ZERO, INFINITY AND NOCER

Zero, infinity and nocer 1

Article type: Research - Science Title: ZERO, INFINITY AND NOCER ('Nature abhors a vacuum' revisited) Author: Gonzalo A. Ordóñez Address: 1775 Fortstone Lane Columbus, Ohio 43228 Telefax: (614) 851 3143 E-mail: [email protected]

Abstract The primary objective of this paper is to discuss two hypotheses: the non-existence of zero and infinity in the real world, and the accumulation limit ('apex'). The starting point is Aristotle's vacuum and the zero (0) and infinity (∞) concepts. The aid of the cer/nocer notions (cerebrations and non-cerebrations) is a method used throughout the exposition in order to have adequate clarity. The apex hypothesis, if shown true in general, could foster an interesting line of theoretical-experimental work. Several speculations are advanced and a conjecture about the density apex is examined and calculated (>=1.5 E28 kg m-3). Two appendices provide additional material; in one of them, an approach for definitions is given and a symbolic analysis is carried out on concepts like perfection, determinism, chance and others.

Key Words Apex hypothesis -- Cer and nocer-- Definitions -- Nozerinf hypothesis -- Vacuum -- Zero and infinity

Abbreviations Cer Cerebration Nocer Noncerebration Massen Mass + Energy Nozerinf Non existence of zero and infinity in nocer Indivison Last indivisible unit of spacetime and/or massen Ak Accumulation Dk De-accumulation K Accumulation apex D Accumulation apex for massenic density

Vitae Gonzalo A. Ordóñez is an Ecuadorian-American engineer.


ZERO, INFINITY AND NOCER

1. INTRODUCTION This article was written in order to look at the implications of assuming a real world without zero or infinity. For this, as for so many other scientific questions, a good starting point is Aristotle.

Aristotle probably would have been astonished should he had known that his contention: void does not exist, in contrast to Democritus': void does exist, was to be an unsolved conundrum along the next 24 centuries at least. By the way, it seems that Aristotle never wrote the celebrated dictum, 'Nature abhors a vacuum'.1 This universally quoted and used aphorism (it can easily get more than 120 000 hits in the internet) is generally attributed to 'ancient Greek philosophers' or, directly but erroneously, to the Stagirite. In reality, maybe one of the earliest appearances in writing of the phrase was in France (Rabelais, 1532): "Natura abhorret vacuum", within a not at all philosophical context, and with the concept Spinoza (c.1660), for instance, concurred: "Since, therefore, there is no vacuum in nature [...]". Aristotle (The Complete Works/Jonathan Barnes) emphatically denied the existence of vacuum or void in nature:

"I (Physics), 213a, 23, p.362: Those who try to show that the void does not exist do not disprove what people really mean by it, but only their erroneous way of speaking; this is true of Anaxagoras and of those who refute the existence of the void in this way. They show that air is something --by straining wine-skins and showing the resistance of the air, and by cutting it off in clepsydras. But people really mean by void an interval in which there is no sensible body. [...]--an interval which divides the whole body so as to break its continuity, as Democritus and Leucippus hold, and many other physicists--or even perhaps as something which is outside the whole body, which remains continuous. "I (Physics), 214a, 17, p.364: Since we have determined the nature of place, and void must, if it exists, be place deprived of body, and we have stated both in what sense it does not, it is plain that on this showing void does not exist, either unseparated or separated; for the void is meant to be, not body but rather an interval in body. [...] But there is no necessity for there being a void if there is movement. It is not in the least needed as a condition of movement in general, for a reason which escaped Melissus; viz. that the full can suffer qualitative change. "I (Physics), 215b, 1, p.366: A, then, will move through B in time C, and through D, which is thinner, in time E (if the length of B is equal to D), in proportion to the density of the hindering body [...] And always, by so much as the medium is more incorporeal and less resistant and more easily divided, the faster will be the movement [...] Now there is no ratio in which the void is exceeded by body, as there is no ratio of nought to a number. For if 4 exceeds 3 by 1, and two by more than 1, and 1 by still more than it exceeds two, still there is no ratio by which it exceeds 0 [...] Similarly the void can bear no ratio to the full, and therefore neither can movement through the one to movement through the other, but if a thing moves through the thinnest medium such and such a distance in such and such a time, it moves through the void with a speed beyond any ratio. "I (Physics), 216b, 20, p.368: It is clear, then, from these considerations that there is no separate void."

1 The style of the quoted phrase is hardly consistent with Aristotle's generally concise and scientific writing style, and the author's impression was confirmed by a PDF search performed by him on the CD edition of the 'Complete Works of Aristotle' (29 books). BNPublishing (www.bnpublishing.com), acquired 2007.


Aristotle denied likewise the existence of infinities in nature (The Works of Aristotle/W. D. Ross):

"II (On The Generation Of Animals), p. 255: [...] and this would have [15] gone on to infinity. But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end."

Also (The Complete Works/Jonathan Barnes): "I (Physics), 206a, 7, p.351: It is plain from these arguments that there is no body which is actually infinite. "I (Physics), 207b, 10, p.353: But in the direction of largeness it is always possible to think of a large number; for the number of times a magnitude can be bisected is infinite. Hence this infinite is potential, never actual: the number of parts that can be taken always surpasses any definite amount. But this number is not separable, and its infinity does not persist but consists in a process of coming to be, like time and the number of time. With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size that it can potentially be, it can actually be. Hence no sensible magnitude is infinite, it is impossible to exceed every definite magnitude; for if it were possible there would be something bigger than the heavens. "I (On the heavens), 273a, 22, p.454: From this is clear that an infinite body is an impossibility; but there is a further point. If there is no such thing as infinite weight, then it follows that none of these bodies can be infinite. For the supposed infinite body would have to be infinite in weight. (The same argument applies to lightness; for if there is infinite weight, there is infinite lightness, if the rising body is infinite). "I (On the heavens), 277a, 26, p.460: There must therefore be some end to locomotion: it cannot continue to infinity. [...] But if movement were infinite speed would be infinite also; and if speed then weight and lightness. For as the lower of two bodies would be quick because of its weight, so infinite increase of weight necessitates infinite increase of speed."

