Chapter:

Translator's
Preface

75

"NUMBER may be defined [classically] in general as a collection of units, or, speaking more

What is [definition of] number?

exactly, as the synthesis of the one and the many. Every number is one, since it is brought before the mind by

a simple intuition and is given a name; but the unity which attaches to it is that of a sum, it covers a multiplicity of parts which can be considered separately."

Note (1): I had already completed the present work when I read in the Critique Philosophique (for 1883 and 1884) F. Pillon's very remarkable refutation of an interesting article by G. Noël on the interconnexion of the notions of number and space. But I have not found it necessary to make any alterations in the following pages, seeing that Pillon does not distinguish between time as quality and time as quantity, between the [conscious] multiplicity of juxtaposition and that of interpenetration. Without this vital distinction, which it is the chief aim of the present chapter to establish, it would be possible to maintain, with Pillon, that number may be built up from the relation of co-existence. But what is here meant by co-existence? If the co-existing terms form an organic whole, they will never lead us to the notion of [classically lisr] number; if they remain distinct, they are in juxtaposition and we are dealing with space. It is no use to quote the example of simultaneous impressions received by several senses. We either leave these sensations their specific differences, which amounts to saying that we do not count them; or else we eliminate their differences, and then how are we to distinguish them if not by their position or that of their symbols? We shall see that the verb "to distinguish" has two meanings, the one qualitative, the other is quantitative: these two meanings have been confused, in my opinion, by the philosophers who have dealt with the relations between number and space.

Return to top of page footnote reference.

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Bergson restarts his footnote counts on each page. So to refer a footnote, one must state page number and footnote number.

Our bold and color highlights follow a code:

• black-bold - important to read if you are just scanning our review
• orange-bold - text ref'd by index pages
• green-bold - we see Bergson suggesting axiomatic memes
• violet-bold - an apparent classical problematic
• blue-bold - we disagree with this text segment while disregarding context of Bergson's overall text
• gray-bold - quotable text
• red-bold - our direct commentary

Similarly to Bergson's manner of starting topic 15, he amazes us here too! He infers a quantum reality where:

Timequanton(quality,quantity), and
                                         
      quanton(subjective_wave,objective_particle)
                                                       
Timequanton(interpenetration,juxtaposition), where

interpenetration is n¤nactuality's c¤mplementary included-middle mediation of multiple comtextual juxtapositions (AKA interrelationships).

We offer some number heuristics based upon his statements about time:

• number is quantum comtextually dependent via mediating juxtaposition,
• number is quantum associative commingling via mediating interpenetration, and
• number is quantum animate as a quanton of both quality and quantity.

Com(n)sider how these three heuristics satisfy Bergson's earlier statements about two classical delusions of reality:

• that reality is stable, and
• that objects in reality are independent.

See our One is the Onliest Number.

Com(n)sider how we can proffer similar quantum analogues for classical concepts of space and mass.

Too, here, we see that, to Bergson, particulate (lisr) objectivity and space are analogous concepts. He shows us how durational (quantum real) reality is classically n¤nnumerable.

Index

"Without attempting for the present any thorough examination of these conceptions of unity and multiplicity, let us inquire whether the idea of number does not imply the representation of something else as well.

"It is not enough to say that number is a collection of units; we must add that these units are

The units which make up a number must be identical.

identical with one another, or at least that they are assumed to be identical when they are counted. No doubt we can count the sheep

in a flock and say that there are fifty, although they are all different from one another and are easily recognized by the shepherd: but the reason is that we agree in that case to neglect their individual differences and to take into account only what they have in common. On the other hand, as soon as we fix our attention on the particular features of objects or individuals, we can of course make an enumeration of them, but not a total [because each sheep is unique/omnifferent]. We place ourselves at these two very different points of view when we count the soldiers in a battalion and when we call the roll. Hence we may conclude that the idea of number implies the simple intuition of a multiplicity of parts or units, which are absolutely alike."

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In Quantonics, in general, we deny any classical concepts of identity and identicalness! Also see difference, omnifference, and one.

