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. We do things then, but that they will get anywhere may be no more than a dream. In any case, 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.”

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.” Throughout my time in Santa Fe, Bell would present slide shows from his series, 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,

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

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

• SALUTATION:
• PREVIEW:
• FROM QUESTION TO METHOD:
• AXIOMS AND FIELDS:
• IDEAS AND CAUSALITY
• SUBSTANCE AND COMPLEMENTARITY:
• WOLF AXIOMS: AN INTERLUDE
• THE GENERATION OF AXIOMS
  • A) Of Quantity
  • B) Of Geometry
  • C) OF Logic
  • D) Of Physics
• FROM LEONARDO TO LA PLACE:
• THE DIPOLE OF ACTION:
• FALLING BODIES:
• THE PENDULUM
• MOMENTUM AND ENERGY
• IMPACT:
• CARNOT:
• POSTLUDE—THE USES OF PARADOX

[0]

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

Parenthesis added by hand in the original.

[1]

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

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.

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.

See also Trotter’s translation of one of Pascal’s Pensées, Sellier 131, Lafuma 97, Brunschvicg 334:

Lust and force are the source of all our actions; lust causes voluntary actions, force involuntary ones.
La concupiscence et la force sont les sources de toutes nos actions. La concupiscence fait les volontaires, la force les involontaires.

This fragment itself may represent such a dipole as Bell will contemplate.

In Herodotus I.8–13, 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. (The queen was indignant because the king had let Gyges see her when she took off her clothes and put them on a chair; the absence of a chair in a hotel room caused me to recall the story of Gyges in “Pyrgos Island.” The confirmation of the sovereignty of Gyges was confirmed by the Pythia, as I recalled in writing “Hypomnesis” about a visit to Delphi.)

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.

(This is on page 325.) There is a tension, which physics has gone on to eliminate (page 327):

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.

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.”

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.

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 Brunschvicg 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, to be taken up again later, whether Bell’s axioms are Collingwood’s absolute presuppositions.

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:

10. 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. An arbitrary property may not define a set of things with that property, but from a given set, the property does separate out, as a set, the elements with that property. In the technical language of Zermelo in “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 M_𝕰 containing as elements precisely those elements x of M for which 𝕰(x) is true.

But Bell is talking about separation of mind and matter. On page 23, he will mention

Leonardo’s axiom of Separation, that no spirit can operate on matter …

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;

and on pages 44–6,

Within one-way logic the axiom of separation, throned over deterministic physics, could only leave mind-body unjoined …

Maxwell had invented a thermodynamic demon who might oppose the drift of entropy.

… 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.

Meanwhile, above, Bell’s formation “pre-fore-or-destinate” would seem to combine “predestine,” “foreordain,” and perhaps “destination.”

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 …

Leibniz states the principe de raison suffisante in the Monadology:

31. 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;

32. 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.

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.

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

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.

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!’

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

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.

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!

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.

Bell would seem to share Collingwood’s criticism of the logician’s “propositional assumption.”

Pascal raises that question in the third person in the long fragment, “Double condition de l’homme” (Sellier 164, Lafuma 131, Brunschvicg 434). He distinguishes between

• the pyrrhoniens (skeptics), who can find no convincing proof of our principles, and
• the dogmatists, who observe that, nonetheless, we cannot doubt these principles.

Pascal would seem to put the last point more precisely:

Je m’arrête à l’unique fort des dogmatistes, qui est qu’en parlant de bonne foi et sincèrement on ne peut douter des principes naturels.

The translations of Trotter and Sellier are:

I notice the only strong point of the dogmatists, namely, that, speaking in good faith and sincerely, we cannot doubt natural principles.

I pause at the dogmatists’ only strong point, which is that, speaking in good faith and sincerely, we cannot doubt natural principles.

These use an ambiguous participial phrase to translate en parlant de bonne foi et sincèrement. Is the English phrase “speaking in good faith and sincerely”

• an ordinary adverb?
• a “sentence adverb”—what is apparently also called a “disjunct” (modalisateur)?