The last centuries and decades of our time have seen instances of pros and cons.

Aguirre, J.B. (c. 1757): "P. 483: Modern philosophers admit the existence of vacuum, following epicureans. [...] Should vacuum not exist, movement would not exist; but movement does exist, then vacuum too. "P. 488: But Aristotle's argument goes against the possibility of coacervate vacuum, and we believe the existence of this vacuum is indeed possible [...] "P. 560: [...] categorematic infinite is given in God's mind; therefore it is possible for it to exist in the reality."

Shu, F.H. (2006): "The words 'nothing,' 'void,' and 'vacuum' usually suggest uninteresting empty space. [...] however, the vacuum has turned out to be rich with complex and unexpected behaviour [...] where quantum fluctuations [...] can lead to the temporary formation of particle-antiparticle pairs."

Eustace, J. (1987): "He [Descartes] accepted the logical consequences of his identification of spatial extension and matter and denied the possibility of a vacuum in nature [...] He also tells us what matter is not - it is not 'res cogitans' or thinking stuff. He cuts the cake in two and gives us 'res cogitans' on the one hand and 'res extensa' on the other."

Kalechofsky, R. (1997): "Aristotle [...] concluded that [...] there can be no vacuum. This error occurs in scientific literature up to the 17th century [...] In spite of important cognitive developments, the idea that 'nature abhors a vacuum,' persists."

Mainly, however, little interest has been displayed in discussing the philosophical subject of vacuum (certainly we are not referring to Torricelli's, Magdeburg's or similar, 'vacuum'


problem). Examples exist of excellent books with virtually no treatment of this issue (Margolis, J. (2006); Kenny, A. (2004); Fischl, J. (1980); Bondy, A.S. (1974)). On the other hand, the void or vacuum has been deemed identical to the number zero. Merriam-Webster's (2006):

"Zero: [...] The arithmetical symbol 0‚ denoting the absence of all magnitude or quantity [...] A state of total absence or neutrality."

Singer, J. (1999): "[...] In the real-number system, 0 is the only number that is neither negative nor positive [...] The Hindu word for zero was sûnya, meaning empty, or void [...]"

In the same line, the infinity in the universe has been identified with the mathematical concept of infinite:

Merriam-Webster's (2006):

"[...] Unlimited extent of time, space or quantity [...] An indefinitely great number or amount [...]"

2. SOME TERMS PUT FORWARD, AND PURPOSE OF THIS PAPER A conceptual device (cer/nocer--more in Appendix A) will, hopefully, clarify semantics and facilitate the discussion hereupon. Cer: Allow the name 'cerebration', or cer (same in singular and plural), for that massless and energyless entity produced by a functioning or working human brain (nervous system included). The 'functioning' quality means properties as determined by the contemporaneous brain science. Cer corresponds in general to what is called, for instance, consciousness or mind, and it is not a substance or an entity or similar; for the author, cer does not have mass or energy. Nocer: Allow the name 'noncerebration', or nocer (same in singular and plural), for everything that is not cer. It corresponds to what is called, for instance, real world, nature or reality. From current knowledge it appears that nocer world stems from a basic substrate constituted by four elementary entities now termed space, time, mass and energy. It can also be said that nocer is made up of only two 'super-entities': spacetime and massenergy ('massen'). Nocer could then be a sort of mesh or lattice structured with indivisible quantum units (let us say, 'indivisons'), which humans think of as spacetime and/or massen. The present paper intends to revisit the vacuum problem by making it precisely correspond to the following question: Does zero (0) exist or not in nocer? Question which poses the complementary one: Does infinity (∞) exist or not in nocer? The rest of this paper explores some of the results obtained when looking at these questions through two hypotheses. 3. THE TWO HYPOTHESES OF THIS MANUSCRIPT 1- Nozerinf hypothesis: in nocer world, both mathematical zero and mathematical infinity do not exist.2 For the sake of brevity this hypothesis will be referred to as nozerinf (no zero-no

2 The 'existence' concept, as used here, is based on its intuitive meaning only. I agree with the view that it belongs to a set of


infinity). Of course, in cer world mathematical zero and mathematical infinity do exist in the sense that they can be defined and handled accurately within the abstract or symbolic realm. (In this paper the terms zero and infinity always mean cer or mathematical or symbolic zero and cer or mathematical or symbolic infinity, unless stated otherwise. A nocer 'zero' is a statistical approximation to the respective cer zero. For the purposes of this paper the mathematical categories of infinity are not relevant, but just the general infinity concept). 2- Apex hypothesis. In a sense this is a nozerinf prediction, and it will be stated later on. 4. ABOUT ZERO AND INFINITY In mathematics, zero and infinity are coupled or interrelated. Therefore, any cer zero implies a cer infinity, and vice versa, given the appropriate symbolic manipulations. (Also, in human language one can generally find, when giving terms meaning zero, e.g.,'null error', corresponding terms meaning infinity, e.g., 'infinite precision'). Assertions on nocer's measurable properties can be verified only statistically because measurements with zero error are not possible and a set of identical measurement results is not achievable.3 By nozerinf, 0.0...% and 100.0...% probabilities (with an infinity of decimal zeros) are cer; ergo, every conceivable nocer event can happen, enough time provided, because it has associated a nonzero probability. This is something of a paradox, because the affirmation that zero does not exist in nocer implies that 0% probability exists of not having zero in nocer. So, is there in nocer a nonzero probability of eventually having a zero massen and/or spacetime, anywhere? Of course, I argue that zero is not a nocer event and that the universe is structurally devoid of zero and infinity. Anyway, maybe there is a nonzero but hugely small probability of having zero spacetime/massen somewhere, locally, even in a structurally nonzero universe. 4.1 Zero, according to references and common knowledge • Merriam-Webster's (2006):

"Etymology: French or Italian; French zéro, from Italian zero, from Medieval Latin zephirum, from Arabic vifr. Date: 1604. The arithmetical symbol 0‚ denoting the absence of all magnitude or quantity. Additive identity; the number between the set of all negative numbers and the set of all positive numbers. A value of an independent variable that makes a function equal to zero. [...] A state of total absence or neutrality."

