les phases et la génération avec une facilité inexplicable et une agilité égale à celle de l'exécutant."

Baudelaire (Les Paradis artificiels)

"As the little world of abstract mathematicians is set a-quiver by some young Frenchman's deductions on the functions of imaginary values—worthless to applied science of the day—so is the smaller world of serious poets set a-quiver by some new subtlety of cadence. Why?"

Ezra Pound (The Wisdom of Poetry) 

"The Divine Spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and not-being, which we call the imaginary root of negative unity."

Leibniz on imaginary numbers as quoted by Morris Kline (who has no understanding whatsoever of Leibniz's nature)

"In the open debate between formalist and intuitionist, since the discovery of the transfinite, mathematicians have become accustomed to summarily designate under the name Platonism any philosophy for which the existence of a mathematical entity is taken as assured, even though this entity could not be built in a finite number of steps. It goes without saying that this is a superficial knowledge of Platonism, and that we do not believe ourselves to be referring to it. All modern Plato commentators on the contrary insists on the fact that Ideas are not immobile and irreducible essences of an intelligible world, but that they are related to each other according to the schemas of a superior dialectic that presides over their arrival. The work of Robin, Stenzel and Becker has in this regard brought considerable clarity to the governing role of Ideas–numbers which concerns as much the becoming of numbers as that of Ideas. The One and the Dyad generate Ideas–numbers by a successively repeated process of division of the Unit into two new units. The Ideas–numbers are thus presented as geometric schemas of the combinations of units, amenable to constituting arithmetic numbers as well as Ideas in the ordinary sense."

Albert Lautman (Essay on the Notions of Structure and Existence in Mathematics, Simon B. Duffy's translation [it is an ill-fated instance of the malignancy of this world that it happens I don't have the French original of Lautman's essays, & shall therefore by my ashes stand forever indebted to the heroine soul (chaste star!) who peradventure send me, with all her fraternity, the French original of just at least these choicest morsels so I could definitely pen them down here as a seasonable kindness to our whole parish!])

"... Ceci montre déjà que notre étude relève de la théorie des permutations entre n éléments, théorie qui remonte à Lagrange et Galois, et qui a été poussée assez loin depuis."

André Weil (Sur l'étude algébrique de certains types de lois de mariage/Lévi-Strauss, Les Structures Élémentaires de la Parenté)

"Es treten in uns fortwährend neue Vorstellungs-massen auf, welche sehr rasch aus unserm Bewusstsein wieder verschwinden. Wir beobachten eine stetige Thätigkeit unserer Seele. Jedem Act derselben liegt etwas Bleibendes zu Grunde, welches sich bei besonderen Anlässen (durch die Erinnerung) als solches kundgiebt, ohne einen dauernden Einfluss auf die Erscheinungen auszuüben. Es tritt also fortwährend (mit jedem Denkact) etwas Bleibendes in unsere Seele ein, welches aber auf die Erscheinungswelt keinen dauernden Einfluss ausübt. Jedem Act unserer Seele liegt also etwas Bleibendes zu Grunde, welches mit diesem Act in unsere Seele eintritt, aber in demselben Augenblick aus der Erscheinungswelt völlig verschwindet. Von dieser Thatsache geleitet, mache ich die Hypothese, dass der Weltraum mit einem Stoff erfüllt ist, welcher fortwährend in die ponderablen Atome strömt und dort aus der Erscheinungswelt (Körperwelt) verschwindet..."

Bernhard Riemann (Neue mathematische Prinzipien der Naturphilosophie)

"For both Riemann and Helmholtz, the problem of hearing was a significant part of their larger enterprises, an intermediate zone in which waves, geometry, and sensation met... questions of hearing must have seemed very important to Riemann if he set them next to or even ahead of his other ambitious projects in electrodynamics, gravitation, and number theory... we can read Riemann’s 'Mechanism of the Ear' as a nascent essay 'On the Hypotheses that Lie at the Foundations of Hearing,' comparable to his earlier work on the hypotheses he considered fundamental to geometry... Though in his 1854 lecture Riemann held that 'color and the position of sensible objects are perhaps the only simple concepts whose instances form a multiply extended manifold,' by 1866 he seems poised to treat hearing as a further example of such a manifold... Where Helmholtz took evidence from hearing and seeing into his geometric investigations, Riemann traversed an opposite course, applying geometric insights to model the functioning of the ear."

Peter Pesic (Helmholtz, Riemann, and the Sirens)

"Riemann is clearly concerned that science at the time—based on Newtonian notions of causality—was heading in the wrong direction. He draws a distinction between Newton’s approach, which he characterizes as 'synthetic', and that of the philosopher Herbart, which he calls 'analytic'. Riemann had in fact studied Herbart’s work closely, and one can say they both subscribed to a similar idealistic outlook, with roots reaching back to Plato... Another idealist who was part of Riemann’s philosophical circle, Gustav Fechner was deeply concerned with trying to understand how all physical stimuli—whether vibration amplitude, sound, light, smell, pressure, or anything else—were transformed into percepts in the mind... Riemann infers that sensations need to be considered in terms of quantitative relations or measurements (that is, making use of Fechner’s law) and that we can learn about per- ception by studying the manifold structure of the higher-dimensional spaces described by these quantities... He lists four physical processes, at the top of the list being 'The absorption of elastic fluids by liquids' and the last of which is 'Galvanic currents,' which no doubt means nerve firings... Put simply, hearing involves the mind reaching out through the ear to per- ceive vibrations in the external world."

Andrew Bell, Bryn Davies, & Habib Ammari (Riemann, the Ear and an Atom of Consciousness)

"Le fondateur de la théorie des fonctions algébriques d’une variable aurait sans doute été Galois s’il avait vécu; c’est ce que permettent de penser les indications qu’on trouve sur ce sujet dans sa célèbre lettre-testament, écrite à la veille de sa mort, d’où on peut conclure qu’il touchait déjà à quelques-unes des principales découvertes de Riemann. Peut-être aurait-il donné à cette théorie une allure algébrique, conforme à l’esprit des travaux contemporains d’Abel et de ses propres recherches d’algèbre pure. Au contraire, Riemann, l’un des moins algébristes sans doute parmi les grands mathématiciens du XIXème siècle, mit la théorie sous le signe du 'transcendant' (mot qui, pour le mathématicien, s’oppose à 'algébrique', et désigne tout ce qui appartient en propre au continu)."

"C’est Dedekind, ami intime de Riemann, mais algébriste consommé, qui devait le premier tirer parti des analogies en question et en faire un instrument de recherche. Il appliqua avec succès, aux problèmes traités par Riemann par voie transcendante, les méthodes qu’il avait lui-même créées et mises au point en vue de l’étude arithmétique des nombres algébriques ; et il fit voir qu’on peut retrouver ainsi la partie proprement algébrique de l’œuvre de Riemann."

André Weil (De la métaphysique aux mathématiques)

"Les travaux qui me valent la bienveillante attention de l’Académie royale datent d’il y a vingt-cinq ans, d’une époque où je faisait partie du milieu scientifique et où je partageais pour l’essentiel son esprit et ses valeurs. J’ai quitté ce milieu en 1970 et, sans renoncer pour autant à ma passion pour la recherche scientifique, je me suis éloigné intérieurement de plus en plus du milieu des scientifiques."

"Or, dans les deux décennies écoulées l’éthique du métier scientifique (tout au moins parmi des mathématiciens) s’est dégradée à un degré tel que le pillage pur et simple entre confrères (et surtout aux dépens de ceux qui ne sont pas en position de pouvoir se défendre) est devenu quasiment une règle générale, et qu’il est en tout cas toléré par tous, y compris dans les cas les plus flagrants et les plus iniques."

"Dans ces conditions, accepter d’entrer dans le jeu des prix et des récompenses serait aussi donner ma caution à un esprit et à une évolution, dans le monde scientifique, que je reconnais comme profondément malsains, et d’ailleurs condamnés à disparaître à brève échéance tant ils sont suicidaires spirituellement, et même intellectuellement et matériellement."

"C’est cette troisième raison qui est pour moi, et de loin, la plus sérieuse. Si j’en fais état, ce n’est nullement dans le but de critiquer les intentions de l’Académie royale dans l’administration des fonds qui lui sont confiés. Je ne doute pas qu’avant la fin du siècle des bouleversements entièrement imprévus vont transformer de fond en comble la notion même que nous avons de la « science », ses grands objectifs et l’esprit dans lequel s’accomplit le travail scientifique. Nul doute que l’Académie royale fera alors partie des institutions et des personnages qui auront un rôle utile à jouer dans un renouveau sans précédent, après une fin de civilisation également sans précédent…"

Alexandre Grothendieck (lettre adressé au secrétaire perpétuel de l’Académie royale des sciences de Suède, publiée en Avril 1988 dans le journal Le Monde)

"Meanwhile, I attempted a partial translation to English [of El continuo peirceano], which generated four articles that I submitted to the Transactions of the Charles S. Peirce Society (2002). After due reception, the articles vanished in some Editor’s disorder and I never heard them again. I only hope that this anecdote amounts just to a one hundredth of the importance of Évariste Galois’ twice lost papers for the Académie des Sciences (1830)..."

Fernando Zalamea (Peirce's Logic of Continuity)

"... we need to acknowledge the true complexity of music. Electronic music, in particular, is not merely a beat machine. It undulates (under envelope control), modulates (under low-frequency oscillator or LFO control), and triggers, as well as beats. These phenomena are the result of many simultaneous clocks and functions of time applied to pitch, timbre, space, etc... Moreover, broken and jagged beats, loops and diced loops are common currency of electronic music discourse. Electronic music can deploy overlapping polyrhythms generated by algorithms and advanced clocking schemes that ambulate around zones of rhythmic morphosis. Moreover, the local temporal context is always functioning within a hierarchy of larger timescales. In all these ways, the art of rhythm is far ahead of science’s ability to track it, much less explain it."

"The epoch of the 14th-century Ars Nova in music, which saw the introduction of notated tempi, was also the age of the earliest mechanical clocks, which divided time into minutes... The promulgation of Newtonian clockwork time coincided with the birth of the classical masters J. S. Bach, G. F. Handel, and D. Scarlatti, around 1685. In their lifetimes, new musical rules for structuring time came into common practice, including time signatures, bar lines, and tempo indications, which divided time according to a uniform grid."

"Fundamental to the modern concept of rhythm is the recognition of the continuum between rhythm and tone (i.e., between the infrasonic frequencies—periodic and aperiodic rhythms—and the audible frequencies— pitched and unpitched tones). As mentioned previously, Henry Cowell described this relationship in 1930... More than two decades later, Karlheinz Stockhausen (1955) formulated another theory of the continuum between rhythm and pitch in the context of serial music."

"Conlon Nancarrow (1912–1997) made amazingly precise music using mechanically driven instruments. Operating a custom-built hole-punching machine, he produced piano rolls that drove two synchronized player pianos. Nancarrow was obsessed with the simultaneous layering of multiple tempo strands, where the tempi were related by a mathematical ratio. For example, in his Study for Player Piano 41a (1965), the tempi are related by an irrational factor, and cascades of notes sweep up and down the keyboard at superhuman speed."

"Complex contrapulsations can lead to chaotic cloud textures whose internal rhythm can only be perceived statistically. A classic example is Ligeti’s Poème Symphonique (1962) for 100 metronomes, where each metronome is set to a different tempo."

"Most European classical and popular music is based on the 12-note ET tuning, in which an octave is divided into 12 equal semitones according to a ratio of approximately 89/84. This translates into a difference between semitones of about 6%. This tuning was invented in the 1600s, but did not achieve dominance in Europe until the mid-1800s. For historical, cultural, and economic reasons, 12-note ET is dominant in global musical culture.

"A formidable advantage of 12-note ET over its predecessors was the equality of its intervals. For example, an ET 'perfect' fifth interval will sound equivalent no matter which pitches are used to form it; this is not generally true of non-ET tuning systems. Such flexibility means that a composer can write functionally equivalent melodies and chord progressions in any key. It also enables harmonic modulation (i.e., a transition from one key to another by means of a chord common to both). The same flexibility fostered the rise of atonal and serial music and the promulgation of increasingly abstract operations on pitch class sets."

"In contrast to 12-note ET, a microtonal tuning system is commonly defined as any tuning that is not 12-note ET; the intervals may be smaller or larger than a semitone. Microtonal music is sometimes referred to as xenharmonic music, the term deriving from the Greek xenia (hospitable) and xenos (foreign or strange)."

"The traditional octaviating tunings cycle is at the 2/1 ratio or octave. In contrast, many non-octaviating tunings do not include the octave interval. Their gamut or repetition cycle (if there is one) is less than or greater than 2/1."

