what is REAL space? what is REAL number? & do fields fluctuate? (the greatest problem of all, now fully illustrated)

Vladimir Favorsky's coverdesign to Pavel Florensky's The Imaginaries of Geometry (1922); 

Somnambule 1 (by A/Z) [I took this video out of Youtube, because I wanted to change it];

Duchamp in Maya Deren's Witch's Cradle (1943); 

The Dimension of Dali (Joan Úbeda & Susi Marquès, 2004);

John Cage performing Water Walk on TV (1960);

Edgard Varèse's Étude pour Espace (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); 

What's a Tensor (Dan Fleisch/Youtube);

Geometry of Linear Algebra (MIT/Youtube); 

Multivariable Functions Lecture (Mathified/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); 

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

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

Orations & disputations from the beginning of Queen Elizabeth's reign down to our own time:

"E pois, te digo, as estrelas, 
no céu imenso espalhadas, 
são a metade e outro tanto 
das mesmas por Deus criadas; 
e, se imaginas que minto 
na quantidade que dei, 
te desafio a contá-las... 
para ver que não errei!"

Cancioneiro Guasca

"But those infinities are perhaps not inevitable..."

Ian Hacking

"... toutes ces grandeurs sont divisibles à l'infini, sans tomber dans leurs indivisibles, de sorte qu'elles tiennent toutes le milieu entre l'infini et le néant."

Pascal (Fragments de l'esprit géométrique)

"Extraordinaire comme les mathématiques vous aident à vous connaitre."

Molloy (man of fine parts and fertile fancy)

"... 'd' merely means 'a little bit of.'"
"Everything depends upon relative minuteness..."
Silvanus P. Thompson (Calculus Made Easy)

"Der phantasiestarke Mathematiker wird den Gebilden seines Denkens auch die Lebendigkeit der Anschauung einzuhauchen wissen, während Geister von schwächerer Flugkraft oder mehr abstrakter Richtung ihm in sein Reich konkreter Schöpfung und Belebung nicht zu folgen vermögen."

Felix Hausdorff (Das Raumproblem/Annalen der Naturphilosophie 1904)

"Mais ce que je veux faire voir, c'est que, dans cette déformation, le monde n'est pas demeuré semblable à lui-même ; les carrés sont devenus des rectangles ou des parallélogrammes, les cercles des ellipses, les sphères des ellipsoïdes. Et cependant nous n'avons aucun moyen de savoir si cette déformation est réelle."

Henri Poincaré (Science et Méthode)

"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)

"Since Einstein introduced discontinuity into the study of light, and de Broglie the continuity of waves into the study of matter, it is impossible to maintain the old idea of domains of physical facts that are separate from one another. The physics of the continuous represents a mode of treatment by differential equations of physical facts. The physics of discontinuity represents a mode of treatment of the same facts by other methods: group theory, calculation of matrices, quantum statistics, etc. There thus exists a certain analogy between contemporary physics and contemporary mathematics, in that they offer each other the spectacle of facts amenable to being studied at the same time by the calculus of the continuous and by the calculus of the discontinuous."

Albert Lautman (Essay on the Unity of Mathematical Sciences, Simon B. Duffy's translation)

"Mathematics is constituted like physics: the facts to be explained were throughout history the paradoxes that the progress of reflection rendered intelligible by a constant renewal of the meaning of essential notions. Irrational numbers, the infinitely small, continuous functions without derivatives, the transcendence of e and of r, the transfinite had all been accepted by an incomprehensible necessity of fact before there was a deductive theory of them. They had the fate of these physical constants like c or h which were essential in an incomprehensible way in the most different domains, up until the genius of Maxwell, Planck and Einstein knew to see in the constancy of their value the connection of electricity and light, of light and energy."

"The idea that... Riemann manifold, which is the Einstein space, would be closed often evokes the image of a closed surface that the intuition could not help but locate in an infinite 3-dimensional space, and yet outside of this surface, by an incomprehensible paradox, there could be no matter, nor even space. The paradox disappears when one realizes that a manifold on which a dsˆ2 of more than two dimensions is defined is in no way amenable to an intuitive comparison with a surface. The notions of intrinsic differential geometry are purely intellectual, they characterize a mode of mathematical exploration of a manifold by following a path on this manifold, in opposition to the extrinsic method which considers this manifold as embedded in a Euclidean space to a sufficient number of dimensions."

"The Mobius ring... only has a single side, and that is an essentially extrinsic property since, to be realized, it is necessary to split the ring and untwist it, which implies a rotation around an axis exterior to the surface of the ring. It is nevertheless possible to characterize this ‘unilaterality’ by a purely intrinsic property. Consider in effect an arrow perpendicular to the line traced on the ring and move this arrow along the line, so that it is always situated on the surface. By presuming the surface transparent enough we realize that the arrow arrives at a point to cover its departure position with an inverse orientation. The surface is said to be non-orientable and this property could be observed by an observer bound to the surface, which would neither split the ring, nor untwist it... This example hints at the philosophical interest of algebraic topology (or even combinatorial topology): the geometric properties of relation to a very large extent let themselves be expressed in intrinsic algebraic properties and, to the extent that the intellectualization of the relations of a figure and of ambient figures is successful, the Kantian distinction between an aesthetic and an analytic is seen to vanish."

Albert Lautman (Essay on the Notions of Structure and Existence in Mathematics, Simon B. Duffy's translation)

"In seeing the sensible thus defined by a mixture of symmetry and dissymmetry, of identity and difference, it is impossible not to recall Plato’s Timaeus. The existence of bodies is based there on the existence of this receptacle that Plato calls the place and whose function consists, as [Albert] Rivaud has shown in the preface to his edition of the Timaeus (1932), in making possible the multiplicity of bodies and their alternation in a single place in the sensible world, just as the role of the Idea of the Other in the intelligible world is to ensure, by its mixture with the Same, both the connection and the separation of types. This reference to Plato enables the understanding that the materials of which the universe is formed are not so much the atoms and molecules of the physical theory as these great pairs of ideal opposites such as the Same and the Other, the Symmetrical and Dissymmetrical, related to one another according to the laws of a harmonious mixture. Plato also suggests more. The properties of place and matter, according to him, are not purely sensible, they are, as Rivaud goes on to say, the geometric and physical transposition of a dialectical theory. It is also possible that the distinction between left and right, as observed in the sensible world, is only the transposition on the plane of experience of a dissymmetrical symmetry which is equally constitutive of the abstract reality of mathematics. A common participation in the same dialectical structure would thus bring to the fore an analogy between the structure of the sensible world and that of mathematics, and would allow a better understanding of how these two realities accord with one another."

Albert Lautman (Symmetry and Dissymetry in Mathematics and Physics, 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!])

"The continuum is one of the most complex concepts that humanity has had to decipher, not only in its technical, internal behavior, inside culture, but also in its constant, external entangling with the cosmos. Like Proteus, the sea god which changed his appearance at will, the continuum moves between the physical world and abstract ideals, between a subtle phaneron and wide hierarchical ramifications in mathematical models, between the most concrete biological evolution and arbitrary completions of discrete breaks. On its hand, a logic of continuity has to propose adequate signs, structures and rules to handle an important part of those passages. The triple action of semeiotics (syntactics, semantics, pragmatics) has to be put at work, in order to understand the global and local forces which shape the ongoing logical passages over the continuum."

Fernando Zalamea (Peirce's Logic of Continuity)

"Some digital audio mixing applications let the user set the grid or time signature on a per-track or even per-clip basis. These temporal grids need not be rigid. Rather, their tempi can be elastic, deforming in the presence of pivotal sound events, analogous to the way that spacetime is warped by the presence of matter (Einstein 1916). In the extreme, the grids may evaporate, leading to free and open temporal spaces."

Curtis Roads, Composing Electronic Music

"It occurs to me that preliminary yagé nausea is motion sickness of transport to yagé state..."
"Fats Terminal came from The City Pressure Tanks where open life jets spurt a million forms, immediately eaten, the eaters canceled by black time fuzz..."
William S. Burroughs

"It is hard to explain, but it bends your thought into a nonlinear, looping sort of format... it pretzels your thoughts into Möbius strips; you see everything inside and out and curling all around itself."