• It is the whole number 0. Some people believe zero is the absence of every number. • It is the real number 0.000000..., with an infinity of decimal zeros. Therefore, the number of decimal zeros notwithstanding, more can always be added. To the right of the point there is nothing but decimal zeros. • It is the symbolic representation of an empty set (although the usual symbol for this is Ǿ).

semantically indefinable notions (as is also 'time', for instance), which cannot yet be clarified in terms of something deeper. Those notions, if attempted at being defined, travel along a chain of dependent definitions or descriptions and finally end in themselves, circularly. 3 Heisenberg's uncertainty principle indicates that any related pair of observables cannot both of them be measured with zero standard deviation; according to nozerinf, it is also not possible in nocer to have null error in one magnitude and infinite error in the other.


• It is the result of the subtraction of two identical numbers: x-x=0. • It is a concept arrived to by 'jump to infinity' in a division by a bigger and bigger number. For instance, let us divide 1 by increasing powers of 10, this way pushing the number 1 of the quotient to the right of the decimal point and interposing in between them more and more zeros. It is said that when the denominator becomes infinite, the quotient will be zero. This would be a zero by infinite divisibility, but in some way it mirrors the intuitive impossibility to reduce to absolutely nothing a piece of matter by divisibility, because always something (the number 1) will remain to be divided again. 4.2 Zero, after author's understanding • It is the total absence of spacetime and massen, or 'perfect vacuum' (so that measurements give an infinity of zeros to the right of the decimal point). Nozerinf hypothesis says that this total absence which, as said above, would be equivalent to a perfect or absolute vacuum, does not exist in nocer.

4.3 Infinity, according to references and common knowledge • Merriam-Webster's (2006):

"Date: 14th century. The quality of being infinite. Unlimited extent of time, space, or quantity: Boundlessness. An indefinitely great number or amount. The limit of the value of a function or variable when it tends to become numerically larger than any pre-assigned finite number.[...] A transfinite number."

• Microsoft Encarta (2004): "The elements of the set [2, 4, 6,..., 2n,...] can be matched in a one-to-one way with the elements of the proper subset [6, 8, 10,..., 2n + 4,...]. A set with this property is called an infinite set. Thus, the set N of all positive integers, the set R of all rational numbers, and the set Z of all real numbers are infinite sets." (Partial quotation).

• Spencer, P. (1997): "In the context of a number system,[...] infinity does not exist. In the context of a topological space,[...] infinity does exist. In the context of measuring sizes of sets,[...] 'infinity' concepts do exist but there are more than one of them."

• Turchin, V. (1991): "[...] we find two concepts of infinity: potential infinity, which is the infinity of a process which never stops, and actual infinity which is supposed to be static and completed,[...] as an object. In this abstraction (potential infinity),[...] at every specific stage the process involves[...] a finite reality; it is infinite only potentially. For actual infinity we have no place in our system of concepts."

• It represents an aggregate of entities always larger than any aggregate previously specified. • It is a concept arrived to by 'jump to zero' in a division by a smaller and smaller number. For instance, let us divide 1 by decreasing negative powers of 10, this way adding more and more zeros to the right of the number 1 of quotient. It is said that when the denominator becomes zero, the quotient will be infinite.

4.4 Infinity, after author's understanding • It is a magnitude of spacetime and massen not having an end or not having any extreme or limit. Nozerinf hypothesis says that infinity does not exist in nocer.

5. ABOUT HYPOTHESIS 1 (NOZERINF) 5.1 Statement


The formulation is very simple: mathematical zero as well as mathematical infinity are cer concepts and they do not exist at all in nocer, i.e., in the "real world". 5.2 General considerations Direct and absolute scientific proof of nozerinf's truth or falsehood is not possible, because it would demand perfect, error-free observations. Of course, this in itself is a petitio principii applied to nocer: to prove no-zero or yes-zero one need yes-zero beforehand, and to prove yes-infinity one would need yes-infinity (in time). Therefore, the scientific way to decide on nozerinf is exclusively by means of testing related predictions or scientific assertions. These can only be verified or falsified to a certain level of certainty, because of nozerinf, but if that level is statistically and theoretically acceptable the verification is deemed as achieved. Also, if zero is not in spacetime, and spacetime is in nocer, then zero is not in nocer. However, the big problem is that 'if'. Nozerinf argues that perfect vacuum does not exist in nocer, but this is a hypothesis to be proved. What we now know as spacetime is indeed part of nocer, but we don't know whether massen is in spacetime or whether both of them, together, constitute a different kind of thing. Spacetime is not necessarily more fundamental than massen. 'Perfect' or 'absolute' vacuum could not exist in a state of huge density in the context of pre-big bang because that would be a contradiction: without massen and spacetime, there is nothing to be compressed. Effectively, compression means to force elemental particles to approach to one another, but 'perfect vacuum' has no particles. Therefore the 'original atom' couldn't contain such a vacuum, along with massen and spacetime. Furthermore, the universe appears to have the property of gradients' equalization, so if a perfect vacuum existed somewhere at some time, massen and spacetime would have 'rushed' to fill in such a vacuum. Is it possible a 'partial vacuum', that is, zero spacetime or zero massen? Nonzero spacetime with zero massen was, in essence, Newton's idea of absolute space. We now know that the entire universe is filled with some type of massen (hadrons, leptons, forces, radiation, dark matter, dark energy...), so nonzero spacetime with zero massen is not possible in nocer. Nonzero massen with zero spacetime would imply massen's null expanse, therefore a dimensionless massen point totally unable to evolve. This is not possible in nocer, either. Therefore, spacetime and massen must coexist, and their permanent interaction is one of the essentials of general relativity. When some set of spacetime/massen interrelation parameters is established, not even one of them can be zero (by nozerinf), hence spacetime and massen would really be just a single physical reality, as yet not understood and without a name. 5.3 Some descriptive predictions derived from nozerinf hypothesis For each of the 'descriptive predictions' the essential item(s) forbidden by nozerinf to be a cer zero or a cer infinity will be identified. (Let's call it nozerinf item). • Spacetime/massen cannot be indefinitely divided. Some elementary level will be reached where the concept of further divisions is not applicable. Thus, point-particles, or dimensionless


