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A Review
Chapter XI
William James'
Some Problems of Philosophy
by Doug Renselle
Doug's Pre-review Commentary
Start of Review

ISM Extremes

Dedication Introduction Note


Move to any Chapter of William James' Some Problems of Philosophy,
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Chapter XI.............Novelty and the Infinite
The Perceptual View


(Most quotes verbatim William James, some paraphrased.)

(Relevant to Pirsig, William James Sidis, and Quantonics Thinking Modes.)


"Kant's and Renouvier's dealings with the infinite are fine examples of the way in which philosophers have always been wont to infer matters of fact from conceptual considerations. Real novelty would be a matter of fact; and so would be the idealistic constitution of experience; but Kant and Renouvier deduce these facts from the purely logical impossibility of an infinite number of conditions getting completed. It seems a very short cut to truth; but if the logic holds firm, it may be a fair cut, [see James' note below] and the possibility obliges us to scrutinize the situation with increasing care.

[James' note:] "Let me now say that we shall ourselves conclude that change completed by steps infinite in number is inadmissible. This is hardly inferring fact from conceptual considerations, it is only concluding that a certain conceptual hypothesis regarding the fact of change will not work satisfactorily. The field is thus open for any other hypothesis; and the one which we shall adopt is simply that which the face of perceptual experience suggests."

{Our bold emphasis. Our bracketed comments and conversions of James' notes. We do not show ED notes from original.}

Ask yourself, "What is purely logical?"

Keep in mind that concepts of infinity arise from extremely basic Aristotelian, substance-based, objective assumptions. For example, we can easily see how any assumption of identity permits subtraction of identicals thus contriving a concept 'zero.' Whence any division by zero inflicts an issue of a possibility of infinity. Leibnitz' and Newton's calculus and its 'limits' contrive similar problematic concepts. In Quantonics we think classical mathematics is in very large trouble. We think it needs to be replaced or rebuilt bottom up.

James' "inadmissible" concludes something similar, but much more general. Note how we may infer his intuition of a minimum Planck length here. And more...

Note how James also speaks of "...the face of perceptual experience..." We wish to infer his intuition of a non-Aristotelian, included-middle, "face of change." James is so, so close to describing what we call a "quanton."

"Proceeding so to do, we immediately find that in the class of infinitely conditioned things, we must distinguish two sub-classes, as follows:

"1. Things conceived as standing, like space, past time, existing beings.

"2. Things conceived as growing, like motion, change, activity.

"In the standing class there seems to be no valid objection to admitting both real existence, and a numerical copiousness demanding infinity for its description. If, for instance, we consider the stars, and assume the number of them to be infinite, we need only suppose that to each several term of the endless series 1, 2, 3, 4.... n . . . , there corresponds one star. The numbers, growing endlessly, would then never exceed the stars standing there to receive them. Each number would find its own star waiting from eternity to be numbered; and this in infinitum, some star that ever was, matching each number that shall be used. As there is no 'all' to the numbers so there need be none to the stars."

James is talking about classical 'standing, stable, immutable' reality.

Classicists assume space, time, and existence 'stand.'

But they also admit some things can achieve motion.

Conceptual numbers can grow as big as they need to be. (James might remind us here, that 'perceptual' numbers are all different and changing via perceptual flux.)

"One cannot well see how the existence of each star should oblige the whole class 'star' to be of one number rather than of another, or require it to be of any terminated number. What I say here of stars applies to the component parts of space and matter, and to those of past time. [see James' note below]

"So long as we keep taking such facts piecemeal, and talk of them distributively as 'any' or 'each,' the existence of them in infinite form offers no logical difficulty. But there is a psychological tendency to slip from the distributive to the collective way of talking, and this produces a sort of mental flicker and dazzle out of which the dialectic difficulties emerge. 'If each condition be there,' we say, 'then all are there, for there cannot be eaches that do not make an all.'

[James' note:] "Past time may offer difficulty to the student as it has to better men! It has terminated in the present moment, paid itself out and made an 'amount.' But this amount can be counted in both directions; and in both, one may think it ought to give the same result. If, when counted forward, it came to an end in the present, then when counted backward, it must, we are told, come to a like end in the past. It must have had a beginning, therefore, and its amount must be finite. The sophism here is gross, and amounts to saying that what has one bound must have two. The 'end' of the forward counting is the 'beginning' of the backward counting, and is the only beginning logically implied. The ending of a series in no way prejudices the question whether it were beginningless or not; and this applies as well to tracts of time as to the abstract regression which 'negative' numbers form."