Curtis Roads, Composing Electronic Music

"Comprendre un concept comme étant un point requiert un formalisme qui dépasse largement le simple concept d’un point de la géométrie classique... un ton est un point dans un espace de paramètres musicaux, et une partition est composée de notes, représentant des symboles pour les tons. Mais dans une compréhension de sons comme elle est pratiquée en informatique musicale, un tel objet est aussi un point dans un espace vectoriel de fonctions, fussent-elles les ingrédients de la représentation de Fourier, de la synthèse FM, des ondelettes ou de la modélisation physique. Mais déjà la compréhension classique de ce qu’est un ton inclut non seulement les notes, mais les intervalles, les accords, ou les motifs mélodiques..."

"Pour les mathématiques du vingtième siècle, le concept de point tel qu’Euclide nous le propose ('punctus est cuius pars nulla est'), était devenu obsolète car l’analyse algébrique de la géométrie avait réduit un point à la liste de ses coordonnées. Plus généralement, tous les objets mathématiques furent réduits aux ensembles dont ils sont composés suivant la fondation ensembliste des mathématiques proposée par Georg Cantor. Il n’existait plus de points au sens euclidien: tout objet mathématique, excepté l’ensemble vide, était dès lors un objet ayant des parties... en 1945, les algébristes Samuel Eilenberg et Saunders Mac Lane inventèrent la théorie des catégories, où les aspects ensemblistes furent radicalement positionnés dans l’arrière-plan de la dynamique conceptuelle. Depuis l’invention des catégories, c’étaient les homomorphismes (i.e. les fonctions respectant les structures auxquelles on s’intéressait), qui jouaient le premier rôle sur la scène des mathématiques modernes. Déjà dans le fameux traité Algebra de van der Waerden (modèle pour l’entreprise de Nicolas Bourbaki), on perçoit le désir de l’auteur de se passer des ensembles constitutifs et de mettre en relief les isomorphismes de structures."

"L'exemple plus élémentaire est la catégorie Ens des ensembles: les objets sont les ensembles, les morphismes sont les fonctions ensemblistes, et l’identité est la fonction identité d’un ensemble. Un exemple typique est la catégorie Grp des groupes: les objets sont les groupes, les morphismes sont les homomorphismes de groupes, la composition est la composition d’homomorphismes de groupes, et l’identité est l’homomorphisme identique d’un groupe. Un autre exemple est la catégorie Top des espaces topologiques: les objets sont les espaces topologiques, les morphismes sont les fonctions continues, la composition est la composition de fonctions, et l’identité est la fonction identique. Un autre exemple est la catégorie GrOr des graphes orientés et de leurs homomorphismes, ou encore la catégorie Schémas des schémas en géométrie algébrique; les exemples sont innombrables et de natures complètement différentes."

Guerino Mazzola (La vérité du beau dans la musique) 

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main hall (out of the common track): 

Niels Henrik Abel
Évariste Galois
(pictures taken from the Internet)
What did they do?
Well, conjure the most remarkable consequences of an old ghost (Vieta's logistica speciosa): "solutions" for the general algebraic equations of degree higher than four.

- Bernhard Riemann
- Kingdon Clifford
(picture taken from the Internet)

What did they do?

Riemann simply solved the following riddle (and by doing so challenged classical and common sense understanding about space, which remains pretty much our understanding till nowadays): what is real space? [tip from Gödelian neoplatonists: the real isn't the actual, the real is also and even more the virtual]. 

Clifford understood that.

"Riemann's allusions were ignored by the majority of contemporary mathematicians and physicists. His investigations were deemed too speculative and theoretical to bear any relevance to physical space, the space of experience. The only one who allied himself firmly to Riemann was the translator of his works into English, William Kingdon Clifford... already in 1870, Clifford saw in Riemann's conception of space the possibility for a fusion of geometry with physics... Clifford conceived matter and its motion as a manifestation of varying curvature [of space]... For Aristotle, space was an accident of substance; for Clifford... substance is an accident of space... These speculations aroused great opposition among academic philosophers..." (Max Jammer, Concepts of Space, Dover 1993);

"Riemann came to a deep philosophical conviction that a complete mathematical theory must be established, which would take the elementary laws governing points and transform them to the great generality of the plenum (by which he meant continuously-filled space)... the Riemann Integral calculus is defined as an infinite sum of integrals of step functions. Such infinite sums became the starting points for the study of infinite by Georg Cantor" (Amir D. Aczel, The Mystery of the Aleph, WSP 2000);

"... sein einsames Leben und dazu körperliche Leiden haben ihn im höchsten Grade hypochondrisch und misstrauisch gegen die Menschen und gegen sich selbst gemacht, wenn er auch äusserlich ganz freundlich erscheint... er ist ein sehr interessanter, wenn auch schroffer, ja zurückstossender Mensch, der schon viel von der Welt gesehen hat... Man muss Alles aufbieten, um einen so vortrefflichen und wissenschaftlich höchst bedeutenden Menschen wie Riemann aus seinem jetzt höchst unglücklichen Zustande herauszureissen; abet er darf die Absicht nicht gar zu deutlich merken; es war von jeher schwer, ihm einen Gefallen zu thun, und nur dann gelang es, ihn zur Annahme irgend einer Gefälligkeit zu bringen, wenn man ihn uberzeugen konnte, dass man ebensowohl aus Rücksicht auf sich selbst als auf ihn handelte; er hasst es andern Menschen Mühe zu machen. Er hat hier die wunderlichsten Dinge gemacht, blos aus solchen Gründen, weil er glaubt, Niemand mag ihn leiden u.s.w."

(Description of Riemann written by Richard Dedekind in a letter to Dedekind's sister [3 August 1857], quoted by Detlef Laugwitz in his Bernhard Riemann)

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fundamental ideas of modern mathematics (extracted from Richard Courant & Herbert Robbins's manual, What is Mathematics): 

The two fundamental operations of arithmetic: addition & multiplication. 

Fundamental properties of addition & multiplication (as applicable to integers): commutative "law" [a + b = b + a; ab = ba], associative "law" [a + (b + c) = (a + b) + c; (ab)c = a(bc)], distributive "law" [a(b + c) = ab + ac] (page 2);

Pascal's triangle and the binomial theorem (page 17).

The class of prime numbers (which is important because any integer can be expressed as a product of primes) (page 22).

Infinity of primes (proved "indirectly", "ad absurdum", "reductio ad impossible" by Euclides) [this should be enough to suggest the narrowness of not accepting the "law" of the excluded middle and double negation in mathematics] (page 22).

Average distribution of prime numbers among the integers (prime number theorem, conjectured by Gauss: asymptotically equal to 1/log n) (this is important because it suggests an intimate connection between two seemly unrelated mathematical concepts, prime number and function) (page 28-30).

Gauss notion of congruence (related to the repeated remainders left after the division by a given number, and related also to the geometrical representation of integers) (Fermat's [little] theorem, concerning primes and division, may be expressed with the use of this notion) (page 32-33, 37).

When going beyond counting and operating with numbers to measure quantities we need to "extend the realm of arithmetic beyond the integers" (page 52). 

Addition and multiplication of rational numbers (which express fractions involving natural numbers) preserve the commutative, associative, and distributive "laws" which hold over the integers).  

It can also be said that as much as the introduction of zero and the negative numbers among the naturals enables unrestricted subtraction, the introduction of the rational numbers (beyond and including the naturals) enables unrestricted division (except in the case of division by zero). 

Sometimes divisions by zero might be denoted "by the symbol ∞ (read 'infinity'), provided that one does not attempt to operate with the symbol ∞ as though it were subject to the ordinary rules of calculation with numbers." 

Addition, subtraction, multiplication and division may be performed unrestrictedly in the domain of the rational numbers and never lead out of it (so this domain is called a field) (page 53-56).

Although rational numbers represent points which are dense on a line, some lengths on it will turn out to be incommensurable with the unit, and they are represented by irrational numbers (such as √2) (page 59-60). 

"... all the mathematical properties of irrational points may [my emphasis] be expressed as properties of nested sequences of rational intervals" (on the basis of such postulations the arithmetic operations and "laws" previously referred can be said to hold also for the irrationals) (p. 69-70).

In Stetigkeit and irrationale Zahlen (1872) and Was sind und was sollen die Zahlen, Dedekind defines irrational numbers through the idea (my emphasis) of a cut in the set of all rational numbers (rather than referring to "specific sequence of nested [rational] intervals") (p. 71).

Cantor defines real numbers as convergent sequences of rational numbers (p. 72).

"Numbers attached to or coordinated with a geometrical object and characterizing this object completely" is the "fundamental idea of analytic geometry" brought about by Descartes and Fermat (p. 73). 

For some basic equations of analytic geometry, see the picture above.

"We cannot include the symbol ∞ in the real number system and at the same time preserve the fundamental rules of arithmetic. Nevertheless the concept of the infinite pervades all mathematics..." (p. 77).

We can identify the equivalence of two finite sets without necessary counting them (by establishing a biunique correspondence). Cantor extended this idea to infinites (p. 78).

Sets containing infinite many objects might be equivalent to subsets of themselves (for example, it's possible to establish a biunique correspondence between the positive integers and the even integers) (p. 79). 

The set of rational numbers is equivalent to the set of the integers (they are both denumerable, as are the rational points in a line) (p. 80).

The set of the real numbers (that is, with irrationals) is non-denumerable (it pertains to an infinite whose type is higher than the infinite of rationals and integers) (proved indirectly, that is ad absurdum, by Cantor, with the help of the ingenious diagonal argument) [together with Euclides proof of the infinity of primes, this should be enough to suggest the narrowness of not accepting the "law" of the excluded middle and double negation in mathematics]. The continuum is also non-denumerable, and even a finite segment of the continuum "contains a non-denumerable infinity of points" (p. 81).

Not all infinite sets are equivalents. There are at least two types of infinity (numerable and non-numerable) (p. 84).

The cardinal number of the set of points in a square is the same as the cardinal number of the set of points in a line, that is, equivalent to the set of real numbers and the set of points in the continuum (they are all non-denumerable sets) (p. 85).

Critics such as Kronecker and Poincaré objected to "the vagueness" of the notion of set and pointed to the non-constructive character of indirect, ad absurdum, proofs (that is, proofs relying on the principle of the excluded middle) (p. 86).

"... the Hypothesis of the Continuum states that there is no set whose cardinal number is greater than that of the set of the integers but less than that of the set of real numbers" (p. 88).

"For many reasons the concept of number has had to be extended even beyond the real number continuum, by the introduction of the so-called complex numbers..." (p. 88).

"The process which first requires the use of complex numbers is that of solving quadratic equations..."  (rational numbers are enough for linear equations, real numbers are necessary but insufficient "to provide a complete theory of quadratical equations") [see figure above] (p. 89).

[***Courant & Robbins say the following in page 89: "of course, this object i, the 'imaginary unit,' has nothing to do with the concept of a number as a mean of counting. It is purely a symbol... and its value will depend entirely on whether by this introduction a really useful and workable extension of the number system can be effected." I appreciate the notion of 'imaginary unit', but what they say on the whole amounts to a very narrow view. First of all, it should apply to the reals themselves. Since Courant and Robbins read Hilbert in a strictly formalistic way (see page 88), and seem to adopt this strictly formalistic view of mathematics themselves, they might accept this consequence (which shows, on the other hand, how strict formalists are in the end completely equivalent to inveterate intuitionists, and we hope their sets are not only both always denumerable but indeed quite finite). Gödelian Platonists like myself have a very different view. I would rather ignore the villainous and pretentious (not to say malignant, blatantly authoritarian, and antidemocratic) things Courant & Robins say about Leibniz (such as in page 434, but see also the mea culpa at pages 518-32), and limit myself to the Deleuzian well-grounded point that notions such as differentiation and derivative have philosophical implications surpassing (although not necessarily contradicting) their strict scientific (or rather positivistic, narrowly-"formalist") formulations (which just by themselves are otherwise: ok).]

Commutative, associative and distributive "laws" hold for complex numbers (and they form a field, since they are closed under subtraction and division besides addition and multiplication, division by zero as well excluded). The field of the complex numbers includes the field of the reals, and a complex number of the form "0 + bi" is considered an imaginary number (that is, it is just "bi") (p. 90).

Complex numbers might be interpreted geometrically. People do so by considering its real part an x-coordinate, and its imaginary part an y-coordinate (points on the x-axis correspond then to "x + 0i" and points on the y-axis to "0 + yi") (p. 92).

Complex numbers are understood as having "conjugates", which are equivalent to other complex numbers that might be represented in the number plane (when complex numbers are interpreted geometrically) by a mirror reflection over the x-axis. 

The distance of a point z (that is, a point represented by a complex number in the number plane) to the origin of the graphic (0,0 coordinates) is called its "module" (given by a real number) [this is easy to understand if you consider that lines can be expressed by quadratic equations, which when solved enable one to substitute for the imaginary unit i (that is, √-1) a real number].

