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"The problem is as to which is the more rational supposition, that of [either] continuous or that of discontinuous additions to whatever amount or kind of reality already exists.

"On the discontinuity-theory, time, change, etc., would grow by finite buds or drops, either nothing coming at all, or certain units of amount bursting into being 'at a stroke.' Every feature of the universe would on this view have a finite numerical constitution. Just as atoms, not half- or quarter-atoms are the minimum of matter that can be, and every finite amount of matter contains a finite number of atoms, so any amounts of time, space, change, etc., which we might assume would be composed of a finite number of minimal amounts of time, space, and change."

"Such a discrete composition is what actually obtains in our perceptual experience. We either perceive nothing, or something already there in sensible amount. This fact is what in psychology is known as the law of the 'threshold.' Either your experience is of no content, of no change, or it is of a perceptible amount of content or change. Your acquaintance with reality grows literally by buds or drops of perception. Intellectually and on reflection you can divide these into components, but as immediately given, they come totally or not at all.

"If, however, we take time and space as concepts, not as perceptual data, we don't well see how they can have this atomistic constitution. For if the drops or atoms are themselves without duration or extension it is inconceivable that by adding any number of them together times or spaces should accrue. If, on the other hand, they are minute durations or extensions, it is impossible to treat them as real minima."

"Each temporal drop must have a later and an earlier half, each spatial unit a right and a left half, and these halves must themselves have halves, and so on ad infinitum, so that with the notion that the constitution of things is continuous and not discrete, that of a divisibility ad infinitum is inseparably bound up. This infinite divisibility of some facts, coupled with the infinite expansibility of others (space, time, and number) has given rise to one of the most obstinate of philosophy's dialectic problems. Let me take.up, in as simple a way as I am able to, the problem of the infinite.

"There is a pseudo-problem, 'How can the finite know the infinite?' which has troubled some English heads. But one might as well make a problem of 'How can the fat know the lean?' When we come to treat of knowledge, such problems will vanish. The real problem of the infinite began with the famous arguments against motion, of Zeno the Eleatic."

"The school of Pythagoras was pluralistic. 'Things are numbers,' the master had said, meaning apparently that reality was made of points which one might number.' Zeno's arguments were meant to show, not that motion could not really take place, but that it could not truly be conceived as taking place by the successive [stoppable/restartable, i.e., like Bergson's Creative Evolution motion picture frames] occupancy of points. If a flying arrow occupies at each point of time a determinate point of space, its motion becomes nothing but a sum of rests, for it exists not, out of any point; and in the point it doesn't move. Motion cannot truly occur as thus discretely constituted.

"Still better known than the 'arrow' is the 'Achilles' paradox. Suppose Achilles to race with a tortoise, and to move twice as fast as his rival, to whom he gives an inch of headstart. By the time he has completed that inch, or in other words advanced to the tortoise's starting point, the tortoise is half an inch ahead of him. While Achilles is traversing that half inch, the tortoise is traversing a quarter of an inch, etc."

(Our bold and color emphasis.)

Reader, we must be careful now. We must begin to distinguish a classical pluralism from a quantum pluralism. Classical pluralism is simply lisr EEMD either/or, Aristotelian excluded-middle, reductionist objects which are treated discretely. Quantum pluralism is quantum~simply EIMA both/and, included-middle, n¤n-ideally lisr quantons which must be omnitored both locally and globally quantum~cohesively.

James discloses one of classical mechanics' many problems with ideal, infinitesimal point-objects. Newton unwisely built his whole classical mechanics on these 'impossibles.'

This 'paradox' neglects reality's many contexts. Achilles has one context, e.g., we may call it Achilles' "step-size" context. Tortoise has another context, e.g., we may call it Tortoise's "step-size" context.

Clearly when one assumes a single context where one-step-size-fits-all, and one racer is given a single step advantage... Of course tortoise wins! How dumb...?

"So that the successive points occupied by the runners simultaneously form a convergent series of distances from the starting point of Achilles. Measured in inches, these distances would run as follows:

"Zeno now assumes that space must be infinitely divisible. But if so, then the number of points to be occupied cannot all be enumerated in succession, for the series begun above is interminable. Each time that Achilles gets to the tortoise's last point it is but to find that the tortoise has already moved to a further point; and although the interval between the points quickly grows infinitesimal, it is mathematically impossible that the two racers should reach any one point at the same moment. If Achilles could overtake the tortoise, it would be at the end of two inches; and if his speed were two inches a second, it would be at the end of the first second; but the argument shows that he simply cannot overtake the animal. To do so would oblige him to exhaust, by traversing one by one, the whole of them, a series of points which the law of their formation obliges to come never to an end."

"Zeno's various arguments were meant to establish the 'Eleatic' doctrine of real being, which was monistic. The 'minima sensibilia' of which space, time, motion, and change consist for our perception are not real 'beings,' for they subdivide themselves ad infinitum. The nature of real being is to be entire or continuous. Our perception, being of a hopeless 'many,' thus is false.