(Our bold emphasis. Our bracketed comments and conversions of James' notes.)

I.e., Pirsigean many truths as many piecemeal facts... implies to James many infinities...
not one global collective infinity! His "flicker, dazzle, and dialectic difficulties," exemplify classical paradice which emerge when one views a plural reality from a monist perspective. James tells us, "Do not do that!"

We want you to distinguish his 'distributive' from its quantum dual, i.e., logically nondistributive from "each to each," but possibly locally distributive within "each." Further denial of any global collective from any dialectic view.

James' "distributive" actually, as used here, implies many contexts, many truths.

Sadly James adheres a classical perspective of sophism. He has not yet progressed to a point of seeing quantum reality as a self-referent (co-aware and plural) recursive and fractal system all of which bodes sophism. Quantum reality is a sophism!

James confuses a classical dichon(one_begin, one_end) as a sophism, which, classically, it certainly is not. From a purely classical perspective, it is a dichotomy a perfect scission, not a sophism. Classicists are "thing-king mode incapable, schism incapable, excluded-middle incapable" of perceiving quantum reality's intrinsic sophist self-reference and recursion. Evidence? Buridan! He claimed, based on classical predicate logic, that all sophisms are FALSE! In spades, Buridan was wrong! Sophistic reality only appears paralogical from a classical, homogeneous, collective perspective.

"Rightly taken, the phrase 'all are there,' means only that 'not one is absent.' But in the mouths of most people, it surreptitiously foists in the wholly irrelevant notion of a bounded total.

"There are other similar confusions. 'How,' it may be asked, in Locke's words, can a
'growing measure' fail to overtake a 'standing bulk?' And standing existence must some time be overtaken by a growing number-series, must be finished or finite in its numerical determination. But this again foists in the notion of a bound. What is given as 'standing' in the cases under review is not a 'bulk,' but each star, atom, past date or what not; and to call these eaches a 'bulk,' is to beg the very point at issue. But probably the real reason why we object to a standing infinity is the reason that made Hegel speak of it as the 'false' infinite. It is that the vertiginous chase after ever more space, ever more past time, ever more subdivision, seems endlessly stupid. What need is there, what use is there, for so much?"

(Our bold emphasis.)

SOM's one global truth in one global context.

This text is incredibly difficult to understand if one does not distinguish classical and quantum views:

  1. Distinguish classical stuff achieving its 'holism' via analytic, homogeneous, continuous concepts. Thus classicism views the monist closed reality, one global/logical truth, one time, one space, an objective and synthetic unity which may be infinitely divided.
  2. Distinguish James', Bergson's, Pirsig's, quantum's, et al's. stuffs achieving their plural 'holism' via, stochastic, heterogeneous, sophist percepts. Thus quantum evolute pluralism views a plural open reality, many islandic truths, many times, many spaces, a multiversal cohesion of quantons both autonomous and commingling.

"Not that any amount of anything is absolutely too big to be; but that some amounts are too big for our imagination to wish to caress them. So we fall back with a feeling of relief on some form or other of the finitist hypothesis. [see James' note below]

"If now we turn from static to growing form of being, we find ourselves confronted by much more serious difficulties. Zeno's and Kant's dialectic holds good wherever, before an end can be reached, a succession of terms, endless by definition, must needs have been successively counted out. This is the case with every process of change, however small; with every event which we conceive as unrolling itself continuously. What is continuous must be divisible ad infinitum; and from division to division here you cannot proceed by addition (or by what Kant calls the successive synthesis of units) and touch a farther limit.

[James' note:] "The reader will note how emphatically in all this discussion, I am insisting on the distributive or piecemeal point of view. The distributive is identical with the pluralistic, as the collective is with the monistic conception. We shall, I think, perceive more and more clearly as this book proceeds, that piecemeal existence is independent of complete collectibility, and that some facts, at any rate, exist only distributively, or in form of a set of eaches which (even if in infinite number) need not in any intelligible sense either experience themselves, or get experienced by anything else, as members of an All."

(Our bold emphasis. Our bracketed comments and conversions of James' notes.)

"Finitist hypothesis" implies SOM.

