Charles Bell’s Axiomatic Drama

Here is an annotated transcription of a 1981 manuscript by Charles Greenleaf Bell (1916–2010) called “The Axiomatic Drama of Classical Physics.” A theme is what Heraclitus observed, as in fragment B49a of Diels, LXXXI of Bywater, and D65a of Laks and Most:

We step and we do not step into the same rivers, we are and we are not.
ποταμοῖς τοῖς αὐτοῖς ἐμβαίνομέν τε καὶ οὐκ ἐμβαίνομεν, εἶμέν τε καὶ οὐκ εἶμεν.

Bell reviews the mathematics, and the thought behind it, of

  1. free fall,
  2. the pendulum,
  3. the Carnot heat engine.

In a postlude called “The Uses of Paradox,” Bell notes:

Forty-five years ago I decided that when reason drives a sheer impasse into an activity which in fact goes on, we have to think of the polar cleavage as both real and unreal.

I like that reference to “an activity which in fact goes on.” In youth it may be hard to recognize that there are activities that do go on. Bell himself goes on:

… that is a job as huge and demanding as Aristotle’s, and for me at 70, just begun.

“Look,” my friends say, “Bell’s been doing the same thing since he was 25. About that time he had a vision of Paradox as paradise, and he’s been stuck there ever since.”

Bell’s picture next to Aristotle’s Physics
The back of Bell’s Five Chambered Heart with
the front of the OCT of Aristotle’s Physics

Charles Bell was a polymath, and I have brought up his work as a poet a couple of times on this blog. Spending the summer of 1986 on the campus of St John’s College, Santa Fe, before my senior year there, I joined Bell and a few other tutors to read Gödel’s paper, “On Formally Undecidable Propositions of Principia Mathematica and Related Systems I.” Bell also had a series of slide shows called Symbolic History Through Sight and Sound; perhaps they could be described as a more detailed version of what Kenneth Clark attempted in Civilisation. It seems a few of Bell’s shows got converted to video and put on YouTube.

My first exposure to Charles Bell was apparently through the text below, delivered by him as a lecture at St John’s College in Annapolis, in the fall semester of 1983. It was a Friday night. This was “the only time in the week when students are lectured to,” according to the promotional material that had helped bring me to the College. I was a freshman, and I had yet to read most of the works that Mr Bell would refer to. Nonetheless, I was impressed by the wide range of the lecture and the intensity of the speaker, who allowed the question period to continue for more than two hours after the lecture.

I can still hear Mr Bell saying, in his breathy voice,

And do not ask of the all if up or down.

This alludes to the fragment of Heraclitus, B60 of Diels, LXIX of Bywater, D51 of Laks and Most,

The way upward and downward: one and the same.
ὁδὸς ἄνω κάτω μία καὶ ὡυτή.

That fragment is indeed in the text below, obtained from a pdf image in the College digital archives. The document is described as “Transcript of a lecture given on December 11, 1981 by Charles Bell at St. John’s College in Santa Fe.” I transferred to that campus in 1984, after my freshman year in Annapolis, where I suppose Bell was asked to come give the same lecture.

Working in Ubuntu Linux, I manipulated the pdf file as follows:

  1. Import pdf to the gimp program
  2. Reverse the order of the pages
  3. Export to tif
  4. Convert tif to txt with the tesseract program
  5. Clean up the txt file
  6. Convert txt to html with pandoc

You can guess that it was at step 5 that I found out step 2 was needed. The tesseract program made a few errors with letters. Its single and double spacing of lines was erratic, and it lost some lines. The mathematical symbolism needed special attention, as did the three diagrams. I obtained two of them with the function “Save a screenshot of an area to Pictures” from the pdf file set at 200% magnification. The third diagram wouldn’t fit my screen that way, so I used gimp to crop it, then mogrify to resize it at 200%.

I have bolded and bracketed the page numbers, which are at the tops of their pages. In the original, page 0 is numbered by hand, page 1 is not numbered, and pages 28 and 29 are transposed.

For my own understanding, I have added

  • highlighting,
  • bullet points;
  • links;
  • italicization of each of Bell’s nine uses of “a priori”;
  • commentary like this, in blue, in a smaller font, and not justified.

There is no end to the possible commentary. I do not know whether I shall want to add more in the future. In any case, there is more work to do on the relation between the real Aristotle and the one that Galileo is supposed to have disproved.

The verse above about “up or down” is not from the lecture, but is the last line of “Five Chambered Heart”:

The first begins to beat like a drop of blood
On the egg-yolk of the world the fifth day
When vessels reach to guide the coded wave
Under brooding wings in the dark of LOVE.

A second cleaves the wish, one on one,
Mounts to spool and gender on its own:
Ruling reptiles upreared in a world
of electric LUST and cycad palm.

The third, of two and one, admits a space
Between self and other, EARTH-manifold,
Where love meanders the sensible,
And what it sees and meets with calls its own.

Four chambers pound with use gone wrong,
That all time, seas and saurians, beast and man
Kindle WASTE by everything we loved;
And lost the flower-turn from four to five.

Heart, infold again, world-fire infold,
SOUL cradled in a spiral swoon;
Still desire in cerements five-fold,
And do not ask of the all if up or down.

This poem introduces Mr Bell’s 1986 book of the same name. I talked briefly of Bell’s reading from it in “Bosphorus Sky,” and I said a bit more of Bell as a poet in “Some Say Poetry.”

In the lecture below, Bell refers to the following artists and thinkers, in this order:

  • Bell himself
  • Leonardo
  • Galileo
  • Huyghens
  • Newton
  • Leibniz
  • Aristotle
  • Carnot
  • Kant
  • Longfellow (implicitly)
  • Kierkegaard
  • Euclid
  • Pascal
  • Archimedes
  • Plotinus
  • Parmenides of Elea
  • Sartre
  • Meno
  • Plato
  • Socrates
  • Descartes
  • Darwin
  • Heraclitus
  • Dante
  • Maxwell
  • Einstein
  • Aeschylus
  • Kepler
  • Lobachevsky
  • Riemann
  • Minkowsky
  • “a Mississippi high school teacher, Miss Hawkins (called Hawkeye)”
  • Shakespeare (implicitly)
  • Poincaré
  • Whitehead
  • Epicurus
  • Democritus
  • La Place
  • Blake
  • Berkeley
  • Bergson (implicitly)
  • Lucretius
  • Taylor
  • Euler
  • William James
  • Boyle
  • Charles
  • Gay-Lussac
  • Clausius
  • Helmholtz
  • Yeats
  • Hume
  • Planck
  • Schroedinger
  • Kelvin
  • George Herbert (implicitly)

Contents

[0]

Charles G. Bell, 1260 Canyon Road, Santa Fe, NM 87501

THE AXIOMATIC DRAMA OF CLASSICAL PHYSICS

SALUTATION:

(Lecture opening)

Parenthesis added by hand in the original.

Two dramas should be enacted tonight:

  • that of axiomatic polarity in the physics of matter and motion, and
  • that of your trying to penetrate the thoughts of Charles Bell.

America has hatched a funny notion of education: that as long as you are sorting over and expressing your own views, you are active; but as soon as a teacher opens up to tell you something, you become passive and your interest must flag.

On the contrary, there is no fiercer activity than the pursuit of what escapes you in the thought of another, of what used to be called a master.

As in physics, the sign of your acceleration must be a proportionate counterforce, which we will feel between us as salutary strain. How else would I know there was a fish on the line? Shall we not be fishers of men?

[1]

Charles G. Bell, 1260 Canyon Road, Santa Fe, NM 87501

THE AXIOMATIC DRAMA OF CLASSICAL PHYSICS

PREVIEW:

Our drama is of axiomatic polarity in the physics of matter in motion. It pursues the question: “What is conserved through dynamic exchanges?” Although history blurs beginnings and endings,

  • we begin with Leonardo on “Force … a spiritual power … an active, incorporeal life …”
  • The middle of our action swells through Galileo, Huyghens and Newton to Leibnitz’ grasp of what we call The First Law of Thermodynamics, that Live Force (our energy) can be neither created nor destroyed.
  • When, by axiomatic necessity, this pooling of active stuff as substratum of the universe, brings it to rest (Aristotle: “Surely the substratum cannot cause itself to move”), our drama ends with Carnot’s Second Law—specter of an energy which, like old Tithonus, cannot die, but always withers away: “consumed with that which it was nourished by.”

Thus no sooner is Leonardo’s mystery of force summed under the logic of substance, than its clamming up is implied.

But since even this outline has invoked Aristotle, with what we have called “axiomatic necessity”, the foreground drama must have a deeper underlay, a kind of metaphysical fate or Anangke.

In Bell’s transliteration of Ἀνάγκη, the two Greek letters γκ become the three Latin letters ngk, reflecting the pronunciation. As a proper noun, the word names the goddess of Necessity; meanings of the common noun include force, constraint, compulsion, and even bodily pain and anguish. The idea is seen in the account by Herodotus (Book I, chapter 11) of how King Candaules of Lydia arranged for his wife’s naked form to be seen by his favorite bodyguard, Gyges. When she understood that Gyges had seen her, the queen told him he must kill Candaules and rule in his stead, with her at his side, or else be killed himself.

ὁ δὲ Γύγης τέως μὲν ἀπεθώμαζε τὰ λεγόμενα, μετὰ δὲ ἱκέτευε μὴ μιν ἀναγκαίῃ ἐνδέειν διακρῖναι τοιαύτην αἵρεσιν. οὔκων δὴ ἔπειθε, ἀλλ᾽ ὥρα ἀναγκαίην ἀληθέως προκειμένην ἢ τὸν δεσπότεα ἀπολλύναι ἢ αὐτὸν ὑπ᾽ ἄλλων ἀπόλλυσθαι: αἱρέεται αὐτὸς περιεῖναι. ἐπειρώτα δὴ λέγων τάδε. ‘ἐπεί με ἀναγκάζεις δεσπότεα τὸν ἐμὸν κτείνειν οὐκ ἐθέλοντα, φέρε ἀκούσω τέῳ καὶ τρόπῳ ἐπιχειρήσομεν αὐτῷ.’

Gyges stood awhile astonished at this; presently, he begged her not to compel him to such a choice. But when he could not deter her, and saw that dire necessity was truly upon him either to kill his master or himself be killed by others, he chose his own life. Then he asked: “Since you force me against my will to kill my master, I would like to know how we are to lay our hands on him.”

First the queen forces Gyges to make a choice; but then he says she forces into the particular choice that he himself makes.

[2] Mostly we think of physics as empirical, not axiomatic. If we hang five steel balls, each from two threads so they swing in a single line, and with each touching the next, we can explore their motions (however the construction was made by someone who knew what he wanted to prove).

  • When I pull the first ball out and drop it, some quantity of motion seems communicated through the chain, so that the last, on the other side, pops out with almost what the first brought in; and so it goes, push-me, pull-you—
  • but not quite: chaos creeps in; the pulse works less and less.

But that is already the history we outlined as enacted through the speculation of three centuries. Is physics so simple-minded? Almost, but not quite. Leibnitz’ presentiment was that what is lost outwardly may be preserved (as heat) in the motions of the minute parts. Curious this insight should become the axiomatic noose on which (in Carnot) deathless energy settles toward its heat death.

As if whoever embarked in ignorance on an axiomatic field—those Renaissance explorers, in the tipsy certainly [sic] of causal logic and algebraic consistency, formulating the vital ambience of the world—must enact, swept on the cross-currents naively denied, the paradox and reversal of the a priori.

So

  • conservation leads to decay,
  • absolute motion to Relativity;
  • causality hatches quantum indeterminacy;
  • last, [3] mathematics, cornerstone of the whole, proves in Goedel its unprovability.