"Our own mathematicians have meanwhile constructed what they regard as an adequate continuum, composed of points or numbers. When I speak again of that I shall have occasion to return to the Achilles-fallacy, so called. At present I will pass without transition to the next great historic attack upon the problem of the infinite, which is the section on the 'Antinomies' in Kant's 'Critique of Pure Reason..'"

"Kant's views need a few points of preparation, as follows: —

"1. That real or objective existence must be determinate existence may be regarded as an axiom in ontology. We may be dim as to just how many stars we see in the Pleiades, or doubtful whose count to believe regarding them; but seeing and belief are subjective affections, and the stars by themselves, we are sure, exist in definite numbers. [Kan't shows his almost too deep naïveté here. Stars are absolutely evolving, dying, being born, etc. Their numbers are, classically speaking, absolutely indefinite. Quantumly and stochastically uncertain. Notice Kan't exhibits hubris via his own self-assessed omniscient certainty. Compare that to Paul Pietsch's "Indeterminacy is the principal feature of intelligence" (Slightly paraphrased.) Doug - 20Dec2006.] 'Even the hairs of our head are numbered,' we feel certain, though no man shall ever count them. Any existent reality, taken in itself, must therefore be countable, and to any group of such realities some definite number must be applicable.

"2. Kant defines infinity as 'that which can never be completely measured by the successive addition of units' — in other words, as that which defies complete enumeration.

"3. Kant lays it down as axiomatic that if anything is 'given,' as an existent reality, the whole sum of the 'conditions' required to account for it must similarly be given, or have been given. Thus if a cubic yard of space be 'given,' all its parts must equally be given. If a certain date in past time be real, then the previous dates must also have been real. If an effect be given, the whole series of its causes must have been given, etc., etc." [In quantum reality Kan't's thing(all_is_given, if_part_is_given) is impossible since anything's quantum c¤mplæmænt is all of reality. That, in a subtle way is just what we mean when we say and write "hologram." We show it in quantonics as quanton(unsaid,said). At best we can say "All is given, but at unfathomably reduced 'resolution.' We doubt that is what Kan't meant. Further said is n¤n objective, thus classically 'unreal.' Too, unsaid is unimaginably n¤n trivial quantum~superpositions of both flux and isoflux. Doug - 20Dec2006.]

Now we know classical "real or objective existence" is not determinate. Indeed, it is stochastic, i.e., probabilistic. Classical induction and classical causation are just more illusions.


And these are called "great minds?"



But Kurt Gödel has shown us that no system is absolute in any sense of being both logically/dialectically complete and consistent simultaneously. No set of axioms is complete. No (nontrivial) set of axioms is consistent. By complete, we mean 'states all truths.' By consistent we mean 'always states the truth.'

Kant's is naïve, classical, causal determinism. Sorry Kant, no such beast!

Kant's largest error, which is kin of most other objectivists' Error, is that reality is flux: absolute flux. Classical 'measurement,' what we refer humourously as 'scalarbation' isn't possible as Kant recan'ts it.

See measure and omnitor.

Red text added 19Dec2006 - Doug.

"But the 'conditions' in these cases defy enumeration: the parts of space are less and less ad infinitum, times and causes form series that are infinitely regressive for our counting, and of no such infinite series can a 'whole' be formed. Any such series has a variable value, for the number of its terms is indefinite; whereas the conditions under consideration ought, if the 'whole sum of them' be really given, to exist (by the principle, 1, above) in fixed numerical amount."

[To make all that even more profoundly challenging, quantum reality is in flux, compared to classical reality which is in 'state,' and stux, quantum reality is emergently changing and evolving at up to Planck rates. To make that phasement apparent ask yourself, "Where am I." Then, "When?" Blow it up (i.e., scale it up)! Where is our Solar System? Sol cycloidally 'rotates' around Milky Way at over 300 kilometers per second. Earth cycloidally 'rotates' around Sol at over 30 kilometers per second. Earth's equator cycloidally 'rotates' around Earth's axis at 500 meters per second. All loci are more and less ambiguous! This nested set physical flux fractals exemplify essence of quantum~uncertainty. See graphic to right... Doug - 19Dec2006.]

(Our bold and color emphasis.)

And these are, amazingly, quantumesque! Renouvier arrives in a quantum realm from an apparently classical pluralism. Let's take these one at a time:

1. absolute novelties -
   All real, superatomic 'stuff' may share similitude with other stuff, but is absolutely novel (via flux).
2. unmediated beginnings -
   Stuff arising from quantum isoflux depends on self and co-awareness of isoflux for self- rather than un-mediated beginnings.
3. gifts -
   Quantum measurement is stochastic, thus sometimes apparently 'giving.'
4. chance -
   Quantum Planck rate change is stochastic, thus chance is intrinsic.
5. freedom -
   While all quantum stuff is cohesive, globally and locally, local 'stuff' may make choices which affect ensemble 'measurement' event outcomes.
6. acts of faith -
   Quantum absolute flux affects natural, moral, better, good change.