Reader, again, we see James' clear insights. However, we sense he misses an important quantum intueme: reality is multitudinous contextually. James appears, as do nearly all classical philosophers, to assume reality is unicontextual. Given that, we assure you that when one considers both zero and infinity in a unit context, one will generate paradice. However, viewed from multitudinous contexts, and even as generators of multitudinous contexts, paradice may be shown to evaporate. 'Begin,' and 'end' are SOM dichons which elicit similar thought.

Kant's "successive synthesis" is just integral calculus an inverse of SOM's classical analytical/differential calculus!

James is very close to perceiving quantum reality here, but his insistence on objective lisr of his 'eaches' denies quantum reality. Unfortunately, Aristotle's syllogisms are deeply buried in most of our psyches. Our Quantonics quantons are comminglings of:

  • both localability and nonlocalability
  • both isolability and nonisolability
  • both separability and nonseparability
  • both reducibility and nonreducibility

We see lisr in our Quantonic symbol as solid and dark green. We see lisr's commingling and compenetrating complement as dashed and light blue.

We probably should expect that James, et al., might have difficulty intuiting these extraordinary Value interrelationships one hundred years ago. But James did intuit flux as essence. Flux is what makes these incredible Value interrelationships possible!

171 "You can indeed define what the limit ought to be, but you cannot reach it by this process. That Achilles should occupy in succession 'all' the points in a single continuous inch of space, is as inadmissible a conception as that he should count the series of whole numbers 1, 2, 3, 4, etc., to infinity and reach an end. The terms are not 'enumerable' in that order; and the order it is that makes the whole difficulty. An infinite 'regression' like the rearward perspective of time offers no such contradiction, for it comes not in that order. Its 'end' is what we start with; and each successive note 'more' which our imagination has to add, ad infinitum, is thought of as already having been paid in and not as having yet to be paid before the end can be attained. Starting with our end, we have to wait for nothing. The infinity here is of the 'standing' variety. It is, in the word of Kant's pun, gegeben, not aufgegeben: in the other case, of a continuous process to be traversed, it is on the contrary aufgegeben: it is a task - not only for our philosophic imagination, but for any real agent who might try physically to compass the entire performance."

Reader, as you can see here, it is incredibly difficult, when using CTMs, to describe quantum reality. CTMs generate paradice and concomitant confusion. Though James does a credible job of it.

Try visualizing 'begin' as one context. Then visualize 'end' as another context. Imagine Value as Quantonic Interrelationships twixt 'begin' and 'end.' Insert 'being' or an equivalent, as a third context twixt 'begin' and 'end.' Consider new Value interrelationships. New light shines!

CTMs lump 'begin,' 'being,' and 'end,' all into a single, homogeneous, unilogical context, and expect practitioners to assess absolute truths about 'each' in [i.e., as classical Aristotelian stoppable 'states' of] that global context! See our more recent Aristotle's apple.

QTMs offer a new way of think-king which are quantumesque and vastly superior to CTMs.
27Apr2000 Doug.


"Such an agent is bound by logic to find always a remainder, something ever yet to be paid, like the balance due on a debt with even the interest of which we do not catch up.

"'Inflnitum in actu pertransiri nequit,' said scholasticism; and every continuous quantum to be gradually traversed is conceived as such an infinite. The quickest way to avoid the contradiction would seem to be to give up that conception, and to treat real processes of change no longer as being continuous, but as taking place by finite not infinitesimal steps, like the successive drops by which a cask of water is filled, when whole drops fall into it at once or nothing. This is the radically pluralist, empiricist, or perceptualist position, which I characterized in speaking of Renouvier (above, pages 164-165). We shall have to end by adopting it in principle ourselves, qualifying it so as to fit it closely to perceptual experience."

(Our bold emphasis.)

"The growing infinite must be treated as discontinuous." (Sidebar from original text.)

Yes! Yes! Yes! steps (packets) of flux!

James and Renouvier intuit Planck's quanta! They intuit quantum reality! This is amazing! We sensed this, but till now were not we are sure...both James and Renouvier both intuited a pluralistic quantal reality! If only they could have intuited quantal reality's included-middle, and its multi-contextualism.