When z lies on the x-axis its module is "its ordinary absolute value".

Complex numbers with module 1 lie on the unit circle (circle with radius 1 and center at the origin of the graphic) (p. 93).

Complex numbers (as represented in a plane) and their conjugates have angles, the angle of a conjugate is the negation of the angle of its correspondent complex number (these angles are multi-determined since the same position on the plane might be represented by adding or subtracting multiples of 360) (complex numbers can be represented in terms of trigonometric functions) (p. 94-95). 

"... in the field of complex numbers there are exactly n different nth roots of 1" (while in the field of the reals there is only 1 root for three to the power of three, in the complex field there are three) (the so-called "De Moivre's Formula" shows this) (these roots are represented by vertices of "n-sided regular polygons inscribed in the unit circle and having the point z = 1) (p. 98). 

"Every algebraic equation of any degree n with real or complex coefficients has solutions in the field of complex numbers" (fundamental theorem of algebra, proved by Gauss in 1799; Cardano, Tartaglia and others had already established this for equations of the 3rd and 4th degrees, which they solved with the help of formulas similar to the quadratic") (p. 101). 

Algebraic numbers are real or complex numbers that are roots of equations with integers coefficients of 2nd or higher degree (even when they cannot be expressed in terms of radicals). Real non-algebraic numbers are called transcendental (Cantor proved their existence indirectly and Liouville gave a prove that enables one to construct them) (p. 104).

Through geometrical constructions with rule and compass [on a plane], one can perform "'rational' algebraic processes" (addition, subtraction, multiplication, and division) "of known quantities" [a single unit length would enable the construction of all the rationals]. The totality does obtained is a number field. The extraction of a square root is what enables one to go beyond [to extend] that [given] field (p. 122). 

By using rule and compass [on a plane], one can connect two points [with the rule, by drawing a straight line],  one can find "the point of intersection of two lines, drawing a circle with a given radius about a point,"  and one can find "the points of intersection of a circle with another circle or with a line" (p. 127). 

By using [the rule and] the compass [just the rule by itself is not enough for this], one can extend a field F "by selecting any number k of F, extracting the square root of k, and constructing [when √k wasn't itself already in F] the field F', consisting of the numbers a + b√k, where a and b are in F." One can reach "n" numbers of fields in this way. All numbers constructed in this way are algebraic [if F are  rationals, F' are roots of quadratic equations (only the ones which were not already rationals) (is it the case, for instance, of √2, which is an algebraic number of degree 2, but also irrational?) (is it the case, for instance, of 1+ √-1, which is what brings about "imaginary" numbers?), F'' are roots of quartic equations (only the ones which are not roots of quadratic equations) etc.] (p. 131-32).

One can understand why it is impossible to solve by straightedge and compass old ancient problems such as doubling the cube, by understanding how a solution to equations involved in such problems cannot be obtained by any extension of fields departing from rationals [would such problems always lead to transcendental irrational numbers?] (p. 134).

"The idea of classifying the different branches of geometry according to the classes of transformation considered was proposed by Felix Klein" (Erlanger Program) (p. 167).

"A decisive discovery of projective geometry" is that "if we have four points A, B, C, D on a straight line, and project these into A', B', C', D' on another line, then there is a certain quantity, called the cross-ratio of the four points, that retains its value under projection" (p. 173).

One might define also the also cross-ratio of four coplanar and concurrent straight lines, and of four coaxial planes (p. 176). 

"If a straight line that intersects another is rotated slowly towards a parallel position, then the point of intersection of the two lines will recede to infinity" (p. 180). 

"... if lines are not parallel, they will intersect in an ordinary point, while if the lines are parallel they will intersect in the ideal point (point at infinity) of the two lines... According to our conventions, a point at infinity is determined or is represented by any family of parallel lines, just as an irrational number is determined by a sequence of nested rational intervals..." [projective geometry] (p. 182).

"In analytic geometry, the 'coordinates' of a geometrical object are any set of numbers which characterize that object uniquely... a point is defined by giving its rectangular coordinates or its polar coordinates, while a triangle can be defined by given the coordinates of its three vertices... a straight line is the geometrical locus of all points whose coordinates satisfy some linear equation ax + by + c = 0 [for instance, when a = 0, b = 1 and c = 0, the line is the x-axis]... quadratic equations define 'conic sections..." (p. 192). 

Centrality of the concepts of function and limit to modern mathematics (p. 272). 

Functions of a continuous variable [when the variable's domain is a continuous interval of the real number axis] "are often defined by algebraic expressions" (p. 274). 

"A mathematical function is simply a law governing the interdependence of variable quantities" (p. 276). 

"The character of a function is mostly clearly shown by a simple geometrical graph" (p. 278). 

Jump discontinuity and infinite discontinuity [of functions] (p. 284-85).

Convergent sequences and limit (definition) (p. 292).

Monotone increasing and monotone decreasing sequences. "Any monotone increasing sequence that has an upper bound must converge to a limit" (relation to the notions of real number and limit) (p. 295-96).

Numbers e and π (p. 298-99). 

Limit of a function (of continuous variable) (p. 303, cf. 311) .

"... a clear understanding and a precise definition of limits had long been blocked by an apparently unsurmountable difficulty... when it comes to checking the existence of a limit in actual scientific procedure it is the (e, ∂)-definition that must be applied... [but both this definition and "the axioms of geometry"] leave out something that is real to the intuition..." (p. 305-306).  

"All polynomials, rational functions, and trigonometric functions are continuous, except for isolated values of x where the function may become infinite" (p. 311).

"In daily life problems of maxima and minima, of the 'best' and the 'worst,' arise constantly... For example, how should a boat be shaped so as to have the least possible resistance to water?" (p. 329).

"In its historic development, the differential calculus was strongly influenced by individual maximum and minimum problems..." (p. 342). 

"It is geometrically evident that at a maximum or minimum of a smooth curve y = f(x) the tangent to the curve must be horizontal, that is, its slope must be equal to zero. Thus we have the condition f'(x) = 0, for the extreme values of f(x)" (p. 342). 

"There are problems of maxima and minima that cannot be expressed in terms of a function of one variable. The simplest such case is that of finding the extreme values of a function z = f(x, y) of two variables... We can represent f(x, y) by the heigh z of a surface above the x, y-plane, which we may interpret, say, as a mountain landscape... a minimum to the bottom of a depression or of a lake" (p. 343).

"... corresponding to types of points with horizontal tangents [such as maxima, minima, and saddle points] we have different types of stationary values f(x, y)" ("intimate connection between the general theory of stationary points and the concepts of topology") (p. 344-45).

"For the distance between a point P and a closed curve there are (at least) two stationary values, a minimum and a maximum. Nothing new occurs if we try to extend this result to three dimensions, so long as we consider a surface C topologically equivalent to a sphere... But new phenomena appear if the surface is of a higher genus, e. g. a torus..." (p. 345-46).

Ergodic trajectories (p. 355).

Riemann's doctoral thesis (1849), solution of extremum problems, existence of a minimum, Dirichlet's principle, Weierstrass' criticism (p. 367-68). 

Independent variables [of a function] which fail to be compact, functions which fail to be continuous (p. 372). 

Euler and Langrange, the calculus of variations (p. 381). 

Fermat, law of refraction of light, principle of geometrical optics (p. 382). 

Hamilton's rediscover and extension of "Euler's variational principles of physics" (p. 383).

Difficulty of calculating areas bounded not by polygons but by curves, crucial question that went from Archimedes (exhaustion method) to the integral calculus (p. 400).

The concept of a derivative as formulated in the 17th century by Fermat (while studying maxima and minima of functions) and others. "To characterize the points of maximum and minimum it is natural to use the notion of tangent of a curve. We assume that the graph has no sharp corners or other singularities and that at every point it possesses a definite direction given by a tangent line"  (p. 415). 

"The differentiability of a function implies its continuity. For, if the limit of ∆x/∆y exists as ∆x tends do zero, then it is easy to see that the change ∆y of the function f(x) must become arbitrarily small as the difference ∆x tends to zero" (p. 422). 

Importance of the notion of derivative to "calculating the rate of change of some quantity f(t) which varies with the time t" (Newton) (velocity as "instantaneous rate of change" in relation to an "average rate", acceleration as "rate of change of velocity") (p. 423).

Differentiation and integration as processes inverse to one another (fundamental theorem of the calculus) (p. 436).

Log x as a "monotone increasing function" (importance of the formula log a + log b = log (ab)) (p. 444). 

Weierstrass' "continuous function whose graph does not have a tangent at any point" (continuous and nowhere differentiable) (p. 464).

Infinite series as limits of sequences of finite partial sums (when the limit exists the series is convergent, when it doesn't the series is divergent) (p. 472).

Power series (p. 473).

Harmonic series (p. 479).

"In 1964 Paul Cohen proved that the truth of the Continuous Hypothesis depends upon which axioms for set theory are chosen. The situation is similar to that for geometry. The truth or falsity of Euclid's parallel axiom depends upon the type of geometry... Earlier Kurt Gödel had proved that the Continuum Hypothesis is true in some axiomatizations of set theory... there is no distinguished choice of axioms that leads to a unique 'natural' theory of sets" (p. 494).  

Hausdorff-Besicovitch dimension of a set (fractal dimension) (need not to be an integer, and by so differs from Poincaré's conception of dimension) (p. 499).

Nonstandard analysis and non-Archimedean numbers systems (p. 519-20).  

more on the imaginary system: 

"What, it may be asked, is the nature of the Geometry in which the coordinates of any point may be complex quantities of the form as x + ix', y + iy', z + iz'? Such a Geometry contains as a particular case the Geometry of real points. From it the Geometry of real points may be deduced" [these only apparently pedestrian sentences are certainly worth of a Gödelian Platonist, and earn many praises to its author]. 

"... given any real straight line I and any real point on it, there are on the straight line an infinite number of pairs of imaginary points. Such a system of points may be termed an imaginary system, base I, centre or mean point 0."

"A real length and an imaginary length are incommensurable, and do not in themselves involve any relative magnitudes. Like two real incommensurable quantities √2 and 3, whose squares are commensurable, the squares of a real and an imaginary quantity may be commensurable."

"The position of a real point on a given base can only be determined graphically when the unit is known in which its distance from a given point on the base is expressed. Similarly, the position of an imaginary point is only known when the unit in which its imaginary distance is expressed is also known. The units in the two cases may be regarded as √+1 and  √-1. As there is no inherent relation as to magnitude between these units, the relative position of real and imaginary points on the base is indeterminate."

"So long as the lengths considered are all real or all purely imaginary each system may be graphed in the same way, the quantities √+1 and √-1 being regarded as units in which the lengths are expressed, the only difference between these units lying in the fact that in one case the square on a line is regarded as positive and in the other negative."

"If is a real point on a straight line and P a point on the straight line at a distance √-1 K from 0, the point P may be taken as the centre of an involution."

"On any real straight line some point may be taken which may be termed a base point. An infinite number of real and an infinite number of imaginary points may be obtained by measuring real or purely imaginary distances from this base point... The imaginary points form a new system of points. An imaginary point of the first system may be taken as a base point and an infinite number of points real with respect to this base point may be obtained and also an infinite number of points imaginary with respect to this base point. The real points with respect to this centre are imaginary points with respect to the base point first taken and are distinct from those obtained from the original base point."

"... imaginary points occur in pairs, viz. in pairs of conjugate imaginary points, and the connector of any pair is the base of the involution of which they are the double points. When an imaginary point is given, the involution of which it is a double point is completely determined, for the centre and constant of the involution are known."

"A given pair of real points are conjugate points of an infinite number of involutions with imaginary double points. The double points of these involutions are harmonic conjugates of the given pair of points. Hence a given pair of real points has an infinite number of pairs of harmonic conjugates which are pairs of conjugate imaginary points."

"Two ranges of real or imaginary points are said to be projective when the anharmonic ratio of four points of one range is equal to the anharmonic ratio of the four corresponding points of the other range... there is in each of two projective ranges, one point termed the vanishing point, which corresponds to the point at infinity in the other range. The vanishing points may be real or imaginary."

- John Leigh Smeathman Hatton, The Theory of the Imaginary in Geometry

from a History of Mathematics (wrote upon strong vellum): 

"... geometry, like counting, had an origin in primitive ritualistic practice. The earliest geometric results found in India constituted what were called the Sulvasutras, or 'rules of the cord.'These were simple relationships that apparently were applied in the construction of altars and temples. It is commonly thought that the geometric motivation of the 'rope-stretchers' in Egypt was more practical than that of their counterparts in India; but it has been suggested that both Indian and Egyptian geometry may derive from a common source—a protogeometry that is related to primitive rites in somewhat the same way in which science developed from mythology and philosophy from theology." 