particles, do not exist in nocer. Recall that, according to nozerinf, infinity and zero are excluded from nocer. In this case nozerinf forbids in nocer an infinity of actions (repeated massen division, for instance), and any infinite division of a field-like massen will theoretically lead to a zero. Nozerinf item: point-particles or dimensionless-particles. • Black holes could not have infinitely large density or be a singularity (zero volume). Nozerinf item: a singularity (zero volume and infinite density). • A zero movement does not exist in nocer, but of course absolute reference points do not exist either. Nozerinf item: a cer zero reference point with a cer zero movement, and distances to the moving point measured with infinite precision. • Absolute (theoretical) zero of temperature will not be reached. Nozerinf item: temperature, whose cer zero value would need an infinite precision measurement to be demonstrated. • Perfect superconductivity does not exist. Some kind of resistance will be present, even a very small one. Nozerinf item: cer zero conductivity, not measurable. • Absolute symmetry is cer (because it implies a set of zero differences) and does not exist in nocer. Some situations of the Standard Model could be related to this. Nozerinf item: parts with pertinent properties having cer zero differences. This is

conceivable only in cer constructs. Therefore physics models with symmetry or supersymmetry would imply some conceptual mistakes.

• No nocer phenomenon obeys a mathematical function in an exact way; for instance, rigorously linear processes do not exist in nocer. Nozerinf item: error in nocer measurements to verify the obedience. This error cannot

be a cer zero. Linear processes are just a particular case. • Completely uniform movement or identical compliance of the laws of physics (as required by special relativity) would not exist in nocer. Nozerinf item: error in nocer measurements to verify compliance. • Perfect simultaneity (taking into account relativity adjustments) does not exist in the real world. Nozerinf item: time interval between 'simultaneous' events cannot be a cer zero. • No elementary particle (or, in the extreme, indivison) can be identical to any other. Nozerinf item: set of differences in all of its physical characteristics cannot be a set of only cer zeros. (See A.4.1). • Cer's identically repetitive movements lead to a periodicity. However, nothing in nocer repeats identically to itself, so in nocer actually there are no exact periodicities or exactly repeating cycles. Nozerinf item: set of differences cannot have only cer zeros. (See A.4.1). • Probability waves and 'wave function collapse' make part of a mathematical model and do not really exist in nocer. Nozerinf item: some or all of quantum state function parameters, which cannot be supposed by the model to go in nocer to a cer zero or a cer infinity. • Quantum mechanics' perfect superposition in nocer would no occur. Rather, nocer's


imperfect nature would assure a universal and permanent decoherence for the particle systems. Nozerinf item: set of differences in all parameters. Perfection is not possible in nocer.

(See A.4.3). • Any cer concept or symbolic model implying zeros or infinities for nocer magnitudes will have to be eventually adjusted because of this. Nozerinf item: nocer magnitudes cannot reach cer zeros or cer infinities. • Nature 'constants' are not such. They vary in spacetime. Nozerinf item: difference in the constant's value on two or more points of time cannot

be a cer zero.

6. ABOUT HYPOTHESIS 2 ('APEX') 6.1 Statement In mathematics and logic, which are cer, the fundamental operation is addition (and its negative equivalent, subtraction), because it is the source of the other operations. I propose that there is not a nocer exact equivalent to these operations. Rather, cer addition corresponds to a phenomenon I will call 'accumulation', ak , and cer subtraction corresponds to 'de-accumulation', dk. Furthermore, these two are the only possible 'operations' in nocer, and their results are also nocer. I also assert that, as a consequence of nozerinf hypothesis, all accumulation or de-accumulation has a specific limit, which can be named 'apex'; that is, the former cannot go up to infinity and the latter cannot go down to zero. It looks as if ak and dk approach almost asymptotically to their respective limit by means of a certain function, which starts above zero and ends under infinity, maybe including some chaotic jumps. Apices in nocer play in some way the roles of cer zero and cer infinity and could be regarded as physical 'constants' in nature. Nocer accumulations would be of types similar to cer's scalar or vectorial additions. 6.2 Subjects of accumulation or de-accumulation If we accept that nocer consists of only four kinds of fundamental entities (space, time, mass and energy), a conclusion is that ak and dk operations must apply to those four entities or to their combinations (call them the 'accumulands'). In my view, ak or dk operate essentially with certain ratios between entities: space/time, massen/space, massen/time. They can be interpreted as keeping virtually constant the denominator (time or space) and varying the numerator (space or massen). (In general, ak's or dk's basic subjects would be the first or higher derivatives). The cer ratio between space (one-dimensional) and time is speed. The relativistic addition of speeds is an accumulation, according to my definition, and their limit is the speed of light. Thus, such accumulation would be a special case of the apex hypothesis. The ratio massen/space (three-dimensional) can be interpreted as a 'massenic' density. Its ak or dk would have a limit, not yet known either in physical characteristics or in amount. The ratio massen/time can be interpreted as a 'massenic' flow. Its ak or dk would have a limit, not yet known either.