"Meanwhile we are challenged by a certain
school of critics who think that what in mathematics is called 'the new infinite' has quashed the old antinomies, and who treat anyone whom the notion of a completed infinite in any form still bothers, as a very naif person. Naif though I am in mathematics, I must, notwithstanding the dryness of the subject, add a word in rebuttal of these criticisms, some of which, as repeated by novices, tend decidedly towards mystification.

"The 'new infinite' and the 'number-continuum' are outgrowths of a general attempt to accomplish what has been called the 'arithmetization' (ariqmoz [ariqmoz] meaning number) of all quantity. Certain quanta (grades of intensity or other difference, amounts of space) have until recently been supposed to be immediate data of perceptive sensibility or 'intuition'; but philosophical mathematicians have now succeeded in getting a conceptual equivalent for them in the shape of collections of numbers created by interpolation between one another indefinitely."

(Our bracketed comments. Our bold emphasis.)

Reader, note that 'antinomies' in classical thought were/are contradictions which arose from paradice which arose from classical dichotomies imposed by SOM's own subject-object schism of reality.

We worry he might call us such! Doug.

James too, sees fundamental flaws in 'modern' mathematics.

One may choose to compare 'ariq' with 'aretê.' Greeks tend(ed) to see number and object as 'excellence.'

James use of "grades of intensity," offers a glimmer of his intuition of an included-middle. But he follows it immediately with "difference," which has a distinct lisr flavor. See our more recent omnifference.

"We can halve any line in space, and halve its halves and so on. But between the cuts thus made and numbered, room is left for infinite others created by using as a divisor, for infinite others still by using 5, 7, etc., until all possible 'rational' divisions of the line shall have been made. Between these it is now shown that interpolation of cuts numbered 'irrationally' is still possible ad infinitum, and that with these the line at last gets filled full, its continuity now being wholly translated into these numbered cuts, and their number being infinite. 'Of the celebrated formula that continuity means "unity in multiplicity," the multiplicity alone subsists, the unity disappears,' [Footnote 1] as indeed it does in all conceptual translations and the original intuition of the line's extent gets treated, from the mathematical point of view, as a 'mass of uncriticized prejudice' by Russell, or sneered at by Cantor as a 'kind of religious dogma.' [Footnote 2]

"So much for the number-continuum. As for 'the new infinite:' that means only a new definition of infinity."

1. H. Poincaré: La science et l' hypothese, p. 30.

2. B. Russell: The Philosophy of Mathematics, i, 260, 287.

(Our bold emphasis and brackets for footnotes.)

"Cuts" refers SOM's great arbitrary scissions of all reality. SOM would do surgery analytically. SOM sees a 'brain' as an object, one parameter of a dichon. But brains, eyes, hearts, et al., are not objects! They are quantons whose interrelationships to their 'body' system are multitudinous and quantum-pathological. SOM denies quantum coherence in living systems! So SOM can cut away, at its pleasure. Numbers and motorcycle parts, too, are objects to SOM. No Static Patterns of Value (SPoVs, or actual complements of quantons) in quantum reality are SOM objective! Numbers are quantons! See our more recent number.

Many have seen that modern classical mathematics' modular induction will ultimately become its nemesis. Note how modular induction is a sophism (self-referent and recursive)! We see classicists worshiping in their hated Straw Man's Multiverse of Reasonings, while pretending they reside in their own Church of (one, unilogical) Reason! Worse, they misused sophism's assumed included-middle by substituting Aristotle's excluded-middle syllogism! Consequences? We can show Peano blew it! Too, we can show mathematics' Axiom of Independence is wrong!
175 "If we compare the indefinitely-growing number-series, 1, 2, 3, 4, - - - n, --- in its entirety, with any component part of it, like 'even' numbers 'prime' numbers, or 'square' numbers, we are confronted with a paradox. No one of the parts, thus named, of the number-series, is equal to the whole collectively taken; yet any one of them is 'similar' to the whole, in the sense that you can set up a one-to-one relation between each of its elements and each element of the whole, so that part and whole prove to be of what logicians call the same 'class,' numerically. Thus, in spite of the fact that even numbers, prime numbers, and square numbers are much fewer and rarer than numbers in general, and only form a part of numbers überhaupt they appear to be equally copious for purposes of counting. The terms of each such partial series can be numbered by using the natural integers in succession. There is, for instance, a first prime, a second prime, etc., ad infinitum; and queerer-sounding still, since every integer, odd or even, can be doubled, it would seem that the even numbers thus produced cannot in the nature of things be less multitudinous than that series of both odd and even numbers which the whole natural series consists." (Our bold emphasis.)