As far as the translators’ English is concerned, Pascal could be doing either of the following:

• making the dogmatists’ own point conditional: if and when we are speaking in good faith, we are implicitly respecting some principle or absolute presupposition;
• telling us how he means what he says: he means it sincerely.

I am not sure that French actually allows the latter interpretation. The former interpretation would be clearer in English if the en of en parlant were translated explicitly: “in speaking … we cannot doubt …,” or perhaps “when speaking …”

On the other hand, en could be the pronom adverbial, so that the phrase should be, “speaking of them …”

Pascal’s omission or use of en as a pronoun raised questions for me in

• Sellier 645, Lafuma 783, Brunschvicg 357, where On se prend à la perfection même seems to have the meaning (“We take issue with perfection itself”) that would seem normally expressed by “On s’en prend …”;
• Sellier 683, Lafuma 431, Brunschvicg 560, where in des Juifs, qui en sont les ennemis irréconciliables, the en could be plural or singular, making Jews enemies of the impious or of religion.

Meanwhile, it makes sense to me that speaking in good faith and sincerely means tacitly accepting some principles, such as the principle that agreement with one’s interlocutor is possible. I would not have thought this made me one of the dogmatists. Nonetheless, Pascal says we have to choose between them and the skeptics:

Voilà la guerre ouverte entre les hommes, où il faut que chacun prenne parti, et se range nécessairement ou au dogmatisme ou au pyrrhonisme, car qui pensera demeurer neutre sera pyrrhonien par excellence. Cette neutralité est l’essence de la cabale. Qui n’est pas contre eux est excellemment pour eux. Ils ne sont pas pour eux‑mêmes, ils sont neutres, indifférents, suspendus à tout sans s’excepter.

Que fera donc l’homme en cet état ? Doutera‑t‑il de tout ? Doutera‑t‑il s’il veille, si on le pince, si on le brûle ? Doutera‑t‑il s’il doute ? Doutera‑t‑il s’il est ? On n’en peut venir là, et je mets en fait qu’il n’y a jamais eu de pyrrhonien effectif parfait. La nature soutient la raison impuissante et l’empêche d’extravaguer jusqu’à ce point.

Maybe Descartes did not have to go to such extremes of doubt to find a principle. Bell would seem to go on to say this now.

The Pascal fragment “Double condition de l’homme” continues from where we left it:

Dira‑t‑il donc au contraire qu’il possède certainement la vérité, lui qui, si peu qu’on le pousse, ne peut en montrer aucun titre et est forcé de lâcher prise ?

Quelle chimère est‑ce donc que l’homme, quelle nouveauté, quel monstre, quel chaos, quel sujet de contradiction, quel prodige, juge de toutes choses, imbécile ver de terre, dépositaire du vrai, cloaque d’incertitude et d’erreur, gloire et rebut de l’univers !

This is a key point: the a priori—in Collingwood’s terms, what is absolutely presupposed—evolves in time. Collingwood spells it out on pages 49–73 of An Essay on Metaphysics:

All metaphysical questions are historical questions, and all metaphysical propositions are historical propositions. Every metaphysical question either is simply the question what absolute presuppositions were made on a certain occasion, or is capable of being resolved into a number of such questions together with a further question or further questions arising out of these.

This is the central point of the present essay …

The essential thing about historical ‘phases’ is that each of them gives place to another; not because one is violently destroyed by alien forces impinging on its fabric from without by war or from within by revolution, but because each of them while it lives is working at turning itself into the next. To trace the process by which one historical phase turns into the next is the business of every historian who concerns himself with that phase. The metaphysician’s business, therefore, when he has identified several different constellations of absolute presuppositions, is not only to study their likenesses and unlikenesses but also to find out on what occasions and by what processes one of them has turned into another.