7. PROBING THE MAGNITUDE OF THE DENSITY APEX


7.1 A conjecture I propose the following conjecture: The accumulation apex D for massenic density is quark's mass density, and this is the same for electrons and neutrinos. Let us proceed with some arithmetic, based on the simple approach of thinking about those particles as solid three -dimensional spheres. (In what follows the X-shaped cross (×) denotes multiplication). Take the stated top size limit for the electron, quark and neutrino (Chart of Fundamental Particles and Interactions, CPEP):

Electron size: < 10-18 m Quark size: < 10-19 m Lightest neutrino (0 - 0.13) × 10-9 GeV/c2 Heaviest neutrino (0.04 - 0.14) × 10-9 GeV/c2

Then electron's volume would be < 5.23599 × 10-55 m3 . Likewise, quark's volume would be < 5.23599 × 10-58 m3 . (We are interpreting 'size' as diameter). Enter the electron's mass (2006 CODATA recommended values):

Electron mass: 9.109 382 15 e-31 kg ; 0.510 998 910 MeV Proton mass: 1.672 621 637 e-27 kg ; 938.272 013 MeV Proton rms charge radius: 0.8768 e-15 m

Then, electron's mass density is >= 1.739763 × 1024 kg m-3 . As to quark mass the situation is fluid as yet (The Mystery of Quark Mass):

"The combination of up and down quarks to form protons [938 MeV] and neutrons [940 MeV] suggests that the u and d are of about equal mass: about 1/3 the mass of a nucleon. But in the extraordinary case of the pion [π+], an up and antidown quark combination has a mass energy of only 140 MeV! Yet the same quark combination in a rho meson [ρ+] has a mass energy of 770 MeV! [...] Georgi comments 'There is good reason to believe that most of the mass of the quark we 'see' in the mass of the proton or the rho is a dynamical effect of quark confinement, that the u and d quarks in the underlying QCD theory actually have masses much smaller than 1/3 the mass of the proton.' "

The 'constituent quark model' usually gives values larger than the 'current ' values (Szczepaniak, A.):

"The size of the momentum space pion wave function in the harmonic oscillator approximation turns out to be of the order of ß ~ 360 [...] and for g3/g2

cr it ~ 1.3 the dynamical quark mass is mq ~ 330 MeV." However, it looks as if the 'current' values are more generally accepted. The u and d quarks, as the mass base of nucleons, are the ones we will consider. Their masses are given with ranges (Yao, W.-M. and others, 2006):

u mass: 1.5 to 3.0 MeV d mass: 3 to 7 MeV Neutrino mass: < 2 eV ,

but let us work with an average (Yao, W.-M. and others/Manohar, 2006): Average u, d mass: 4.4 ± 1.5 MeV

Then: Quark mass (average u, d mass): 4.4 MeV = 4.4 × 106 × 1.782 661 731 × 10-36 kg = 7.8437116 × 10-30 kg . Mass density of quark (using volume given above): >= 1.4980379 × 1028 kg m-3 ~ 1.5 × 1028 kg m-3 . By comparison with the electron's mass density, quark's is the largest, so my conjecture turns out to be that 1.5 × 1028 kg m-3 , or bigger, is the density apex D in nocer. Should we apply


this apex to the electron mass we could get another possible electron diameter: <= 4.876778 × 10-20 m, that is to say ~ 20 times smaller than suggested in the literature. Other calculations: Proton volume, given a charge radius (2006 CODATA recommended values): 2.8235157 × 10-45 m3 = 2.8235157 fm3 . Mass density of proton given its mass (2006 CODATA recommended values): 5.9238971 × 1017 kg m-3 , which is close to 11 orders of magnitude lower than the putative apex. Heaviest neutrino mass, upper limit (Chart of Fundamental Particles and Interactions, CPEP): 0.14 × 10-9 GeV c-2 = 2.495726 × 10-37 kg . Applying the density apex, neutrino volume is <= 1.663817 × 10-65 m3 . Therefore, neutrino diameter could be <= 3.16739 × 10-22 m .

Thus, comparing diameters: electron would be about 0.5 times the quark size. Quark would be ~ 315 times bigger than neutrino. Electron would be ~154 times bigger than neutrino. Proton would be > 17 536 times larger than quark but 35 958 times larger than electron.

And comparing volumes: proton would be 5.39 × 1012 times bigger than quark! Atom diameter, in average: 10-10 m ; so, volume would be 5.23599 × 10-31 m3 . Comparing diameters, atom could be 57 025 times larger than proton. Comparing atom's volume to proton's, atom could be about 1.85 × 1014 times bigger.

Along the same line, let us conjecture that apex D is the maximum black hole's density Consider a huge one accumulating 2.5 × 106 solar masses (as the postulated black hole in the center of the Milky Way and with an event horizon of some 49 seconds-light diameter). Therefore such black hole's mass could be about 5 × 1036 kg and its volume (with D) would be 3.3 × 108 m3 . Imagining it as a sphere, its diameter could be some 860 m .

7.2 General expression

K = accumulation apex n = total number of 'accumulands', less one (starting from the second,

because accumulation begins with two values) si = magnitude of each 'accumuland', i=1,2,3,...,n akn = magnitude of resulting accumulation Then, in general, akn = f [si , n, K]. (1)

7.3 Special case for the accumulation of two speeds For the accumulation of two parallel, same direction speeds, values are: K = c n = 1 si = v1 , v2 ak1 = resulting speed=V1


V1 = [v1 + v2]/{1+ [v1 × v2 c-2]} (2) Equation (2) comes of course from Einstein's special theory of relativity. 7.4 Generalization of a particular accumulation case This generalization for the accumulation of magnitudes assumes that for each consecutive two the applicable formula is as in equation (2). Using the notation given in 8.2 above and equation (1), one gets:

ak1 = [s1 + s2]/{1 + [s1 × s2 K-2]} A simple heuristic generalization leads to:

akn= xn / yn , in which xn = K2 × [akn-1 + sn+1] yn = K2 + [akn-1 × sn+1]

For the special case of having the same magnitude for all the 'accumulands' s one arrives to the following:

xn = SUM |i=1; 2; n+2| {C |n+1; i| [si× K(2n - i – 1)]} yn = SUM |i=0; 2; n+2| {C |n+1; i| [si× K(2n - i – 2)]} If i> n+1, C=0. Here,

SUM |i=a; b; e| is the summation (over index i) of the argument (in brackets), from a to e , step b.

