Smallest Pieces and Modern Physics: Arguments 2-7 Against Anaxagoras, Locke and Ability

Orientation #

The recording begins mid-sentence, already inside the numbered arguments of Reading 9. He refers back to “the first difference” between natural philosophy and mathematics “that we saw in our reading on the difference between natural philosophy and mathematics,” and to the Democritus texts “that we saw before” and the fragments of Anaxagoras and Empedocles. He says near the end why “I attach this ninth reading to the reading we had before”: it lets us go further into first matter as pure ability. The previous class had already begun Reading 9 on prime matter and the eightfold critique of Anaxagoras.

The class, in order #

1. Arguments 2, 3, 4 are linked; the “second difference” (“the second difference between quantity in pure math”) #

Point: arguments 2, 3 and 4 form one chain (what 2 proves is the premise of 3 and 4), and all three depend on a discovery Aristotle was first to make.

• His grouping: arguments 8, 5 and 1 share the principle of fewness, common to all physical science “from Galileo, Kepler, Newton, to Einstein and beyond”; 2, 3 and 4 share the second difference, which enters physical science only in the twentieth century.
• Terminology of the physicists (Einstein, de Broglie, Bohr, Heisenberg): “modern physics” = twentieth century; “classical” or “Newtonian physics” = seventeenth to nineteenth century (“classical” does not mean the Greeks; Newton is the model because he united Galileo and Kepler).
• First difference recalled: natural philosophy considers number, shape, length, width, depth only as the number or shape of natural bodies; the geometer and arithmetician consider them in separation from all sensible bodies. The geometrical sphere has no matter, is neither heavy nor light, hard nor soft, has no color, sound or taste. Few after Aristotle saw this clearly, “though these good students like Thomas had seen it.”
• Second difference, his statement: natural quantities have limits, in the direction of the large or the small or both, due to the natures of these things; limits discovered through sense experience, but not foreseeable from pure math, which considers quantity in separation from natures. Seeing the first difference is what opened Aristotle to the second.

2. Second argument: no infinitely small parts (“the if and what syllogism”) #

Point: Aristotle refutes the claim that parts of flesh, bone etc. can fall below any size, using the refuting form of the if-then syllogism.

• Form he gives: if A is so, then B is so; B is not so; therefore A is not so (contrasted with the obvious form: if A then B, A, therefore B).

1. If the parts of an animal or plant can fall below any size, the whole can fall below any size (the whole is composed of the parts; “if you understand the relation of part and whole, the if-then statement is kind of obvious”).
2. But the whole cannot: cats, elephants, each kind has limits above and below which you don’t find that kind, with variation within the limits; trees here vs. redwoods/sequoias in California; grass never as tall as trees. Different natures, different limits.
3. Therefore the parts cannot fall below any size: there must be a smallest piece of flesh, bone, etc.

• Student answer that Anaxagoras held “it’s made of parts that are infinitely small” confirmed as the position overthrown.
• Aside: “domino theory”: Anaxagoras reached his position by steps, so knocking over the last knocks over the earlier ones.

3. Third argument: same ratio vs. same amount (“there is a smallest of the small”) #

Point: given a smallest piece of flesh, you cannot take flesh out of a finite blade of grass forever, so Anaxagoras’ “infinity of pieces of everything inside everything” fails.

• Contrast: a mathematical line can be halved, then the remainder halved, forever, because the pieces fall below any given size (same ratio); but taking the same amount each time, however small, from a finite line cannot go on forever, since multiplying the same amount infinitely eventually exceeds any finite quantity. Math can take the same ratio forever because, as Anaxagoras said in a fragment, there is “no smallest of the small” there; in nature there is one, so you must take out at least that amount, and generation would stop, contrary to what all the Greeks held.
• Example 1 (mathematical retail): Berquist selling mathematical lines: half to the first customer, half of what’s left to the next, never has to reorder, “never runs out”; but “they keep on getting smaller, smaller, smaller.”
• Example 2 (canteen in the desert): giving half the water, then half of what’s left; Berquist eventually runs out, because there is a smallest piece of water, “maybe the molecule,” which has a definite size, so a canteen cannot contain an infinity of molecules.