Bell’s terminology in the last clause, as I understand it:

one-way quantification

measuring things without being able to recover the things from the measurements;

narrowing of thought

assuming that you can recover the things, or at least everything about them that matters.

Did anybody really narrow thought too much? Perhaps they only took up what Collingwood traces to the “Christian fathers,” namely the “principle of the limited objective.” In The New Leviathan (1942), which has perhaps been overlooked as a work in the philosophy of science, Chapter XXXI is “Classical Physics and Classical Politics”:

31. 61. All modern science recognizes what I will call the principle of the limited objective. That is the most fundamental difference between the modern sciences and the sciences of ancient Greece.

31. 62. Ancient sciences aimed at an unlimited objective. They defined their aims by asking questions like: ‘What is Nature?’ ‘What is Man?’ ‘What is Justice?’ ‘What is Virtue ?’ A question of this sort was to be answered by a definition of the thing. From this definition, which had to state the ‘essence’ of the thing defined, implications could be derived, each implication being the statement of some ‘property’.

31. 7. The classical physics obeyed the principle of the limited objective, limiting its explanatory efforts to such facts as admitted of mathematical treatment. The unlimited objective, the hope of understanding Nature at large, was abandoned. In its place was put the limited objective, understanding so much of Nature as could be measured, weighed, or in some other way treated mathematically.

31. 71. The principle of limited objective, applied to physics with memorable results by Galileo, was not first laid down by Galileo. It was first expounded by those too little known writers (if they were better known the main lines of European history would be better understood) whom we call the Christian Fathers. The sciences with unlimited objectives went bankrupt with what is called the collapse of ancient paganism. The Christian sciences (nostra philosophia, one of the Fathers calls them) are sciences with limited objectives.

The Christian Father to whom Collingwood alludes may be Augustine. “Nostra philosophia” is the first part of the title of Chapter XI of Christianity and Classical Culture (1957/1940), by Charles Norris Cochrane, who like Collingwood was born in 1889, and who ends his Preface by saying,

I am particularly grateful for the help I have received from Professor R. G. Collingwood and Mr. R. Syme.

This is dated at Oxford, July 1939, when Collingwood was off on the Mediterranean cruise that he would recount in The First Mate’s Log. In his own Chapter VI, Cochrane writes on pages 231–2,

The evolution of a specifically Christian philosophy was to some extent promoted by theoretical attacks against the faith such as those levelled by Celsus and Porphyry in the third, and by Julian and his circle in the fourth, century. It was stimulated also by events in the world of action, such as the persecutions from Decius to Diocletian and, subsequently, the Caesaro-papism of the New Monarchy. It was, indeed, this latter which gave point and significance to the theological controversies which intervened between the adoption of the Nicene Creed in 325 and its confirmation, fifty years later, at Constantinople. And it was not until she had undergone these experiences that the Church was in a position to ‘spoil the Egyptians’; that is to say, until she could construct out of the dismantled fragments of Romanitas a system of thought designed to supplement and reinforce the appeal of naïve Christianity, and thus secure its final victory. But in this respect her shortcomings were in 313 still painfully apparent. Accordingly the day was yet remote when a Christian could write: ‘Can paganism, I ask you, [232] produce anything equal to ours, the one true philosophy?’ ¹ Yet this was the moment when the emperor Constantine made his astonishing gamble with fortune by calling upon the Christians to provide an immediate and specific remedy for the ills of an expiring world.

¹ Augustine, Contra lulianum, iv. 14. 72: ‘obsecro te, non sit honestior philosophia gentium quam nostra Christiana, quae una est vera philosophia.’