C |p; q| is the non-repeating combination of p elements taken q at a time. Numerical example: K=300 000 n=6 (i.e., there are in all 7 values to accumulate, each of them equal to s) s=40 000 Then, x6= 1.801907369 × 1060 y6= 8.174946643 × 1054 ak6=220 418.2422 7.5 Accumulation of two massenic densities In order to accumulate two massenic densities a not yet known function is needed, but let us suppose it has the same form as for speeds. In this case:

K = D = density apex n = 1 si = d1, d2 ak1 = resulting density = dr dr = [d1 + d2]/{1 + [d1 × d2 D-2]} (3)

Two numerical examples using equation (3): First d1 = d2 = 8 g cm-3 D = 1.5 × 1028 kg m-3

Then, dr = 16 000/(1.000...<some 48 zeros>) = 16 000 kg m-3. That is to say, two


low-density elements yield essentially the same result as an arithmetic addition. Second d1 = d2 = 1026 kg m-3 D = 1.5 × 1028 kg m-3 Then, dr = (d1 + d2)/1.000044444 = 1.99991 × 1026 kg m-3 .

D value is so huge that, when one accumulates two densities, their magnitude has to be almost as big as the apex for some decrease in the addition to be noticed. However, this decrease may be greater if n is large, which would be the case when accumulating many units. 7.6 Rough outline for possible experimental testing As far as I know, direct experimental verifications for the accumulation of speeds lower than c are few, if at all. A classical checking in this context is Michelson-Morley's and equivalents, which usually deal with two 'accumulands': c and Earth's orbital velocity. An experiment--for various 'accumulands'--could make use of velocity measurements with an arrangement similar to having an nth rocket fired by an nth-1 rocket fired by an nth-2 rocket fired by an nth-3 rocket, ..., a second rocket in turn fired by a first rocket. An experimental test for density accumulation would require measuring masses and volumes with very high precision and accuracy. For instance, a heavy gas could be compressed in a vessel. Let us have a number of these vessels with carefully measured gas mass for each one. Proceed then to (meticulously) accumulate all the masses by compression in just one vessel. Finally, the total mass would be measured, and this should be less than the plain arithmetic addition. Temperature and other thermodynamic conditions should be kept adequately controlled. 8. SOME SPECULATIONS REGARDING THE APEX HYPOTHESIS • It looks as if apex D is much bigger than the density required for a black hole to be born. Maybe all of nocer (including ourselves) is built by means of enormous amounts of something like ultra small 'black holes' born with the big bang (the quarks and leptons?) with density D (therefore presumably without the capacity to catastrophic growth), and this would guarantee a stable microcosm because such elements couldn't fuse and would rebound when colliding (or annihilate, if they are a particle-antiparticle pair). • To increase density up to the apex, perhaps an energy amount bigger than any one achievable in nocer could be required. • The imagined 'big crunch' could be an extreme situation of massen and spacetime accumulation, and conceivably a jump to the apex would lead to a new big bang. • The origin of part of a black hole's generated radiation is due maybe to partial massen's 'rebound' as black hole's density is increasing towards the apex. • It appears as if apices are either cer and assumed/calculable (as the absolute zero of temperature, in the realm of de-accumulation), or else nocer and observable/calculable (as light velocity, in the realm of accumulation).

APPENDIX A: PHILOSOPHICAL IMPLICATIONS OF CER, NOCER AND NOZERINF


It is known that Renè Descartes wrote very straight statements on the mind-body problem (Descartes, R. /Haldane and Ross, 1952):

"Meditation VI, 101. [...] that there is a great difference between mind and body, inasmuch as body is by nature always divisible, and the mind is entirely indivisible. "Meditation II, 79. [...] I find here that thought is an attribute that belongs to me; it alone cannot be separated from me. I am, I exist, that is certain. But how often? Just when I think [...] "Arguments, 130. [...] VI. That substance in which thought immediately resides, I call Mind. [...]",

but his formation and cultural environment prevented him from assigning to mind purely worldly characteristics:

"Reply to Second Objections, 127. [...] But this is sufficient to let us conclude that the mind [...] is immortal.",

or to tie a mind to its body:

"Meditation VI, 98. [...] it is certain that this I [that is to say, my soul by which I am what I am], is entirely and absolutely distinct from my body, and can exist without it. "Arguments, 133. [...] But mind and body are substances [...] that can exist apart from each other [...] Hence there is a real distinction between mind and body."

Furthermore, he was unclear at separating the material world from mind's subjects, as mathematics:

"Meditation VI, 96. [...] to inquire whether material things exist. And certainly I at least know that these may exist in so far as they are considered as the objects of pure mathematics [...]"

My basic intent with cer/nocer is to have my conceptual and communication channels free from such ambiguities. A.1. On the workings of definitions Philosophical understanding of zero and infinity in nocer could demand a test onto nocer objects, which therefore need to be clearly defined. Is required a fundamental cer notion, definition, that is, a means we employ to set out the boundaries of something cer or nocer in order to employ it with low ambiguity for our human purposes. Such a means cannot be totally precise because it must be stated in nocer language (inaccurate), contains statistical uncertainties and it may depend on a chain of other definitions. An ideal or perfect definition is unequivocal and contains zero of any other alien idea or definition. However, perfect definitions could only be the ones given in mathematics and logic. Mutually excluding nocer states, like detection / no detection, live / dead, presence / absence, etc., appear as implying zero because we suppose one of the two states contains zero of the other. However, for this to be true the implied objects would have to be managed as whole numbers with infinite precision, so that when one is 'subtracted' from the other, zero results. The problem with asserting that there is a cer zero of something nocer lies in how to define in an absolute and unequivocal manner what that something is. If using the common terms (A.4.6) of everyday life, of course one can affirm, for instance, that a bus has zero passengers; but in the context of nozerinf one would have first to define what is a nocer