From this larger cycle, we draw the sportive interlude of classical physics.

Of the four points above, Bell will take up mainly the first, and never again the last. As for the third, I have a post “On Causation” (recently edited). I do not know whether anybody uses the abstract nouns “causality” and “causation” with clearly distinct meanings. I think the basic idea of a cause is expressed in a lecture called “The Politics of Hell,” by Urban Hannon, “delivered at the Pro Civitate Dei summer school in La Londe-les-Maures, France on June 12, 2022”:

Satan was the highest of the angels who fell … He is even, in some way, the cause of the rest of their sin—not by compulsion, which would make their choice involuntary and thus not a choice at all, but by suggestion or exhortation.

Candaules’s queen caused Gyges to kill her husband, but Gyges was still guilty of murder, because he could have chosen not to do the deed, albeit on pain of death. But how can it be established that something could have happened otherwise, when in fact it did not? Causation feeds into compulsion, or necessitation, as in Newton’s First Law (here in the translation of Cohen and Whitman; Bell will quote all three Laws later):

Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.

As Collingwood puts it in the essay on causation (extracted, typeset, and annotated by me) that forms one of the examples in An Essay on Metaphysics (1940),

when we come to Newton … we find him using as a matter of course a whole vocabulary which, taken literally, ascribes to ‘causes’ in nature a kind of power which properly belongs to one human being inducing another to act as he wishes him to act.

There is a tension here, which physics has gone on to eliminate:

Of the two Newtonian classes of events, (a) those that happen according to law [and] (b) those that happen as the effects of causes, class (a) has expanded to such an extent as to swallow up (b).

Perhaps the notion of law here is what Bell means by causality, and the concept of “quantum indeterminacy” was too young or obscure for Collingwood to make sense of.

How subtly, in that action, the Tao and Yin-Yang complementarities, commonplace in the new physics, have enlivened the enterprise from the start.

FROM QUESTION TO METHOD:

What does it mean that Kant was able to draw from reason itself antinomies of “yes” and “no” to those four crucial questions of metaphysics?—proving at once:

  1. That the world has, and has not, a beginning in time and a limit in space;
  2. That the composite must, and cannot be, made of simple parts;
  3. That within the laws of nature there both is and is not a causality of freedom;
  4. That an absolutely necessary being (God) exists and does not exist.

In Kant’s enlightened comedy, that crisis of Pure Reason finds a solution in the Moral Will. As if the outcrop of contradiction were a salutary warning that speculation has climbed too high and must descend to empirical pastures, handing the metaphysical banner to Practical Reason: “Excelsior!”

Evidently Bell alludes to the Longfellow poem whose first stanza is,

The shades of night were falling fast,
As through an Alpine village passed
A youth, who bore, ’mid snow and ice,
A banner with the strange device,
      Excelsior!

The youth does climb too high.

In Latin, excelsior is the comparative of excelsus, which seems to have the same meaning as celsus: lofty, high, sublime. Apparently celsus is, in origin, a participle of an unattested verb, *cello. The compound excello “excel” is attested. The Indo-European root gives us

  • through Germanic, “hill” and “holm”;
  • through Greek, “colophon”;
  • through Latin, not only “excel,” but also “culminate” and “column.”

[4] But what if the moral imperative should abnegate (in Fear and Trembling) reason’s categorical ground? What if the empirical should still gape with “yes-and-no”?

  • Is the first antinomy clearly separable from ours of steady-state against the Big Bang?
  • Or the second from our Bevatron search for Quarks?
  • The third from the present convergence of biochemistry, neurology, cybernetics and fractals?

Does not the horn of dilemma bare itself in every seed-plot and fish-pool of knowledge—as

  • in science, that of deterministic cause;
  • in arithmetic, of the unit base of integers,
  • in geometry,
    • the dimensionless point of Euclid’s First Definition
    • (not to speak of the Parallel Postulate).

Through all Kant’s a priori, cricket voids are chirping: “both impossible and necessary”, and “as sure as unthinkable”.

I’m not sure, but regarding science, arithmetic, and geometry, Bell could be alluding to the following.

  • As in the notes above, cause is not originally deterministic, since the person who was caused to do something could have done otherwise.

  • In studying cardinality, Cantor glossed over the problem that if the elements of a set are indistinguishable from one another, it is not clear how they can be oconsidered many and not one. In a paper “On Commensurability and Symmetry” (Journal of Humanistic Mathematics, Volume 7, Issue 2 [July 2017], pages 90–148; DOI: 10.5642/jhummath.201702.06), I argued that von Neumann solved this problem by defining each number as the set of lesser numbers. However, the first number then is the empty set, a paradoxical concept in itself.

  • By Euclid’s definition, Σημεῖόν ἐστιν, οὗ μέρος οὐθέν, “A point is that which has no part”; but then how does a point manage to be anything? If we “point out” that Euclid never actually uses the definition of point, so that we may just leave the concept undefined, letting it be determined by postulation, as Euclid practically does by saying

    Ἠιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν
    Let it have been postulated, from any point to any point, a straight line to draw,

    and Hilbert by saying,

    For every two points A, B there exists a line a that contains each of the points A, B

    —if we do this, I think we have just swept the problem under the rug.

  • How can we postulate anything, such as Euclid’s parallel postulate, or Lobachevski’s parallel postulate? Mathematics cannot tell us, and I think that is why it, along with the other arts (τέχναι), corresponds only to the third (or the second, if one counts from the other end) of the four sections of the Divided Line in Plato’s Republic, examined by me last fall.

Paradox has never been contained in the comfortable category of warning. When Pascal took it head-on, he sensed reason’s overthrow. But did he panic too soon, forgetting his own deepest insight: “since the opposite principles are also true”? Instead of reeling to the creed-wager, he might have run into the street crying the Eureka of Paradox: “The leap which is our calling.” So stripped for action, would he not have seen the Probability  of an Odyssean METHOD, such a calculus of symbolic fields as Leibnitz would glimpse in his youth and almost articulate in the Monadology—though [5] hampered by the Axiom of Separation which would “pre-fore-or-destinate” his Harmony?

The Monadologie (the Monadology) of Leibniz begins where the Elements does, Euclid’s point having become a monade:

  1. La Monade, dont nous parlerons ici, n’est autre chose qu’une substance simple, qui entre dans les composés ; simple, c’est-à-dire sans parties.
    The Monad, of which we shall here speak, is nothing but a simple substance, which enters into compounds. By ‘simple’ is meant ‘without parts.’

Meanwhile Bell has referred, as he will again, to the following fragment of the Pensées of Pascal:

Tous leurs principes sont vrais, des pyrrhoniens, des stoïques, des athées, etc., mais leurs conclusions sont fausses, parce que les principes opposés sont vrais aussi.
All their principles are true, the skeptics’, the stoics’, the atheists’, and so forth; but their conclusions are false, because the opposite principles are also true.

This is number 22 of the 37 fragments of Dossier II of the Pensées diverses and is numbered by Sellier 512, by Lafuma 619, by Brunschwicg 394. I don’t know what Bell means by an “Odyssean method,” but Collingwood’s 1933 Essay on Philosophical Method would seem to be a head-on response to the paradox that Pascal recognized.

For Collingwood, the paradox is the “overlap of classes,” named on page 31 of the Essay:

The specific classes of a philosophical genus do not exclude one another, they overlap one another … the overlap of classes is to serve as a clue to discovering the peculiarities that distinguish philosophical thought from scientific … although it is familiar in philosophy and also … to common sense, it is a paradox from the point of view of science

Collingwood resolves the paradox with the “scale of forms.” This assessment is prompted by James Connelly and Giuseppina D’Oro, who, as the editors of the 2005 edition of Collingwood’s Essay, report on the “paradox of analysis” as what Collingwood resolves, or avoids:

A further merit of Collingwood’s methodological approach, according to Michael Beaney, is that it avoids the so-called ‘paradox of analysis’, a problem which he identified several years before the phrase was coined. The paradox is that either the analysandum is the same in meaning as the analysans or it is different. In the first case the analysis is true but trivial; in the second it is interesting and informative but false. From this it would seem to follow that an analysis cannot be both correct and informative. Collingwood’s solution lay in his conception of the scale of forms of progressively more adequate and comprehensive knowledge.

Beaney’s article “Analysis” in the Stanford Encyclopedia of Philosophy mentions Collingwood and the connection of the paradox of analysis to the paradox of Meno (which Bell will mention later):

Phenomenology is not the only source of analytic methodologies outside those of the analytic tradition. Mention might be made here, too, of R. G. Collingwood, working within the tradition of British idealism, which was still a powerful force prior to the Second World War. In his Essay on Philosophical Method (1933), for example, he criticizes Moorean philosophy, and develops his own response to what is essentially the paradox of analysis (concerning how an analysis can be both correct and informative), which he recognizes as having its root in Meno’s paradox. In his Essay on Metaphysics (1940), he puts forward his own conception of metaphysical analysis, in direct response to what he perceived as the mistaken repudiation of metaphysics by the logical positivists. Metaphysical analysis is characterized here as the detection of ‘absolute presuppositions’, which are taken as underlying and shaping the various conceptual practices that can be identified in the history of philosophy and science.

I have written a lot in this blog about absolute presuppositions, perhaps most extensively in “Nature.” It is a question whether it is they that Bell means by axioms.

Meanwhile, in his “Analysis” article, I think Beaney could have mentioned also The Principles of Art (1938), where Collingwood names the components of what is effectively the paradox of analysis. First comes “the grammatical transformation of language,” whereby language is conceived as being cut up into words, which are grouped into sentences. Next comes the logical transformation of language:

  • Some sentences are propositions, which make statements: this is the “propositional assumption.”
  • One sentence may be interchangeable with another, or with a group of them; this is the “principle of homolingual translation.”
  • Of two interchangeable sentences or groups of them, one may have the feature of “logical preferability.”

We are in § 8 (of nine), “The Logical Analysis of Language,” of Chapter XI (of fifteen), “Language,” which belongs to Book II (of three), “The Theory of Imagination.” Says Collingwood,

Like the grammarian’s modification of language, the logician’s modification of it can be to a certain extent carried out. But it can never be carried out in its entirety. When the attempt is made to do this, what happens is that language is subjected to a strain tending to pull it apart into two quite different things, language proper and symbolism. If the division could be completed, the result would be the state of things which Dr. Richards is presumably trying to describe when he distinguishes ‘the two uses of language’ … according to Dr. Richards there is a complete division between the ‘scientific use of language’, i.e. its use for the making of statements, true or false, and a purely aesthetic quasi-musical ‘use’ which is the extreme case of what he calls the ‘emotive use’, i.e. its use to evoke emotion …

Is this distinction a real one, or is it only a statement of the two forces between which a tension, but a not altogether disruptive tension, is introduced into language by the attempt to intellectualize it? I shall try to show that the latter alternative is the truth; that language intellectualized by the work of grammar and logic is never more than partially intellectualized, and that it retains its function as language only in so far as the intellectualization is incomplete.

Meanwhile, here is how Collingwood himself describes the scale of forms, on pages 57–8 of An Essay on Philosophical Method (as in Bell’s essay proper, the bullet points are mine, as is the bolding, above and now):

The combination of differences in degree with differences in kind implies that a generic concept is specified in a somewhat peculiar way. The species into which it is divided are so related that each

  • not only embodies the generic essence in a specific manner,
  • but also embodies some variable attribute in a specific degree …

whenever the variable, increasing or decreasing, reaches certain critical points on the scale, one specific form disappears and is replaced by another.

  • A breaking strain,
  • a freezing-point,
  • a minimum taxable income,

are examples of such critical points on a scale of degrees where a new specific form suddenly comes into being. A system of this kind I propose to call a scale of forms.