All of this depends on SOM's one, homogeneous, unilogical, homological, global context. Move to a multitudinous contextual (quantum) logical system, and these paradice evaporate. See 'Connections' at our Buridan review. Doug.

Note how these mappings, done by 'mathematicians,' require sophism! They require self-reference! By classical definition, none of these mappings may be tautologies! How's that grab you, eh? So modern mathematics is on firm ground, eh? Just makes one giddy think-king about it!
27Apr2000 Doug.

And professional 'academics' with their bureaucracies, c.v.'s, and titles go on teaching this subterfuge. Shriek heterogeneity! Shriek, "Not time for change, people, but times for changes!" It's Millennium III, folks! Times for changes! Can you hear foundations creaking...?

"These paradoxical consequences result, as one sees immediately, from the fact that the infinity of the number-series is of the
'growing' variety (above, page 170). They were long treated as a reductio ad absurdum of the notion that such a variable series spells infinity in act, or can ever be translated into standing or collective form. [see James' note below] But contemporary mathematicians have taken the bull by the horns. Instead of treating such paradoxical properties of indefinitely growing series as reductiones ad absurdum, they have turned them into the proper definition of infinite classes of things. Any class is now called infinite if its parts are numerically similar to itself.

[James' note:] "The fact that, taken distributively, or paired each to each, the terms in one endlessly growing series should be made a match for those in another (or 'similar' to them) is quite compatible with the two series being collectively of vastly unequal amounts. You need only make the steps of difference, or distances, between the terms much longer in one series than in the other, to get numerically similar multitudes, with greatly unequal magnitudes of content. Moreover the moment either series should stop growing, the 'similarity' would cease to exist."

(Our bracketed addition of James' footnotes.)

Note how 'growing' requires self-reference! There is no classical ontology for 'growing.' There is a quantum ontology for 'growing.' Note how growth intrinsically alters context. Note how these contexts become local/nonlocal quantum islands of both dynamis and stasis as quantum complements! We see Pirsig's MoQquanton(DQ,SQ)!

Note that quantons are not classically 'distributive.'

"If its parts are numerically dissimilar, it is finite. This definition now separates the conception of the class of finite from that of infinite objects.

Next, certain concepts, called 'transfinite numbers,' are now created by definition. They are decreed to belong to the infinite class, and yet not to be formed by adding one to one ad infinitum, but rather to be postulated outright as coming after each and all of the numbers formed by such addition. [see James' note below] Cantor gives the name of 'Omega' to the lowest of these possible transfinite numbers. It would, for instance, be the number of the point at which Achilles overtakes the tortoise if he does overtake him by exhausting all the intervening points successively. Or it would be the number of the stars, in case their counting could not terminate.

[James' note:] The class of all numbers that 'come before the first transfinite' is a definitely limited conception, provided we take the numbers as eaches or anys, for then any one and every of them will have by definition to come before the transfinite number comes even though they form no whole and there be no last one of them, and though the transfinite have no immediate predecessor. The transfinite is, in a word, not an ordinal conception, at least it does not continue the order of entire numbers."

(Our bracketed addition of James' footnotes.)

More classical 'cuts.' SOM wields its knife, as Pirsig told us, well and at will!

Cantor is, as a good classicist, guilty of his own paradice! I.e., "The Set of All Sets!" See our Sophism Connection.

"Or again it would be the number of miles away at which parallel lines meet if they do meet. It is, in short, a 'limit' to the whole class of numbers that grow one by one, and like other limits, it proves a useful conceptual bridge for passing us from one range of facts to another.

"The first sort of fact we pass to with its help is the number of the number-continuum or point-continuum described above (page 173) as generated by infinitely repeated subdivision. The making of the subdivisions is an infinitely growing process; but the number of subdivisions that can be made has for its limit the transfinite number Omega just imagined and defined; thus is a growing assimilated to a standing multitude; thus is a number that is variable practically equated (by the process of passing to the limit) with one that is fixed; thus do we circumvent the law of indefinite addition, or division which previously was the only way in which infinity was constructable, and reach a constant infinite at a bound. This infinite number may now be substituted for any continuous finite quantum, however small the latter may perceptually appear to be."