passenger, with infinite precision, which is not possible. In that case one will not be able to categorically affirm that a bus has cer zero passengers. In other words, every nocer definition is carried out on sets of elements like molecules, atoms, energy quanta and so on, such sets being usually huge and subject to random fluctuations. Therefore, the extent of the realm governed by the definition is essentially inaccurate, as one or more of such elements could be put in, taken out or left behind, purposely or at random. In short, nocer definitions do not admit of zero and infinity, thus rendering inconclusive the rigorous comparison of mutually excluding nocer states. A.2 Additional considerations on cer Let's take a working car. It moves and transports people. These, among others, are the functions of the car, and they do not have mass or energy; they would be the 'car's cer', in a way. Of course, mass and energy – the car's chassis, motor, doors, roof, fuel, etc. -- are the essential nocer producing that 'cer'. When, for example, the car has moved long time, probably it will need to refuel or to have some maintenance; this is the way its 'cer' exerts an action over its 'nocer'. Something similar can be said of any other mechanism (computers, for instance) or organism, including the human being. By exclusion, cer is not heat, bioelectric impulses, electromagnetic fields, secretions or other physical-chemical entities scientifically measurable. By inclusion (possibly repetitive but not necessarily exhaustive), cer is: thoughts, reasoning, ideas, concepts, abstraction, imagination; knowledge and information; sensations, emotions, dreams, beliefs, morality, doctrines. In particular, mathematics is cer, can be expressed in nocer through symbols and allows some kind of total accuracy as well as notions like zero and infinity. These notions are lost, however, as soon as we move into nocer. In cer, for instance, 1+1 = 2, with an infinity of decimal zeros in addends and in the total, but 1 nocer apple+1 nocer apple is not equal to 2 nocer apples. Indeed, in the first place a nocer apple would have to be exactly defined. If an apple has randomly lost 10 atoms or one million atoms, let us say by friction, does it continue to be 1 apple, or is it now only 0.999999... apple? Thus, we cannot be completely sure about what is really being added, and the result will not be 2 with an infinity of decimal zeros. Whole numbers do not exist in nocer, because otherwise it could be possible to make subtractions resulting in zero. Quantities in nocer could better be expressed by real numbers, although maybe nocer 'numbers' do not have any cer exact equivalent. Nocer world appears as analogue to cer, and the digitizing (which is cer) used in electronics and other fields is based in a sort of statistical rounding of nocer phenomena. All affirmations or qualifications about nocer are cer, and every science also is cer. Of course, some sciences deal with cer, others with nocer, and others with cer-nocer interrelationships. A.3 More on the relations of cer and nocer Human brain is nocer, and it is the cer generating entity. Likewise, human senses are nocer,


what we know through them is nocer if they are working properly, and they could be regarded as 'transducers' of cer to nocer or vice versa. Different person's cer cannot be directly contrasted to each other but only through nocer intermediation. I know in a direct way just my own cer, but as a reasonable hypothesis I admit the rest of living/functioning people (and other beings) have them too. To perceive the other persons' cer I have to examine the nocer events (e.g., spoken or written language, music, drawings, etc.) with which such persons try to describe their cer to me. Human language is nocer, hence intrinsically inaccurate and able to give rise to many interpretations (of legal instruments, for one). Logical-mathematical language, which is cer, can be rendered totally unambiguous, but to be employed by humans it needs to be translated to nocer language. We have proposed that cer does not have massen; therefore, it cannot directly act on nocer. Nevertheless, cer can originate actions over the brain because, in fact, those work as physical-biochemical reactions of brain cells over brain cells. Being massenless, cer cannot sustain supposed phenomena like telepathy, telekinesis and similar. Symbolic / mathematical models (cer, of course) are one way to translate nocer to cer. Model truthfulness or accuracy depends on the extent of its agreement with the corresponding nocer properties. Note that the law of massen conservation, and any other zero-sum law, are statistical in nature, therefore not being verifiable beyond certain error limits and not being possible to demonstrate they imply a cer zero. A.4 Some epistemological implications A.4.1 Identical objects. Let us say that for each nocer knowable object Nobi a cer set Cnobi can be constructed with all of its physical characteristics p (which would imply a number of measured magnitudes and its conditions of measurement, as distance, area, volume, mass, time intervals, temperature, type and number of molecules or atoms, field intensity, frequency, charge, etc.), thus: Cnobi = (pi1, pi2, ..., pin, ..., piz), where pi1 could be, for instance, temperature, ..., and piz represents the last of the characteristics. Now let us have another Nobk whose set Cnobk = (pk1, pk2, ..., pkn, ..., pkz) contains precisely the same type of measured magnitudes, ordered in the same sequence. Construct the set of differences Dnobik = Cnobi - Cnobk = [(pi1- pk1), (pi2- pk2), ..., (pin - pkn), ..., (piz - pkz)] . If then we find Dnobik = [(0), (0), ..., (0), ..., (0)] , with all the zeros being cer zeros, i.e. with an infinity of decimal zeros to the right, then we will say that Nobi and Nobk are identical to each other. However, by nozerinf, such a set of zeros cannot exist in nocer and, besides, we cannot be totally sure Cnobi contains all of p. Thus, no one thing in nocer can be exactly identical to another one. Corollary: symmetries in nocer are not perfect or absolute, because when symmetry operations are applied to some nocer entity, say Nobi , the resulting 'symmetrical' entity, say Nobk , cannot be (exactly) identical to the Nobi . A.4.2 Identity Law (everything is equal to itself). We return to the object Nobi and its cer set Cnobi . The physical characteristics p are probably measured at different times, but in a