''The Egyptians early had become interested in astronomy and had observed that the annual flooding of the Nile took place shortly after Sirius, the dogstar, rose in the east just before the sun. By noticing that these heliacal risings of Sirius, the harbinger of the flood, were separated by 365 days, the Egyptians established a good solar calendar made up of twelve months of thirty days each and five extra feast days..."

"Recognition by the Egyptians of interrelationships among geometric figures, on the other hand, has too often been overlooked, and yet it is here that they came closest in attitude to their successors, the Greeks."

"The fertile Nile Valley has been described as the world's largest oasis in the world's largest desert. Watered by one of the most gentlemanly of rivers and geographically shielded to a great extent from foreign invasion, it was a heaven for peace-loving people who pursued, to a large extent, a calm and unchallenged way of life."

"One is almost tempted to see in it [Babylonian ratio calculations] the genuine origin of geometry, but it is important to note that it was not so much the geometric context that interested the Babylonians as the numerical approximations that they used in mensuration. Geometry for them was not a mathematical discipline in our sense, but a sort of applied algebra or arithmetic in which numbers are attached to figures."

"It is indeed difficult to separate history and legend concerning the man [Pythagoras], for he meant so many things to the populace—the philosopher, the astronomer, the mathematician, the abhorrer of beans, the saint, the prophet, the performer of miracles, the magician, the charlatan."

"One of the tantalizing questions in Pythagorean geometry concerns the construction of a pentagram or star pentagon... In each case a diagonal point divides a diagonal into two unequal segments such that the ratio of the whole diagonal is to the larger segment as this segment is to the smaller segment. This subdivision of a diagonal is the well-known 'golden section' of a line segment, but this name was not used until a couple of thousand years later... The construction required is equivalent to the solution of a quadratic equation... Pythagoras could have learned from the Babylonians how to solve this equation algebraically. However, if a is a rational number, then there is no rational number x satisfying the equation. Did Pythagoras realized this?"

"... when the five diagonals of a regular pentagon are drawn, these diagonals form a smaller regular pentagon, and the diagonals of the second pentagon in turn form a third regular pentagon, which is still smaller. This process can be continued indefinitely, resulting in pentagons that are as small as desired and leading to the conclusion that the ration of a diagonal to a side in a regular pentagon is not rational. The irrationality of this ratio is, in fact, a consequence of the argument... in which the golden section was shown to repeat itself over and over again. Was it perhaps this property that led to the disclosure, possibly by Hippasus, of incommensurability?"

"The Pythagoreans had assumed that space and time can be thought of as consisting of points and instants; but space and time have also a property, more easily intuited than defined, known as 'continuity.'"

"... Plato in the Thaetetus says that his teacher, Theodorus of Cyrene—of whom Thaetetus also was a pupil—was the first to prove the irrationality of the square roots of the nonsquare integers from 3 to 17 inclusive."

"Plato looked upon the dodecahedron as composed of 360 scalene right triangles, for when the five diagonals and five medians are drawn in each of the pentagonal faces, each of the twelve faces will contain thirty right triangles. The association of the first four regular solids with the traditional four universal elements provided Plato in the Timaeus with a beautifully unified theory of matter according to which everything was constructed of ideal right triangles... He emphasized that the reasoning used in geometry does not refer to the visible figures that are drawn but to the absolute ideas that they represent..."

"Inasmuch as he hesitated to follow Platonic mathematicians into the abstractions and technicalities of the day, Aristotle made no lasting contribution to the subject. He is said to have written a biography of Pythagoras, although this is lost..."

"Among Euclid's lost works are also one on Surface Loci, another on Pseudaria (or fallacies), and a third on Porisms... The loss of Euclidean Porisms is particularly tantalizing, for it may have represented an ancient approximation to an analytic geometry."

"Book X of the Elements was, before the advent of early modern algebra, the most admired—and the most feared. It is concerned with a systematic classification of incommensurable line segments... Today we would be inclined to think of this as a book on irrational numbers... Book X contains 115 propositions—more than any other—most of which contain geometric equivalents of what we now know arithmetically as surds."

"No account of Ptolomy's work would be complete without mention of his Tetrabiblos (or Quadripartitum), for it shows us a side of ancient scholarship that we are prone to overlook..."

"The full title of [Boethius's] Expositio indicates, in fact, that it is an exposition of mathematical matters useful to an understanding of Plato. It explains, for example, that the tetractys consisting of the numbers 1, 2, 3, and 4 contains all the musical consonances inasmuch as it makes up the ratios 4:3, 3:2, 2:1, 3:1 and 4:1."

"In a sense it is not fair to criticize Diophantus for being satisfied with a single answer, for he was solving problems, not equations. In a sense the Arithmetica is not an algebra textbook, but a problem collection in the application of algebra."

"The idea of negative numbers seems not to have occasioned much difficulty for the Chinese since they were accustomed to calculating with two sets of rods—a red set for positive coefficients or numbers and a black set for negatives."

"Yang Hui's works included also results in the summation of series and the so-called Pascal triangle, things that were published and better known through the Precious Mirror of Chu Shih-chieh..." 

"The Mayas of Yucatan, in their representation of time intervals between dates in their calendar, used a place value numeration, generally with twenty as the primary base and with five as an auxiliary (corresponding to the Babylonian use of sixty and ten respectively)."

"It should be mentioned also that the Hindus, unlike Greeks, regarded irrational roots of numbers as numbers. This was of enormous help in algebra... Hindu algebra is especially noteworthy in its development of indeterminate analysis, to which Brahmagupta made several contributions... It is greatly to the credit of Brahmagupta that he gave all integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation."

"... ultimately the scheme of numeration making use of the Hindu numerals came to be called simply algorism or algorithm, a word that, originally derived from the name al-Khwarizmi, now means, more generally, any peculiar rule of procedure or operation—such as the Euclidean method for finding the greatest common divisor."

"Omar Khayyam (ca. 1050-1123), the 'tent-maker,' wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree."

"The book in which Fibonacci described the new algorism is a celebrated classic, completed in 1202, but it bears a misleading title—Liber abaci (or Book of the Abacus). It is not on the abacus; it is a very through treatise on algebraic methods and problems in which the use of Hindu-Arabic numerals is strongly advocated."

"Perhaps through the Arabs Fibonacci had learned what we call 'Horner's method,' a device known before this time in China. This is the most accurate European approximation to an irrational root of an algebraic equation up to that time..."

"Even more imaginative than Oresme's notations was his suggestion that irrational proportions are possible."

"In 1545, the solution not only of the cubic but of the quartic as well became common knowledge through the publication of the Ars magna of Geronimo Cardano (1501-1576)."

"By birth illegitimate, and by habit an astrologer, gambler, and heretic, Cardan nevertheless was a respected professor at Bologna and Milan, and ultimately he was granted a pension by the pope."

"The solution of cubic and quartic equations [by Cardano] was perhaps the greatest contribution to algebra since the Babylonians, almost four millennia earlier, had learned how to complete the square for quadratic equations."

"Another immediate result of the solution of the cubic was the first significant glance at a new kind of number. Irrational numbers had been accepted by the time of Cardan, even though they were not soundly based, for they are readily approximated by rational numbers. Negative numbers afforded more difficulty because they are not readily approximated by positive numbers, but the notion of sense (or direction on a line) made them plausible. Cardan used them even while calling them numeri ficti..."

"Whenever the three roots of a cubic equation are real and different from zero, the Cardan-Tartaglia formula leads inevitably to square roots of negative numbers. The goal was known to be a real number, but it could not be reached without understanding something about imaginary numbers."

"Physics and astronomy had reached the point where there was increasing need for arguments concerning the infinitely large and small—the subject now known as analysis. Viète had been one of the first to use the word 'analysis' as a synonym for algebra, but he was one of the earliest analysts also in the more modern sense of one who studies infinite processes."

"Stevin, Kepler, and Galileo all had need for Archimedean methods, being practical men, but they wished to avoid the logical niceties of the method of exhaustion. It was largely the resulting modifications of the ancient infinitesimal methods that ultimately led to the calculus, and Stevin was one of the first to suggest changes."

"The infinitely small was of more immediate relevance to Galileo than the infinitely large, for he found it essential in his dynamics."

"Salviati then concludes that by bending the line segment into the shape of a circle, he has 'reduced to actuality that infinite number of parts into which you claimed, while it was straight, were contained in it only potentially,' for the circle is a polygon of an infinite number of sides... Galileo had intended to write a treatise on the infinite in mathematics, but it has not been found." 

"... the world remembers Cavalieri [disciple of Kepler] for one of the most influential books of the early modern period, the Geometria indivisibilibus continuorum, published in 1635."

"For a number of years Descartes traveled about in conjunction with varied military campaigns, first in Holland with Maurice Prince of Nassau, then with Duke Maximillian I of Bavaria, and later still with the French army at the siege of LaRochelle... The entire universe, he postulated, was made up of matter in ceaseless motion in vortices, and all phenomena were to be explained mechanically in terms of forces exerted by contiguous matter."

"The work of Descartes far too often is described simply as the application of algebra to geometry, whereas actually it could be characterized equally well as the translation of the algebraic operations into the language of geometry."

"... it is appropriate to follow Laplace in acclaiming Fermat as the discoverer of the differential calculus, as well as a codiscoverer of analytic geometry. Obviously Fermat was not in possession of the limit concept, but otherwise his method of maxima and minima parallels that used in calculus today..."

"The result was one of the most unsuccessful great books ever produced. Even the ponderous title was repulsive—Brouillon projet d'une atteinte aux événements des rencontres d'un cone avec un plan (Paris, 1639).... Moreover Girard Desargues used a bizarre new vocabulary full of terms borrowed from botany, a terminology that repelled scholars and practitioners alike."

"... in 1640, the young Pascal, then sixteen years old, published an Essay pour les coniques. This consisted of only a single printed page—but one of the most fruitful pages in history. It contained the proposition, described by the author as mysterium hexagrammicum, which has ever since been known as Pascal's theorem."

''On the night of November 23, 1654, from 10:30 to about 12:30, Pascal experienced a religious ecstasy which caused him to abandon science and mathematics for theology."

"Pietro Mengoli had rediscovered Oresme's conclusion, arrived at by a grouping of terms, that the ordinary harmonic series does not converge, a theorem usually attributed to Jacques Bernoulli in 1689; he also showed the convergence of the reciprocals of the triangular numbers, a result for which Huygens usually is given credit."

"Even proportion, the stronghold of ancient geometry, Wallis held to be an arithmetic concept. In this his attitude represented the tendency of mathematics for at least the following century, but it should be remarked that such a movement was without a solid foundation, since real numbers had not been defined."

"[In Arithmetica infinitorum, published in 1655] Wallis arithmetized the Geometria indivisibilibus of Cavalieri, as he had arithmetized the Conics of Apollonius... Wallis abandoned the geometric background after having associated the infinitely many indivisibles in the figures with numerical values. If, for example, one wishes to compare the squares of the indivisibles in the triangle with the squares of the indivisibles in the parallelogram, one takes the length of the first indivisible in the triangle as zero, the second as one, the third as two, and so on up to the last, of length n - 1, if there are n indivisibles."

"[Among Wallis] most important contributions in infinitesimal analysis [was his anticipation] of Euler on the gamma or factorial function."

" Thomas Hobbes (1588-1679) was foremost among those who criticized Wallis' arithmetization of geometry, objecting strenuously to 'the whole herd of them who apply their algebra to geometry' and referring to the Arithmetica infinitorum as 'a scab of symbols.'"

"... it is likely that it was in Italy, through Mengoli and Angeli, that James Gregory came to appreciate the power of infinite series expansions of functions and of infinite processes in general."

"Newton's first discoveries, dating from the early months of 1665, resulted from his ability to express functions in terms of infinite series—the very thing that Gregory was doing in Italy at about that time."

"... Newton's indirect approach... made clear to him that one could operate with infinite series in much the same way as with finite polynomial expressions... he had found that the analysis by infinite series had the same inner consistency and was subject to the same general laws as the algebra of finite quantities. Infinite series were no longer to be regarded as approximating devices only; they were alternative forms of the functions they represented... he wrote in De analysi per aequationes numero termonorum infinitas, composed in 1669: '... albeit we Mortals whose reasoning Powers are confined within narrow Limits, can neither express, nor so conceive all the Terms of these Equations as to know exactly from thence the Quantities we want...'"

"As was the case with Newton, infinite series played a large role in the early work of Leibniz."

"... it was on reading the letter of Amos Dettonville on Traité des sinus du quart de cercle that Leibniz reported that a light burst upon him. He then realized, in about 1673, that the determination of the tangent to a curve depended on the ration of the differences in the ordinates and abscissas, as these became infinitely small, and that quadratures depended on the sum of the ordinates or infinitely thin rectangles making up the area. Just as in the arithmetic and harmonic triangles the processes of summing and differencing are oppositely related, so also in geometry the quadrature and tangent problems, depending on sums and differences respectively, are inverses of each other. The connecting link seemed to be through the infinitesimal or 'characteristic' triangle, for where Pascal had used it to find the quadrature of sines, Barrow had applied it to the tangent problem."