It is a conception with a long history in philosophical thought. It is a favourite with Plato, who has various scales or attempts at

  • a scale of the forms of knowledge:
    • nescience, opinion, knowledge;
    • conjecture, opinion, understanding, reason;
    • poetry, mathematics, dialectic;
  • a scale of the forms of being, from nothing through half-being to true being;
  • scales of the forms of pleasure,
    • those of the body and those of the soul, the latter more truly pleasures than the former, or
    • the impure and the pure, or
    • a gradation from pain through quiescence to true pleasure;
  • scales of the forms of political constitutions,

and so on almost endlessly …

Nor is it a conviction peculiar to Plato …

  • Aristotle recognizes the same type of logical structure, for example when he distinguishes the vegetable, animal, and human ‘souls’ as three forms of life arranged on a scale so that each includes its predecessor and adds to it something new.
  • Locke classifies his main types of knowledge explicitly into ‘degrees’.
  • Leibniz attempted to make of it a central principle of philosophical method, as the law of continuity.
  • Kant, whether through the influence of Leibniz or rebus ipsis dictantibus, reverts to it again and again, even at the cost of apparent or real inconsistency …
  • The positivists and evolutionists of the nineteenth century were no less emphatic in their belief that knowledge was specified into grades on a scale of abstraction, and nature into specific forms on a scale of development.

Bell will allude later to the “law of continuity” that Collingwood mentions. Leibniz would seem to express it in the Monadology thus:

  1. Je prends aussi pour accordé que tout être créé est sujet au changement, et par conséquent la Monade créée aussi, et même que ce changement est continuel dans chacune.
    I assume also as admitted that every created being, and consequently the created Monad, is subject to change, and further that this change is continuous in each.

As for the “axiom of separation” that Bell mentions, Zermelo introduced an axiom of that name in 1908, by way of resolving the Russell Paradox. I quote Zermelo from “Investigations in the foundations of set theory I” in From Frege to Gödel A Source Book in Mathematical Logic, 1879-1931, edited by Jean van Heijenoort:

AXIOM III. (Axiom of separation [Axiom der Aussonderung].) Whenever the propositional function 𝕰(x) is definite for all elements of a set M, M possesses a subset M𝕰 containing as elements precisely those elements x of M for which 𝕰(x) is true.

Perhaps Bell does not have this in mind. Later he will mention “Leonardo’s axiom of Separation, that no spirit can operate on matter.” Meanwhile, above, his formation “pre-fore-or-destinate” would seem to combine “predestine,” “foreordain,” and perhaps “destination.”

AXIOMS AND FIELDS:

What is clearest (and least clarified) about axioms, those inner necessities of thought, Aristotle’s famous first principles, common to all, which do not require or admit of proof (though Kant seems to be proving them), is that they tend to come in polar pairs, forcing us to the question: are such certainties

  • Categorical,
  • Hypothetical (if such-and-such, then so-and-so), or
  • Disjunctive (less Kant’s either-or, than a complementary yes-no)?

It would be strange to say “all in one”; since the categorical excludes if  and no, and how have first principles proclaimed themselves but categorically?

Bell’s axioms, Aristotle’s first principles, would seem to be the absolute presuppositions of Collingwood, who writes in An Essay on Metaphysics (page 61),

Aristotle knew well enough that the science he was creating was a science of absolute presuppositions, and the text of his Metaphysics bears abundant witness to the firmness with which he kept this in mind and the perspicacity with which he realized its implications; but Aristotle is also responsible for having initiated the barren search after a science of pure being, and for the suggestion that a science of pure being and a science of absolute presuppositions were one and the same. This perplexity has never been overcome.

In 1938–9, Bell was a Rhodes Scholar at Oxford, but Collingwood was away from October to April, cruising to and from the Dutch East Indies, writing the words above and the following, from pages 75–6:

A civilization does not work out its own details by a kind of static logic in which every detail exemplifies in its own way one and the same formula. It works itself out by a dynamic logic in which different and at first sight incompatible formulae somehow contrive a precarious coexistence …

The same characteristic will certainly be found in any constellation of absolute presuppositions; and a metaphysician who comes to his subject from a general grounding in history will know that he must look for it. He will expect the various presuppositions he is studying to be consupponible only under pressure, the constellation being subject to certain strains and kept together by dint of a certain compromise or mutual toleration having behind it a motive like that which causes parties to unite in the face of an enemy. This is why the conception of metaphysics as a ‘deductive’ science is not only an error but a pernicious error …

At a liberal arts college there was a soulful tutor who got so wrapped up in Plotinus—

From such a unity as we have declared the One to be, how does anything at all come into substantial existence, any multiplicity, dyad, or number? (and) Why should the One overflow?

he used to meet his math class with a look beyond infinity. “Now let us contemplate the One” he would say, and drop his face into his hands. For ten or fifteen minutes, silence would reign. Before long the students went to the Dean. Even at a Platonic academe, they wanted a stir of becoming. Special as that tutor was, he had to move on; which showed [6] how essential it is for the One (however incomprehensibly) to overflow.

Suppose we drop our heads into our hands and contemplate the oldest and deepest assurance mind gives us—what we may call the “Eleatic Axiom”—of the ultimate rational identity of substance, prototype for Plotinus, for causality, for mathematical equation: that Being is what it is. Parmenides of Elea:

It is necessary both to say and to think that Being is; for … it is impossible that non-being is … And … Being is without beginning and indestructible; it is universal, existing alone, immovable and without end … From what did it grow, and how? … If it came into existence, it is not Being … So its generation is extinguished, and its destruction is proved incredible … Further it is unchanged … for if it lacked anything it would lack everything …

Nor is there nor will there by anything apart from Being … Wherefore all these things which mortals determined in the belief that they were true will be but a name: that things arise and perish, that they are and are not, that they change their position and vary in colour …

If this is an axiom of mind (and there is no more venerable one), its eternizing truth evacuates the actual. When the One is drunk neat (Substance identified with the cosmos itself), its logic is at peak, its practicality almost nil. Aristotle: “Though these opinions seem to follow in a dialectical discussion, yet to believe them seems next door to madness when one considers the facts.” That Eleatic axiom takes no account of particularity or change, of incarnate becoming. In absolute verity and total [7] distortion, it must cast, like a shadow, some existential contrary—why not the most obvious and neglected rule of Philosophy? “If a thing is, it is possible.” How far from the Eleatic “by the logic of Being, change cannot be” is this chessboard advance of the merely factual pawn: “Since change exists it can be.” (Sartre of the Existentialists: “They think that existence precedes essence”.)

Simple as Aristotle’s other conclusion in the Posterior Analytics, when confronted with the Meno paradox of learning (to seek for what you have no notion of), he traces First Principles up an organic stair from animal Perception through Memory to the universals of Science—soaring to his greatest (perhaps his only) simile:

It is like a rout in battle stopped by first one man making a stand and then another, until the original formation  has been restored.

That wonderful bow to Plato’s Recollection (“the original formation”) suspends the organic climb over a priori Paradox. To which Aristotle’s answer is acceptance of the field (as if he had said “if a thing is, it is possible”): “The soul”, he writes, “is so constituted as to be capable of this process.”

IDEAS AND CAUSALITY

How could reason after Parmenides (or even in Parmenides himself) do anything but slack off from the Eleatic absolute? Yet—as reason—it has still to [8] reconstitute substantial identity over factual denial. The Doctrine of Ideas is one such recovery. For Socrates and Plato the dilemma of Particularity is unavoidable: that Being, contrary to its charter, appears subdivided all over the place. Subsume it, then, under the intelligible species of eternity by giving each configuration, from table and man to beauty and good, its immortal Idea. Though for that recovery of the lost One under the alienation of many, the original question continually revives: how can the primary and changeless have any connection with the fleeting and relative? As Socrates confessed when asked in the Parmenides whether dirt and dung have Forms: “Sometimes I think there is nothing without an idea … and then I think I may fall into a bottomless pit of nonsense and perish.”

Suppose now the focus should shift from the spatial  adversary of Being, Particularity, to the temporal  adversary, to that Flux which Socrates himself rapturously parodies in the Theaetetus :

in the language of nature all things are being created and destroyed, coming into being and passing into new forms; nor can any name fix or detain them … O Theaetetus, are not these speculations sweet as honey?

Must not Eleatic Being, fished up from the river of change, take the form of Causality, the axiom of sufficient reason: [9] that what was, bears the causal identity of what is and will be?

Leibniz states the principe de raison suffisante in the Monadology:

  1. Nos raisonnements sont fondés sur deux grands principes, celui de la contradiction, en vertu duquel nous jugeons faux ce qui en enveloppe, et vrai ce qui est opposé ou contradictoire au faux.
    Our reasonings are grounded upon two great principles, that of contradiction, in virtue of which we judge false that which involves a contradiction, and true that which is opposed or contradictory to the false;

  2. Et celui de la raison suffisante, en vertu duquel nous considérons qu’aucun fait ne saurait se trouver vrai ou existant, aucune énonciation véritable, sans qu’il y ait une raison suffisante, pourquoi il en soit ainsi et non pas autrement, quoique ces raisons le plus souvent ne puissent point nous être connues.
    And that of sufficient reason, in virtue of which we hold that there can be no fact real or existing, no statement true, unless there be a sufficient reason, why it should be so and not otherwise, although these reasons usually cannot be known by us.

Such is Descartes’ proof of God:

since I am a thinking thing, and possess in myself an idea of God, whatever … be the cause of my existence, it must of necessity be admitted that it is likewise a thinking being, and that it possesses in itself the idea and all the perfections I attribute to Deity … since what is cannot be produced by what is not.

It is rabbits out of a hat—that if you pull them out, there must have been rabbits in there. So God can be proved a rabbit, or an eternal rabbit-thinking fool. Though for all the causal axiom, Darwin pulls rabbits out of some hat or other, where no rabbits were before. It seems evolution can’t have worked by the logic of equation. So the causal axiom also requires its existential contrary—operation bootstraps, a conditioned coming-to-be of what was not—complement of categorical causality: The Axiom of Emergence:—For time to be real (Kierkegaard’s moment … decisive), the future cannot be contained in predictive causality. [10]

SUBSTANCE AND COMPLEMENTARITY:

It was in fact just before Parmenides, that Heraclitus had celebrated the counterpole of Eleatic Being—Flux:

  • The way up and the way down is the same.
  • Fire lives in the death of air and air lives in the death of fire; water lives in the death of earth, earth in that of water …
  • We do not step twice in the same rivers, for other waters are always flowing on: we are and we are not.

I identified the third and first fragments earlier. The second fragment is apparently found in Maximus of Tyre; it is Bywater XXV. Bell’s translation is nearly the translation of Jones, as included in volume IV of the Loeb edition of Hippocrates; according to this version, there are are two interchanges,

  • of fire and air, and
  • of water and earth.

However, the fragment is also the first part of Diels B76 and is given there as

ζῆι πῦρ τὸν γῆς θάνατον καὶ ἀὴρ ζῆι τὸν πυρὸς θάνατον, ὕδωρ ζῆι τὸν ἀέρος θάνατον, γῆ τὸν ὕδατος.

This is the version that Guy Davenport translates, in Herakleitos and Diogenes (San Francisco: Grey Fox Press, 1990), as

The life of fire comes from the death of earth. The life of air comes from the death of fire. The life of water comes from the death of air. The life of earth comes from the death of water.

Here there is a single cycle, earth to fire to air to water and back to earth. Jones has a note on the difference:

In the texts ἀέρος and γῆς are transposed. Diels reads as above; Bywater retains the old order.