179 "When I spoke of my 'mystification,' just now, I had partly in mind the contemptuous way in which some enthusiasts for the 'new infinite' treat those who still cling to the superstition that 'the whole is greater than the part.' Because any point whatever in an imaginary inch is now conceivable as being matched by some point in a quarter-inch or half-inch, this numerical 'similarity' of the different quanta, taken point-wise, is treated as if it signified that half-inches, quarter-inches, and inches are mathematically identical things anyhow, and that their differences are f acts which we may scientifically neglect. I may misunderstand the newest expounders of Zeno's famous 'sophism,' but what they say seems to me virtually to be equivalent to this." (Our bold emphasis.)

If, classically, we can cut whole reality into pieces (classical analysis), and we put the pieces back together (classical synthesis), how could reality before analysis be different from reality after synthesis? This is pure SOMthing-king, pure classical Boole!

Aristotelian identity is impossible in quantum reality!

To classicist Zeno, any meme which was not a dichon was a sophism. And Zeno knew sophisms are bad! How? Plato and Aristotle told him so. More recently we revise our views of Zeno. It appears that his four paradice anicipated many of our Quantonics views of quantum reality. See our Zeno's Paradice. Doug - 24Mar2003.

"Mr. Bertrand Russell (whom I do not accuse of mystification, for Heaven knows he tries to make things clear!) treats the Achilles-puzzle as if the difficulty lay only in seeing how the paths traversed by the two runners (measured after the race is run, and assumed then to consist of nothing but points of position
coincident with points upon a common scale of time) should have the same time-measure if they be not themselves of the same length. But the two paths are of different lengths; for owing to the tortoise's head-start, the tortoise's path is only a part of the path of Achilles. How, then, if time-points are to be the medium of measurement, can the longer path not take the longer time?

"The remedy, for Mr. Russell, if I rightly understand him, lies in noting that the sets of points in question are conceived as being infinitely numerous in both paths, and that where infinite multitudes are in question, to say that the whole is greater than the part is false. For each and every point traversed by the tortoise there is one point traversed by Achilles, at the corresponding point of time; and the exact correspondence, point by point, of either one of the three sets of points with both the others, makes of them similar and equally copious sets from the numerical point of view."

Carefully considered, one may see a real problem here is classicists view reality as a quantifiable, infinitely divisible, homogeneous continuum.

But we must learn to see that quantum reality is a qualitative, indivisible, heterogeneous cohesion.

Former is SOMthink. Latter is Quantonic/MoQthink.

"There is thus no recurrent 'remainder' of the tortoise's head-start with which Achilles cannot catch up, which he can reduce indefinitely, but cannot annul. The books balance to the end. The last point in Achilles' path, the last point in the tortoise's, and the last time instant in the race are terms which mathematically coincide. With this, which seems to be Mr. Russell's way of analyzing the situation, the puzzle is supposed to disappear.

"It seems to me however that Mr. Russell's statements dodge the real difficulty, which concerns the 'growing' variety of infinity exclusively, and not the 'standing' variety, which is all that he envisages when he assumes the race already to have been run and thinks that the only problem that remains is that of numerically equating the paths. The real difficulty may almost be called physical, for it attends the process of formation of the paths."

Views path and time time as homogeneous continua.

Views 'time instant' as infinitesimal division of homogeneous time.

Whole cannot grow without mutation, self-reference, evolute and holistic extension. Is classic whole reality evolute or not? Is it mutable or not? (SOMesque questions.)
182 "Moreover, two paths are not needed that of either runner alone, or even the lapse of empty time, involves the difficulty, which is that of touching a goal when an interval needing to be traversed first keeps permanently reproducing itself and getting in your way. Of course the same quantum can be produced in various manners. This page which I am now painfully writing, letter after letter, will be printed at a single stroke. God, as the orthodox believe, created the space continuum, with its infinite parts already standing in it, by an instantaneous fiat. Past time now stands in infinite perspective, and may conceivably have been created so, as Kant imagined, for our retrospection only, and all at once. 'Omega' was created by a single decree, a single act of definition in Prof. Cantor's mind. But whoso actually traverses a continuum, can do so by no process continuous in the mathematical sense." (Our bold emphasis.)
183 "Be it short or long, each point must be occupied in its due order of succession; and if the points are necessarily infinite, their end cannot be reached, for the 'remainder,' in this kind of process, is just what one cannot 'neglect.' 'Enumeration' is, in short, the sole possible method of occupation of the series of positions implied in the famous race; and when Mr. Russell solves the puzzle by saying as he does, that 'the definition of whole and part without enumeration is the key to the whole mystery,' he seems to me deliberately to throw away his case."
184 "After this disagreeable polemic, I conclude that the new infinite need no longer block the
way to the empiricist opinion which we reached provisionally on page 172. Irrelevant though they be to facts the 'conditions' of which are of the 'standing' sort, the criticisms of Leibnitz, Kant, Cauchy, Renouvier, Evellin and others, apply legitimately to all cases of supposedly continuous growth or change. The 'conditions' here have to be fulfilled seriatim; and if the series which they form were endless, its limit, if 'successive synthesis' were the only way of reaching it, could simply not be reached."