4. Fourth argument: not everything is in everything (“in the smallest piece of flesh”) #

Point: the smallest piece of flesh can contain no bone, so the original conclusion of Anaxagoras is overthrown.

1. If part of the smallest piece of flesh were bone, only part of it would be flesh, so there would be flesh smaller than the smallest: contradiction. Same for the smallest piece of bone containing flesh.
2. Therefore not everything is actually inside everything. He recaps how 2, 3, 4 chain together.

• Qualification: there is a way everything is in everything, “in ability”; making it actually there runs into the second difference.
• Modern chemistry is “based on Aristotle’s idea that there’s a smallest piece”: atom for each element (different atoms different sizes, every hydrogen atom the same size), molecule for each compound; something smaller than a water molecule exists, but it is not water.
• Elementary particles: if every particle were composed of all the rest, you would have the fifth argument’s difficulty over again, particles smaller and smaller; contrary to lab experience that all electrons have the same mass. So each has the rest in it only in ability. Heisenberg saw this, but the “natural” way of speaking, as if composed and you get them out, is easier for the imagination.
• de Broglie’s parallel: nineteenth-century physicists would say white light striking a prism is composed of the colors, which the spectrum merely separates; now we say the colors exist in white light only as a possibility, which would surprise the nineteenth-century physicist.
• Two fallacies involved: false imagination (you cannot imagine something in something without imagining it actually there); equivocation (two senses of “in”), “the first fallacy in language”; and simply / in some respect, “the second one outside of language.” Are there chairs in the trees? Without qualification, no; qualified, in ability, yes. Is a cube in the clay sphere? Only in ability.

5. Twentieth-century physics: limits small and large (“modern physics begins officially in December of 1900”) #

Point: the two best-established parts of modern physics are each the recognition of a limit in natural quantity.

• Quantum theory: Planck, December 1900, black body radiation; Newtonian physics gave absurd consequences (infinite energy); his hypothesis: energy is given or received only in a smallest amount or its multiples, nothing less, nothing between. Einstein 1905: light not understood without the quantum; Bohr 1913: the atom; perfected by the late 1920s. Newtonian physics assumed energy in any amount (half, a quarter, an eighth). A limit in the direction of the small.
• Monetary example: two friends, two beers each and one 25-cent bag of potato chips; the departing friend’s share is 12½ cents, but there is no way to pay it: a penny or its multiples, nothing between one and two pennies. Energy “comes in pennies,” except the penny is arbitrary and the quantum natural.
• Special relativity, 1905: the speed of light as the maximum speed in the universe; Newtonian physics has no limit to speed. A limit in the direction of the large.
• Newtonian physics as “a mathematical science of nature”: applying pure math, which implies what is true of pure math is true of nature (4 − 2 = 2: a dog with two legs cut off has two left; likewise a chair); largely true, but anything happening to quantity due to natures escapes the pure mathematician. In pure math there is no shortest or longest line; Euclid’s fourth book (inscribing and circumscribing circles and squares) implies figures can get smaller and bigger without limit.
• “The discovery of these limits is a sign that modern science is getting closer to the natures of things.” Heraclitus, “the great common teacher”: nature loves to hide. It also explains why for centuries math was assumed true of natural quantity: the first difference was not seen.

6. Democritus, then the frontiers: minimum length and finite cosmos (“Democritus arrived at his position by a thought experiment”) #

Point: Democritus may already have felt the difficulty behind the second difference; the current frontiers (elementary particles, cosmology) are also marked by limits.