The Diels reading that Jones is talking about must be the one in a footnote, explained in German; for Jones’s translation reflects his own text,

ζῇ πῦρ τὸν ἀέρος θάνατον, καὶ ἀὴρ ζῇ τὸν πυρὸς θάνατον· ὕδωρ ζῇ τὸν γῆς θάνατον, γῆ τὸν ὕδατος.

In short, there is no agreement on what Heraclitus is talking about. Of the remaining two parts of Diels B76, the first is from Plutarch:

πυρὸς θάνατος ἀέρι γένεσις, καὶ ἀέρος θάνατος ὕδατι γένεσις.

It is this that is selected to represent Diels B76 in a Turkish edition, Herakleitos, Fragmanlar (Kabalcı Yayınları, 2009); the translation by Cengiz Çakmak is,

Ateşin ölümü havanın doğumudur; havanın ölümü suyun doğumudur,

that is, “Fire’s death is air’s birth; air’s death is water’s birth.”

The remaining part of Diels B76 is from Marcus Aurelius:

ὅτι γῆς θάνατος ὕδωρ γενέσθαι και ὕδατος θάνατος ἀέρα γενέσθαι καὶ ἀέρος πῦρ καὶ ἔμπαλιν.
that death for the earth is to become water, death for water is to become air, and for air, to become fire; and inversely.

This is the only part of B76 that Laks and Most include, and they include it as part of R54, which is a longer quotation from Marcus Aurelius.

Yet Heraclitus was not just a Flux man (his arrow of change: fire-to-air-to-water-to-earth—rounds, either way, to a wheel of fire):

Not on my authority, but on that of the truth (logos), it is wise for you to accept that all things are one.

This is Bywater I, Diels B50, Laks–Most D46. Diels includes the words of Hippolytus that accompany the quotation of Heraclitus.

He had hymned the sacred opposition of One and many, Being and becoming:

  • From what draws apart results the most beautiful harmony … as in the bow and the lyre.
  • All things take place by strife …
  • Changing (the all) finds rest. (Metaballon anapauetai .)

These are as follows.

  • Bywater XLV, Diels B51, Laks–Most D49, also paraphrased in R32, which is from Plato’s Symposium and corresponds pretty well with what Bell gives.

  • Part of Bywater LXII, Diels B80, Laks–Most D63

  • Diels B84a, Laks-Most D58, from Plotinus:

    Changing, it remains at rest.
    μεταβάλλον ἀναπαύεται.

The complementarity was not just of Parmenides against Heraclitus, or even of thought against world; since changeless One and transforming many are at once limits of the phenomenal and yoked cognitions of mind.

If the Categorical, however, precipitates its polar pair, it must be logically Disjunctive. Consider “The whole is equal to the sum of its parts.”

Who can deny it? But grind some old college table to sawdust. In what sense will the unarguable truth prove true? Or try a costlier experiment. Take a pair of [11] freshmen (not to discriminate, boy and girl) and dissolve them in acid. When the police come, tell them “The whole is equal to the sum of its parts.” If the watchdogs are clever, they will say: “Your axiom is disjunctive. It is true; but in those words of Pascal:

les conclusions sont fausses, parce que les principes opposes sont vrays aussy.

“No whole,” they will say, “in so far as it is whole, can be simply a sum of parts.” So they take you off, gloating as Dante’s devil did over Montefeltro: “Maybe you didn’t know I was a logician?”—“Forse tu non pensavi ch’io loico fossi!

We saw the Pascal earlier. The Dante is from Canto XXVII, lines 112–23, of the Inferno, here in the translation of Allen Mandelbaum (1980; Everyman’s Library, 1995):

  Then Francis came, as soon as I was dead,
for me; but one of the black cherubim
told him: ‘Don’t bear him off; do not cheat me.
  He must come down among my menials;
the counsel that he gave was fraudulent;
since then, I’ve kept close track, to snatch his scalp;
  one can’t absolve a man who’s not repented,
and no one can repent and will at once;
the law of contradiction won’t allow it.’
  O miserable me, for how I started
when he took hold of me and said: ‘Perhaps
you did not think that I was a logician!’

Does every axiom cast the shadow of its own antinomy?

Here, like Aristotle, we must trace axiomatic assurance back through nature and animal perception into the latency of a vanishing intuition. At that point it will appear that what seemed Categorical and then Disjunctive, must also be Hypothetical.

Since all a priori assurances spring from an archetypal root—the intuition of a medium of linear covariance, of a causal jell [sic] we may call Substance. It undergirds thought, perception, motion, even space-time, with the form of a homogeneously ordered and reliably continued symmetry—as if, swimming our incarnate breaststroke, we could depend on a fluid congruence, responsive alike to thrust and counterthrust—no space-pockets, no discontinuities, but [12] such measurable substantiality as inertial devices use to keep spaceships on course.

I don’t know what Bell has in mind for “linear covariance,” but he is about to give some evidence, and more after that.

WOLF AXIOMS: AN INTERLUDE

To exhibit how deep that ground of intuitive knowledge, I have set down the axioms of covariance as seen by wolf or coyote:—

  • “Is” means “is”, at least for a while.
  • If the signs are rabbit, it won’t turn to bear.
  • You know what you know all round. It may hide, it can’t disappear.
  • A half eaten rabbit is not a rabbit by half.
  • A push of the right sends you left; of the left symmetrically.
  • Plot your course; no sinkholes in space.
  • Rocks don’t move themselves; when things move, hunt for why.
  • If wind moves itself, bay the wind.
  • One and one are one and one—better but harder to catch.
  • You know straight from crooked, if you don’t know why.
  • If you roll over, you’re upside down.
  • If you jump you fall.
  • If you run forward, you keep going.
  • If you hit something there’s a jolt—the bigger the bigger, the faster the bigger still. [13]
  • Pant for heat, drink for thirst.
  • Dead wolf, cold wolf; no hot turd.
  • Wolf on wolf makes little wolves.
  • Did big wolf begin it? Bay the unknowables: fire, wind, sun, moon, sky.

Though I never heard wolf, coyote, or dog announce these truths, I am sure they couldn’t operate without such awareness. What is it but of Substance: a consistently ordered and reliably continued linear symmetry—yet of something also beyond that, something mysterious to be bayed?

We humans do many things unawares. We find this out when we worry that we haven’t done them—closed the windows and locked the door for example. We go back and check, and lo and behold, we must have done them, though we cannot remember.

Nonetheless, if Bell thinks we cannot operate by truths that we are not aware of, perhaps he would agree with Margaret Wertheim:

I’d like to propose that sea slugs and electrons, and many other modest natural systems, are engaged in what we might call the performance of mathematics. Rather than thinking about maths, they are doing it.

This is in “How to play mathematics” (Aeon, February 7, 2017). By Wertheim’s account, a nudibranch does mathematics by being a model of hyperbolic plane geometry; and she reports,

Schrödinger equations … are so complicated that, when Feynman was alive, the best supercomputers could barely simulate even the simplest orbits. So how could a brainless electron be effortlessly doing what it was doing? Feynman wondered if an electron was calculating its Schrödinger equation.

To this way of thinking, it seems to me, all of nature does mathematics, simply by respecting the laws of physics, which are mathematical. But there’s a big difference.

  • Nature cannot break the laws of physics; otherwise they would not be laws.
  • Humans do break human laws; otherwise there would be no judicial system.

Likewise, nature cannot get mathematics wrong; and yet a nudibranch will not undertake the task of showing how to construct a hyperbolic triangle with three given angles whose sum is less than two right angles. A human may undertake this task and fail.

We may judge that animals undertake some tasks, at which they sometimes fail. Perhaps Bell is thinking on this level. I do not recall seeing coyotes in New Mexico, though I may have heard them. I have seen a cat misjudge its ability to leap from the floor to a countertop.

I assume amoeba hunting paramecium has like assurances. Evolution must have actualized these frames of sensibility, which Emanuel Kant reflected back on the world. Maxwell seems shrewder. “The only laws of matter,” he writes, “are those which our minds must fabricate” (that’s Kant); “and the only laws of mind are fabricated for it by matter” (that’s a bolder Darwin).

The Maxwell quote is popular, but as usual, the quote sites don’t source it. Wikiquote is the honorable exception, and there I found a link to the source article, Maxwell’s “Analogies in Nature.” Bell’s selection comes at the end, but I do not see how the article justifies it.

Nudibranchs cannot teach us hyperbolic geometry, not in the sense of showing us how theorems follow certain axioms. We have to do that ourselves. We fabricate the understanding whereby a nudibranch models a hyperbolic plane.

Being made of matter, the brain follows the laws of matter; but the laws of the mind would seem to be the laws of logic.

I do not really understand Maxwell’s article. The author seems to start out sensibly, saying for example

… no question exists as to the possibility of an analogy without a mind to recognise it—that is rank nonsense. You might as well talk of a demonstration or refutation existing unconditionally. Neither is there any question as to the occurrence of analogies to our minds.

The following may be a fair account of the development of the notion of causation.

We cannot, however, think any set of thoughts without conceiving of them as depending on reasons. These reasons, when spoken of with relation to objects, get the name of causes, which are reasons, analogically referred to objects instead of thoughts. When the objects are mechanical, or are considered in a mechanical point of view, the causes are still more strictly defined, and are called forces.

Some thoughts give us reason to think others. By a kind of anthropomorphism, we detect such influences out in nature, now calling them “causes” or, in the refined setting of mathematical physics, “forces.” This seems to be what Maxwell is saying; but it must not be, since it yields a conclusion contrary to the one Maxwell goes on to draw:

When we consider voluntary actions in general, we think we see causes acting like forces on the willing being … Some had supposed that in will they had found the only true cause, and that all physical causes are only apparent. I need not say that this doctrine is exploded.

I was satisfied that the doctrine had been re-established, by Collingwood if nobody else!

THE GENERATION OF AXIOMS

All axiomatic absolutes, all trusted first principles, arise when the intuition of some such pre-cognitive ground is applied at different levels of impacted and abstracted experience. Thus the categorical becomes hypothetical, its truth bearing a conditionality mostly ignored:—If this, then that: if the covariance of substance be posited of [14] such and such realm, such and such common notions and deductive proofs must follow. Nothing, however, is assured about the fitness of the intuition or of its generated postulates to any actual field. How should so conditioned a certainty not promote dialectical polyvalences?

I suppose Euclid then shows the passage from categorical to hypothetical, or at least the beginning of the passage. His postulates give us a toolbox containing stylus, ruler, compasses, and set square. The compasses are collapsing, and will not hold an angle. The set square cannot be used for drawing, but only confirms that all right angles are equal, wherever the set square can be carried. When one straight line crosses two others, we can also use the set square to check whether the interior angles on the same side are together less then two right angles. If they are, then we can use the ruler and stylus to extend the two lines on that side until they intersect. Even if we have no more, then we can establish the traditional categorical truths of geometry: Euclid shows this. By the time of Lobachevsky, we understand that a different toolbox can give us different truths. Eventually somebody figures that the nudibranchs knew this all along.

A) Of Quantity

Of the great scientific extensions, let the first (and least empirical) be to quantity  itself. All the premises of magnitude and number, equals to equals, ratio, the commutative, associative and distributive laws of arithmetic, crop out inevitably; though of course their relevance to a world of organized material stands under probability and contradiction—Einstein’s “As far as the Propositions of mathematics refer to reality, they are not certain, and insofar as they are certain they do not refer to reality.”

We have already toyed with the irreversible reduction of whole students to the pseudo-identity of molecular parts. And what could show better than the physics we are to dramatize, the eviction of perception and response from a universe stripped to the determinants of equational cause?