"Either we must stomach logical contradiction, therefore, in these cases; or we must admit that the limit is reached in these successive cases by finite and perceptible units of approach drops, buds, steps, or whatever we please to term them, of change, coming wholly when they do come, or coming not at all. Such seems to be the nature of concrete experience, which changes always by sensible amounts, or stays unchanged. The infinite character we find in it is woven into it by our later conception indefinitely repeating the act of subdividing any given amount supposed. The facts do not resist the subsequent conceptual treatment; but we need not believe that the treatment necessarily reproduces the operation by which they were originally brought into existence.

"The antinomy of mathematically continuous growth is thus but one more of those
many ways in which our conceptual
transformation of perceptual experience makes it less comprehensible than ever."

Logical contradiction is possible only via classical reasoning.

See our more recent contradict, negate.

I.e., via a quantum approach!

Classical mathematics is doomed as is classical Aristotelian/Newtonian ontology which it and classical logic spawned. Spawn of Iliadic wrath, spawn of Platonic antisophism hate.

186 "That being should immediately and by finite quantities add itself to being, may indeed be something which an onlooking intellect fails to understand; but that being should be identified with the consummation of an endless chain of units (such as 'points'), no one of which contains any amount whatever of the being (such as 'space') expected to result, this is something which our intellect not only fails to understand, but which it finds absurd. The substitution of 'arithmetization' for intuition thus seems, if taken as a description of reality, to be only a partial success. Better accept, as Renouvier says, the opaquely given data of perception, than concepts inwardly absurd." Bravo! We agree! (Our bold emphasis.)

"So much for the 'problem of the infinite,' and for the interpretation of continuous change by the new definition of infinity. We find that the picture of a reality changing by steps finite in number and discrete, remains quite as acceptable to our understanding and as congenial to our imagination as before; so, after these dry and barren chapters, we take up our main topic of inquiry just where we had laid it down. Does reality grow by abrupt increments of novelty, or not? The contrast between discontinuity and continuity now confronts us in another form. The mathematical definition of continuous quantity as 'that between any two elements or terms of which there is another term' is directly opposed to the more empirical or perceptual notion that anything is continuous when its parts appear as immediate next neighbors, with absolutely nothing between.

[Portion of a James footnote which started on page 186] "...Extremes meet; and although Russell and Zeno agree in denying perceptual motion, for the one a pure unity, for the other a pure multiplicity takes its place. It is probable that Russell's denial of change, etc. is meant to apply only to the mathematical world. It would be unfair to charge him with writing metaphysics in these passages, although he gives no warning that this may not be the case."

(Our bracketed comments. Our bold emphasis.)

Here is a major problem with mathematics! Reality is not a continuous quantity! Physics too, based upon this misguided mathematical definition, is misguided.

If both Russell and Zeno agree, then we know what we can say about both Russell and Zeno: both are foole Boole.
188 "Our business lies hereafter with the perceptual account, but before we settle definitively to its discussion, another classic problem of philosophy had better be got out of the way. This is the 'problem of causation.'"
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©Quantonics, Inc., 2000-2011 Rev. 5May2009  PDR Created: 7Apr2000  PDR
(10Dec2001 rev - Add top of page frame-breaker.)
(17Nov2002 rev - Add anchor to page 171.)
(9Jan2003 rev - Add
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(24Mar2003 rev - Repair minor typos. Swap a smiley GIF for its wingdings font. Ditto north-vertical arrows.)
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