• Two texts on Democritus: in the De Generatione (later book, not specified) Aristotle speaks as if the atoms were points; elsewhere as having size and shape. Thought experiment recalled: a body cut every way it can be cut; something uncut must remain (you cannot make something of nothing); “atom” means uncut. Berquist’s diagnosis: an ambivalence; thinking mathematically, the only indivisible is the point; but a line cannot be made of points (shown before), so atoms need size, hence a limit in the small: very small, not infinitely small.
• Fragment attributed to Democritus (Loeb Greek mathematical works): a cone (he first says pyramid) cut parallel to the base; are the two faces equal? If equal everywhere, a stack of equals is a cylinder; if unequal, the side goes “chit-chit-chit” in steps. Shows the paradox of a body as a pile of surfaces, as a line as points: lines composing a surface must touch, and touching, having no width, they coincide; surfaces composing a body likewise, having no depth.
• Elementary-particle physicists (Heisenberg, Gell-Mann) hypothesised a minimum length, 10⁻¹³ cm; “not established,” but a sign their thinking has changed, since a minimum length is “so contrary to pure math.”
• General relativity (1915, confirmed after the First World War) opened cosmology; the universe now seems finite, as Aristotle thought, rather than infinite as the early Greeks and Renaissance physicists held (Koyré’s title recalled). Expansion and the big bang: the universe perhaps limited in time too; Planck particle, Planck time mentioned. Both frontiers may be characterised by discovering limits, small and large. Aristotle “would be less surprised than anybody else in the twentieth century.”
• Recap of the grouping: 8, 5, 1 (fewness) and 2, 3, 4 (second difference).

7. Arguments 6 and 7: substance and accident; like to like (“a mixture of substance and accident”) #

Point: 6 and 7 are more particular difficulties, but 6 reveals Anaxagoras’ failure to see substance vs. accident.

• Sixth: in the fragments Anaxagoras mixes flesh, blood, bone with color, taste, odor, and mind separates them. One can separate chicken skin from bone (both substantial), but not the bone from its color; “you could put my arm in there and my leg in there, but you couldn’t put the shape of my arm in there and my arm in there.” A mind separating accidents from substances is “a kind of stupid mind.”
• The two main divisions of being: being in act vs. being in ability; substance vs. accident (quantity, quality, “more besides”). Anaxagoras confuses act and ability (putting what is in the ability of matter actually there), hence matter and form (form is act, matter ability), and also substance and accident.
• Bertrand Russell’s remark: Mr Smith’s accidents have no more need of a substance to inhere in than the Earth of an elephant to rest upon. Reply: an accident that exists by itself is a substance; he does not know what an accident is. Substance disappears in modern philosophy: Berkeley (only mind and thought), Hume (no substance). “Only substances exist” makes some sense; “only accidents exist” makes none. Moderns saying every particle is composed of all the rest confuse act and ability similarly.
• Student: doesn’t Heisenberg see that? He does; the “well-known formula” phrase he uses several times is a common way of speaking; a chemist (Pindell? [unclear in recording]) said it is easier to speak that way. It is easier to see how animal is in man than how man is in animal.
• Seventh: Anaxagoras spoke as if all coming to be were addition of like to like (more pieces of man to get a man). Aristotle: sometimes unlike things are brought together, as wood screwed into cement in a basement.
• Root of all the difficulties: he wants everything that comes to be out of something to be actually in there, so must make it infinitely small; the physicist does the same with the particle.
• Student links this to the big bang (everything compact, huge density). Reply: the interesting point is that physicists now entertain the universe as limited in time. Aristotle thought it always was; Thomas, in the De Aeternitate Mundi and elsewhere, examines all arguments for and against and finds none demonstrative; by faith he holds a beginning. The big bang is not conclusive but more in accord with a beginning.

8. The same mistake in logic, math, theology: act and ability (“all of these in ability, but none in act”) #

Point: Anaxagoras’ error is proportional to Locke’s, the pantheist’s, and the schoolchild’s line of points; the common root is the difficulty of understanding ability.