B) Of Geometry

The second application takes in dimensionality, also deeply bedded in experience. But what experience of curved images on a curved retina, where spherical perspective turns receding train rails to a lens-shape meeting on the horizon [15] before and behind—what experience ever showed anybody a straight line? Here too there is an unexpressed condition: If space conform to the linear homogeneity of what we have called Substance, Euclidean Geometry, its definitions, postulates and propositions ensue, mere spellings out of unbiased dimensionality.

This is not to deny that paradoxes (as between points and line) will lurk under Euclidean clarity—as in Aeschylus the Chthonic Furies are enshrined under Athena’s Acropolis. Nor is it guaranteed that such a geometry will hold at high velocities, or in the force-fields of nuclei, dwarf stars, Black Holes—will hold, indeed, anywhere absolutely. Yet those very anomalies must be assimilated to reason by applying again, under transformation, the consistencies of covariance. Thus Kepler would bring circular eccentricities into the concord of the ellipse.

What can be claimed is this: that

  • if we apply the intuition of homogeneous symmetry to a dimensionality conceived as linear, we will come up with Euclidean Geometry.
  • If the homogeneity is conceived as of uniform negative curvature, the issue will be the geometry of Lobachevsky;
  • if spherically or elliptically positive, that of Riemann;
  • if as involved with time in hyperbolic curvature, it will be that of Einstein out of Minkowsky.

In each case the formulabilities of covariance are reinvoked under the perturbation of deeper denial. [16]

C) Of Logic

The third classical application is to the language of concept. Its hypothesis is this: if expressed thought is a predicational ordering of isolable and equationally reliable units (which nothing expressed in words has ever been; but if expression should be that predicational ordering of integral concepts around which you can draw circles and pretend “this does not merge with that”; so that virtue’s being knowledge precludes its being habit)—then Aristotelian logic, with its laws of non-contradiction and the excluded middle, necessarily results (since if the terms are reliable, is  precludes is not  and yes  does not shade into no).

To which, memory opposes a Mississippi high school teacher, Miss Hawkins (called Hawkeye) who to every searching question would answer: “Well, yes, and then again, no.” In the end she was wiser. Let Descartes exhibit the risk of logical inference: (from the Method)

In the act of thinking all was false, it came to me as necessary that I, who did the thinking, must be real. Thus observing that the truth: I think, therefore I am, was so certain that no extravagance of the skeptics could ever shake it, I took it boldly for the first principle of the Philosophy I was in search of … I thence concluded that I was a substance whose whole essence or nature consists only in thinking, and which, that it may exist, has need of no place, nor is dependent on any material thing …

(which Pascal postscripts: “Shall I doubt that I doubt?”) [17]

Descartes’ certainty (of a similar proof) is stressed in the Meditations:

I consider the demonstrations of which I here make use to be equal or even superior to the geometrical in certitude and evidence.

Let us call them equal, since their certitude rest [sic] on the same conditionality: “If the properties of Substance apply, this follows.” But the if  here seems iffier—stretched from the old ratios of embodied act to the quickenings of symbolic thought: if mind, world, God are formulable, without paradox, in the equational logic of words, then Cartesian doubt, equated with Thought, may draw from the Eleatic axiom self as eternal thinking substance, world as formulable and extended substance, both secured by God as infinite substance and eternal rational cause—all with the unanswerability of Euclid, so long as the operations are not held to the sliding ambiguities of any actual self, doubt, or world. But how can the axiom of substance be applied to a consciousness born in time and subject to the fluctuation of doubt? That “I ” is too amorphous not to be put in quotation marks. So too the word “think”, of which the immediate evidence was doubt. You’d as soon say: “I nightmare, therefore I am”—and prove by the axiom of sufficient cause that God is a damned nightmarer. And what of the copular of Being, “I am”? Its eternity crowns dim adumbration. It also has to go into quotation marks. Is Descartes’ “geometric certitude” a shrewd hunch of some [18] conscious activity? The truth is more radical, both better and worse than that. What Descartes has exhibited is the indisputable pole of Eleatic “I am”, between which and the equally indisputable contrary of sceptic nothingness, thought and experience hang. It is Hamlet’s “paragon of animals … quintessence of dust”, Pascal’s “gloire et rebut”—“pride and garbage of the universe.”

D) Of Physics

After number, geometry and logic, the fourth extension of Eleatic reason is to matter in motion. When the motions, actual and potential, of gravitating bodies are treated as a quantity which, by the axiom of substance, is continuously preserved, modern physics results. Poincaré objects that the conservation of motion can hardly be a priori if the Greeks failed to see it; and surely Aristotle chose conservation of place, by which mass tries to stand still. But place is not a quantity; and here, as before, the axiom of substance, conquering a new realm, has let the a priori evolve in time.

It has been common to call the new science empirical—Leonardo’s “experience, true mistress”, Galileo’s “brute fact” after what Whitehead calls the “unbridled rationalism” of the Medieval. But Leonardo continues: “Experience always proceeds from accurately determined first principles … Understand the cause and you will have no need of the experiment.” From Galileo and Descartes to Newton [19] and Leibnitz, everybody’s first principle is the same. It is in the inborn axiom of Being, applied now to some assumed “quantity of motion”. Descartes: “That God is the first cause of motion, and that He always conserves an equal quantity of it in the universe.” The advance of the century was chiefly in mathematizing the variants (velocity and acceleration, momentum, force, energy) of that ubiquitous Motion. And what made those seekers into nature vulnerable to the backlash of the a priori, was not that they looked too much at matter and too little’s [sic] at thought’s necessity; but that they narrowed thought too much—taking the one-way quantifications of Substance for unambiguous truth.

That the closed-system logic they employed could only formulate the inert, did not prove the cosmos inert. When they found at last that energy itself must subside to a stagnant pool, what had the computer done but to grind out the problem it had been fed; while the programmers took its printout for the starry universe?

FROM LEONARDO TO LA PLACE:

Leonardo had praised the ambiguity of force, animistic mover of inert mass:

Force I define as an incorporeal agency, an invisible power, which … constrains all created things to change of form and position … It is born in violence and dies in liberty; and the [20] greater it is the more quickly it is consumed … It desires to conquer and slay the cause of opposition, and in conquering destroys itself … Without force nothing moves.

In this live enigma, the First and Second laws of Thermodynamics are precursively fused.

With what clarity of “divide and conquer” Newton’s three laws have rationalized Leonardo’s mystery.

Law I:
Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by force impressed upon it.
Law II:
The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
Law III:
To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts—

Yet nature will creep where it cannot go. Laired under these three laws are the denial of causality and the paradox of inertia:

  • first, of causality: that motive force  cannot come into being but as validated by the inertial counterforce of acceleration, its assumed result; so cause withdraws from physics into the consciousness which has willed “I’m going to move that rock”;
  • second of inertia: that mass resists acceleration only by accelerating (as: “I resist sin by sinning”). [21]

Meanwhile even Leonardo, in that pre-dawn of rational consistency, had not only hymned the riddle of force, but had set the traps of material cause. When he wrote:

a spirit cannot have either sound or form or force … where there are neither nerves nor bones there cannot be any force exerted in any movement made by imaginary spirits—

how could he not know he had turned the universe to Leibnitz’ Mill, into which, entering, we

would find only pieces working one on the other, but never anything to explain Perception?

Had not the whole issue been debated by classical atomists: Epicurus, finding in the determinism of Democritus no loophole for what we experience as volition, and so, to give a ground for the choice on which philosophy itself was reared, introducing atomic swerve, each atom, as Lucretius would take it up, making its clinamen “at no fixed places and at no fixed time”—a margin which quantum indeterminacy would unwittingly restore. But it was exactly such metaphysical speculation those new men of science wanted to avoid: Newton, in the Principia, Preface—

I wish we could derive the rest of the phenomena of Nature by the same kind of reasoning from mechanical principles—

fathering, for all its author’s theological scruples, the determinism of La Place:

to comprehend in one formula the movements of the largest bodies … as of the minutest atom … nothing uncertain, … past, present, or future—

[22] against which Blake was already in rebellion: “Without contraries is no progression …” and: “The same dull round even of a universe would soon become a mill with complicated wheels.”

THE DIPOLE OF ACTION:

The truth is, those Promethean conquerors of nature—even as they trusted the one-way application of Eleatic Substance to matter and motion—had turned by choice or necessity from the axiomatic antinomies their genius could so easily have formulated: those extensions of the Greek One-many, that

  • nothing can be conceived but as one, nothing but as composed and modified in space and time;
  • nothing but as substance, nothing but as accident;
  • nothing but as absolute, nothing but as relative; or (the root of quantum ambivalence)
  • nothing but as particle, nothing but as field (so that even impact, magnified, becomes also action at a distance).

It had been clear to the Renaissance speculators that some power had to be invoked to stir up matter, long conceived as inert. What was not clear was that mind was grappling with a metaphysical dipole which physics could not escape, the opposed axioms of cause, springing from

  • the One  as motionless, and
  • the activating other  as disparate, unable to contact the One.

That is the reversing field in which [23] Dynamics would be whirled. As I phrased it almost 50 years ago (though no one has paid much mind):

  1. The one  cannot stir itself (so in Parmenides, Being comes to rest).

  2. If the dual is invoked for action (Heraclitus “all things take place by strife”), then contraries cannot interact, insofar as they are contraries. (It is Leonardo’s axiom of Separation, that no spirit can operate on matter).

If we try to bridge the dipole (as nature does such complements: my mind voicing these words, my will stirring this hand), then the I that both thinks and moves, knows by the logic it also contains that such a self-opposed and self-activating unity lives in the mystery of paradox. The fact that experience everywhere attests such powers, does not make them less recalcitrant to quantitative method.

For the first dictum of the di-pole, we have already cited Parmenides, with Aristotle’s “the substratum cannot cause itself to move”. It was the second which led Plato and the whole late classical world to deny the commerce of God with man: (Symposium)

For God mingles not with man; but through Love all the intercourse and converse of God with man … is carried on …

(as if a middle term could solve such an antinomy). And though the God-man double was joined in Christian faith, it would cleave again with the revival of Classical reason. Leonardo’s powerless spirits point to the Cartesian separation of extended matter and non-extended mind, which [24] Descartes’ caprice of the pineal gland could hardly fuse. In Leibnitz the logical denial of a communion obviously experienced—

For it is not possible to conceive how one (soul) can have an influence on the other (body)—

necessitates the God-ordained harmony of mechanist mill and windowless monads. The same impossibility led Berkely [sic] to a monism in which esse is  percipe—all externals thinned to ideas in the mind of God.

Of course this solution, by the first law of the dipole, must produce an actionless universe. The inference could have been dodged by calling God a transrational mystery, both one and many, at peace and at war with himself, but that is just what Berkeley’s 18th century reason will not admit. “Impossible,” he writes, “even for an infinite mind to reconcile contradictions”—oblivious that his rational and Eleatic One, like energy later, can only fall into the mill and determinism of Blake’s “dull round”.

As for the resolution of the di-pole, if physicist, astronomer, biologist (unwittingly) should make energy that tensile one, both substratum and elan vital [sic], it will turn out indistinguishable from spirit, the purposive paradox crowning a phenomenal field. Leibnitz and Pascal went further than anyone of the time (almost than anyone since) in suspending complementarity,—Pascal driven to vertigo, [25] Leibnitz on the verge of Method. As voiced near the beginning of the Monadology:

The passing state, which involves and represents a multitude in unity or in the simple substance, is nothing else than what is called perception;

and toward the close:

According to this system bodies act as if (to suppose the impossible) there were no souls, and souls act as if there were no bodies, and yet both body and soul act as if each were influencing the other.