Asimov reviews the history in Asimov’s New Guide to Science (1984), pages 299 and 397–9:

Lord Kelvin (William Thomson) and the English physicist James Prescott Joule had shown that, even in the gaseous state, a gas can be cooled simply by letting it expand and preventing heat from leaking into the gas from outside, provided the temperature is low enough to begin with …

… The French physicists Jean Baptiste Joseph Fourier, in 1822, and Nicholas Leonard Sadi Carnot, in 1824, studied the flow of heat and made important advances. In fact, Carnot is usually considered the founder of the science of thermodynamics (from Greek words meaning “movement of heat”). He placed the working of steam engines on a firm theoretical foundation …

Joule spent thirty-five years converting various kinds of work into heat … He found that a given amount of work, of any kind, always produces a given amount of heat. Joule had, in other words, determined the mechanical equivalent of heat.

Since heat could be converted into work, it must be considered a form of energy (from Greek words meaning “containing work”). Electricity, magnetism, light, and motion can all be used to do work, so they, too, are forms of energy. And work itself, being convertible into heat, is a form of energy.

These ideas emphasized something that had been more or less suspected since Newton’s time: that energy is conserved and can be neither created nor destroyed. Thus, a moving body has kinetic energy (“energy of motion”), a term introduced by Lord Kelvin in 1856. Since a body moving upward is slowed by gravity, its kinetic energy slowly disappears. However, as the body loses kinetic energy, it gains energy of position, for, by virtue of its location high above the surface of the earth, it can eventually fall and regain kinetic energy. In 1853, the Scottish physicist William John Macquorn Rankine named this energy of position potential energy. It seemed that a body’s kinetic energy plus its potential energy (its mechanical energy) remain nearly the same during the course of its movement; this constancy was called conservation of mechanical energy. However, mechanical energy is not perfectly conserved: some is lost to friction, to air resistance, and so on.

What Joule’s experiments showed above all was that such conservation could be made exact when heat is taken into account …

It was not till 1847 that a sufficiently respectable academic figure put this notion into words. In that year, Heinrich von Helmholtz enunciated the law of conservation of energy: whenever a certain amount of energy seems to disappear in one place, an equivalent amount must appear in another. This is also called the first law of thermodynamics. It remains a foundation block of modern physics …

Now, although any form of work can be converted entirely into heat, the reverse is not true. When heat is turned to work, some of it is unusable and is unavoidably wasted …

The capacity of any system to perform work is its free energy. The portion of the energy that is unavoidably lost as non useful heat is reflected in the measurement of entropy—a term first used in 1850 by the German physicist Rudolf Julius Emmanuel Clausius.

Clausius pointed out that, in any process involving a flow of energy, there is always some loss, so that the entropy of the universe is continually increasing. This continual increase of entropy is the second law of thermodynamics, sometimes referred to as the “running-down of the universe” or the “heat-death of the universe.”

According to Bell, how did cause get into physics in the first place, if indeed it was there? I have followed Collingwood in thinking cause is originally a feature of consciousness. Chapter XIV of The New Leviathan is “Reason”:

14. 3. Reason is distinguished into theoretical reason and practical reason: i.e. reason for ‘making up your mind that’ (reason for what logicians call a proposition) and reason for ‘making up your mind to’ (reason for what moralists call an intention). We shall see that, of these two, practical reason is the prior: it is the original form of reason, theoretical reason being a modification of it; and by the Law of Primitive Survivals a practical element is always present in a case of theoretical reason.

14. 5. Practical reason is prior to theoretical reason, which is a modification of it (14. 3).

14. 51. Were it not so, there would be no accounting for a tendency which, everyone knows, besets the work of theoretical reason: the tendency to anthropomorphism.

14. 52. We reason anthropomorphically when we seek reasons for the behaviour of things other than ourselves on the analogy of the reasons we have already found for our own behaviour.

14. 53. Thus when a fly-rod hooks me in the ear, a hammer hits me on the thumb, or a bicycle throws me into the ditch, I think of them as maliciously thwarting my endeavours to control them. In myself, malice of this kind is a familiar object; I cannot help ascribing it to these non-human agents.

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

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.