James St. James

"Paysages dentelés, horizons fuyants, perspectives de villes blanchies par la lividité cadavéreuse de l'orage, ou illuminées par les ardeurs concentrées des soleils couchants, — profondeur de l'espace, allégorie de la profondeur du temps."

Baudelaire (Les Paradis artificiels)

"Qual o peso da luz?"

A Hora da Estrela (narrador)

"La métaphorisation de la métaphore, sa surdéterminabilité sans fond, semble inscrite dans la structure de la métaphore, mais comme sa négativité. Dès qu'on admet que dans une relation analogique tous les termes sont déjà pris, un à un, dans une relation métaphorique, tout se met à fonctionner non plus en soleil mais en étoile, la source ponctuelle de vérité ou de propriété restant invisible ou nocturne. Renvoyant en tout cas, dans le texte d'Aristote, au problème du nom propre ou de l'analogie de l'être."

Derrida (La mythologie blanche)

"... il n'y a pas d'espace homogène, soumis à un seul et même type de technique et d'économie... y a-t-il, à l'intérieur d'un espace auquel se rapporte un seul et même corps «propre», des facteurs d'hétérogénéité et par conséquent des impératifs économiques différents, voire incompatibles, entre lesquels il faut choisir et parmi lesquels des sacrifices sont nécessaires, et une organisation des hiérarchies."

Derrida (De la grammatologie)

"Avant l'actualisation, d'un part — une fois que l'individuation a eu lieu —,  de l'autre: il nous faudra ainsi changer de métaphysique, ou plutôt renoncer à la métaphysique (celle de l'être éternel opposé au devenir, ou de l'être absolu opposé au à l'apparence), pour que, passant par cette autre distinction première, mais qui, cette fois, se refuse à la séparation (elle fait ressortir au contraire la constante transition du réel), on puisse entrer dans cette logique de la processivité (même la distinction aristotélicienne puissance/acte ne saurait cadrer, et même elle n'a rien à voir)..."

François Jullien (Traité de l'efficacité) 

"The great square has no corners; 

The great completion is not completed..."

Daodejing/41 (Edmunds Ryden's translation)

"... that there is no present and no past, that there are no dates, that time—and language which is time's expression—is a series of coincidences which are general all over humanity."

Richard Ellmann

"Procurei demonstrar-lhe a insuficiência da caracterização tradicional da poesia como arte que se realiza no tempo, acentuando que as artes evoluíam no sentido de uma superação das noções de tempo e espaço..."

Augusto de Campos (Memória e Desmemória)

"... l'originalité du cubisme a trait à sa représentation, sans précédent connu, de la profondeur. Franquissant les limites conventionnelles de la troisième dimension, utilisées depuis la Renaissance... une certaine manière de représenter la durée... une connaissance nouvelle des intervalles, chargé d'autant d'énergie que les figures qui les déterminent. Dans la peinture cubiste, ce sont les objets déterminés par le peintre, ou plutôt leur dynamisme, qui crée l'espace..."

André Masson (Peindre est une gageure/Écrits, anthologie établie par Françoise Levaillant)

"Depuis Cézanne (bien qu'il se soit appuyé sur la perspective traditionnelle; en partie) l'inquiétude spatiale ne nous a plus quittés, et pourquoi? — Qui ne le sait: nous ne pouvons plus séparer le concept de l'espace de celui de la durée."

André Masson (Depuis Cézanne.../Écrits, anthologie établie par Françoise Levaillant)

"La touche employée comme il convient sert à prononcer les différents plans des objets. Fortement accusée elle les fait venir en avant: le contraire les recule."

Delacroix (quoted by Masson, Le Maitre des Orages/Écrits, anthologie établie par Françoise Levaillant)

"L'espace pour le peintre d'Asie, n'est ni extérieur, ni intérieur, il est jeu d'énergies — surgissement pur. Il est l'insituable... un camp magnétique où se rencontrent et s'enchevêtrent des forces —un lieu où s'ébattent sillages et trajectoires. Et renoncer au foyer unique... Ces considerations sur l'émancipation spatiale, et l'expression des forces élémentaires ne se limitaient pas à la peinture à l'encre ou à l'eau, elles s'étendaient à la pratique de l'huile, dans le sens de la fluidité et de l'emportement. Ainsi, le Turner de la dernière période y trouverait sa place."

André Masson (Une peiture de l'essentiel/Écrits, anthologie établie par Françoise Levaillant)

"[Sounds that are not noted] appear in the written music as silences, opening the doors of the music to the sounds that happen to be in the environment... The glass houses of Mies van der Rohe... There is no such thing as an empty space or an empty time."

John Cage (Experimental Music)

"... ein vollkommnes Ausser-sich-sein mit dem distinktesten Bewusstsein einer Unzahl feiner Schauder und Überrieselungen bis in die Fusszehen; eine Glückstiefe, in der das Schmerzlichste und Düsterste nicht als Gegensatz wirkt, sondern als bedingt, als herausgefordert, sondern als eine nothwendige Farbe innerhalb eines solchen Lichtüberflusses; ein Instinkt rhythmischer Verhältnisse, der weite Räume von Formen überspannt..."

"... das Süßeste, Leichtfertigste und Furchtbarste strömt aus Einem Born mit unsterblicher Sicherheit hervor. Man weiß bis dahin nicht, was Höhe, was Tiefe ist; man weiß noch weniger, was Wahrheit ist."

Nietzsche

"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."

Carl B. Boyer (A History of Mathematics)

"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..."

"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."

"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."

"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."

"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."

"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

"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)

"... a mathematics attentive to the phenomenon of shifting, but with the capacity to detect invariants behind the flux, a mathematics that goes against the grain of supposedly ultimate foundations, absolute truths, unshakable stabilities, but which is nevertheless capable of stabilizing asymptotic webs of truth. In Grothendieck’s work, objects tend to be situated over certain ‘bases’ (the sheaf over its underlying topological space, the scheme over its spectrum), and many important problems arise when base changes are carried out."

"... the dialectic of discovery and invention in Grothendieck... cannot be reduced to either of its two poles... a position as much realist (‘discovery’) as idealist (‘invention’) is indispensable in advanced mathematics, once the latter goes beyond a certain threshold of complexity for structures, languages and the transits/obstructions at stake."

Fernando Zalamea (Synthetic Philosophy of Mathematics)

"... it is the subjective act of transcendental synthesis which transforms the chaotic array of sensual impressions into 'objective reality.' Shamelessly ignoring the objection that we are confounding ontological and empirical levels, here we must invoke quantum physics: it is the collapse of the quantum waves in the act of perception which fixes quantum oscillations into a single objective reality. And, furthermore, this point must be universalized: every figure of reality is rooted in a determinate standpoint. Even at a level closer to us, we know how different 'reality' appears to a hog or a bird, starting with the different tapestry of colors: each living being perceives (and interacts with) its own 'reality.' And one should push this insight to the extreme of Cartesian doubt: the very notion of greatness should be relativized. How do we know that our Milky Way is not just a speck of dust in another universe?"

Slavoj Zizek (Haunted by A/Z, Feyerabend with vengeance & the New Wave)

**************************************************************

BOTH depend on arrays of matter, on arrays of aggregates, that is, ultimately, on arrays of forces, on informational differences, the manifold turnings of uncountable autonomous spectral Möbius strips, Mallarmé's coup de dés:

real space (since the creation of the world until its final destruction, if ever detected):

"In mathematical space, and even in physical space, absolute measurement seemed to elude us, since in view of the continuity of space it appeared impossible to proceed with an enumeration of points. But in the case of the retina, its surface is no longer homogeneous; it possesses a heterogeneous structure like all tissues, probably a discrete one forming a pattern. In all such cases a definite metrics suggests itself naturally, just as on a net, in the absence of a ruler, we would compare lengths instinctively by counting the holes separating our points..."