Though this vital mystery seems to defy the claimed first law of Leibnitz’ logic: “that whatever implies a contradiction is false.” Since what else but radiant contradictions are his multitude in the unity, transience in the eternal, trivalence of physics, spirit and biology?

How prophetic the 17th century might have been if, instead of mouthing classical logic, it had applied the polar suspensions of physics to metaphysics as well. All those scientists were skillful at balancing opposite trends (as inertia and gravity) where exact solution required it. They knew that without force and counterforce no real action, elastic limit or material boundary can exist. What follows? If an object so bounded were a single substance, it would be transrational, a counter-tensile one (Lucretius’ atoms axiomatically mystical); so every raindrop, sun, galaxy, is to be conceived under the dialectic of one-many (in which nature happily concurs): the sun as one matter is drawn in; as particles accelerated by that fall together, it [26] thrusts out. But now every formed thing, down to the particles called primary, must be dissolved in its turn, or become (like the damped-wave model of the electron) some quantum mystery, subsuming paradox and therefore indeterminate. In the 17th century only Pascal—

In this abridged atom … an infinity of universes … without end and without cessation—

and Leibnitz—

organic bodies are still machines in their smallest parts, ad infinitum

intuited that bottomless universe—Leibnitz delighted, Pascal clutching for the handrail. If Leibnitz, skilled in infinites and infinitesimals, had pursued Pascal’s contraries all the way, would not matter and energy have been seen from the first as suspended over entropic decay by anti-entropic buttressing, without which no form could resist the chute of imperative nothingness?

FALLING BODIES:

When we raised the first of our steel balls and let it descend along the arc of a pendulum, we accepted its motion, without asking what the axiom of Substance implied of falling bodies. In this case Leonardo had held to the old view:

Where there is the greater weight there is the greater desire, and that thing which weighs the most, if it is left free falls most rapidly …

[27] Though thought, apart from observation could have told Leonardo: every unit of weight is not only pulled by an equal desire but restrained by an equal inertia. So for Galileo to state the new principle of constant acceleration did not depend on dropping bodies off the leaning tower of Pisa (the results, using cotton and lead, might have been misleading); mind had only to grasp what the axiom of Substance requires—a classical continuity the opposite of quantum a-causality.

Conceive two like bodies. Dropped separately they should fall (the Aristotelians said) with a speed (they meant acceleration) proportional to their weight. But what if we bring them gradually together until they are in contact, and then glue them? At some indeterminate moment the new body, having twice the weight, would have to fall twice as fast. How is it to know when? The inert continuity of equation is defied. So bodies in constant force fields can only accelerate equally, and uniformly, whether one or many, heavy or light.

Bell does not say who the “Aristotelians” are, but the closest to what Bell attributes to them that I have found in the works of Aristotle himself is in Book IV, Chapter 8, of the Physics, Bekker 215a25–9. In the old translation of Hardie and Gaye:

We see the same weight or body moving faster than another for two reasons, either because there is a difference in what it moves through, as between water, air, and earth, or because, other things being equal, the moving body differs from the other owing to excess of weight or of lightness.

Bodies are conceived here as travelling through a medium, and there is no suggestion of acceleration, only speed. The translation above is found in The Basic Works of Aristotle of 1941, edited by McKeon, but dates back to the Oxford translation of the complete works of Aristotle of 1931. Here is the Greek, from the Oxford Classical Text of Ross (1950, corrected 1982):

ὁρῶμεν γὰρ τὸ αὐτὸ βάρος καὶ σῶμα θᾶττον φερόμενον διὰ δύο αἰτίας, ἢ τῷ διαφέρειν τὸ δι’ οὗ, οἷον δι’ ὕδατος ἢ γῆς ἢ δι’ ὔδατος ἢ ἀέρος, ἢ τῷ διαφέρειν τὸ φερόμενον, ἐὰν τἆλλα ταὐτὰ ὐπάρχῃ διὰ τὴν ὑπεροχὴν τοῦ βάρους ἢ τῆς κουφότητος.

Aristotle does not simply list three possible media, but he gives them by pairs: “through water or earth, or through water or air.” Of particular interest to us is τὸ βάρος, which occurs twice:

  • τὸ αὐτὸ βάρος καὶ σῶμα “the same weight and body”;
  • διὰ τὴν ὑπεροχὴν τοῦ βάρους ἢ τῆς κουφότητος “through an excess of weight or lightness.”

We might understand the latter occurrence of “weight” as density. It doesn’t really matter, since other things—presumably including volume—are taken to be the same.

Some manuscripts omit the former occurrence of “weight,” along with the conjunction “and.” The translators make this “or,” as perhaps we do when given the choices of tea and coffee, and we know to take one or the other, but not both. Richard Feynman made the reverse move in the anecdote that gives his popular book its name:

“Would you like cream or lemon in your tea, Mr. Feynman?” It’s Mrs. Eisenhart, pouring tea.

“I’ll have both, thank you,” I say, still looking for where I’m going to sit, when suddenly I hear “Heh-heh-heh-heh-heh. Surely you’re joking, Mr. Feynman.”

Hippocrates Apostle has a note on the conjunction in his own translation, first published in 1969:

We observe that the same weight or body travels faster for two reasons, either because there is a difference in the medium through which it travels, as through water or earth or air, or because, other things being the same, the travelling body has an excess of density [or weight] or of lightness.

The bracketed alternative is by Apostle, who suggests in his note on “weight or,” now inserted below in brackets by me,

Perhaps the expression [“weight or”] should not be omitted, for “weight” indicates a body by nature going down, “body” includes a body which by nature goes up, and “same” indicates other things being equal … Thus the common belief that Aristotle posits the velocity of a body as proportional to its weight may be erroneous.

In any case, the passage from the Physics does not allow Bell’s facile refutation. Aristotle’s points would seem to be two:

  • The same body travels more slowly through the denser medium.
  • Through the same medium, the denser body travels more slowly.

These points would seem also to be correct.

Bell asks us to conceive two like bodies. They are not like in weight, for they have different weights. Perhaps they are like in shape and volume; but then, if we glue them together, the new body is like neither of the original ones. Bell seems to have knocked down a straw man.

In the article “Aristotelian physics,” Wikipedia currently (June 19, 2022) says,

Aristotle proposed that the speed at which two identically shaped objects sink or fall is directly proportional to their weights and inversely proportional to the density of the medium through which they move.

This is close to my interpretation of the passage in the Physics, except that there Aristotle does not name any proportions. Wikipedia cites not Aristotle himself, but Simon (or Semyon Grigorevitch) Gindikin, Tales of Mathematicians and Physicists (1988), page 29:

Galileo was above all interested in free fall, one of the most common forms of motion in nature. At the time, one had to begin with what Aristotle said on the matter. “Bodies having a greater degree of heaviness or lightness but in all other respects having the same shape, traverse an equal space more rapidly in the same proportion as the quantities mentioned.” Thus, according to Aristotle, the velocity of a falling body is proportional to its weight. A second assertion is that velocity is inversely proportional to “the density of the medium.” This assertion led to complications, since in a vacuum, whose “density” is zero, the velocity should have been infinite. As to this, Aristotle declared that a vacuum cannot exist in nature (“nature abhors a vacuum”).

I do not know why Gindikin calls free fall “one of the most common forms of motion in nature,” since it would seem to me that most motions in nature, such as walking, slithering, waving up and down as the sea does, waving in the breeze as trees do, or flying through the air as birds do, are not falling; and when things like raindrops or snowflakes fall, we see them held at terminal velocity by the air, so they not free in the sense of being affected only by gravity. If we want to include the translunary world in nature, then perhaps most objects are affected only by gravity; but this does not seem to be Galileo’s concern.

Gindikin gives no source for his Aristotle quotes. In any case, as far as I understand, the terminal velocity of a body falling through a resisting medium is indeed proportional to its weight in the medium, all else being fixed. In a vacuum, velocity would increase without bound, thus being “infinite” by one interpretation of the term.

I found some other relevant passages of Aristotle. In On the Heavens, Book IV, Chapter 1, beginning around 308a28, in the translation of J. L. Stocks (which is also in the Basic Works), Aristotle says,

By absolutely light, then, we mean that which moves upward or to the extremity, and by absolutely heavy that which moves downward or to the centre. By lighter or relatively light we mean that one, of two bodies endowed with weight and equal in bulk, which is exceeded by the other in the speed of its natural downward movement.

In the next chapter, beginning around 308b5, Aristotle would seem to be aware of the use, in Plato’s Timaeus, of the thought behind Bell’s putative refutation:

One use of the terms ‘lighter’ and ‘heavier’ is that which is set forth in writing in the Timaeus [63c], that the body which is composed of the greater number of identical parts is relatively heavy, while that which is composed of a smaller number is relatively light. As a larger quantity of lead or of bronze is heavier than a smaller—and this holds good of all homogeneous masses, the superior weight always depending upon a numerical superiority of equal parts—in precisely the same way, they assert, lead is heavier than wood. For all bodies, in spite of the general opinion to the contrary, are composed of identical parts and of a single material. But this analysis says nothing of the absolutely heavy and light. The facts are that fire is always light and moves upward, while earth and all earthy things move downwards or towards the centre.

In such a descent, whether free or along a frictionless plane, Galileo’s triangles of successive times and velocities, with their areas equal to the distance covered (equal also to the rectangle of time by average velocity, which is half the terminal height) enable us to write a priori these necessary laws: [28]

Galileo’s Plot of Uniform Acceleration:

Right triangle, horizontal leg representing time t; vertical leg, terminal velocity v. Rectangle superimposed with base t and height v/2

at any moment,
s  (the space or distance covered) = (v/2) ⋅ t  (average velocity times the time).

since vt, and since a  (the acceleration, or slope) = v/t, therefore s = ½ at ²

It is also clear, by the principle of conservation, that whatever the falling weight realizes in quantity of motion, must equal what we put into it by raising it up. Clear too that to lift it another equal height would give it as much again, and that to lift other weights equally would multiply the quantity in the same ratio. Therefore the product of weight times height lifted (which is force-times-distance) must equal the total realized motion or work done, which already, from Galileo’s triangle, we might guess to be ½ mv ².

For Aristotle, everything moves through a medium; but for us now the medium is air, and the bodies moving through it are so heavy that the air does not impede them. By the “axiom of substance,” gravity induces the same acceleration on all bodies, and it doesn’t matter whether the bodies are already moving. So we can call the acceleration a, and we get the equation that Bell does,

s = ½ at ²,

for the distance a dropped body falls over time t. We go on to figure that the body has an intrinsic “mass,” which Bell has now summoned up with the letter m. The body also has a weight, which let us denote by w; however, we suppose this is proportional not only to m, but to the effect of gravity, namely the acceleration a. In particular, we suppose

w = ma.

The products of the corresponding sides of our two equations yield the equation that Bell alludes to,

ws = ½ mv ²,

since at = v.

THE PENDULUM

But that triangle was based on free fall or descent along a frictionless plane. And since our suspended weight falls down the always shifting incline of a pendulum’s arc, [29]
to axiomatize it requires a new mathematics harnessing the luminous paradox of vanishing ratios.


Pendulum

Conceive three triangles: above, the space triangle of the pendulum’s swing; below, the force triangle of the weight components; between them, the small triangle of height and arc (displacement, approximating the tangent drawn at the bob, which is the restitutional component of its weight). All three triangles approach similarity as the amplitude is made less and less. Once that is grasped, we can simply read off the laws of the pendulum from the vanishing triangles.