"Leibniz by 1676 had arrived at the same conclusion that Newton had reached several years earlier, namely, that he was in possession of a method that was highly important because of its generality. Whether a function was rational or irrational, algebraic or transcendental (a word that Leibniz coined), his operations of finding sums and differences could always be applied."

"Whereas Descartes' geometry had once excluded all nonalgebraic curves, the calculus of Newton and Leibniz showed how essential is the role of these in their new analysis... Moreover, Leibniz seems to have appreciated, as did Newton, that the operations in the new analysis can be applied to infinite series as well as to finite algebraic expressions."

"The ambivalent status of complex numbers is well illustrated by the remark of Leibniz, who was also a prominent theologian, that imaginary numbers are a sort of amphibian, halfway between existence and nonexistence."

"The fourth and last part [of Jacques Bernoulli's Ars Conjectandi] contains the celebrated theorem that now bears the author's name, and on which Bernoulli and Leibniz had corresponded: the so-called 'Law of large numbers.'"

"Berkeley [in The Analyst] did not deny the utility of the techniques of fluxions nor the validity of the results obtained by using these; but he had been nettled on having a sick friend refuse spiritual consolation because Halley had convinced the friend of the untenable nature of Christian doctrine."

"Euler spent almost all of the last seventeen years of his life in total darkness."

"It may fairly be said that Euler did for the infinite analysis of Newton and Leibniz what Euclid had done for the geometry of Eudoxus and Theaetetus, or what Viète had done for the algebra of Al-Khwarizmi and Cardan. Euler took the differential calculus and the method of fluxions and made them part of a more general branch of mathematics which ever since ha been known as 'analysis'—the study of infinite processes... From 1748 [the time of publication of Euler's Introductio in analysin infinitorum] onward the idea of 'function' became fundamental in analysis. It had been adumbrated in the medieval latitude of forms, and it had been implicit in the analytic geometry of Fermat and Descartes, as well as in the calculus of Newton and Leibniz."

"The first volume of the Introductio is concerned from start to finish with infinite processes—infinite products and infinite continued fractions, as well as innumerable infinite series. In this respect the work is the natural generalization of the views of Newton, Leibniz, and the Bernoullis, all of whom were fond of infinite series."

"Opposing the views of Leibiniz and Euler, d'Alembert insisted that 'a quantity is something or nothing; if it is something, it has not yet vanished; if it is nothing, it has literary vanished... d'Alembert held that the differential notation is merely a convenient manner of speaking that depends for its justification on the language of limits. His Encyclopédie article on the differential referred to Newton's De quadratura curvarum, but d'Alembert interpreted Newton's phrase 'prime and ultimate ratio' as a limit rather than as a first or last ratio of two quantities just springing into being... D'Alembert denied the existence of the actually infinite, for he was thinking of geometric magnitudes rather than of the theory of aggregates proposed a century later."

"In his selection of the unifying principle, however, Carnot made a most deplorable choice. He concluded that 'the true metaphysical principles' are 'the principles of the compensation of errors.' Infinitesimals, he argued, are 'quantités inappréciables" which, like imaginary numbers, are introduced only to facilitate the computation and are eliminated in reaching the final result."

"The whole motive of [Lagrange's function theory] was not to try to make the calculus more utilitarian, but to make it more logically satisfying... In 1767 Lagrange published a memoir on the approximation of roots of polynomial equations by means of continued fractions; in another paper in 1770 he considered the solvability of equations in terms of permutations on their roots. It was the latter work that was to lead to the enormously successful theory of groups and to the proofs by Galois and Abel of the unsolvability, in the usual terms, of equations of degree greater than four." 

"The new branch of geometry that Gauss initiated in 1827 is known as differential geometry, and it belongs perhaps more to analysis than to the traditional field of geometry. Ever since the days of Newton and Leibniz, men had applied the calculus to the study of curves in two dimensions, and in a sense this work constituted a prototype of differential geometry. Euler and Monge had extended this to include an analytic study of surfaces; hence they sometimes are regarded as the fathers of differential geometry. Nevertheless, not until the appearance of the classical treatise of Gauss, Disquisitiones circa superficies curvas, was there a comprehensive volume devoted entirely to the subject."

"Roughly speaking, ordinary geometry is interested in the totality of a given diagram or figure, whereas differential geometry concentrates on the properties of a curve or a surface in the immediate neighborhood of a point on the curve or the surface. In this connection Gauss extended the work of Huygens and Clairaut on the curvature of a plane or gauche curve at a point by defining the curvature of a surface at a point... Gauss proceeds to show that the properties of a surface depend only on E, F, and G. This leads to many consequences. In particular, it becomes easy to say what properties of the surface remain invariant. It was in building on this work of Gauss that Bernhard Riemann and later geometers transformed the subject of differential geometry." 

"Oddly enough, however, no one before Wessel and Gauss took the obvious step of thinking of the real and imaginary parts of a complex number a + bi as rectangular coordinates of points in a plane."

"Functions no longer needed to be of the well-behaved form with which mathematicians had been familiar. Lejeune Dirichlet, for instance, in 1837, suggested a very broad definition of function: If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x. This comes close to the modern view of a correspondence between two sets of numbers, but the concepts of 'set' and 'real number' had not at that time been established. To indicate the completely arbitrary nature of the rule fo correspondence, Dirichlet proposed a very 'badly behaved' function: When x is rational, let y = c, and when x is irrational, let y = d ≠ c."

"Rejecting the Taylor's theorem approach of Lagrange, Cauchy made the limit concept of d'Alembert fundamental, but he gave it an arithmetic character of greater precision. Dispensing with geometry and with infinitesimals or velocities, he gave a relatively clear-cut definition of limit... Where many earlier mathematicians had thought of an infinitesimal as a very small fixed number, Cauchy defined it clearly as a dependent variable..."

"The history of mathematics teems with cases of simultaneity and near simultaneity of discovery, some of which have already been noted. The work by Cauchy that we have just described is another case in point, for similar views were developed at about the same time by Bernhard Bolzano (1781-1848), a Czechoslovakian priest whose theological views were frowned upon by his church and whose mathematical work was most undeservedly overlooked by his lay and clerical contemporaries.... The similarity in their arithmetization of the calculus and of their definitions of limit, derivative, continuity, and convergence was only a coincidence. Bolzano in 1817 had published a book, Rein analytischer Beweis, devoted to a purely arithmetic proof of the location theorem in algebra, and this had required a nongeometric approach to the continuity of a curve or function. Going considerable further in his unorthodox ideas, he disclosed some important properties of infinite sets in a posthumous work of 1850, Paradoxien des Unendlichen."

"From Galileo's paradox on the one-to-one correspondence between integers and perfect squares, Bolzano went on to show that similar correspondences between the elements of an infinite set and a proper subset are commonplace. For example, a simple linear equation, such as y = 2x, establishes a one-to-one correspondence between the real numbers y in the interval from 0 to 2, for example, and the real numbers x in half this interval. That is, there are just as many real numbers between 0 and 1 as between 0 and 2, or just as many points in a line segment 1 inch long as in a line segment 2 inches long. Bolzano seems even to have recognized, by about 1840, that the infinity of real numbers is of a type different from the infinity of integers, being nondenumerable. In such speculations on infinite sets the Bohemian philosopher came closer to parts of modern mathematics than had his better-known contemporaries. Both Gauss and Cauchy seem to have had a kind of horror infiniti, insisting that there could be no such thing as a completed infinite in mathematics."

"The name of Cauchy appears today in connection with a number of theorems on infinite series, for, despite some efforts on the part of Gauss and Abel, it was largely through Cauchy that the mathematician's conscience was pricked concerning the need for vigilance with regard to convergence."

"Through a point Clying outside a line AB there can be drawn more than one line in the plane and not meeting AB. With this new postulate Lobachevsky deduced a harmonious geometric structure having no inherent logical contradictions. This was in every sense a valid geometry, but so contrary to common sense did it appear, even to Lobachevsky, that he called it 'imaginary geometry.'"

"Bolyai and Lobachevsky were far removed from Paris and Göttingen. Still, presence in Paris did not guarantee success to even the brightest young mathematical minds of the day. The most illustrious examples of men who felt frustrated by their failure to find the recognition they sought in paris are the Norwegian Niels Henrik Abel (1802-1829) and the Frenchman Évariste Galois (1812-1832)."

"Abel found Paris inhospitable... and wrote home to a friend: 'Every beginner has a great deal of difficulty in getting noticed here. I have just finished an extensive treatise on a certain class of transcendental functions... but Mr Cauchy scarcely deigns to glance at it'... Abel had returned to his native Norway; increasingly weakened by tuberculosis, he kept sending more material to Crelle. He died in 1829, scarcely aware of the interest his publications were creating."

"Young geniuses whose lives were cut short by death from dueling or consumption are part of the real and fictional literary tradition of the Romantic Age. Someone wishing to present a mathematical caricature of such lives could do no better than to create the characters of Abel and Galois."

"By the age of sixteen Galois knew what his teachers had failed to recognize—that he was a mathematical genius. He hoped, therefore, to enter the school that had nurtured so many celebrated mathematicians, the École Polytechnique, but his lack of systematic preparation resulted in his rejection. This disappointment was followed by others: A paper Galois wrote and presented to the Academy when he was seventeen was apparently lost by Cauchy..."

"His letter to Chevalier, published in September 1832, had contained an outline of the main results of the memoir that had been returned by the Academy. Here Galois had indicated what he considered to be the essential part of his theory. In particular, he stressed the difference between adjoining one or all of the roots of the resolvent, and related it to the decomposition of the group G of the equation. In modern terminology, he indicated that an extension of the given field is normal if and only if the corresponding subgroup is a normal subgroup of G."

"Inspired by Abel's proof of the unsolvability by radicals of the quintic equation, Galois discovered that an irreducible algebraic equation is solvable by radicals if and only if its group—that is, the symmetric group o its roots—is solvable... Lagrange had already shown that the order of a subgroup must be a factor of the order of the group; but Galois went deeper and found relations between the factorability of the group and an equation and the solvability of the equation. Moreover, to him we owe the use in 1830 of the word group in its technical sense in mathematics."

"Among Jakob Steiner's unpublished discoveries are those relating to the fruitful geometric transformation known as inversive geometry... Inasmuch as there is no outside point P' corresponding to P when P coincides with the center O, one has in a sense a paradox similar to that of Bolzano. The inside of every circle, no matter how small, contains, as it were, one more point than the portion of the plane outside the circle. In a exactly analogous manner one readily defines the inverse of a point in three-dimensional space with respect to a sphere."

"Another discoverer was A. F. Möbius (1790-1860), also a student of Gauss... He introduced his 'barycentric coordinates' by considering a given triangle ABC and defining the coordinates of a point P as the mass to be placed at A, B, and C so that P is eh center of gravity of these masses. Möbius classified transformations according to whether they were congruences (leaving corresponding figures equal), similarities (corresponding figures similar), affine (corresponding figures preserving parallel lines, or collineations (lines going into lines), and suggested the study of invariants under each family of transformations."

"The notations and patterns of reasoning of the four inventors of homogeneous coordinates [Plücker, Gergonne, Lamé & Möbius (?)] differed somewhat, but they all had one thing in common—they made use of three coordinates instead of two to locate a point in a plane."

"Plücker chowed that every curve (other than a straight line) can be regarded as having a dual origin: It is a locus generated by a moving point and enveloped by a moving line, the point moving continuously along the line while the line continues to rotate about the point."

"Cayley in 1843 had initiated the ordinary analytic geometry of n-dimensional space, using determinants as an essential tool."

"Non-Euclidean geometry continued for several decades to be a fringe aspect of mathematics until it was thoroughly integrated through the remarkably general views of G. F. B. Riemann (1826-1866). The son of a village pastor, Riemann was brought up in very modest circumstances, always remaining frail in body and shy in manner."

"[Riemann's Habilitationsschrift] urged a global view of geometry as a study of manifolds of any number of dimensions in any kind of space. His geometries are non-Euclidean in a far more general sense than is Lobachevskian geometry, where the question is simply how many parallels are possible through a point. Riemann saw that geometry should not even necessarily deal with points or lines or space in the ordinary sense, but with sets of ordered n-tuples that are combined according to certain rules."

"... the fundamental change in geometric thought that Riemann's Habilitationsschrift brought about was Riemann's suggestion of the general study of curved metric spaces, rather than of the special case equivalent to geometry on the sphere, that ultimately made the theory of general relativity possible."