In quantum reality, what we can do (theoretically) is make a Bose-Einstein Con(m)densate (BEC) of them. In such a case, all 'objects' would share common quantum phasicities. In a sense many classical 'objects' could become ¤ne quantum co-here-nt quanton. Comsider how BECs eliminate quantonic interrelationships among condensed quantons. This process is ~reversible; however, when 'reversed' each classical 'object' would, initially share common quantum phasicities as ensemble "absolutely alike" initial comditions which would then, at up to Planck-rates, evolve omnifferently from thence on.

Bergson apparently did n¤t anticipate our quantum description above. He just said that since all sheep are, indeed, quantumly omnifferent ¤ne another, we cann¤t classically total them, ideally-mathematically, as a sum of cloned identical classical numeric units. Ten sheep as a total is n¤t ten classically 'identical' sheep!

Index

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Now we see Bergson actually anticipate BECs! Taken literally, Bergson's "merge into a single unit" is quantum coherence, or more specifically Bose-Einstein Con(m)densation (BEC).

Can you see what an impact this has on classical mathematics? All numerical sums, to be quantum physially real, must represent BECs! Personally, we had never intuited that Bergsonian meme prior to now (16Apr2002).

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But, but, but, ... Einstein has told us time and space are identical!!! He said, "Reality is a space-time identity!" Is this what Feynman and Dyson meant when they conjectured Einstein had become a manipulator of mathematics and numbers and thus had lost touch with concrete (i.e. real) physical reality?

In quantonics, we can easily answer this issue using QTMs. All pairs of meme interrelationships in quantonics are quantons. All quantons are uncertainty interrelationships. So we just say, "quanton(time,space)." Spaces always have temporalities and temporalities always have spaces. How much of each depends upon which we are attempting to measure. This meme is analogous a Poisson bracket of a quanton's position and momentum. Space and time are quantum c¤mplementary! Quantons are paired quantum c¤mplementary interrelationships! We can use them to express general quantum systems, because general quantum systems are nestings of animate quantum c¤mplementary interrelationships.

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Here, via his "...in time only," Bergson denies our view of quantonic uncertainty interrelationships among measurables in reality. Doesn't this deny his own statements about classicists' two greatest delusions that:

1) reality is stable, and
2) objects in reality are independent?

But then he shows us how "terms" in duration are unstoppable. (Note how a classical concept of "waiting" AKA "stoppability" is necessary for mathematical induction, which is at heart of Peano's counting principle which Bergson is discussing here. Unstoppable duration (quantum reality) is, as we have shown elsewhere and as Bergson is showing us in his works, n¤n-inductive, n¤ndeterministic, and thus n¤ncausal!) Study that link carefully! Doug - 25Mar2004.

We might use photographs as cinematographical analogies here. They do n¤t really make duration "wait," either! We just tend to classically thingk that they do. Classical scientists do exactly this with their mathematics and their measurement and their digital computers! All they do is take 'photographs' of reality, process those 'photographs' analytically, state-ically, dichotomously, anachronistically, inanimately and call their results "real."

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Quantum reality is n¤t classically synthetic. It is, rather, durationally emerging.

Classical number exclusively complements unity and multiplicity; however, classical number denies quantum requisite c¤mplementarity of animacy and inanimacy (quantum requisite absence of classical stability), and quantum requisite c¤mplementarity of included-middle and excluded-middle (quantum requisite absence of "exclusive" classical objective independence). To be able to understand and differentiate classical reality from quantum reality, we cann¤t over emphasize importance of what we just wrote! Doug - 17Apr2002.

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Index

"But we soon perceive that, while all multiplication implies the possibility of treating any number whatever as a provisional unit which can be added to itself, inversely the units in their turn are true numbers which are as big as we like, but are regarded as provisionally indivisible for the purpose of compounding them with one another. Now, the very admission that it is possible to divide the unit into as many parts as we like, shows that we regard it as extended.

"For we must understand what is meant by the discontinuity of number. It cannot be denied

Number in process of formation is discontinuous but, when formed, is invested with the continuity of space.

that the formation or construction of a number implies discontinuity. In other words, as we remarked above, each of the units with which we form the number 3 seems to be indivisible while we are dealing with it, and we pass abruptly from one to

the other. Again, if we form the same number with halves, with quarters, with any units whatever, these units, in so far as they serve to form the said number, will still constitute elements which are provisionally indivisible, and it is always by jerks, by sudden jumps, so to speak, that we advance from one to the other. And the reason is that, in order to get a number, we are compelled to fix our attention successively on each of the units of which it is compounded."

