But

*those*infinities are perhaps not inevitable...
Ian Hacking

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 Möbius strips, Mallarmé's

*coup de dés*:

###
*real space:*

"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."]

[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,

***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

*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:*

"... 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

"... 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

*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:*

*****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]."

###

###

More on non-locality (and complementarity):

"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]."

"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);
"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,

*Seven Experiments that could change the world*(Riverhead 1995);
"... 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

"Dedekind was deeply impressed by the characterization of simply infinite sets as those that can be mapped into some of their subsets, and thought

"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,

"Paul Schützenberger holds that integers exist

- 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,*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,

*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,*Why is there Philosophy of Mathematics*.###
*do fields fluctuate?*

This question was proposed by Rupert Sheldrake. 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*).###
*corollary &/or FICTION* is REAL*:*

*******And here you will find a paper I presented originally at the Joint Annual Conference of the Society for European Philosophy and the Forum for European Philosophy Annual Conference ("Philosophy After Nature", Utrecht University, 2014).

*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):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,

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

========================================*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.

*Somnambule 1*:
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