"The unification of geometry that Riemann had achieved was especially relevant in the microscopic aspect of differential geometry, or geometry 'in the small.' Analytic geometry, or geometry 'in the large,' had not been much changed."

"The program that Felix Klein gave, which became known as the Erlanger Programm, described geometry as the study of those properties of figures that remain invariant under a particular group of transformations. Hence, any classification of groups of groups of transformations becomes a codification of geometries."

"Euclidean geometry in Klein's view, is only a special case of affine geometry. Affine geometry in its turn becomes only a special case of a still more general geometry—projective geometry."

"Analysis, the study of infinite processes, had been understood by Newton and Leibniz to be concerned with continuous magnitudes, such as lengths, areas, velocities, and accelerations, whereas the theory of numbers clearly has as its domain the discrete set of natural numbers."

"Gudermann had impressed upon the young Weierstrass what a useful tool the power series representation of a function was, and it was in this connection that Weierstrass produced his greatest work, following in the footsteps of Abel."

"... perhaps no one is more deserving to be known as the father of the critical movement in analysis than is Weierstrass."

"The importance of work such as that of Weierstrass is felt particularly in mathematical physics, in which solutions of differential equations are rarely found in any form other than as an infinite series."

"[The Frenchman H. C. R. Charles Méray, the four Germans Karl Weierstrass, H. E. Heine, Georg Cantor and J. W. R. Dedekind]" in a sense represented the climax in half a century of investigation into the nature of function and number that had begun in 1822 with Fourier's theory of heat and with an attempt made in that year by Martin Ohm to reduce all of analysis to arithmetic..."

"It was in seeking to liberalize Dirichlet's conditions for the convergence of a Fourier series that Riemann developed his definition of the Riemann integral; in this connection he showed that a function f(x) may be integrable in an interval without being representable by a Fourier series. It was the study of infinite trigonometric series that led also to the theory of sets of Cantor..."

"We have seen that the revolution in geometry took place when Gauss, Lobachevsky, and Bolyai freed themselves from preconceptions of space. In somewhat the same sense the thoroughgoing arithmetization of analysis became possible only when, as Hankel foresaw, mathematicians understood that the real numbers are to be viewed as 'intellectual structures' rather than as intuitively given magnitudes inherited from Euclid's geometry."

"Bolzano during the early 1830s had made an attempt to develop a theory of real numbers as limits of rational number sequences, but this had gone unnoticed and unpublished until 1962."

"Weierstrass sought to separate the calculus from geometry and to base it upon the concept of number alone. Like Méray, he also saw that to do this it was necessary to give a definition of irrational number that is independent of the limit concept, inasmuch as the latter had up to this point presupposed the former. To correct Cauchy's logical error, Weierstrass settled the question of the existence of a limit of a convergent sequence by making the sequence itself the number limit."

"... Now, Dedekind pointed out, the fundamental theorems on limits can be proved rigorously without recourse to geometry. It was geometry that had pointed the way to a suitable definition of continuity, but in the end it was excluded from the formal arithmetic definition of the concept. The Dedekind cut in the rational number system, or an equivalent construction of real number, now has replaced geometrical magnitude as the backbone of analysis."

"The life of Cantor was tragically different from that of his friend Dedekind. Cantor was born in St. Petersburg of parents who had migrated from Denmark, but most of his life was spent in Germany, the family having moved to Frankfurt when he was eleven. His parents were Christians of Jewish background—his father had been converted to Protestantism, his mother had been born a Catholic. The son Georg took a strong interest in the finespun arguments of medieval theologians concerning continuity and the infinite, and this militated against his pursuing a mundane career in engineering as suggested by his father. In his studies at Zurich, Göttingen, and Berlin the young man consequently concentrated on philosophy, physics, and mathematics—a program that seems to have fostered his unprecedented mathematical imagination."

"Cauchy and Weierstrass saw only paradox in attempts to identify an actual or 'completed' infinity in mathematics, believing that the infinitely large and small indicated nothing more than the potentiality of Aristotle—an incompleteness of the process in question. Cantor and Dedekind came to a contrary conclusion."

"The set of perfect squares or the set of triangular numbers has the same power as the set of all the positive integers, for the groups can be put into one-to-one correspondence. These sets seem to be much smaller than the set of all rational fractions, yet Cantor showed that the latter set also is countable or denumerable, that is, it, too, can be put into one-to-one correspondence with the positive integers, hence has the same power... The real numbers can be subdivided into two types in two different ways: (1) as rational and irrational or (2) as algebraic and transcendental. Cantor showed that even the class of algebraic numbers, which is far more general than that of rational numbers, nevertheless has the same power as that of the integers. Hence, it is the transcendental numbers that give to the real number system the 'density' that results in a higher power."

"More surprising still is the fact that dimensionality is not the arbiter of the power of a set. The power of the set of points in a unit line segment is just the same as that of the points in a unit area or in a unit volume—or, for that matter, all the three-dimensional space. (Dimensionality, however, retains some measure of authority in that any one-to-one mapping of points in a space of unlike dimensionality is necessarily a discontinuous mapping.) So paradoxical were some results in point-set theory that Cantor himself on one occasion in 1877 wrote to Dedekind, 'I see it, but I don't believe it'; and he asked his friend to check the proof."

"Dedekind and Cantor were among the most capable mathematicians, and certainly the most original, of their day; yet neither man secured a top-ranking professional position. Dedekind spent almost a lifetime teaching on the secondary level..."

"Kronecker is said to have asked Lindemann of what use was the proof that pi is not algebraic inasmuch as irrational numbers are nonexistent. Sometimes it is reported that his movement died of inanition. We shall see later that it can be said to have reappeared in a new form in the work of Poincaré and Brouwer."

"David Hilbert... described the new transfinite arithmetic as 'the most astonishing product of mathematical thought, one of most beautiful realizations of human activity in the domain of the purely intelligible.'"

"Liouville... constructed an extensive class of nonalgebraic real numbers. The particular class that he developed are known as Liouville numbers, the more comprehensive set of nonalgebraic real numbers being called transcendental numbers."

"Here, finally, was the answer to the classical problem of the quadrature of the circle. In order for the quadrature of the circle to be possible with Euclidean tools, the number pi would have to be the root of a algebraic equation with a root expressible in square roots. Since pi is not algebraic, the circle cannot be squared according to the classical rules. Emboldened by his success, Ferdinand Lindemann later published several purported proofs of Fermat's last theorem, but they were shown by others to be invalid."

"Boole carried the formalism to its conclusion. No longer was mathematics to be limited to questions of number and continuous magnitude. Here for the first time the view is clearly expressed that the essential characteristic of mathematics is not so much its content as its form. If any topic is presented in such a way that it consists of symbols and precise rules of operation upon these symbols, subject only to the requirement of inner consistency, this topic is part of mathematics."

"Boole's Investigation of the Laws of Thought of 1854 is a classic in the history of mathematics, for it amplified and clarified the ideas presented in 1847, establishing both formal logic and a new algebra, known as Boolean algebra, the algebra of sets, or the algebra of logic."

"De Morgan was a lover of conundrums and witticisms, many of which are collected in his well-known Budget of Paradoxes, a delightful satire on circle-squarers edited after his death by his widow."

"Just as Lobachevsky had created a new geometry consistent within itself, by abandoning the parallel postulate, so Hamilton created a new algebra, also consistent with itself, by discarding the commutative postulate for multiplication... Hamilton, quite naturally, always regarded the discovery of quaternions as his greatest achievement. In retrospect it is clear that it was not so much this particular type fo algebra that was significant, but rather the discovery of the tremendous freedom that mathematics enjoys to build algebras that need not satisfy the restrictions imposed by the so-called 'fundamental laws,' which up to that time, supported by the vague principle of permanence of form, had been invoked without exception."

"The concept of an n-dimensional vector space had received detailed treatment in Hermann Grassmann's Ausdehnungslehre, published in Germany in 1844. Grassmann also was led to his results by studying the geometric interpretation of negative quantities and the addition and multiplication of directed line segments in two and three dimensions. He emphasized the dimension concept and stressed the development of an abstract science of 'spaces' and 'subspaces' which would include the geometry of two and three dimensions as special cases."

"C. S. Peirce continued his father's work in this direction by showing that of all these algebras there are only three in which division is uniquely defined: ordinary real algebra, the algebra of complex numbers, and the algebra of quaternions."

"In 1870 William Kingdon Clifford wrote a paper 'On the Space-Theory of Matter' in which he showed himself to be a stauch British supporter of the non-Euclidean geometry of Lobachevsky and Riemann."

"The concept of field was implicit in work by Abel and Galois, but Dedekind in 1879 seems to have been the first one to give an explicit definition of a number field—a set of numbers that form an Abelian group with respect to addition and with respect to multiplication (except for the inverse of zero) and for which multiplication distributes over addition. Simple examples are the system of rational numbers, the real number system, and the complex number field."

"Not satisfied to leave the basic concepts of arithmetic, hence of algebra, in so vague a state, the German logician and mathematician F. L. G. Frege was led to his well-known definition of cardinal number. The basis his views came from the theory of sets of Boole and Cantor."

"Frege undertook to derive the concepts of arithmetic from those of formal logic, for he disagreed with the assertion of C. S. Peirce that mathematics and logic are clearly distinct."

"Peano's aim was similar to that of Frege, but it was at the same time more ambitious and yet more down to earth. He hoped in his Formulaire de mathématiques to develop a formalized language that should contain not only mathematical logic but all the most important branches of mathematics."

"Poincarés doctoral thesis had been o differential equations... which led to one of his most celebrated contributions to mathematics—the properties of automorphic functions."

"... as a date for the beginning of the subject [of topology] none is more appropriate than 1895, the year in which Poincaré published his Analysis situs."

"Poincaré referred to Cantor's Mengenlehre as a disease from which later generations would regard themselves as having recovered."

"Hilbert's Grundlagen opened with a motto taken from Kant: 'All human knowledge begins with intuitions, proceeds to concepts, and terminates in ideas,' but Hilbert's development of geometry established a decidedly anti-Kantian view of the subject."

"... the first of [Hilbert's] twenty-three problems... concerned the structure of the real number continuum. The question is made up of two related parts: (1) Is there a transfinite number between that of a denumerable set and the number of the continuum; and (2) can the numerical continuum be considered a well-ordered set? The second part asks whether the totality of all real numbers can be arranged in another manner so that every partial assemblage will have a first element. This is closely related with t he axiom of choice named for the German mathematician Ernst Zermelo..."

"Efforts to solve the second query of Hilbert led in 1931 to a surprising conclusion on the part of a young Austrian mathematician, Kurt Gödel. Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system."

"In a sense Gödel's theorem, sometimes regarded as the most decisive result in mathematical logic, seems to dispose negatively of Hilbert's second query. In its implications the discovery by Gödel of undecidable propositions is as disturbing as was the disclosure by Hippasus of incommensurable magnitudes, for it appears to foredoom hope of mathematical certitude through use of the obvious methods."

"If any one book marks the emergence of point set topology as a separate discipline, it is Hausdorff's Grundzüge. It is interesting to note that although it was the arithmetization of analysis that began the train of thought that led from Cantor to Hausdorff, in the end the concept of number is thoroughly submerged under a far more general point of view. Moreover, although the word 'point' is used in the title, the new subject has a little to do with the points of ordinary geometry as with the numbers of common arithmetic. Topology has emerged in the twentieth century as a subject that unifies almost the whole of mathematics, somewhat as philosophy seeks to coordinate all knowledge."

"The first volume of Bourbaki's Éléments appeared in 1939, the thirty-first in 1965..." 

- Carl B. Boyer, A History of Mathematics: Revised by Uta C. Merzbach; 

"Thus, Cardan, in chapter 37 of Ars Magna (1545), sets up and solves the problem of dividing 10 into two parts whose product is 40. The equation is x(l0 - x) = 40. He obtains the roots 5 + √-15 and 5 - √-15, then says, 'putting aside the mental tortures involved,' multiply 5 + √-15 and 5 - √-15; the product is 25 - (-15) or 40."

"The biggest improvement in arithmetic during the sixteenth and seventeenth centuries was the invention of logarithms."

"Euler had already defined logarithms as exponents and in 1728, in an unpublished manuscript (Opera Posthuma, II, 800-804), introduced 'e' for the base of the natural logarithms."

"It would seem that Vieta's successors would immediately have been struck by the idea of general coefficients. But as far as one can judge, the introduction of letters for classes of numbers was accepted as a minor move in the development of symbolism."

"Cardan's proof is geometrical; for Cardan, 't' and 'u' were volumes of cubes whose sides were the cube root of 't' and the cube root of 'u', and whose product was a rectangle formed by the two sides whose area was m/3... There is a difficulty with Cardan's solution of the cubic, which he observed but did not resolve. When the roots of the cubic are all real and distinct, it can be shown that t and u will be complex... Vieta, in De Aequationum Recognitione et Emendatione, written in 1591 and published in 1615, was able to solve the irreducible case of cubics by using a trigonometric identity, and so avoided the use of Cardan's formula. This method is also used today."