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Index

83

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dichon(continuity, discontinuity)

Quantum real emergent number! In "...process of formation."

Classical unreal state-ic number, "...once formed."

Index

"Thus, a complex feeling will contain a fairly large number of simple elements; but, as long as these elements do not stand out with perfect clearness, we cannot say that they were completely realized, and, as soon as consciousness has a distinct perception of them, the psychic state which results from their synthesis will have changed for this very reason. But there is no change in the general appearance of a body, however it is analysed by thought, because these different analyses, and an infinity of others, are already visible in the mental image which we form of the body, though they are not realized: this actual and not merely virtual perception of subdivisions in what is undivided is just what we call objectivity. It then becomes easy to determine the exact part played by the subjective and the objective in the idea of number. What properly belongs to the mind [quantum stage] is the indivisible process by which it concentrates attention successively on the different parts of a given space; but the parts which have thus been isolated [quantum islandicity] remain in order to join [quantum included-middle holism] with the others, and, once the addition is made, they may be broken up in any way whatever. They are therefore parts of space, and space is, accordingly, the material, with which the mind builds up number, the medium in which the [classical] mind places it.

"Properly speaking, it is arithmetic which teaches us to split up without limit the units of which number consists. Common sense is very much inclined to build up number with indivisibles."

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Here, Bergson offers us his rendition of quantum uncertainty. It is analogous a Poisson bracket of change and perfect clarity.

Classically:

    dichon(subjective_opposite, objective_opposite)

Quantumly:

   quanton(subjective_c¤mplement,objective_c¤mplement)

Bergson explains how we impose tendentious classicism on a quantum reality.

Bergson explains a source classicism's grand delusion of analytic reducibility: "...a cuisinart is our best means of understanding biological constructs."

Readers might also wish to comtemplate our take on classical "common sense." Doesn't common sense imply that we all have to think alike in order to agree on what we observe? Isn't common sense an analogue of mental cloning? What do modern academic institutions all do at Millennium III's commencement? Don't they do their best to clone minds on various subjects?

Isn't it classically considered good that classical 'scientists' all share one paradigm, one disciplinary matrix, one set of exemplar hermeneutics?

Would any truly extraordinary and pioneering scientist ever want his-her mind cloned or to be/have a cloned mind of a socially constructed paradigm?

Cloned minds are but ideal social concepts.

If we want our cultures to be diverse, why would we not want minds within our paradigms to be 'diverse.' We must realize that paradigms are state-ic patterns of social value, and as such, and happily, they are their own worst threat to survival.

We believe that social value AKA "common sense" should seldom be permitted to rule intellectual, intuitional, and instinctual patterns of value! Resist, yæs! Rule n¤! Fortunately, it does n¤t and ultimately it cann¤t.

One more comment: "Common sense," to us, is font and grail of denigration trumpeted against novel memes and their pioneers who threaten those trapped in devolving paradigms. Those trapped call their threats, "...idiots, insane, fools, charlatans, fraudulent, equivocal, prevaricative, liars, etc." When we hear those denigrations, we know those emitting such denigrations are, indeed, weak and fearful. Extraordinary scientists understand that n¤vel memes are precursors of exceptional scientific breakthroughs.

See our review of Thomas Kuhn's The Structure of Scientific Revolutions.

Index

"And this is easily understood, since the provisional

It follows that number is actually thought of as a juxtaposition in space.

simplicity of the component units is just what they owe to the mind, and the latter pays more attention to its own acts than to the material on which it works. Science confines itself, here,

to drawing our attention to this material: if we did not already localize number in space, science would certainly not succeed in making us transfer it thither. From the beginning, therefore, we must have thought of number as of a juxtaposition in space. This is the conclusion which we reached at first, basing ourselves on the fact that all addition implies a multiplicity of parts simultaneously perceived.