"But... our intuitive visual appreciations yield results which differ from results obtained with rods, not merely accidentally as a result of the imperfection of human observation, but systematically... A coin that measures out as round will appear flattened to the eye. This phenomenon is illustrated by the well-known optical illusion wherein two rigid rulers which coincide when placed side by side, appear of unequal magnitude when placed horizontally and vertically, respectively. It is a well-known fact that the vertical appears to be longer than the horizontal. For this reason, vertical stripes on a cloth cause the wearer to appear thinner and taller..."

"... the mere fact that we have agreed to accept such discrepancies as due to optical illusions rather than to untrustworthiness of our rods proves that we deliberately reject our intuitive judgment of shape and size in favour of more sophisticated rules of measurement. In other words, we have abandoned direct intuition for physical determinations, hence for convenient but conventional standards..."

"... it is the physical behaviour of material bodies and light rays which is in the final analysis responsible for our natural belief in absolute shape..."

"... the geometry the physicist credits to space is contingent on his acceptance of a number of physical laws; and by varying these laws in an appropriate way he could still account for observed facts and credit corresponding types of geometry to space... the real space of physicists [is] the space to which he is led when he seeks to co-ordinate the phenomena of the physical world with the maximum of simplicity... the various material bodies we encounter are by no means identical in nature; some are light, others are heavy, and their chemical and molecular constitutions are certainly not the same. And yet in every case, whether our rods be of wood, of stone, or of steel, we obtain the same Euclidean results provided we operate as far as possible under the same conditions of temperature and pressure... Then again, there are the dynamical properties of space, which we cannot afford to neglect. If physical space were amorphous, all paths through space should be equivalent, and yet centrifugal force and forces of inertia manifest themselves for certain paths and motions and not for others... real space appears to be permeated by an invisible field, the Metrical Field, endowing it with a metrics or structure."

"Riemann did not attribute this structure of space to the presence of some invisible medium, the ether, possessing a structure of its own... He felt that the metrical field of space should be compared to a magnetic or an electric field pervading space... Riemann searched for the physical cause of the metrical field... he found it in the matter of the universe... a redistribution of the star matter in the universe, altering as it would the lay of the metrical field, would produce deformations in the shape of a given body and variations in the paths of light rays... Any non-homogeneous distribution of matter would then entail a variable structure or geometry for space from place to place."

"... Einstein had been led to recognise that space of itself was not fundamental. The fundamental continuum whose non-Euclideanism was to be investigated was one of Space-Time, a four-dimensional amalgamation of space and time possessing a four-dimensional metrical field governed by the matter distribution."
[And why Space-Time? Ultimately because of the very strange peculiar behaviour of light rays, which do not vary in speed when going from one frame of reference to another (this speed is not affected by the movement of bodies in the frames, while it would be expected that the rays suffered some kind of friction). "... the importance of Einstein definition lies... in its enabling us to co-ordinate time reckonings in various Galilean frames in relative motion. So long as we restrict our attention to space and time computation in our frame, we may, as before, appeal to vibrating atoms for the measurement of congruent time-intervals and to rigid rods for the purpose of measuring space. It is when we seek to correlate space and time measurements as between various Galilean frames in relative motion that astonishing consequences follow."]

- A. D'Abro, The Evolution of Scientific Thought: from Newton to Einstein (Dover, 1950);
***Also: "The victory over the concept of absolute space or over that of the inertial system became possible only because the concept of the material object was gradually replaced as the fundamental concept of physics by that of the field. Under the influence of the ideas of Faraday and Maxwell the notion developed that the whole of physical reality could perhaps be represented as a field whose components depend on four space-time parameters. If the laws of this field are in general covariant, that is, are not dependent on a particular choice of coordinate system, then the introduction of an independent (absolute) space is no longer necessary," Albert Einstein's foreword to Max Jammer's Concepts of Space (Dover 1993).

real number (as far out of the high-way of thinking, not excepted):

"... if two sequences are asymptotic to a third, they are asymptotic to each other, and, furthermore, if one converges to a certain rational number as a limit, the same is true of the other... a great number of sequences may, in spite of their difference in form, represent the same number... an evanescent geometrical sequence will always converge towards a rational limit, and... any rational number can be regarded as the limit of some rational geometrical series..." 

"... and so Cantor extended the idea of convergence, which hitherto applied only to those sequences which were asymptotic to some rational repeating sequence, by identifying the two terms self-asymptotic and convergent... he extended the idea of limit by regarding the self-asymptotic sequence as generating a new type of mathematical being which he identified with what had long before him been called real number."

"Inasmuch as any real number can be expressed by infinite convergent rational sequences, the rational domain, reinforced by the concepts of convergence and limit, will suffice to found arithmetic, and through arithmetic the theory of functions, which is the cornerstone of modern mathematics... this capital fact is of just as great importance in applied mathematics. Since any rational sequence can be represented as an infinite decimal series, all computations may be systematized."

"We do not confine ourselves any more to using infinity as a figure of speech, or as shorthand for the statement that no matter how great a number there is one greater: the act of becoming invokes the infinite as the generating principle for any number; any number is now regarded as the ultra-ultimate step of an infinite process; the concept of infinity has been woven into the very fabric of our generalized number concept."

"... wether we use a ruler or a weighing balance, a pressure gauge or a thermometer, a compass or a voltmeter, we are always measuring what appears to us to be a continuum, and we are measuring it by means of... the aggregate of numbers at our disposal; we are tacitly admitting an axiom which plays within this continuum the role which the Dedekind-Cantor axiom plays for the straight line."
"... while Galileo dodged the issue by declaring that the attributes of equal, greater, and less are not applicable to infinite, but only to finite quantities, Cantor takes the issue as a point of departure for his theory of aggregates. And Dedekind goes even further: to him it is characteristic of all infinite collections that they possess parts which may be matched with the whole... The reader will remember Liouville's discovery of transcendentals. This existence theorem of Liouville was re-established by Cantor as a sort of by-product of his theorem that the continuum cannot be denumerated... the algebraic and the transcendental... the power a of the aggregate of natural numbers... the power c of the continuum... And here too, in this domain of real numbers, the part may have the power of the whole... a segment of a line, no matter how short, has the same power as the line indefinitely extended, an area no matter how small has the power of the infinite space of three dimensions..."

- Tobias Dantzig, Number: the language of science (Plume, 2007);

***Also: "[Galileo] argues that by bending a line segment into the shape of a circle one has 'reduced to actuality that infinite number of parts into which, while it was straight, were contained in it only potentially," Amir D. Aczel, The Mystery of the Aleph (WSP, 2000); "[in Galileo's On Two New Sciences] Salviati sets up a one-to-one correspondence between all the integers and all the squares of integers and says 'we must conclude that there are as many squares as there are numbers'... Galileo found that infinite sets are very different from their finite counterparts: an infinite set can be shown to have 'the same number of elements' as a proper subset of itself," Aczel, The Mystery; "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," Aczel, The Mystery; "most numbers on the number line are transcendental... Choosing a rational number, or an algebraic one—even though there are infinitely many of them—[on the real line] is just too unlikely because of the preponderance of the transcendental numbers," Aczel, The Mystery; "As far as infinity goes, dimension does not matter. Any continuous space, whether a line or a plane or a n-dimensional space, has as many points as the continuum. All these spaces are uncountable," Aczel, The Mystery. 

real space & real number (to let us a vagary some millions of miles into the very heart of the planetary system):

*****Cabalistic, hermetic and neoplatonic (& other related peculiar) notions about space & number:

"... the Lord is the dwelling-place of His world but His world is not His dwelling-place..."

"... adding the squares of the numbers corresponding to the letters of the holy name one gets the sum of the numbers that correspond to the letters of the word 'place'..."

"... God is the center of everything, whose circumference is nowhere to be found [Fludd/Trismegistus]."

"... Deus creaturus mundos contraxit praesentiam suam [Luria]."