Since arc (spatial displacement) and the restitutional tangent converge, S ∝ −F  (or −a), that is s ∝ −, (displacement varies as its own second derivative oppositely [30] directed), which is the condition for sin-wave or simple harmonic oscillation. Since the vanishing arc approximates a parabola, the height, or subtence [sic] varies as the base (chord, arc, all indistinguishable) squared. But the force varies with that space (hence velocity with distance); so the period (T) must be constant for all amplitudes (so long as they are zero). Since for longer pendula the drawn figure increases symmetrically, but v  as a square, the period will vary as the root of pendulum length. Everything falls out on inspection, if one makes the right pact with the Lucifer of vanishing ratios.

Thus the logic of wave motion throughout (Taylor, Euler, etc.) rests on the wave’s not waving.

The string of the pendulum making angle θ with the vertical, g being acceleration due to gravity, the force on the bob in the direction of its motion is given by

F = mg tan θ.

The string having length l, horizontal displacement of the bob is given by

s = l tan θ.

Hence Bell’s proportion

s ∝ −F,

but it is only approximate, since the direction of F is not quite horizontal. As Bell says, an exact proportion would yield simple harmonic motion. The period depends on the constant of proportionality, which is independent of the amplitude of the motion. In particular, since in magnitude

F = mgs/l,

the square of the period varies as l.

That is everything Bell concludes, but he also has another derivation. The vertical height h of the bob from its lowest point varies roughly as the square of the horizontal displacement, since a circle is roughly a parabola:

hs ².

If we now fix h as the maximum vertical height, then, from the previous computations with freely falling bodies, v being the speed of the bob at its lowest point,

hv ².

Thus maximum speed varies with maximum horizontal displacement. This all shows that acceleration varies with horizontal displacement, that is, simple harmonic motion obtains.

The period of such a motion varies directly with amplitude, inversely with maximum speed. If we fix everything but l, then this varies with amplitude and v ², and so period varies with √l.

MOMENTUM AND ENERGY

Having deduced something about the ratio of height to the square of vanishing arc and the square of velocity, we drop the mass again. Our axiom has equated the work of lifting it against gravity, with the final realized quantity of motion. But how shall we define quantity of motion? Is it the directional product of velocity times mass, called momentum; or the absolute quantity of one half the mass times the velocity squared, to be called energy?

In any transfer of motion, action and counter-action, as consentaneous, must occupy the same time. So whatever is communicated by force (mass times acceleration) acting [31] through time (which multiplied, ms/t ² times t  reduces to ms/t, mass times velocity to be called Momentum) that must be equal for both bodies, equal and opposite. Thus for the compound whole, the progress of the center of mass cannot be changed by any action or energy transformation internal to the system (which is the first principle of motion, applied to a complex body).

If a shell is exploded in a cannon, the massive cannon is kicked back, the shell is sped forward. The same explosive pressure acting through the same time must produce the same Momentum (mv), which means, if the cannon is a thousand times as heavy, the velocity of the shell will be a thousand times that of the cannon’s backward kick. So the center of gravity of the system remains as before, however much charge has been exploded.

But the chemical explosion delivered live force or energy, some dissipated as heat, the rest turning into motion. How much of that energy  did the cannon get and how much the shell?

The action occurred not only in time but in space, and if the time was equal for cannon and shell, the spaces clearly were not, since the cannon was barely kicked back, while the shell was being thrust through the length of the bore. So the product of force through space has to be a thousand times as great for the shell, covering, at a thousand times the speed, a thousand times the distance. [32] What is that product of force times the space through which it acts? Examine the dimensionalities.

Where force times time reduced to mass times velocity, force (ms/t ²) times distance puts another s  in the numerator, yielding the dimensions of mass times velocity squared. It is this quantity (with a ½ thrown in, as from Galileo’s triangular graph of accelerated motion) which Leibnitz would call Living Force, and we Work or Energy. And whereas cannon and shell sucked up equal portions of momentum—of this absolute quantity, the shell has sucked up a thousand times as much as the cannon (luckily for those at the breech). But nature had pointed to this conserved quantity before, in the lifting and falling of a weight: that the distance lifted (the area of Galileo’s triangle) varies as half the terminal velocity squared.

IMPACT:

Again, in act or thought, we lift our elastic sphere and let it fall. It strikes, and again the last ball on the other side pops out with comparable velocity. Why can’t four balls move off with one fourth the velocity? It would answer for momentum; but not for that other quantity where velocity is squared. But if we glued them, they would have to stay together. How to predict the result? We drop two. Two pop out. We drop three. The middle one carries through. Conceive the two and the three (indeed any ratio [33] of masses) joined into unequal spheres approaching and overtaking at whatever speeds—can we draw a general solution from our first principle of motion conserved?

Descartes tried. “If the bodies were exactly equal, moving with equal speeds in a straight line toward each other … they would rebound equally.” Good. Reason requires it and the elastic spheres seem to agree.

“Second,” Descartes writes: “If one were just a little greater, then only the lesser would rebound … since the lesser cannot compel the greater.” That nominalist logic turns to nonsense under physical question: “How much less; how much greater? Not compel it at all?” Let the difference diminish to the weight of a hair, the weight of a molecule. A crisis must develop: “To move or not to move?”

Descartes protests:

The demonstrations of all this are so certain that although experience may seem to show us the contrary, we should nonetheless be compelled to put more trust in our reason than in our senses.

That would be fine, if Descartes had used the right kind of reason. Indeed, no experiment is needed; only the intuition of substance, which implies continuity. Leibnitz: “it cannot come about that two bodies are perfectly equal and alike” and “no event takes place by a leap”. What Descartes fumbled was the quantitative axiom itself.

How beautifully Huyghens, starting from the common first principle that “motion will be continued uniformly in [34] a straight line if no impediment is interposed”, imagines the impact as occurring on a moving frame of reference—a boat, seen from the shore; so that by changing the boat’s motion each case can be adjusted to one of symmetry, where, the center of gravity stilled, equal momenta are exchanged.

But as Leibnitz would point out, that visualization was not required. We had only to reason from the applied principle of substance: that the motion of a body is conserved. But what is a body? It is not a mystical one, but a one-many: in this case the summed whole and a rebounding two.

  • If the whole, around which we could draw an isolating circle, must, with its center of mass, continue as before; and
  • if the parts must also preserve whatever motion they had relative to each other (as an elastic ball would bounce off a wall),

Leibnitz’ three equations automatically arise (“Essay on Dynamics”):—the lineal  equation:

v₁ − v₂ = v₂′ − v₁′

(that the difference of velocities before impact equals the difference after, under a change of sign); the plane  equation:

mv₁ + mv₂ = mv₁′ + mv₂′

(that the net Momentum of the whole, or vector sum, must be the same after impact as before); but these two may be compounded (since one is plus and the other minus and with a little shuffling of terms they can be multiplied) to produce the third or Solid Equation, that the sum of the masses [35] times the velocities squared also remains unchanged: that is, mv₁ ² + mv₂ ² must equal the prime values of the same.

According to the lineal equation, the velocity of one body with respect to the other only changes sign, not magnitude, in the collision. I do not recall learning this, but may have forgotten. It is suggested in the Scholium after the Proof of Proposition 9 in Leibniz’s Essay de Dynamique as printed in an appendix of Leibniz and Dynamics: The Texts of 1692, by Pierre Costabel, translated by R.E.W. Maddison (1973), available at the Internet Archive.

In any case, the third or solid equation is

mv₁ ² + mv₂ ² = mv₁′ ² + mv₂′ ²,

expressing the conservation of kinetic energy. Rearrange and factorize:

mv₁ ² − mv₁′ ² = mv₂′ ² − mv₂ ²,
m₁(v₁ − v₁′)(v₁ + v₁′) = m₂(v₂′ − v₂)(v₂′ + v₂).

Rearrange and factorize also the equation of momenta, or plane equation:

mv₁ − mv₁′ = mv₂′ − mv₂,
m₁(v₁ − v₁′) = m₂(v₂′ − v₂).

Division now yields

v₁ + v₁′ = v₂′ + v

and then the lineal equation. This work is done in the Wikipedia article, Elastic collision. One can reverse the steps and derive solid equation from the plane and lineal, as Bell says.

Marvelous, that by multiplying the two aspects of a complex motion, the progress of the whole and the inner velocity of the parts, we get the absolute, non-directional quantity, work, which had already appeared as force times distance, or as that ½ mv ² acquired in gravitational fall from a measured height.

A deeper marvel if suddenly it dawns upon us, that what held of this body, which was a complex of two spheres bouncing against each other, must hold of the billions of interbouncing Lucretian parts within all moving masses.

So when the relative velocities subside from inner friction, when even the momentum of the whole dies away by attrition in the strings and with the air (while the axiom attests: all motion lost is lurking somewhere)—what if the analysis into one  and many  has already pointed the way?—that what has visibly disappeared must remain as a sum of squares in the heat dance of the constituent molecules?

With what triumphant insight Leibnitz wrote of that frictional going-under of energy:

But this loss of the total force, this failure of the third equation, does not detract from the inviolable truth of the law of the conservation of the same force in the world. For that which is absorbed by the minute parts is not absolutely lost for the universe, though lost for the total force of the concurrent bodies. [36]

Ironic, that in Leibnitz’ claim of live-force as indestructible (our First Law of Thermodynamics) the Second Law is also implied: “though lost for the total force of the concurrent bodies.” No Newtonian System of the World would save the 18th century from a Deist Clockwork which can only run down hill. No calculus of determinist quantity could avert the dark encroachment of Entropy, or shake its axiomatic ground: that what had been introduced to keep the cosmos alive has become an equational pool of which it might be asked: why should it flow from here to there?

The irony looks back to the oldest distinction of Greek thought, Being and Becoming. Which is more real?

  • Being, said the Eleatics and summed it into changelessness. Totaled.
  • Becoming, said Heraclitus, and wondered at the life of fire.

When Calculus took it up,

  • the integral stood for Being, a summed function;
  • the derivative for Becoming, a rate of change.

And again Being came to rest. The integrating Sigma of force-through-distance made a squared eternal out of change itself, Leonardo’s evanescent Force charmed to inert quantity: action that cannot act, energy ceasing to energize. So Newton, trying to see relative motion as absolute, points to the centripetal effects in the spun bucket of water, and says: “The thing is not altogether desperate.” But Newton was deceived. The rate of change is not the thing itself.

  • Just so his light-bearing AEther would be withdrawn, leaving only its [37] fluctuant modifications, fields and waves, more real than the medium they were supposed to agitate.
  • So too Jamesian Psychology would dissolve Mind into passing states of its own vanished being.

As for Leibnitz’ Absolute Force, like an ocean without a wind, it would require a new potential to keep the stuff of flow flowing—or besoul itself to that omnipresent contriver which Newton, on speculative tack (Optics), made of a space conceived as God’s sensorium. Without such admission of animating spirit, the all-mover settles to the causal and statistical gradient of Entropy—what gravitational analogy supplies to keep the process physical.

CARNOT:

Anyone skilled in the axiomatic field should have learned from Leibnitz what was going to happen; just as one could have fathomed without Carnot that you can’t get a heat engine to operate between two bodies at the same temperature, since axiomatically motion is by disparity. The truth is, you can’t prove anything but what you already know. Proof is the rhetoric of rigor. But what a glorious proof Carnot has given in his invention of an ideal engine (pub. 1824). [38]


Garnot’s Ideal Engine

The engine (a piston in a cylinder of gas) is outlined here beneath the graph of its own action above, where the pressure of the gas is plotted against its volume. Each hyperbola of the graph fulfills Boyle’s law, that for an ideal gas, at constant temperature, pressure times volume remains the same. There are two curves, matching the two temperatures over which the engine operates. They exhibit the law of Charles and Gay-Lussac, that pressure times volume varies as Absolute Temperature (pv = T/273). In this engine of pure thought, the likes of which nobody could ever build, we imagine the cylinder as being applied to the source of heat. The piston goes out just fast enough for the absorbed heat to keep the gas, otherwise cooled by [39] expansion, at the same temperature. If we started at A, we follow the graph down the isothermal hyperbola to B, at a larger volume and lower pressure. At B we remove the heat, but let the piston go on out, expanding and cooling the gas (adiabatic, no heat flowing in or out). So it reaches the lower isothermal at C. Here we put the cylinder on the cold source and let the piston move back to the left, condensing the gas. Against its tendency to heat, we refrigerate it just enough to track the isotherm back to point D. There we remove the cold source but let the gas go on being compressed, adiabatic again, no heat flowing in or out; so it warms up until, by ideal management, it returns to point A where it began. Since every change of temperature has been directly associated with a change of volume, this engine has an ultimate efficiency beyond which no actual heat engine can go. And since every area on the graph is pressure times volume, or work, the net work done by the heat flow is simply the shaded area of the trapezoid or queer parallelogram ABCD.