"Now, if this conception of number is granted, it will be seen that everything is not counted in the

Two kinds of multiplicity: (1) material objects, counted in space; (2) conscious states, not countable unless symbolically represented in space.

same way, and that there are two very different kinds of multiplicity. When we speak of material objects, we refer to the possibility of seeing and touching them; we localize them in space. In that case, no effort of the inventive faculty or of symbolical

representation is necessary in order to count them; we have only to think them, at first separately, and then simultaneously, within the very medium in which they come under our observation. The case is no longer the same when we consider purely affective psychic states, or even mental images other than those built up by means of sight and touch."

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Yes, we agree. In our view, quantum reality shows us that reality is n¤t substantial, material, objective, spatially immutable, etc. Rather quantum reality is flux. Space is n¤t extensible number. Quantum space is flux wavelength!

Quantum space's smallest (currently known) wavelength is what we call a "Planck length." It apparently limits quantum flux's maximum frequency, and its minimum wavelength.

Quantum space's longest wavelength is ~unlimited depending upon whether and how quantum philosophy and science ultimately discover and decide what most closely represents percept zero in Nature. It apparently limits quantum flux's minimum frequency, and its maximum wavelength.

When we grasp quantum essence that quantum flux is not classically, spatially limited, we see that a current simple, classical univalent concept of number derived from it is wholly naïve. And that is just what Bergson is attempting to show us. We emerse a n¤vel meme that any quantum number is omnivalent. We might even think of quantum number as a qubit. E.g., an atom may be QTM-thought of as a qubit of qubits (e.g., its electrons and nuclei).

Bottom line: a naïve concept of classical number is innately incapable of representing a quantum reality whose quantons' numbers are essentially omnivalent qubits.

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In his Creative Evolution, Bergson refers this kind of (more quantumesque) think-king as "...think being directly." See our coined thibedir.

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As we said above, "this medium" is pragmabsolute quantum flux. We can show that space, time, mass, and gravity are all functions of quantum flux. Bergson did n¤t k-now this. (Our use of "k-now" is quantonics lingo for Bergson's "...think being directly..." and our comtraction of it: thibedir. Where classicists 'know' via classical reality's state-ic, rote information on yuck-stuck status quo is the way to go know-ledges, students of Quantonics are k-now-ing via direct experience of an animate, plural, c¤mplementary quantum reality.)

Bergson's "two kinds of multiplicity" are still both classical since they are both spatial. To extend his kinds of multiplicity into a quantum realm we must first realize that quantum flux is crux. Then we can begin to fathom that, in a sense, quantum reality is unlimited multiplicity — beyond any naïve classical categorization.

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Impenetrability is a classical code word for Aristotle's excluded-middle. Classical impenetrability describes SOM's wall. But as students of Quantonics, we k-now that Quantum flux compenetrates quantum flux. Thus quantum reality is, indeed, penetrable. William James would say it is "compenetrable." Fritjof Capra would say it is "interpenetrable." Sadly, classicists, though they refer themselves 'scientists,' are just like us naught but amateurs on Nature's quantum real stage. Decline and fall of their ugly classical deign to feign is nigh. A quantum tsunami of Millennium III change shall wash our Earth clean of them. Doug - 18Apr2002.

And here, Bergson implies that classical logic misinterprets physical (to us, physial) reality. If that is what he intends, then we agree. Of course, reader, you understand that Quantonics denies classical syllogistic logic as n¤nquantum. Quantum Bose-Einstein Condensation physically demonstrates, "[~Unlimited] bodies [can] occupy the same place at the same time."

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Here, Bergson puts us in pure awe of him! This is so beautiful, so prescient, so quantum...

If matter is impenetrable and we use number to describe and measure matter, then how can number be penetrable? If number is impenetrable, how can scientists use number to do sums and products, both of which we may view as penetrable quantum condensates of classically physical description? Heads up, readers! Doug - 18Apr2002.

Amazing! Bergson shows us that concepts of classical number, and classical material impenetrability classically contradict one another! Simply Awesome! Wow!

Why has science ignored this immense classical problematic! We have not tumbled to this Bergsonian perspective prior to now. To us, this is an outright denial of classical material science and its 'valid' uses of classical mathematics. Also, we see this as another tell of reality's underlying quantum nature.

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