"Although Newton does not explicitly draw the conclusion that centrifugal forces determine absolute motion which in its turn determines absolute space, it is clear that this was his intention."

"... by existing always and everywhere, [God] constitutes duration and space [Newton]."

"... when the immediate cause of change is in the body, that body is truly in motion; and then the situation of other bodies, with respect to it, will be changed consequently, though the cause of that change be not in them [Leibniz]."

"Kant finds the clue to the riddle of left and right in transcendental idealism. The mathematician sees behind it the combinatorial fact of the distinction of even and odd permutations [Hermann Weyl]."

"Space, as a pure form of intuition, leads, according to Helmholtz, to a single conclusion: that all objects of the external world must necessarily be endowed with spatial extension. The geometric character of this extension, however, is in his view purely a matter of experience."
"Newton's experiment with the rotating vessel of water simply informs us, that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotation with respect to the mass of the earth and the other celestial bodies [Mach]."

"If the ether as an absolute system could be demonstrated, the notion of absolute space could be saved. Indeed, one of the most important experiments to this end, the Michelson-Morley experiment, was in 1904 interpreted by Lorentz in this sense."

"Gauss seems to have recognized the logical possibility of a non-Euclidean geometry even before Lobachevski and Bolyai came out with their sensational discoveries."

"Gauss great contribution to differential geometry rests in his proof that the curvature of a surface, which is determined as a reciprocal product of the two principal radii, can be expressed in terms of intrinsic properties of the surface."

"A continuous n-dimensional manifold is called a Riemannian space, if there is given in it a fundamental tensor."

"... the concept of length or distance is foreign to the amorphous continuous manifold and has to be put in or impressed from without."

"Riemann selected the simplest hypothesis, namely, that ds is the square root of a homogeneous function of the increments of the second degree. He was fully aware of the arbitrariness in his determination of the length of the line element and emphasized the possibility of other expressions, as, for instance, the fourth power of ds as a biquadratic form of the coordinate differentials. The problem is of course connected with the question of the validity of the Pythagorean theorem in the vicinity of a point."

"... being essentially a geometry of infinitely near points, Riemann's theory of space conforms to the Leibnizian idea of the continuity principle, according to which all laws are to be formulated as field laws..."

"Riemann's geometry can be compared with Faraday's field interpretation of electrical phenomena... as a strictly homogeneous magnetic or electrostatic field is never encountered in reality, so a homogeneous metrical field of space is only an idealization... as the the physical structure of the magnetic or electrostatic field depends on  the distribution of magnetic poles or electric charges, so the metrical structure of space is determined by the distribution of matter..." 

"Es muss also entweder das dem Raume zu Grunde liegende Wirkliche eine discrete Mannigfaltigkeit bilden, oder der Grund der Massverhältnisse ausserhalb, in darauf wirkenden bindenden Kräften, gesucht werden [Riemann]."

"The only one who allied himself firmly to Riemann's 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 the varying curvature [of space]."

"... this property of being curved or distorted is continually  being passed on from one portion of space to another after the manner of a wave... this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial... in the physical world nothing else takes place [Clifford]."

"For Aristotle, space was an accident of substance; for Clifford, so to speak, substance is an accident of space."

"Gravitation, as understood by the theory of general relativity, is to be comprehended in the geometric structure of space-time."

"... the idea of a fourth dimension was cordially welcomed in spiritual circles. Henry More had already applied this notions for his spiritualistic conception of spissitudo essentialis... J. K. F. Zöllner..."

"Cantor's famous one-to-one correspondence between the points of a line and the points of a plane... showed deficiencies of the traditional definition of dimensionality."

"The distance between two particles [in microphysics] is determined by the minimal number of particles necessary to form a chain of coincidences between the given particles."

"A profound epistemological analysis of certain quantum-mechanical principles seems to suggest that the traditional conceptions of space and time are perhaps not the most suitable frame for the description of microphysical processes... In his discussion of electron transitions between stationary states within the atom, Niels Bohr already called such processes 'transcending the frame of space and time.'"

"Les données de nos perceptions nous conduisent à construire un cadre de l'espace et du temps où toutes nos observations peuvent se localiser. Mais le progrès de la Physique quantique nous amènent à penser que notre cadre de l'espace et du temps [as classically or ordinarily understood] n'est pas adéquat à la véritable description des réalités de l'échelle microscopique" [Louis de Broglie].

"... it has been suggested by Riemann and Clifford, and later ingeniously corroborated by Einstein in his theory of general relativity, that the metric of space structure is a function of the distribution of matter and energy."

"Mach's Principle, as originally announced, claimed the intrinsic dependence of every local inertial system, that is, a local coordinate system in which Newton's laws hold, upon the distribution of mass in the universe."

"Should it become evident that Mach's Program cannot be satisfied within the general theory of relativity perhaps merely because the energy-momentum tensor which caracterizes matter presupposes already metrical magnitudes: in other words, because matter cannot be understood apart from knowledge of space-time, then matter as the source of the field will become part of the field. On the basis of such a unified field theoretic conception as proposed for example by J. Callaway, the field itself would constitute the ultimate, and in this sense absolute, datum of physical reality" [***this statement shows that the dichotomy referred in the book's last chapter between "relational" and "absolutist"  conceptions of space (space doesn't exists vs. space does absolutely exist) is actually a pseudo-problem, with no interest whatsoever outside the monopoly of academic philosophical priesthood; that is, what is important is the concept of field, and that there is some probabilistic distribution over it, independently of the ontological status of the much weaker classical and ordinary concepts of matter and space].

"During the past few years some attempts have even been made, as by Takao Tati, to formulate the fundamental laws of interactions between elementary particles without using the concept of space-time, which thus becomes a statistical notion, like 'temperature' in statistical mechanics..."

"As Salecker and Wigner rightly remarked, the concepts of rigid reference frames or of (practically) rigid rods as conventionally defined and taken as the basis for the construction of space-time metric and for the physical interpretation of the Lorentz covariance cannot be meaningfully applied in the quantal world of elementary particles." 

- Max Jammer's Concepts of Space (Dover 1993);

****Also (to dive deep enough into the first causes of human ignorance and confusion):"The term 'field' was first introduced into science by Michael Faraday in the 1840s, in connection with electricity and magnetism. His key insight was that attention should be focused on the space around a source of energy, rather than on the source of energy itself. In the nineteenth century the field concept was confined to electromagnetism and light. It was extended to gravitation by Einstein in his general theory of relativity in the 1920s. According to Einstein, the entire universe is contained within the universal gravitational field, curved in the vicinity of matter. Moreover, through the development of quantum physics, fields are now thought to underlie all atomic and subatomic structures... Fields are inherently holistic. They cannot be sliced up into bits, or reduced to some kind of atomistic unit; rather, fundamental particles are now believed to arise from fields," Rupert Sheldrake's Seven Experiments that could change the world (Riverhead 1995);

"The uncertainty principle of Werner Heisenberg suggests... that the vacuum is not empty. Accord­ing to the principle, uncertainty about the energy of a system increases as it is examined on progressively shorter time scales. Particles may violate the law of the conservation of energy for unobservably brief instants; in effect, they may materialize from nothing­ness. In QED [Quantum Electrodynamics] the vacuum is seen as a complicated and seething medium in which pairs of charged "virtual" parti­cles, particularly electrons and posi­trons, have a fleeting existence. These ephemeral vacuum fluctuations are polarizable just as are the molecules of a gas or a liquid. Accordingly QED predicts that in a vacuum too electric charge will be screened and effectively reduced at large distances," Chris Quigg's Elementary Particles and Forces (Scientific American, vol. 252, n. 4, 1985, pp. 84-95);

"For if in the von Neumann formulation one does seek to determine the cause of the ‘free choice’ within the representation of the physical brain of the chooser, one finds that one is systematically blocked from determining the cause of the choice by the Heisenberg uncertainty principle, which asserts that the locations and velocities of, say, the calcium ions, are simultaneously unknowable to the precision needed to determine what the choice will be. Thus, one is not only faced with merely a practical unknowability of the causal origin of the ‘free choices’, but with an unknowability in principle that stems from the uncertainty principle itself, which lies at the base of quantum mechanics," Jeffrey M. Schwartz, Henry P. Stapp and Mario Beauregard's "Quantum physics in neuroscience and psychology: a neurophysical model of mind–brain interaction," Philosophical Transactions of the Royal Society (2005).