If now, instead of that one engine operating over a temperature gap from T2 down to Tl, we would conceive an infinite number, each operating over an infinitesimal gap, the most work we would ever get out would approach the area between the two hyperbolae and the straight lines dropped from A and B. But there’s all that other energy between the lower isothermal and absolute zero, diffused and spread [40] around like lukewarm water. And Carnot, after such a leap of insight and proof, is simply telling us: This machine can’t get anything out of that lukewarm water. Such the general preposition of The Motive Power of Heat: “its quantity is fixed solely by the temperature of the bodies between which the flow is effected.”

From which Clausius, who invents the term entropy  for the measure of the spent or useless energy in any system, extrapolates (from the divorced abstraction of the closed and inert)—“The entropy of the universe tends toward a maximum” (though the closure of the universe rests under paradox). So the cosmic optimism of Leibnitz, to which Helmholtz as late as 1847 would subscribe:

The universe possesses, once and for all, a store of energy which is not altered by any change of phenomena, can neither be increased nor diminished, and which maintains any change which takes place on it—

that euphoria of the First Law would bring forth after 1850 (and from the same causal womb) Kelvin’s apocalyptic thunder of the Second:

Within a finite period of time past, the earth must have been, and within a finite period of time to come the earth must again be, unfit for the habitation of man as at present constituted, unless operations have been, or are to be performed, which are impossible under the laws to which the known operations going on at present in the material world are subject.

Out of the most rational and equational science of all, the physics of matter in motion, has come this [41] contradiction: Work will endure forever, but not as work. To which the new Heraclitean would have been laughing, like Yeats’ old Rocky Face, in tragic joy, having foreseen from the start the inevitable reduction of a priori quantity syllogistically applied to an organic universe of polyvalent energy exchanges.

And indeed, how anyone could have let the axiomatic subsidence of a merely conceptual substratum disturb him, in a cosmos peppy after fifteen billion years, is perhaps obscure.

William Buter Yeats, “The Gyres”:

THE GYRES! the gyres! Old Rocky Face, look forth;
Things thought too long can be no longer thought,
For beauty dies of beauty, worth of worth,
And ancient lineaments are blotted out.
Irrational streams of blood are staining earth;
Empedocles has thrown all things about;
Hector is dead and there’s a light in Troy;
We that look on but laugh in tragic joy.

What matter though numb nightmare ride on top,
And blood and mire the sensitive body stain?
What matter? Heave no sigh, let no tear drop,
A-greater, a more gracious time has gone;
For painted forms or boxes of make-up
In ancient tombs I sighed, but not again;
What matter? Out of cavern comes a voice,
And all it knows is that one word ‘Rejoice!’

Conduct and work grow coarse, and coarse the soul,
What matter? Those that Rocky Face holds dear,
Lovers of horses and of women, shall,
From marble of a broken sepulchre,
Or dark betwixt the polecat and the owl,
Or any rich, dark nothing disinter
The workman, noble and saint, and all things run
On that unfashionable gyre again.

POSTLUDE—THE USES OF PARADOX

I have traced the Tao of the old physics here, as if I did not need the New, yet I grew up in the curved space of Einstein’s relativity, trying, as Dante had done, to conceive a three dimensional sphere turned inside out. Before I knew of Heraclitus I had welcomed complementarity and indeterminacy. These broke for me the “misplaced concreteness” of the old determinist law.

Forty-five years ago I decided that when reason drives a sheer impasse into an activity which in fact goes on, we have to think of the polar cleavage as both real and unreal.

  • Real, as mind’s measure of the mystery of world-process, of its incom­men­surate­ness with the contrary truth of logic;
  • unreal as in no way precluding the operation so denied.

I found in paradox two rightful uses, as of night and day:

  • of [42] night, soul’s guide toward noumenal reverence;
  • of day, thought’s X and Y for mapping, clarifying, ultimately for navigating the experiential field.

But that is a job as huge and demanding as Aristotle’s, and for me at 70, just begun.

“Look,” my friends say, “Bell’s been doing the same thing since he was 25. About that time he had a vision of Paradox as paradise, and he’s been stuck there ever since.” Can I deny it? You get a notion how to juggle when you’re young and you go through life skirting Charybdis and Scylla on Odyssean voyage. They say: “That juggling comes too easy. ‘Look, Ma, no hands’; and you’re riding through the universe as if contradiction were fun and contraries could be fused whenever you wanted.” But it’s not as if a lifetime of labor hadn’t gone into the articulation of that dangerous insight. Because for us the Heraclitean content must comprise the whole of modern knowledge, science, history, the creative arts.

Nor is it a glib or even a personal choice to rear thought over the horns of dilemma. Only on the precipitated axes of polar opposites (that modern replacement for the ambiguity of Platonic dialogue) can reason stretch the hyperbolic spectrum of knowledge, without arrogating beyond its own involvement in the shifting emergence of inner and outer, self and world. The philosophy which ignores this [43] may well be the glib one, deceiving itself and sometimes others.

Let Hume be our pot-bellied example. There is a picture of him in Edinburgh, flopped at a desk of papers, completely flabbergasted; you can’t tell whether he’s gorged on haggis or stunned by gout. Yet it may be the moment when, having posited in his great work, Concerning Human Understanding, which occupied him three years in France, nothing but an atomism of simple impressions, with ideas as their dim images—though anyone might ask if simple impressions can exist, except as nameable fixes, red  and the like, in the amorphous and relative, ingatherings already tinged with one and many, will and idea—he discovers, how touchingly, that he has no way, true to the logic he has played, of accounting for any composition, self or substance, even to the power of thinking or saying “I am”. As he writes in the appendix to those ambitious tomes:

If perceptions are distinct existences … no connections are … discoverable … all my hopes vanish when I come to explain the principles that unite our successive perceptions in our thought or consciousness … For my part, I must plead the privilege of a sceptic, and confess that this difficulty is too hard for my understanding.

There’s honesty. There’s experience. And surely honest experience is a good. But could not the axiomatic syllogism, which pulls from the hat only the rabbits you put in, have tipped him off: “You expect conscious deliberation [44] from a grab-bag of perceptions stripped of organic coherence?”

Of course, not even Kant could spring the cage of Newtonian phenomenology. Within one-way logic the axiom of separation, throned over deterministic physics, could only leave mind-body unjoined. In field-thought, too, abstract polarity remains, but turned, like clouds at sunset, into light. Since both the evolution of consciousness from matter, and the appearance, in primal matter itself, of quantum uncertainty, have fused the separate realms in existential flame.

Yet still one hears, after almost a century: “isn’t indeterminacy a cop-out, enthroning ignorance of the causes?” Well, since the regress of cause will always be bottomless, why make a faith-claim for a billiard-ball determinism incommensurate with the search we experience? Why insist on a universe in which the moment can never be decisive? It’s kicking a fart out of a dead donkey.

“But this indeterminacy,” they say, “only defines the limits of observation. What has that to do with free will?” No doubt when quantum theory surfaced, it was hailed from religious circles: “Now we’ve got a Holy Ghost because there’s a swerving at the base of nature.” Then came the counterswing: “How can you fetch volition from an experimental probability with a constant named after Planck?” Yet it was not a preacher but a scientist, [45] Schroedinger, who set anti-entropy against entropy, bringing choice from quantum resonance up the nerve and genetic thresholds of the organic stair—although not noting that to rationalize organic building by giving it also a causal gradient was to endow energy with contrary impulses, stretching the whole over the paradox of spirit: “If wind move itself, bay the wind.”

In any case, it is scientists today who have begun to realize that the precipitation of polarities is not the shipwreck of reason, but the delineation of its natural field. What is probability but the weft between absolute law and indeterminacy? So before Schroedinger, before quantum theory, Maxwell had invented a thermodynamic demon who might oppose the drift of entropy.

Conceive two boxes of gas, an opening between them large enough for a molecule, a frictionless trapdoor, in charge of a little demon. Say one box started hot and the other cold. The demon has gone to sleep. Soon Carnot would despair of running an engine between those to boxes. But the demon wakes up, like Lord Kelvin, shocked at entropy. The molecules still have a Gaussian distribution of motion. Every time a faster one comes he opens the trap; a slower one, claps her to. Inside a demon’s working day, you’ll have it hot as blazes on one side and cold on the other (as indeed chemicals are concentrated all the time in bodily tissues, against inorganic probability—retu [sic] mirabile). [46] All you need is a frictionless intelligence. But that had been ruled out by Leonardo’s axiom of separation: that a spirit can no more stir matter than Peter Ibbetson’s astral self would be able to pluck a feather off a sleeve. Though surely the universe is reared over this contradiction.

It was clever of Maxwell to take his demon myth from the deeply rooted faith-world of pre-science. But can science plant demons at nature’s trapdoors without defying the axiom of physicality: if they are disembodied, how can they work the traps?

Like all polarity, once precipitated, it is insoluble. We have only the trans-logical leap of conceiving (across the poles) through all organized structures (and what is unorganized?) an increment of perceptive purpose, both transcendental and inseparable from energy itself: some “virtue of a virtue”, peaking up, through the feedback thresholds of living structure, to conscious will; as down the aggregates of the mineral it settles almost to the inertia of predictive law. To stretch this spectrum between the abstracted poles is not our choice but what is required of us.

The cosmos does not feed the mind on pabulum, but on the self-vaulting knot of contradiction. Those who quarrel with antinomy quarrel with gods and world. Of course, reason has planted the quarrel in us and it must go on; but within the embrace and celebration of generative enigma—[47] as walking weds stasis and falling, Descartes’ assurance, Pascal’s vertigo.

Since this much we see clearly and distinctly: that the quantitative and equational (the logic of Eleatic substance applied to things) is always determinate, like any resolution of mathemable forces (whether balanced or unbalanced); so the only axiomatic source of the exploratory, of probability, life and evolution, is the mated mystery of Sic et Non, Yes and No. It is the breathing of Paradox which besouls the worlds.

You must sit downe, sayes Love, and taste my meat:
     So I did sit and eat.

George Herbert, “Love (III)”:

LOve bade me welcome: yet my soul drew back,
                    Guiltie of dust and sinne.
But quick-ey’d Love, observing me grow slack
                    From my first entrance in,
Drew nearer to me, sweetly questioning,
                    If I lack’d any thing.

A guest, I answer’d, worthy to be here:
                    Love said, You shall be he.
I the unkinde, ungratefull? Ah my deare,
                    I cannot look on thee.
Love took my hand, and smiling did reply,
                    Who made the eyes but I?

Truth Lord, but I have marr’d them: let my shame
                    Go where it doth deserve.
And know you not, sayes Love, who bore the blame?
                    My deare, then I will serve.
You must sit down, sayes Love, and taste my meat:
                    So I did sit and eat.

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