Pascal may have felt vertigo in contemplating the “Disproportion de l’homme” (Sellier 230, Lafuma 199, Brunschvicg 72): how the world about us is “a sphere of which the center is everywhere, the circumference nowhere,” while in every “abridged atom” (as Bell will have it below, but “shrunken atom” could do too: raccourci d’atome), there is “an infinity of universes.” However, I do not find the word vertige in Pascal’s Pensées. Baudelaire used it in his sonnet about Pascal:

LE GOUFFRE

Pascal avait son gouffre, avec lui se mouvant.
— Hélas ! tout est abîme, — action, désir, rêve,
Parole ! et sur mon poil qui tout droit se relève
Mainte fois de la Peur je sens passer le vent.

En haut, en bas, partout, la profondeur, la grève,
Le silence, l’espace affreux et captivant…
Sur le fond de mes nuits Dieu de son doigt savant
Dessine un cauchemar multiforme et sans trêve.

J’ai peur du sommeil comme on a peur d’un grand trou,
Tout plein de vague horreur, menant on ne sait où ;
Je ne vois qu’infini par toutes les fenêtres,

Et mon esprit, toujours du vertige hanté,
Jalouse du néant l’insensibilité.
— Ah ! ne jamais sortir des Nombres et des Êtres !

I was led to that poem when contemplating the pensée that I also quoted earlier, Sellier 131, Lafuma 97, Brunschvicg 334, itself an example of complementaries:

La concupiscence et la force sont les sources de toutes nos actions. La concupiscence fait les volontaires, la force les involontaires.

In any case, Leibniz had two principles, of contradiction and of sufficient reason, in ¶¶ 31 and 32 of the Monadologie, quoted earlier: “Nos raisonnements sont fondés sur deux grands principes, celui de la contradiction, en vertu duquel nous jugeons faux ce qui en enveloppe …” Here are the other paragraphs from which Bell now quotes:

14. L’état passager, qui enveloppe et représente une multitude dans l’unité ou dans la substance simple, n’est autre chose que ce

    • qu’on appelle la Perception,
    • qu’on doit distinguer de l’aperception ou de la conscience, comme il paraîtra dans la suite.

    Et c’est en quoi les cartésiens ont fort manqué, ayant compté pour rien les perceptions, dont on ne s’aperçoit pas. C’est aussi ce qui les a fait croire

    • que les seuls Esprits étaient des Monades et
    • qu’il n’y avait point d’Âmes
      • des Bêtes ni
      • d’autres Entéléchies, et
    • qu’ils ont confondu avec le vulgaire un long étourdissement avec une mort à la rigueur, ce qui
      • les a fait encore donner dans le préjugé scolastique des âmes entièrement séparées, et
      • a même confirmé les esprits mal tournés dans l’opinion de la mortalité des âmes.

81. Ce système fait que les corps agissent comme si (par impossible) il n’y avait point d’âmes, et que les âmes agissent comme s’il n’y avait point de corps, et que tous deux agissent comme si l’un influait sur l’autre.

In ¶ 14, I don’t think the que of qu’ils ont confondu is really parallel with any other.

Pascal, “Disproportion de l’homme” (Sellier 230, Lafuma 199, Brunschvicg 72):

Je veux lui faire voir là‑dedans un abîme nouveau. Je lui veux peindre non seulement l’univers visible, mais l’immensité qu’on peut concevoir de la nature dans l’enceinte de ce raccourci d’atome ; qu’il y voie une infinité d’univers, dont chacun a son firmament, ses planètes, sa terre, en la même proportion que le monde visible, dans cette terre des animaux, et enfin des cirons dans lesquels il retrouvera ce que les premiers ont donné, et trouvant encore dans les autres la même chose sans fin et sans repos, qu’il se perde dans ces merveilles aussi étonnantes dans leur petitesse, que les autres par leur étendue, car qui n’admirera que notre corps, qui tantôt n’était pas perceptible dans l’univers imperceptible lui‑même dans le sein du tout, soit à présent un colosse, un monde ou plutôt un tout à l’égard du néant où l’on ne peut arriver ?