"Bohm and Hiley (1984) generalized the Penrose twistor theory to a Clifford algebra, paving the way for a description which allows continuous space-time to emerge from a deeper pre-space they call an implicate order. Bohm (1986) further proposed that the “ex-plicate” space and time that we consciously experience is likewise projected from its enfoldment in deeper implicate orders. In neural terms what becomes interesting here is Pribram’s (1991) holographic theory of neural memory, for the hologram (where information about the whole is stored in each part) is a paradigmatic example of an implicate order (of course, similar ideas have been explored in connection with artificial neural networks)," Paavo Pylkkanen's "Cognition, the implicate order and rainforest realism", Futura, 31, 2012/2, pp. 74-83.

"But that, that’s a very important point, because most people think that Bohm, and maybe myself are determinists and we’re not... start with the structure of process idea. Now, in that structure of process idea I have built into it the idea of an implicate order. What we can do from the implicate order? We can project from it into an explicative order. And in the explicate order you can construct trajectories. So, you’re actually constructing your position and your momentum and your trajectories from this deeper order. You’re not assuming it is there a priori and then generalizing the order... Is the holomovement something which is local or nonlocal? My answer would be, it’s neither" Basil Hiley interview.

"Piaget describes how, when children are asked to draw a map of the local area around their home and are asked to position the playground, school, ice cream shop, dentist etc., they place the ice cream shop and the playground close to home, but will place the dentist and the school far from home, regardless of their actual physical distance from their home. The children are using a neighbourhood relation which has to do with pleasure and not physical distance. Thus by generalising the notion of neighbourhood, it is possible to have many different orders on the same set of points depending on what is taken to be the relevant criterion for the notion of neighbourhood in a particular context. Thus our description becomes context dependent and not absolute. This is very important both for thought and quantum theory... The importance of context in quantum theory has only recently begun to emerge, although it was always implicit in Bohr’s notion of wholeness. However in the Bohm interpretation context dependence becomes crucial," Basil J. Hiley's "Mind and matter: aspects of the implicate order described through algebra" (in K. H. Pribram's and J. King's Learning as Self-Organisation, New Jersey, Lawrence Erlbaum Associates, 1996, pp. 569-86);

"Bohm notes that if we try to analyze a thought into smaller and smaller elements we eventually come to a point where further analysis seems impossible. Analogously, some of the essential properties of a quantum system (e.g. whether it is a wave or a particle) depend on its indivisible and incompletely controllable connections with surrounding objects. This suggests that both thought and quantum phenomena are characteristic of a radical type of wholeness, unanalyzability and context-dependence," Paavo Pylkkänen's "Fundamental Physics and the Mind – Is There a Connection?" (in H. Atmanspacher's, C. Bergomi's, T. Filk's, K. Kitto's, Quantum Interaction 2014: 8th International Conference, Filzbach, Switzerland, June 30 - July 3, 2014);
"... in the 1950s one reaction against the failure of field theory in strong-interaction physics was to turn away from intractable Feynman diagrams towards the transition probabilities themselves—perhaps at this level of analysis some sense could be made of the strong interaction... the entire array of probabilities covering transitions between all conceivable initial and final states was known as Scattering- or S-matrix," Andrew Pickering's Constructing Quarks (Chicago, 1984);

"... the two tendencies, intuitionist and platonist, are both necessary; they complement each other, and it would be doing oneself violence to renounce one or the other... the idea of the continuum is a geometrical idea which analysis expresses in terms of arithmetic... the concept of number appears in arithmetic. It is of intuitive origin, but then the idea of the totality of numbers is superimposed... in geometry the platonistic idea of space is primordial," Paul Bernays, as quoted by Ian Hacking in Why there is Philosophy of Mathematics at all (Cambridge, 2014);

"Dedekind was deeply impressed by the characterization of simply infinite sets as those that can be mapped into some of their subsets, and thought the characterization of number might begin there," Hacking's Why is there Philosophy of Mathematics;

"Benacerraf observed that numbers cannot be identical to any one of their analyses in set theory, for those analyses are not identical. At best the integers can be the shared structure of all sound analyses," Hacking's Why is there Philosophy of Mathematics;

"Paul Schützenberger holds that integers exist only in physics... He calls this a Pythagorean thesis. What impresses him is that, for example, in crystallography there are certain intrinsic whole number relationships. That's where integers live," Hacking's Why is there Philosophy of Mathematics;

do fields fluctuate (at certain intervals and changes of the moon)? 

This question was proposed by Rupert Sheldrake (chaste star!). Actually, his question was about fundamental physical constants (such as the Gravitational constant, the velocity of light, and Planck's constant). Do they fluctuate? But since these constants govern fields, the point is basically the same:

"There has been little consideration of the third possibility, which is the one I am exploring here, namely the possibility that constants may fluctuate, within limits, around average values which themselves remain fairly constant. The idea of changeless laws and constants is the last survivor from the era of classical physics in which a regular and (in principle) totally predictable mathematical order was supposed to prevail at all times and in all places... What I propose is a series of measurements of the universal gravitational constant to be made at regular intervals—say monthly—at several different laboratories all around the world, using the best available methods. Then, over a period of years, these measurements would be compared. If there were underlying fluctuations in the value of G, for whatever reason, these should show up at the various locations. In other words, the 'errors' might show a correlation..." (Sheldrake, Seven Experiments).
"If nature is constructed of knowables, then the acts of knowing with which we are familiar should be special cases of a pervasive set of similar acts: the world should somehow be constructed of such acts, and of a substrate that is suited to be acted upon by such acts, but that supports, as a matter of principle, only what can become known by other acts. Acts of knowing become, then, the primitives of nature, along with the substrate upon which they act" (Jeffrey M. Schwartz, Henry P. Stapp and Mario Beauregard, "Quantum physics in neuroscience and psychology: a neurophysical model of mind–brain interaction," Philosophical Transactions of the Royal Society).

corollary &/or FICTION is REAL, complex-imaginary prevails, Riemann with revenge! (this is to serve for parents and governors instead of a whole volume upon the subject):

*******And here you will find a paper I presented originally at the Joint Annual Conference of the (known by the name of) Society for European Philosophy and the Forum for European Philosophy Annual Conference ("Philosophy After Nature", Utrecht University, 2014). As notable and curious a dissertation as ever engendered in the womb of speculation, it addresses problems in the foundation of statistic classifications in relation to the topological notion of continuum and concepts coming from post-structuralist thinking and literary studies: Jakobson's notion of shifter in particular, and Blanchot's reading of Henry James' The Turn of the Screw. As I argue, to think about reality, literary notions such as shifter are much more rich and interesting than classical philosophical notions such as Kripke's rigid designator.

The paper has been accepted as a submission by several journals, but has never been really published, on account of its interdisciplinary and supposedly at the same time "flippant" and "demanding" style (the journals include Journal of Historical Fictions, Studies in the Novel, University of Toronto Quarterly, Mosaic, Victorian Review, OLR, Configurations, The Journal of Popular Culture, Explorations: a journal of language and literature, among others):

- Interactive while Indifferent—Kinds & Phantasmagoria circa 1900;

More on non-locality (and complementarity): 

"For D'Espagnat, it is exactly if we consider seriously and realistically certain results of physics that we have to open the two complementary perspectives. D'Espagnat welcomed these complementary perspectives grounding himself on the many violations of CHSH-Bell inequalities obtained experimentally since the 1980s," Alessandro Zir, Luso-Brazilian Encounters of the Sixteenth-Century: A Styles of Thinking Approach (Fairleigh Dickinson Univ. Press, 2011, p. 66).