"Of course the cubic equation has three roots. It was Leonhard Euler who, in 1732, gave the first complete discussion of Cardan's solution of the cubic, in which he emphasized that there are always three roots and pointed out how these are obtained."

"Descartes, in the third book of La Geometrie, said that an equa­tion can have as many distinct roots as the number of dimensions (degree) of the unknown. He said 'can have' because he considered negative roots as false roots. Later, by including imaginary and negative roots for the pur­pose of counting roots, he concluded that there are as many as the degree."

"The binomial theorem for positive integral exponents was known by the Arabs of the thirteenth century... The arrangement of numbers in which each number is the sum of the two immediately above it, already known to Tartaglia, Stifel, and Stevin, was used by Pascal (1654) to obtain the coefficients of the binomial expansion... Despite the fact that this arrangement was known to many predecessors, it has been called Pascal's triangle."

"The work on permutations and combinations is connected with another development, the theory of probability, which was to assume major im­portance in the late nineteenth century but which barely warrants mention in the sixteenth and seventeenth centuries."

"Fermat's work in the theory of numbers determined the direction of the work in this area until Gauss made his contributions. Fermat's point of departure was Diophantus."

"The dependence of algebra on geometry began to be reversed somewhat when Victa, and later Descartes, used algebra to help solve geometric con­ struction problems. The motivation for much of the algebra that appears in Vieta's In Artem Analyticam lsagoge, is solving geometric problems and systematizing geometrical constructions."

"Algebra for Vieta meant a special procedure for discovery; it was analysis in the sense of Plato, who opposed it to synthesis."

"While algebra was to Vieta largely a royal road to geometry, his vision was great enough to see that algebra had a life and meaning of its own."

"The first differentials dy and dx of a function y = f(x) were legitimized in the nineteenth century, but the higher-order differentials, which the eighteenth-century men used freely, have not been put on a rigorous basis even today."

"Euler showed how to use complex functions to evaluate real integrals... Euler remarks that every function of z which for z = x + iy takes the form M + iN, where M and N are real functions, also takes on, for z = x - iy, the form M - iN... This he states is the fundamental theorem of complex numbers.

"A vital step that made the erection of a theory of functions of a complex variable more intuitively reasonable was the geometrical representation of complex numbers and of the algebraic operations with these numbers... Cotes, De Moivre, Euler, and Vandermonde... thought of the solutions as the vertices of a regular polygon."

"A somewhat different geometric interpretation of complex numbers was given by a Swiss, Jean-Robert Argand (1768-1822). Argand, who was also self-taught and a bookkeeper, published a small book, Essai sur une maniere de representer les quantiles imaginaires dans les constructions geometriques... Argand, like Wessel, showed how complex numbers can be added and multiplied geo­metrically and applied these geometrical ideas to prove theorems of trigonometry, geometry and algebra."

"Gauss was more effective in bringing about the acceptance of complex numbers. He used them in his several proofs of the fundamental theorem of algebra. In the first three proofs (1799, 1815, and 1816) he presupposes the one-to-one correspondence of points of the Cartesian plane and complex numbers. There is no actual plotting of x + iy, but rather of x and y as coordinates of a point in the real plane. Moreover, the proofs do not really use complex function theory because he separates the real and imaginary parts of the functions involved. He is more explicit in a letter to Bessel of 1811, wherein he says that a + ib is represented by the point (a, b) and that one can go from one point to another of the complex plane by many paths."

"Gauss also introduced some basic ideas about functions of a complex variable. In the letter to Bessel of 1811... Gauss pointed out the necessity of taking imaginary (complex) limits into account."

"Though these observations of Gauss and Poisson are indeed significant, neither published a major paper on complex function theory. This theory was founded by Augustin-Louis Cauchy."

"Abel knew the work of Euler, Lagrange, and Legendre on elliptic integrals and may have gotten suggestions for the work he undertook from remarks made by Gauss, especially in his Disquisitiones Arithmeticae."

"The other discoverer of elliptic functions was Carl Gustav Jacob Jacobi (1804-51). Unlike Abel, he lived a quiet life. Born in Potsdam to a Jewish family, he studied at the University of Berlin and in 1827 became a professor at Konigsberg."

"The im­ portant point about the periods is that there are two periods (whose ratio is not real), so that the elliptic functions are doubly periodic. This was one of Abel's great discoveries."

"The successes achieved in the study of the elliptic integrals and the corresponding functions encouraged the mathematicians to tackle a more difficult type, hyperelliptic integrals."

"The study of a generalization of the elliptic and hyperelliptic integrals was begun by Galois, but the more significant initial steps were taken by Abel in his 1826 paper."

"Though Abel did not carry the study of Abelian integrals very far, he proved a key theorem in the subject. Abel's basic theorem is a very broad generalization of the addition theorem for elliptic integrals."

"A new period of discovery in the theory of algebraic functions, their integrals, and the inverse functions is due to Riemann. Riemann actually offered a much broader theory, namely, the treatment of multiple-valued functions, thus far only touched upon by Cauchy and Puiseux, and thereby paved the way for a number of different advances."

"It seems very likely, on the basis of evidence given by Felix Klein, that Riemann's ideas on complex functions were suggested to him by his studies on the flow of electrical currents along a plane."

"The key idea in Riemann's approach to multiple-valued functions is the notion of a Riemann surface..."

"It is not possible to represent Riemann surfaces accurately in three dimensional space."

"Riemann thought of his surface as an n-sheeted duplication of the plane, each replica completed by a point at infinity. However, it is difficult to follow all of the arguments involving such a surface by visualizing in terms of the n interconnected planes. Hence mathematicians since Riemann's time have suggested equivalent models that are easier to contemplate. It is known that a plane may be transformed to a sphere by stereographic projection."

"It is important to note that for Riemann the domain of u was any part of a Riemann surface, including possibly the whole surface. In his doctoral thesis he considered surfaces with boundaries and only later used closed surfaces, that is, surfaces without boundaries, such as a torus."

"... if a surface is simply connected, no cross-cut is necessary and the surface has connectivity connectivity (Grundzahl) 1. A surface is called doubly con­nected if by one appropriate cross-cut it is changed into a single simply connected surface. Then the connectivity is 2. A plane ring and a spherical surface with two holes are examples. A surface is called triply connected when by two appropriate cross-cuts it is converted into a single simply connected surface. Then the connectivity is 3. An example is the (surface of a) torus with a hole in it."

"The analysis of Abelian integrals sheds light on what kinds of functions can exist on a Riemann surface. Riemann treats two classes of functions; the first consists of single-valued functions on the surface whose singularities are poles. The second class consists of functions that are one-valued on the surface provided with cross-cuts but discontinuous along each cross-cut."

"There are also everywhere finite functions on the surface. One can represent such a function by means of functions of the first of the above classes. One can also build up algebraic functions on a surface by combining integrals of the second and third kind. Thus Riemann showed that algebraic functions can be represented as a sum of transcendental functions. Also single-valued functions algebraically infinite in a given number of points can be represented by rational functions."

"Weierstrass also worked on Abclian integrals during the 1860s. But he and the other successors of Riemann in this field set up transcendental functions from the algebraic functions, the reverse of Riemann's procedure."

"Galois not only created the first significant coherent body of algebraic theory but he introduced new notions which were to be developed into still other broadly applicable theories of algebra. In particular the concepts of a group and a field emerged from his work and Abel's."

"Abel read Lagrange's and Gauss's work on the theory of equations and while still a student in high school tackled the problem of the solvability of higher degree equations by following Gauss's treatment of the binomial equation. At first Abel thought he had solved the general fifth degree equation by radicals. But soon convinced of his error he tried to prove that such a solution was not possible (1824-26). First he succeeded in proving the theorem: The roots of an equation solvable by radicals can be given such a form that each of the radicals occurring in the expressions for the roots is expressible as a rational function of the roots of the equation and certain roots of unity. Abel then used this theorem to prove the impossibility of solving by radicals the general equation of degree greater than four."

"In this last work Abel introduced two notions (though not the terminology), field and polynomial irreducible in a given field. By a field of numbers he, like Galois later, meant a collection of numbers such that the sum, difference, product, and quotient of any two numbers in the collection (except division by 0) are also in the collection. Thus the rational numbers, real numbers, and complex numbers form a field. A polynomial is said to be reducible in a field (usually the field to which its coefficients belong) if it can be expressed as the product of two polinomials of lower degrees and with coefficients in the field. If the polinomial cannot be so expressed it is said to be irreducible."

"After Abel's work the situation was as follows: Although the general equation of degree higher than four was not solvable by radicals, there were many special equations, such as the binomial equations x to the power of p = a, p a prime, and Abelian equations that were solvable by radicals. The task now became to determine which equations are solvable by radicals. This task, just begun by Abel, was taken up by Evariste Galois (1811-32)."

"Like Lagrange, Galois makes use of the notion of substitutions or permutations of the roots."

"Let R be the field formed by rational expressions in p and q and with coefficients in the field of rational numbers... One says with Galois that R is the field obtained by adjoining the letters or indeterminates p and q to the rational numbers. This field R is the field or domain of rationality of the coefficients of the given equation and the equation is said to belong to the field R. Like Abel, Galois did not use the terms field or domain of rationality but he did use the concept."

"... the group of an equation with respect to a field R is the group or subgroup of substitutions on the roots which leave invariant all the relations with coefficients in R among the roots of the given equation (wether general or particular)."

"We may sec from the above discussion that the group of an equation is a key to its solvability because the group expresses the degree of indistinguish­ability of the roots. It tells us what we do not know about the roots."

"If all of the successive resolvents are binomial equations, then, in view of Gauss's result on binomial equations, we can solve the original equation by radicals."

"One other development of the middle nineteenth century is both note­ worthy and instructive. Arthur Cayley, very much influenced by Cauchy's work, recognized that the notion of substitution group could be generalized. Cayley introduced the notion of an abstract group."

"In 1822 Gauss won a prize offered by the Danish Royal Society of Sciences for a paper on the problem of finding the analytic condition for transforming any surface conformally onto any other surface... Gauss did not answer the question of whether and in what way a finite portion of the surface can be mapped conformally onto the other surface. This problem was taken up by Riemann in his work on complex function."

"Until this work surfaces had been studied as figures in three-dimensional Euclidean space. But Gauss showed that the geometry of a surface could be studied by concentrating on the surface itself... given these u and v coordinates on the surface and the expression for dsˆ2 in terms of E, F, and G as functions of u and v, all the properties of the surface follow from this expression... the surface can be considered as a space in itself because all its properties are determined by the dsˆ2."

"Riemann's idea was that by relying upon analysis we might start with what is surely a priori about space and deduce the necessary consequences. Any other properties of space would then be known to be empirical. Gauss had concerned himself with this very same problem but of this investigation only the essay on curved surfaces was published. Riemann's search for what is a priori led him to study the local behavior of space or, in other words, the differential geometric approach as opposed to the consideration of space as a whole as one finds it in Euclid or in the non­ Euclidean geometry of Gauss, Bolyai, and Lobatchevsky."

"Prior to and during the work on non-Euclidean geometry, the study of projective properties was the major geometric activity. Moreover, it was evident from the work of von Staudt that projective geometry is logically prior to Euclidean geometry because it deals with qualitative and descriptive properties that enter into the very formation of geometrical figures and does not use the measures of line segments and angles. This fact suggested that Euclidean geometry might be some special­ization of projective geometry."

"Independently of Laguerre, Cayley made the next step. He approached geometry from the standpoint of algebra, and in fact was interested in the geometric interpretation of quantics (homogeneous polynomial forms)..."

"Cayley's work proved to be a generalization of Laguerre's idea. The latter had used the circular points at infinity to define angle in the plane. The circular points are really a degenerate conic. In two dimensions Cayley introduced any conic in place of the circular points and in three dimensions he introduced any quadric surface. These figures he called the absolutes. Cayley asserted that all metric properties of figures are none other than projective properties augmented by the absolute or in relation to the absolute. He then showed how this principle led to a new expression for angle and an expression for distance between two points. He starts with the fact that the points of a plane arc represented by homogeneous coordinates. These coordinates are not to be regarded as distances or ratios of distances but as an assumed fundamental notion not requiring or admitting explanation."

"It will be noted that the expressions for length and angle involve the algebraic expression for the absolute. Generally the analytic expression of any Euclidean metrical property involves the relation of that property to the absolute. Metrical properties are not properties of the figure per se but of the figure in relation to the absolute... Cayley's idea was taken over by Felix Klein (1849-1 925) and generalized so as to include the non-Euclidean geometries."