• Locke: his difficulty is logical. Genus is to difference as matter to form (genus taken from matter, difference from form, because the things we know have something like matter and form). Making matter composed of the forms it can receive is like putting the differences actually into the genus. Matter “is all of these things … but in a way, it’s none of them.” Locke’s general triangle, equilateral or isosceles or scalene, right or obtuse, “all and none”: he cannot see it is all in ability, none in act; the difference is actual, in the genus only in ability.
• Porphyry’s “species-making difference” goes back to Aristotle, Topics book six, eidopoios; eidos/species names a form you can see; in English “form” is used for species: democracy as one form of government; drama, novel, short story, epic as forms of fiction.
• Pantheism: everything is in God’s ability to produce; the pantheist puts it actually in God, so God becomes a composition of all things (“pan means all things”), a different kind of ability, an active one, but the same mistake.
• Nixon on the Democrats’ program: any way you slice it, baloney, so made of baloney; cut a line anywhere and you get a point, so made of points? Points touching must coincide; a million or an infinity coinciding have no more length than one, which is none. A line has an infinity of points in ability; children are taught it is composed of an infinity of points, making actual what is there only in ability.
• “The inability of man’s minds … to understand ability is a common, common inability”: in math, in logic, in natural philosophy. This is why Reading 9 was attached to the previous reading: first matter is pure ability, divisible [unclear in recording], hard to understand.
• Closing caution: that hydrogen and oxygen can be gotten out of water is not enough to say water is composed of them; if it were, every particle would be composed of all the rest, absurd especially when what comes out is bigger than the original. The Greeks (explicitly Anaxagoras and Empedocles) understood every change as change of place, which needs no radical grasp of ability: what changes place already actually exists.

His words #

• “second difference” — limits in natural quantities, large or small, due to natures, unforeseeable from pure math.
• “first difference” — natural philosophy considers quantity as of natural bodies; math in separation from sensible bodies.
• “if and syllogism” — if A then B; not B; therefore not A [ed.: modus tollens].
• “principle of fewness” — what arguments 8, 5, 1 share.
• “ability” — [ed.: potency]; “in ability” vs. “actually in there.”
• “same ratio” vs. “same amount” — halving forever vs. removing a fixed quantity.
• “no smallest of the small” — Anaxagoras’ fragment, true in math, false in nature.
• “simply / in some respect” — second fallacy outside language; “equivocation” the first in language.
• “species-making difference” — Porphyry, from Aristotle’s eidopoios; eidos = a form you can see.
• “domino theory” — overthrowing Anaxagoras’ last step overthrows the earlier ones.

Texts #

Read in class

• Aristotle, Physics I, Reading 9, arguments 2, 3, 4, 6, 7 against Anaxagoras (handout “Note for Book One, Reading Nine”, p. 2, verified)
• Handout “The First Definition of Reason”, p. 103 (verified; passage not identifiable from transcript)
• Handout “On the Natural Fragments”, p. 33 (verified; Anaxagoras fragment “no smallest of the small”)

Mentioned

• Anaxagoras, fragments (mixture of flesh, blood, bone, color, taste; mind separating)
• Empedocles, fragments (change as change of place)
• Aristotle, De Generatione, a later book (Democritus’ atoms as points)
• Loeb, Greek Mathematical Works, fragment attributed to Democritus (cone cut parallel to base)
• Aristotle, Topics VI (eidopoios); Porphyry (species-making difference)
• Euclid, Elements IV (inscribing/circumscribing) (inferred: “the fourth book”)
• Heraclitus, “nature loves to hide”
• Thomas Aquinas, De Aeternitate Mundi
• Koyré, From the Closed World to the Infinite Universe
• Bertrand Russell (Mr Smith’s accidents); Locke (general triangle); Berkeley; Hume; de Broglie; Heisenberg; Gell-Mann; Planck; Bohr; Einstein

His questions #

• If the parts can fall below any size, what about the whole? Then the whole can fall below any size.
• Do we find that different kinds of animals and plants have just any size? No; each nature has limits above and below.
• Can there be an infinity of the same amount of flesh in a finite blade of grass? No; multiplying the same amount infinitely eventually exceeds the finite.
• Can Berquist go on giving water forever? No; there is a smallest piece of water with a definite size.
• In the smallest piece of flesh, is there any bone? No; otherwise there would be flesh smaller than the smallest.
• Are there chairs in the trees out there? Not simply; in ability, yes.
• If you cut a cone parallel to the base, are the two faces equal or unequal? Equal gives a cylinder; unequal gives a stepped side; either way a paradox.
• Any time you cut a line you get a point; must it be made of points? No; touching points coincide and have no length.
• Is getting hydrogen and oxygen out of water enough to say water is composed of them? Left open (“maybe, maybe a little bit careful there”).