cer context we could say that they were all measured at the same time t0 , thus giving us a frozen cut t0Cnobi of Nobi at t0 . But at indivison level time passes or goes by, in average, just in a unidirectional way (as is the general human understanding up to now), towards what we call future (although in my opinion what we humans call past and future are cer, present only being nocer). At t0 + (1 time indivison) we could get another set t0+1Cnobi . According to the Identity Law, the set of differences Dnobit = t0Cnobi - t0+1Cnobi will be Dnobit = [(0), (0), ..., (0), ..., (0)] , with cer zeros. However, by nozerinf, this is not possible; besides, we cannot be totally sure t0Cnobi contains all p. Therefore, the Identity Law is not valid in nocer but only in cer (if so we choose). In effect, at t0 + (1 time indivison) all of p have changed even if by a tiny amount, for instance because of random modifications in the position and intensity of particles and fields. A.4.3 Perfection. Now we take again the object Nobi and its cer set Cnobi , but this time the object could also be a cer, i.e. it is a desired or designed or potential object. Anyway, its physical characteristics p are cer, viz, requirements or standards or specifications we want Nobi to fulfill now or once it becomes nocer. One of these two events occurred, we proceed to measure each of p and construct the set mCnobi . Then we get the set of differences Dnobi = Cnobi - mCnobi . If we get Dnobi = [(0), (0), ..., (0), ..., (0)] , with all the zeros being cer zeros, then we will say that Nobi is perfect, because it is totally and absolutely fitted to its set of requirements. However, by nozerinf, this is not possible, so nothing in nocer can be deemed perfect; the notion of perfection is cer. The perfect would then be that entity embracing or containing everything of a preset something, this however not being rigorously verifiable in nocer. Magnitude comparison is not perfect, because it would need perfect definitions, and only statistical approximations can be achieved. Corollary: perfect order or disorder does not exist in nocer. A.4.4 Determinism. Absolute determinism for nocer means that, for a given process or evolving nocer system i , the cer set t0Csysi describing all its states and parameters at time t0 is known with complete precision, up to the last indivison; then, provided we are able to apply to that system a symbolic model absolutely exact, we could determine at time t0 + t the cer set t0+tCsysi to describe or predict the future situation of system i . But, by nozerinf, it is not possible (a) to have a perfect inclusion (A.4.3) of all states and parameters (hence the hidden variables concept does not work); (b) to measure with complete precision, i.e., with null error, the values of t0Csysi , and (c) to devise a symbolic model able to predict with null error. Thus, absolute determinism is only a cer concept and cannot exist in nocer. Corollary: a symbolic model of a nocer process cannot be fulfilled in a totally exact way. A.4.5 Chance. Let us deal now with the cer set Cevei which contains all the states and parameters characterizing event i as we expect it to be at time t , once we perform or repeat certain measurements or observations. After measurements, we will have the cer set tCevei . In a deterministic world Cevei ≡ tCevei . However, totally identical nocer objects or events, and determinism, are not possible (A.4.1 and A.4.4). When cer is not able to predict exactly some nocer (as in here), humans say that in such nocer there is chance, randomness or chaos, and nocer world for them appears as only probabilistically knowable.


So, by nozerinf, chance is for cer an inescapable nocer trait. Our predictions, of course, might be very accurate, within error margins. Chance and non-determinism are, in essence, the same cer, and also is cer the so-called 'objective randomness' of quantum mechanics, all of them stemming from the interaction cer-nocer. Should we take cer out of the picture then chance disappears given that no one will be trying to know or predict; nocer just exists and changes. A.4.6 Other reflections. • Causality: Anything happening in nocer has a nocer cause, because by nozerinf a null or zero cause can only be cer. I posit that nocer causes must arise from one or more massen indivisons acting on one or more massen indivisons, within a spacetime lattice. By A.4.4, effects can be linked to causes only statistically. • The conceptual difficulties of human beings (scientific or philosophical, mainly) arise when they try to adapt nocer things to some cer. Nocer world never has been and, very likely, never will be exactly describable by any cer. • Every paradox is cer. Paradoxes cannot exist in nocer world. • In nocer world nothing totally 'good' or 'bad' exists; this could happen only in the cer world. • Any nocer object, event or process has identifiable advantages and disadvantages. Relative weight of good aspects versus bad aspects differs from one case to the next, thus making it possible optimization processes. • Commonly used terms in everyday life, like accurately, exactly, identical, instant, never, always, perpetual, perfection, all, nothing, absolutely, countless, boundless, impossible, zero, infinity, and similar, are approximate concepts which have a limit or cutoff point as demanded by nozerinf. For instance, a perfection notion based in rough, limited or incomplete parameters / verifications is much employed in nocer. APPENDIX B: LIMITS The problem of limits or boundaries of things is not simple, and is the realm of Mereology (Varzi, A., 2004):

"Euclid defined a boundary as 'that which is an extremity of anything' (Elements Bk I, Df 13), and Aristotle made this more precise by defining the extremity of a thing x as 'the first thing outside of which no part [of x] is to be found, and the first thing inside of which every part [of x] is to be found.' (Metaphysics 1022) [...] Aristotle mereological definition [...] only seems to apply to a realm of continuous entities [...] On closer inspection, the spatial boundaries of physical objects are imaginary entities surrounding swarms of subatomic particles [...] Most realist theories about boundaries, construed as lower-dimensional entities, share the view that such entities are ontological parasites [...] 'I define the limit of a body as the aggregate of all the extreme (äusserst) ether-atoms which still belong to it [...]' [Bernard Bolzano Paradoxes of the Infinite § 66 (1851: 167-68)] [...] 'There is no line which sharply divides the matter composing [Mount] Everest from the matter outside it. Everest's boundaries are fuzzy. Some molecules are inside Everest and some molecules outside [...]' [Michael Tye, Vague Objects (1990: 535)]."

My intent now is only to offer some thoughts pertinent to the issues expounded in this paper. Of course, nozerinf forbids a continuum in nocer because it would imply divisibility ad infinitum and no individual limits, and also forbids discontinuity, i.e., totally separated spacetime / massen discrete, limited particles because it would imply absolute vacuum among them. Clarification of this ancient dilemma still pertains to the future, but it could be


approached by postulating that nocer is totally filled with spacetime and massen indivisons, as a 'grainy' continuum, whose limits would touch, overlap or interlace permanently without leaving any zero emptiness amongst them. So, nocer indivison's sets have possibly identifiable and measurable limits, these however being something like dynamical or permanently changing fractals. By nozerinf the whole numbers (or 'natural' or enumerator numbers) exist only in cer, and consequently its application in nocer is statistical because the nocer 'whole' number is a (usually huge) set of elemental entities, and this set is what in average appears as a whole to us. And no one nocer element, even at indivison level, can be represented by a whole number because its limits are not specifiable with infinite precision. In addition, everything in nocer is subject to limits or restrictions. Assume that nocer has the equivalent of three spatial dimensions, plus the dimension called time. Thus, objects with one or two dimensions, and a one- or two-dimension boundary, are only cer and do not exist in nocer because, by nozerinf, such objects would imply one or more zero dimensions.

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