Leibniz, again from the Monadologié:

64. Ainsi chaque corps organique d’un vivant est une espèce de machine divine, ou d’un automate naturel, qui surpasse infiniment tous les automates artificiels. Parce qu’une machine, faite par l’art de l’homme, n’est pas machine dans chacune de ses parties. Par exemple, la dent d’une roue de laiton a des parties ou fragments, qui ne nous sont plus quelque chose d’artificiel et n’ont plus rien qui marque de la machine par rapport à l’usage où la roue était destinée. Mais les machines de la nature, c’est-à-dire les corps vivants, sont encore machines dans leurs moindres parties jusqu’à l’infini. C’est ce qui fait la différence entre la Nature et l’Art, c’est-à-dire, entre l’art divin et le nôtre.

The balancing of desire and restraint is illustrated in Paul G. Hewitt, Conceptual Physics, 3rd edition (1977), page 24, which is opposite the first page of Chapter 3, “Newton’s Laws of Motion.”

In a photograph, two little girls point up to a large rock and a small rock, which are held up by a man, who is looking at us. Speech balloons give the girls and the man these words:

When you drop the two rocks simultaneously, the big one will hit the ground first because it is heavier—which means that its [sic] pulled with a greater gravitational force which in turn produces a greater acceleration.

Not so! The big rock has more mass than the little rock, which means it has more inertia and will be more sluggish in its motion—by the time it gets around to fully responding to gravity, the small rock will already have hit the ground!

What does happen … and why?

I first saw this picture in 1977, when Hewitt’s book was the text for a rather freeform seventh-grade science course. Five years later, another physics teacher pointed out that Hewitt’s book had been intended for a university course. Nonetheless, the twelfth edition, of 2015, looks like a children’s book.

There are lots of colors, and bits of text here and there, as in a book from my childhood: Richard Scarry, Best Word Book Ever. My family must have had the first edition, of 1963, but the spread below, from the new revised edition at the Internet Archive, does not look any different from what I remember.

A new feature in the 12th edition of Conceptual Physics is QR codes for videos. I’m not sure what the benefits are. In § 7 of “One & Many,” I wrote about a physics film by Hume and Ivey, shown to us by Robert Morse, the high-school physics teacher I have already mentioned. In “Reading Ourselves to Death” (New Atlantis, Number 68, Spring 2022, pp. 73–79), Kit Wilson says we do more reading now than ever before, and

Reading encourages us to put outside reality on hold, to construct a parallel world in our minds, and retreat into it.

In a seeming echo of Bell’s earlier comments about “taking the one-way quantifications of Substance for unambiguous truth,” Wilson says,

First, we developed language, which, as Nietzsche pointed out, led us to believe that our words were the same thing as the reality to which they referred. Then we invented the written word, which codified and solidified language even more. Then we created the modern world, in which we now readily assume that models of reality encoded in computer language are all there is.

I don’t know who the “we” are who do this assuming. I am getting the sense that students today do not want to read, but would rather have everything on video. It is not clear that they should be indulged.

In Hewitt’s 12th edition, I could not find anything corresponding to the picture I selected above from the 3rd edition, the picture that illustrates Bell’s remark, “every unit of weight is not only pulled by an equal desire but restrained by an equal inertia.”

Three drawings in the 3rd edition that illustrated the concept of inertia have been given color in the 12th, but also a smaller size on the page.

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 215^a25–9. 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 308^a28, 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 308^b5, 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.

Galileo’s Plot of Uniform Acceleration:

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

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

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

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

since a t = v.

Pendulum

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

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

Hence Bell’s proportion

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

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:

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,

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.

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.

v₁ − v₂ = v₂′ − v₁′

m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′

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

expressing the conservation of kinetic energy. Rearrange and factorize:

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

Division now yields

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.

Garnot’s Ideal Engine

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.

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.