"Klein's major idea was that by specializing the nature of Cayley's absolute quadric surface (if one considers three-dimensional geometry) one could show that the metric, which according to Cayley depended on the nature of the absolute, would yield hyperbolic and double elliptic geometry. When the second degree surface is a real ellipsoid, real elliptic paraboloid, or real hyperboloid of two sheets one gets Lobatchevsky's metric geometry, and when the second degree surface is imaginary one gets Riemann's non­ Euclidean geometry (of constant positive curvature)."

"Klein's success in subsuming the various metric geometries under projective geometry led him to seek to characterize the various geometries not just on the basis of nonmetric and metric properties and the distinctions among the metrics but from the broader standpoint of what these geometries and other geometries which had already appeared on the scene sought to accomplish... Klein's basic idea is that each geometry can be characterized by a group of transformations and that a geometry is really concerned with invariants under this group of transformations."

"Klein also projected the study of invariants under one-to-one continuous transformations with continuous inverses. This is the class now called homeomorphisms and the study of the invariants under such transformations is the subject matter of topology."

"The computation ofalgebraic invariants did not end with Hilbert's work. Emmy Noether (1882-1935), a student of Gordan, did a doctoral thesis in 1907 'On Complete Systems of lnvariants for Ternary Biquadratic Forms.' She also gave a complete system of covariant forms for a ternary quartic, 331 in all. In 1910 she extended Gordan's result to n variables."

"This view of real curves with complex points was already familiar from the work in projective geometry. To the theory of birational transformations of the surface corresponds a theory of birational transforma­ tions of the plane curve... Higher multiple points on curves also correspond to other peculiarities of Riemann surfaces."

"This work replaces Riemann's determination of the most general algebraic function having given points at which it becomes infinite. Also the Brill-Noether result transcends the projective viewpoint... for the first time the theorems on points of intersection of curves were established algebraically. The counting of constants as a method was dispensed with."

"The topic which has received the greatest attention over the years is the study of singularities of plane algebraic curves."

"The subject of algebraic geometry now embraces the study of higher­ dimensional figures (manifolds or varieties) defined by one or more algebraic equations. Beyond generalization in this direction, another type, namely, the use of more general coefficients in the defining equations, has also been undertaken. These coefficients can be members of an abstract ring or field and the methods of abstract algebra are applied. The several methods of pursuing algebraic geometry as well as the abstract algebraic formulation introduced in the twentieth century have led to sharp differences in language and methods of approach so that one class of workers finds it very difficult to understand another."

"The critical investigation of geometry extended beyond the reconstruction of the foundations. Curves had of course been used freely. The simpler ones, such as the ellipse, had secure geometrical and analytical definitions. But many curves were introduced only through equations and functions. The rigorization of analysis had included not only a broadening of the concept offunction but the construction ofvery peculiar functions, such as continuous functions without derivatives."

"However, Jordan's definition of a curve, though satisfactory for many applications, was too broad. In 1890 Peano discovered that a curve meeting Jordan's definition can run through all the points of a square at least once."

"These examples show that the definition of a curve Jordan suggested is not satisfactory because a curve, according to this definition, can fill out a square. The question of what is meant by a curve remained open. Felix Klein remarked in 1898 that nothing was more obscure than the notion of a curve."

"By 1900 no one had proved that every closed plane curve, as defined by Jordan and Pcano, encloses an area. Helge von Koch (1870-1924) complicated the area problem by giving an example of a continuous but non-differentiable curve with infinite perimeter which bounds a finite area."

"...The idea that motivated the creation of functional analysis is that all of these operators could be considered under one abstract formulation of an operator acting on a class of functions. Moreover, these functions could be regarded as elements or points of a space. Then the operator transforms points into points and in this sense is a generalization of ordinary transfor­mations such as rotations. Some of the above operators carry functions into real numbers, rather than functions. Those operators that do yield real or complex numbers are today called functionals and the term operator is more commonly reserved for the transformations that carry functions into functions."

"... Hilbert did not regard these sequences as coordinates of a point in space, nor did he use geometrical language. This step was taken by Schmidt and Frechet... functions were represented as points of an infinite-dimensional space... Such a space has since been called a Hilbert space."

"The central part of functional analysis deals with the abstract theory of the operators that occur in differential and integral equations. This theory unites the eigenvalue theory of differential and integral equations and linear transformations operating in n-dimensional space."

"Though a Banach space is more general than a Hilbert space, because the inner product of two elements is not presupposed to define the norm, as a consequence the key notion of two elements being orthogonal is lost in a Banach space that is not also a Hilbert space. The first and third sets of conditions hold also for Hilbert space, but the second set is weaker than conditions on the norm in Hilbert space..."

"Quantum mechanical research showed that the observables of a physical system can be represented by linear symmetric operators in a Hilbert space and the eigenvalues and eigenvectors (eigenfunctions) of the particular operator that represents energy are the energy levels of an electron in an atom and corresponding stationary quan­tum states of the system."

"... a general theory unifying Hilbert's work and the eigenfunction theory for differential equations was missing. The use of operators in quantum theory stimulated work on an abstract theory of Hilbert space and operators; this was first undertaken by John von Neumann (1903-57) in 1927."

"Tensor analysis is often described as a totally new branch of mathematics, created ab initio either to meet some specific objective or just to delight mathematicians. It is actually no more than a variation on an old theme, namely, the study of differential invariants associated primarily with a Riemannian geometry."

"Dedekind's under­ standing of the value of abstraction is noteworthy. He saw clearly in his work on algebraic number theory the value of structures such as ideals and fields. He is the effective founder of abstract algebra."

"By 1880 new ideas on groups came into the picture. Klein, influenced by Jordan's work on permutation groups, had shown in his Erlanger Programm that infinite transformation groups, that is, groups with infinitely many elements, could be used to classify geometries. These groups, moreover, are continuous in the sense that arbitrarily small trans­ formations are included in any group or, alternatively stated, the parameters in the transformations can take on all real values."

"Sophus Lie, who had worked with Klein around 1870, took up the notion of continuous transformation groups, but for other purposes than the classification of geometries. He had observed that most of the ordinary differential equations that had been integrated by older methods were invariant under classes of continuous transformation groups, and he thought he could throw light on the solution of differential equations and classifythem."

"Though the structures rings and ideals were well known and utilized in Dedekind's and Kronecker's work on algebraic numbers, the abstract theory is entirely a product of the twentieth century."

"In the late nineteenth century a great variety of concrete linear associa­tive algebras were created. These algebras, abstractly considered, are rings, and when the theory of abstract rings was formulated it absorbed and generalized the work on these concrete algebras. This theory of linear associative algebras and the whole subject of abstract algebra received a new impulse when Wedderburn, in his paper "On Hypercomplex Numbers," took up results of Elie Cartan (1869-1951) and generalized them."

"The theory of rings and ideals was put on a more systematic and axio­ matic basis by Emmy Noether, one of the few great women mathematicians, who in 1922 became a lecturer at Gottingcn. Many results on rings and ideals were already known when she began her work, but by properly formulating the abstract notions she was able to subsume these results under the abstract theory. "

"Lie algebras arose out of Lie's efforts to study the structure of his continuous transformation groups. To do this Lie introduced the notion of infinitesimal transformations. Roughly speaking, an infinitesimal trans­formation is one that moves points an infinitesimal distance." 

"In his thesis, Sur la structure des groupes de transformationsfinis et continus, Cartan gave the complete classification of all simple Lie algebras over the field of complex values for the variables and parameters."

"... abstract algebra has subverted its own role in mathematics. Its concepts were formulated to unify various seemingly diverse and dis­ similar mathematical domains as, for example, group theory did. Having formulated the abstract theories, mathematicians turned away from the original concrete fields and concentrated on the abstract structures. Through the introduction of hundreds of subordinate concepts, the subject has mush­ roomed into a welter of smaller developments that have little relation to each other or to the original concrete fields. Unification has been succeeded by diversification and specialization. Indeed, most workers in the domain of abstract algebra are no longer aware of the origins of the abstract structures, nor are they concerned with the application of their results to the concrete fields."

"The man who first formulated properly the nature of topological investigations was Mobius, who was an assistant to Gauss in 1813. He had classified the various geometrical properties, projective, affine, similarity, and congruence and then in 1863 in his "Theorie der elementaren Verwand­ schaft" (Theory of Elementary Relationships), he proposed studying the relationship between two figures whose points are in one-to-one correspond­ence and such that neighboring points correspond to neighboring points. He began by studying the geometria situs of polyhedra... In 1858 he and Listing independently discovered one-sided surfaces, of which the Mobius band is best known."

"The greatest impetus to topological investigations came from Riemann's work in complex function theory. In his thesis of 1851 on complex functions and in his study of Abelian functions, he stressed that to work with functions some theories of analysis situs were indispensable. In these investigations he found it necessary to introduce the connectivity of Riemann surfaces."

"Since the structure of Riemann surfaces is complicated and a topo­logically equivalent figure has the same genus, some mathematicians sought simpler structures... Klein suggested another topological model, a sphere with handles." 

"The complexity of even two-dimensional closed figures was emphasized by Klein's introduction in 1882 of the surface now called the Klein bottle."

"The projective plane is another example of a rather complex closed surface... It can also be formed by pasting the edge of a circle along the edge of a Mobius band (which has just a single edge), though again the figure cannot be constructed in three dimensions without having points coincide that should be distinct."

"The man who made the first systematic and general attack on the combinatorial theory of geometrical figures, and who is regarded as the founder of combinatorial topology, is Henri Poincare (1854- 1912)... Though he used curved cells or pieces of his figures and treated manifolds, we shall formulate his ideas in terms of complexes and simplexes, which were introduced later by Brouwer." 

"Poincare introduced next the important quantities he called the Betti numbers (in honor of Enrico Betti). For each dimension of possible simplexes in a complex, the number of independent cycles of that dimension is called the Betti number of that dimension (Poincare actually used a number that is 1 more than Betti's connectivity number.) Thus in the case of the ring, the zero-dimensional Betti number is 1."

"In his efforts to distinguish complexes, Poincare introduced (1895) one other concept that now plays a considerable role in topology, the funda­mental group of a complex, also known as the Poincare group or the first homotopy group. The idea arises from considering the distinction between simply and multiply connected plane regions. In the interior of a circle all closed curves can be shrunk to a point. However, in a circular ring some closed curves, those that surround the inner circular boundary, cannot be shrunk to a point, whereas those closed curves that do not surround the inner boundary can be shrunk to a point."

"... Another change in the formulation of basic combinatorial properties made during the years of 1925 to 1930 by a number of men and possibly suggested by Emmy Noether, was to recast the theory of chains, cycles and bounding cycles into the language of group theory."

- Kline, Morris. Mathematical Thought: From Ancient to Modern Times.

See also:
- actual infinite falling (against Carlo Rovelli's pseudo-problem);
- the dogma of semantic uniformity & Python Gored Naturalism;
- the odd transformation of Der Herr Warum (Gödel & Resnais);
- the only three types of ingenuity (with Cantor & Dedekind);
- self-help books (with Rupert Sheldrake);
- what is REAL space? what is REAL number?
- Timothy Leary in the 1990s;
- 5G?! Get real...
- list of charming scientists/engineers;
- pick a soul (ass you wish);
- view from Berthe Trépat's apartment;
- list des déclencheurs musicaux;
- Dark Consciousness (with Yasuo Yuasa);
- The Doors of Perception (with Huxley);
- Structuralism, Poststructuralism (with Julia Kristeva);
- List des figures du chaos primordial (Deleuze);
- Brazilian Perspectivism (Viveiros de Castro vs. Haroldo de Campos);
- Piano Playing (with Kochevitsky);
- L'Affirmation de l'âne (review of Smolin/Unger's The Singular Universe);
And also:
- Dogen with Hagakure;
- L'articulation (Maurice Blanchot);
- L'intelligence des fleurs;

- Trondheim (Norway); 
- Spooky Blue;
- Interactive while indifferent—Kinds & Phantasmagoria circa 1900;

- Mathematics is Metaphysics (Fernando Zalamea); 

- Synthetic Philosophy of Mathematics (Fernando Zalamea); 

- La vision unificatrice de Grothendieck (Mathieu Bélanger); 

- Deleuze et les mathématiques (Anna Longo); 

- The Mathematical Continuum: A Haunting Problematic (Elizabeth de Freitas); 

- Infinitesimal (review of Amir Alexander's book);

- Círculo Vicioso y Revolución Crítica en la Teoría de los Problemas (Gonzalo Santaya); 

- El cálculo transcendental: Gilles Deleuze y el cálculo diferencial (Gonzalo Santaya); 

*****extremely useful site for reviewing almost all mathematical concepts and skills in a systematic and organized way (Sal Khan's Academy): https://www.khanacademy.org/math;