What makes this book difficult to understand and accept (it received just a few although favourable reviews), is an insight that classical dichotomies such as subjective/objetive, fictional/real, should be understood as the mere result of displacements of boundaries in an overall indeterminate context (borrowing from Joyce, you could call this context a chaosmos) which is always somewhat "excessive." This context is neither literary nor scientific, strictly speaking. Or to say it in a positive way: the reality we inhabit is both literary and scientific (which doesn't mean these two perspectives should be simply collapsed), it encompasses non-fiction and fiction (Mallarmé's conception of the book, for instance), and fiction is real (in a deadly serious way, not as some fanciful proposition). This is a key idea in deconstruction (and deconstruction goes back to centuries before Derrida, as he himself willingly admitted). I found it in the first encounters between Europeans and the land, fauna, flora and people of Brazil. But very unfortunately scholars have emphasized only the more superficial political implications of deconstruction while forgetting its "metaphysical" dimension (and both things have to go together). Post-colonial studies suffers conspicuously from this reductionist and much impoverished approach. 

More on David Bohm, Niels Bohr & Wolfgang Pauli: 

"During this period which I found somewhat depressing as far as physics was concerned, we got a student, Donald Schumacher, was quite brilliant. He came from America, but he was not very stable mentally. He took a great interest in Niels Bohr... This fellow Schumacher had some insight which made it much more clear what Bohr was about. What he said could be summed up by saying that the form of the experimental conditions and the meaning or content of the results are a whole, not further analyzable. This question of the observer and the observed, which is fascinated by quantum mechanics, is very hard to put consistently without confusion... it would have been a very repellant philosophy to me had I understood that in the early days. To say we have nothing but the appearance and we have a mathematical algorithm which enables us to compute the probability of a certain phenomenon. Looking at it more carefully, Bohr did not deny there is reality, but he denied you could say anything about it. Now, Pauli did not do that. He said quite a bit about it. He said that the mind is involved, the mind of the observer is involved in that reality. I would not have gone on with Pauli that way, but there were a great many other points about Pauli I probably would have gone on... Pauli is pretty elusive too..." David Bohm's conversation with Maurice Wilkins;
"Bohr argued that we should not regard the coupling between entangled pairs as arising from a mechanical force. Rather he talks of 'an influence on the very conditions which define the possible types of prediction regarding the future development of the system'. Bohr felt this was a key point because he italicised the phrase in his original paper and repeated it word for word in his Discussions with Einstein. In a mechanical explanation all 'influences' must be mechanical, but our proposals suggest a more 'organic' picture where information is playing a dynamic and objective role, namely, it is active. Thus once again rather than contradicting Bohr, I argue that the Bohm interpretation actually offers some clarification of his position!" Basil Hiley's "From the Heisenberg Picture to Bohm: a New Perspective on Active Information and its relation to Shannon Information" (in A. Khrennikov, Proc. Conf. Quantum Theory: reconsideration of foundations, Sweden, Växjö University Press, pp. 141-162,  2002);

More things about relativity: 

"... since our judgment of straightness is contingent on the disclosures of experience, even the geometry of the space in which we actually live cannot be decided upon a priori... But today, thanks to Einstein, we have definite reasons for believing that ultra-precise observation of nature has revealed our natural geometry arrived at with solids and light rays to be slightly non-Euclidean and to vary from place to place. So although the non-Euclidean geometers never suspected it (with the exception of Gauss, Riemann and Clifford), our real world happens to be one of the dream-worlds whose possible existence their mathematical genius foresaw..."
"In other words, the geometry the physicist credits to space is contingent on his acceptance of a number of physical laws; and by varying these laws in an appropriate way he could still account for observed facts and credit corresponding types of geometry to space."

"... a first reason for rejecting the concept of an amorphous space arises when we find that a large number of different methods of investigation all point to the same definite metrics for space."

"Then again, there are the dynamical properties of space, which we cannot afford to neglect. If physical space were amorphous, all paths through space should be equivalent, and yet centrifugal force and forces of inertia manifest themselves for certain paths and motions and not for others."

"If Riemann's ideas are accepted we can understand how a redistribution of the star matter of the universe, altering as it would the lay of the metrical field, would produce deformations in the shape of a given body and variations in the paths of light rays."

"The fundamental continuum whose non-Euclideanism was to be investigated was therefore not one of space but one of Space-Time, a four-dimensional amalgamation of space and time possessing a four-dimensional metrical field governed by the matter distribution."

"These views were confirmed by the behaviour of physical phenomena known to classical science; hence the uniqueness of time and the absolute character of simultaneity had been accepted. In other words, the simultaneity and the order of succession of two spatially separated events were assumed to constitute facts transcending our choice of a system of reference... But the point we wish to stress is that if perchance experiment were ever to suggest that simultaneity was not absolute, no rational argument could be advanced to prove the absurdity of this opinion. As we shall see, Einstein's interpretation of certain refined electromagnetic experiments consists precisely in recognising the relativity of simultaneity and the ambiguity of duration... As is the case with space, mathematical time is an amorphous continuum presenting no definite metrics. The concept of congruence or of the equality of successive durations has no precise meaning; and a definition of time-congruence can be obtained only after we have imposed some conventional measuring standard on an indifferent duration... What is commonly called the sense of rhythm, which manifests itself in the primitive music of drum beats and in the dances of savages, is nothing but a consequence of this intuitive understanding of time-congruence."

"In other words, it is the principle of inertia coupled with an understanding of spatial congruence that yields us a definition of congruent stretches of absolute time."

"... a near approach to ideal conditions would appear to be given by the phenomenon of the earth's rotation. And so the rotation of the earth, defining astronomical time, was regarded as furnishing science with the most reliable objective criterium of congruent time-intervals that it was possible to obtain."

"Römer in 1675... noticed that Jupiter's satellites appeared to emerge from behind the planet's disk several minutes later than was required by the law of gravitation. Rather than cast doubts on the accuracy of this law, he ascribed a finite velocity to the propagation of light."

"... a great variety of methods for determining time have been considered. Some were mechanical, others astronomical, others optical, while still others appealed to radioactive phenomena... all the methods of time-congruence mentioned lead to exactly the same determinations so far as experiment can detect. It would appear as though a common understanding existed between the rates of evolution of the various phenomena mentioned."

"... coincidences constitute the most exact form of observation... It is when we seek to correlate space and time measurements as between various Galilean frames in relative motion that astonishing consequences follow."

"If we wish to define the position of a point on the plane, we must refer it to some system of reference. Three centuries ago Descartes devised a method whereby this result could be accomplished... It would be just as feasible, in place of ours horizontals and verticals, to select two families of intersecting curves... This generalization of Descartes method was introduced by Gauss, and for this reason curvilinear mesh-systems are also called Gaussian mesh-systems..."

"... the method reduces to an application of the differential calculus to geometrical problems, and for that reason is named differential geometry... we may conceive of spaces where ∆s would not tend to a limit, and where, however tiny the area of our surface, we should still be faced with waves within waves ad infinitum. The situation would be similar to that presented by curves with no tangents at any point."

"... a fundamental problem of theoretical physics is to determine how the equation of a phenomenon referred to a frame B, which is in motion with respect to a frame A, can be obtained from its equation referred to the frame A, by purely mathematical means... transformations of this caracter are termed space and time transformations. The entire problem reduces, therefore, to the discovery of those space and time transformations which hold in this world of ours. This transformations cannot be guessed a priori."

"Unless we were to assume that this extraordinary coincidence in the values of these two characteristic velocities, that of electromagnetic induction and that of light waves, was due to blind chance, there was no other alternative but to recognise that what we commonly called a ray of light was nothing else than a series of oscillations in the electromagnetic field, propagated from point to point."

"... this means that the variable v which enters into the Lorentz transformations now stands for the relative velocity of one frame with respect to the other, and no longer for velocity through the ether... In Lorentz theory there was a privileged observer situated in the frame at rest in the ether."

"... without appealing directly to the equations of electrodynamics, we can deduce the Lorentz-Einstein transformations merely by taking one of their consequences into consideration, namely, the invariant velocity of light, and combining it with the relativity of Galilean motion."

"... the colour of an opal has no meaning. It is red from here, green from there, blue from elsewhere, and yet the opal has not changed. It is our position with respect to the opal which has changed, and the color of the opal is indeterminate until such time as we have specified our relative position. In other words, length and the colour of the opal both express relations and not immanent characteristic. Similar considerations apply to the slowing down of time. Duration and time are mere relatives, mere expressions of relationships, and have no absolute significance per se. This does not mean that the duration we sense is a myth..."

"... upon first inspection, the fact that whatever our Galilean motion may be, experiments conducted in our frame will always yield exactly the same results, would seem to relegate the ether to the realm of ghosts, making it a useless hypothesis. If this were the case we could no longer conceive of electromagnetic fields and light rays as expressing states of the ether, but should have to regard these fields as constituting independent realities of some new category, differing from matter but susceptible of existence in space without the support of a carrier, or without being the mere manifestations of its states."

"The revised laws [of the Special Theory of Relativity] were found to entail the relativity of mass. By this we mean that the mass of a given body could no longer be regarded as an invariant, independent of the body's relative motion... Since part of the mass of a body when in motion was a manifestation of its kinetic energy, it appeared probable that the entire mass, the rest-mass as well as the added mass, should be identified with energy... Suppose then a particle of matter, whether at rest or in motion, were to be completely annihilated. Material mass would then be converted into non-material mass, or energy, called loose energy... it is easy to realise the tremendous liberation of energy and the dire consequences that would result from the complete disintegration of even a small parcel of matter..."

"Since light is a form of energy, light should possess mass and momentum and exert a pressure over bodies on which it impinges."

"... physical simultaneity and duration, which were considered absolutes by classical science, must now be considered relatives having no absolute universal significance... But it is most important to realise that this relativity of simultaneity holds only for events occurring in different places and does not apply to events occurring in the same place... in contradistinction to the simultaneity of events occurring in different places, which is essentially a relative depending on the choice of our frame of reference, a coincidence of events is an absolute, that is, remains a coincidence or a simultaneity in all frames of reference... Einstein's [Special] theory still permits us to retain our belief in the absolute nature of a coincidence of events, that which are copunctual and which occur at the same time in any given frame. But it denies the absoluteness of the simultaneity of spatially separated events."

"Just as each frame had its own space, so also would it now have its own time and its own definition of simultaneity. We may summarise these various discoveries by stating that there can exist no universal definition of practical congruence either for space or for time; there can exist, therefore, no geometry for a universal space or for a universal time. The philosophical consequence of these new points of view is to deprive universal space and time of the objective significance with which they were formerly credited. Henceforth, in the words of Minkowski, space and time, by themselves, sink to the position of mere shadows."

"The achievement of this supreme synthesis of the points of view of all observers [an impersonal and hence common objective understanding of nature] was accomplished by Minkowski in 1908. He succeeded in obtaining an invariant definition of congruence by combining any give observer's definitions of practical congruence for space and for time, and by showing that this combination possessed an invariant value holding equally for all other observers. Of course, the type of congruence obtained was one neither of space nor of time. It was a combination of both."

"While it is true that a world-line which measures out as straight from a Cartesian system may appear curved when referred to a curvilinear or Gaussian one, yet, on the other hand, intersections or non-intersections of world-lines are absolute, in contrast to simultaneities of events at spatially separated points... Thus, if two billiard balls kiss in the observation of one man, they will continue to kiss in the observation of all other men, regardless of the relative motion of these men. The kissing of the balls constitutes a coincidence, an intersection of the world-lines of the two balls; hence it is an absolute. On teh other hand, fi two balls hit different cushions simultaneously in te observation of one man, they will not in general hit the cushions simultaneously in the observation of a man in motion with respect to the first."

"We can also understand how it comes that the discovery of space-time permits us to account for the duality in the nature of motion, relative when translationary and uniform, absolute when accelerated or rotationary... these accelerated motions violate... the flat space-time structure; and for this reason forces of inertia will always be generated by them. On the other hand, when we consider Galilean or uniform or translational motions, we see that, as in the case of three-dimensional space, they will be represented by straight lines. These motions will then stand in perfect harmony with the flat space-time structure, and no forces of inertia will arise."

"Not only si time continuously passing, but it is ever flowing in the same direction. To this mystery of the unidirectional passage of time, they theory of relativity contributes no new information..."

"To be sure, it would be possible to reverse the process, but only through the medium of some intelligent activity sorting out the particles and distributing them according to states of lesser probability. The action of a demon of this sort, Maxwell's demon, would cause the direction of teh irreversible phenomena to be reversed, so that the direction of time would appear to change..."

"... by basing his deductions on the kinetic theory of gases, Boltzmann succeeded in giving a more concrete representation to entropy, defining it as the logarithm of a probability... the principle of entropy becomes one of maximum probability, losing thereby its status of absolute validity and assuming a statistical significance... We may note that it was by following this method, or rather by combining the two definitions of entropy [Clausius' and Boltzmann's], that Planck established his quantum radiation formula, and was thus led to his quantum theory... it is a principle governing the chaos, or again, it is a statistical principle..."

"When we perceive an object, say a telegraph pole, we see it under a certain angle, which we may call the visual angle. Is this visual angle real?... It is real but it expresses a relationship... Visual angle in classical science is analogous to length in Einstein's theory."

"The departure of our twin brother and his return to earth constitute two definite events. The duration separating two events being robbed of any definite value by the theory of relativity, there is no cause to be surprised that this duration should manifest different magnitudes to different observers... We need not be surprised at these very real effects produced by acceleration, for we must remember that acceleration is an absolute and not a relative like mere velocity."

"The precise type of force we have mentioned is called the Coriolis force; in addition, there also exists another type of force, the better-known centrifugal force. Both these types of force are called forces of inertia. It is their ensemble which constitutes the field of inertial force existing in a rotating frame... All these different illustrations show us that whereas, in Galilean frame, no field of force exists, yet a field springs into existence automatically as soon as we place ourselves in any accelerated frame."

"With these new views, the force of gravitation acting at a point loses its attributes of absoluteness. Just like a force of inertia, a force of gravitation betrays a relationship existing between the frame of reference selected and the surrounding conditions of the world. Just as a force of inertia can be annulled by changing the motion of our frame, so now can a force of gravitation be annulled."

"... in a region of space-time curvature it would be quite impossible to rid ourselves of a field of force throughout space. We might annul it at the point and in its immediate neighbourhood, but the field of force would reappear for more distant regions."

"... the equality of two vectors at a point of space constitutes an equality which a change in our co-ordinate system can never destroy... If these conditions are satisfied, we see that equations between vectors and tensors, often called vector equations and tensor equations, exhibit the remarkable property of remaining unaffected by a change of mesh-system... The simplest physical illustration of a tensor at each point of space is afforded by the state of compression and tension at each point of an elastic medium which has been distorted... we see that the magnitudes in which classical science was chiefly interested were either invariants, vectors or tensors... laws of nature must be vector or tensor equations..."

"... in a general way, we see why it is that when we leave the special theory, where space-time is flat and where the observer, being Galilean, abides by Cartesian mesh-systems, and when we investigate the general theory, where the observer is accelerated or space-time si curved, the Lorentz-Einstein transformations must give place to a more general type. Under these circumstances the tensor calculus becomes a mathematical instrument of great power... the special theory is merely a particular limiting case of the general one... It is here that Riemann's discoveries fit in with Einstein's requirements; for Riemann and his successors had discovered tensors built up from the fundamental structural tensors of a continuum..."

"... this would imply that at an infinite distance from matter, space-time would become rigorously flat, as we had assumed it to be in the special theory... where λ represents an arbitrary constant... The presence of this additional term will imply that even for very distant regions, space-time would still manifest a trace of residual spherical curvature so that the universe would close round on itself and be finite."

"Gravitation would thus preserve youth, and, more generally, two men living on two different stars of unequal mass would not age at the same rate. It is to be noted that it is not the force of gravitation itself which is responsible for this slowing down of time as we near the sun. It is rather the decrease in the value of the potential..."

"This means that one second in time corresponds in our formulae to the distance which light covers during one second, namely, 186,000 miles long. It follows, therefore, that if we measure time in seconds we must, in order to co-ordinate results, measure space in terms of unit rods 186,000 miles long."

"... the curvature of time alone is able to produce the major effect of Newtonian attraction, such as weight, whereas the effects due to the curvature of space remain imperceptible until we consider the motion of Mercury, which is moving fairly fast with respect to the inertial frame of the sun. When we consider the propagation of light rays, the effects due to the curvature of space reassert themselves fully, and as a result the double bending of a ray of light can be attributed in equal proportion to the curvature of time and to the curvature of space."

"... we see that the principle of the invariant velocity of light is accurately true only in free space far from matter, and, even then, only when computed with reference to our Galilean frame. In an accelerated frame, as in the neighbourhood of matter, the cornerstone of the special theory of relativity no longer applies."

"Calculation then showed that the mass of a body would become infinite when the velocity of the body reached that of light; for this reason no material body could ever move as fast as light."

"A distinction must be made between material mass (or energy) and radiation mass (or energy)."

"We thus find that there exists a progressive curvature of space-time as we pass from empty space-time far removed from all matter and energy to empty space-time in the neighbourhood of matter, then to regions where there are electromagnetic fields but no matter, and finally to regions filled with matter or electrons. If we regard the successive gradations of curvature of space-time as corresponding to successive grades of materialisation, starting from emptiness, we must place electromagnetic fields, or loose energy, at a stage of materialisation preceding that of matter proper, or bound energy."

"In the course of time astronomers discovered that there were millions of these spiral nebulae strewn more or less uniformly through space as far as the telescope could explore."

"Newton objectivises space... The opposite stand was taken by Mach, who endeavoured to account for the dynamical manifestations of acceleration and rotation without appealing to that suprasensible entity, 'absolute space.'"

"... when we examine the implications necessitated by Einstein's cylindrical universe, we find that Mach's views may turn out to be correct after all... [with the gravitational equations] Mach's mechanics received at last the rational justification which until then had been lacking; and the first inkling of a possible physical connection between the origin of inertia and gravitation was obtained."

"... al vectors such as velocities and forces, and all tensor such as Maxwell's stresses of the electromagnetic field, erstwhile expressed in three-dimensional space, must now be extended and supplemented with additional components so as to yield vectors and tensors in four-dimensional space-time... when we represent these intensities in terms of space-time, we find that they are given by the various components of one same space-time tensor... Electromagnetic induction, discovered by Faraday, the additional electrical term introduced tentatively by Maxwell, radio waves, everything in the electromagnetics of the field, could have been foreseen at one stroke of the pen... It is also interesting to note that the four separate relations which constitute Maxwell's equations of electromagnetics are now merged into two; and the cumbersome aspect of the equations gives place to forms of great simplicity... Still another simplification which the discovery of space-time has conferred upon us is to be seen when we study the principles of the conservation of energy and of momentum. We find that these two distinct principles constitute in reality but one. Conservation of energy is given by the time component, while conservation of momentum is given by the three space components..."

"It is usual to assume that the curvatures are produced by those concrete somethings which we call mass, momentum, energy, pressure. In this way, we must concede a duality to nature; there would exist both matter and space-time, or, better still, matter and the metrical field of space-time. Einstein, when he elaborated his hypothesis of the cylindrical universe, attempted to remove this duality by proving that it was possible to attribute the entire existence of the metrical field, hence of space-time, to the presence of matter. This attitude led to a matter moulding conception of the universe, elevating matter over the metrical field of space-time. And as we recall, only when this attitude was adhered to could Mach's belief in the relativity of all motion be accepted. Eddington's attitude is just the reverse. He prefers to assume that the equations of gravitation are not equations in the ordinary sense of something being equal to something else. In his opinion they are identities. They merely tell us how our senses will recognise the existence of certain curvatures of space-time by interpreting them as matter, motion, and so on. In other words, there is no matter; there is nothing but a variable curvature of space-time."

"... Now it is possible to remove this restriction imposed by Riemann without falling into any logical inconsistency; and we may perfectly well assume that two unit rods which coincide at a point A may cease to coincide when brought together at another point B, provided they have followed different routes while moving from A to B... length is non-transferable. In view of the indeterminateness that surrounds the comparison of lengths in different places, we must confine ourselves to the comparison of lengths at any one place or at points separated by infinitesimal intervals. We therefore agree to consider that at every point in space there is fixed a rod which is to serve as unit of length when we measure lengths situated by its side. The totality of these unit rods constitutes what is known as the gauge-system..."

"... it is a remarkable fact that these purely geometrical relations or equations which Weyl had discovered, expressing as they do the value of the Weylian curvature, are none other than Maxwell's equations of the electromagnetic field... These views led Weyl to assume that when electromagnetic fields were present, the Weylian characteristics of space-time manifested themselves, whereas, in regions free of electromagnetic fields, space-time resumed its classical characteristics and became once again the space-time of Einstein's theory."

"In order to distinguish these magnitudes from ordinary tensors, which transcend solely our choice of mesh-system, Eddington calls them in-tensors and in-invariants, and indicates them by asterisks. Henceforth it will be to these in-tensors and invariants that we shall have to appeal in order to express those magnitudes which (depending neither on mesh-system nor on gauge) are intrinsic to the world itself... This modification in the form of the Einsteinian laws will affect only the mechanical and gravitational laws, since the electromagnetic ones defining the Weylian curvature are already in-tensor equations... However, we may assume that the Einsteinian tensor laws, in teh form in which they were given by Einstein, are in harmony with some particular gauge-system; and this Weyl calls the natural gauge. We shall have to conceive of this natural-gauge system as one imposed upon us by nature and defined by certain intrinsic properties of the universe... Weyl defined the natural gauge at every point as given by a unit of length equal to the radius of curvature of the universe at that point or to some definite fraction therof."

"... in Weylian space-time not only would the transference of length of an Einsteinian interval yield indeterminate results, but that the same would apply automatically to the transference of spatial, as also of temporal, magnitudes... It would be as though the size of an atom and its frequency of vibration depended on its previous life history..."

"Since objects in equilibrium, such as rigid objects, electrons and atoms, all adjust themselves to the radius of curvature of the universe, it is only natural that results of greatest simplicity should be obtained when we select this radius or some definite fraction therof as a gauge at every point."

"... we were led to Weyl's generalisation, which permits the electromagnetic field to enter into the general synthesis, no longer as a foreign adjunct, but as a constituent element of space-time structure. But, besides the field, there is the electron, and the field alone does not seem to afford us the possibility of building up the electron; we cannot account for atomicity or for the quanta of action. Therein resides the problem of matter, and it remaisn to-day an outstanding challenge to science... Field physics, together with the principle of action, encounters further difficulties when we consider the quantum phenomena and the problem of the atom..."

A. D'abro (The Evolution of Scientific Thought)

  

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);

- the most auspicious tetrahedron;

- 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);

- What is quantum mechanics trying to tell us? (N. David Mermin); 

And also:

- Dogen with Hagakure; 

- L'articulation (Maurice Blanchot); 

- L'intelligence des fleurs; 

- Spooky Blue;

- Dante and the 3-Sphere (Mark Peterson, American Journal of Physics, Dec. 1979); 

- 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);

- Helmholtz, Riemann, and the Sirens (Peter Pesic); 

- Rhythm and Textual Temporality (Sha Xin Wei & Garrett Laroy Johnson);

- Meter networks: a categorical framework for metrical analysis (Alexandro Popoff & Jason Yust, 2020);

- Pulse as Dynamic Attending: Analysing Beat Bin Metre in Neo Soul Grooves (Anne Danielsen, 2018);

- Time and Meter in Mande Drumming from Mali (Rainer Polak & Justin London, MTO, 2014);

- 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;