Anaxagoras Refuted: Fewness, Infinity and the Limits of Natural Quantity (Part 1)

Orientation #

He opens by closing the previous discussion (“we should stop here before we go into reading nine”) and says reading nine is attached “for a better understanding of this first matter.” He ends mid-sentence, about to begin the second, third and fourth arguments against Anaxagoras. The previous class treated prime matter as knowable only by proportion, and the next treats Aristotle’s second to fourth arguments against Anaxagoras as a linked sequence.

The class, in order #

1. Angels as spirits and flame of fire (“He’s made his angels spirits”) #

Aside before the reading: Thomas on Hebrews 1 (Catena Aurea), two names for the angels.

• Messenger / spirit (from breath, air): a messenger receives a message and passes it on; air quickly receives sound and colour and carries them to eye and ear, and is swift.
• Minister / flame of fire: three things about fire — most active of the four elements (fitting their doing things), it warms (their role in encouraging charity), the flame goes upward (they refer everything to God’s glory).
• Student: fire also enlightens. He: that belongs to the messenger/teacher likeness, not the minister one. A metaphor is adapted by the quality used: lion for Christ (courageous, triumphs, king of beasts) and lion for the devil (Peter: prowling, seeking whom he may devour); fire for charity, or for anger or concupiscence.
• Also: the last leaf of the “cat” example from last time — letters in their order are parts of the word before one sees body-and-soul, “only proportional.”

2. Is the first matter an actual substance? (“substance in ability only”) #

The point: the first matter must be a substance in ability only.

1. The pre-Aristotelian Greeks made the first matter some actual substance (earth, water, air, fire, or Empedocles’s four).
2. If there is really change of substance, the underlying subject cannot be an actual substance; otherwise that substance would be eternal and all change would be accidental modification of it.
3. They could not understand how the first matter could be ability only (ability is hard to understand; we imagine it as something actual) — and Anaxagoras shows the difficulties of making first matter a combination of actual substances.

3. Man is one thing: the argument from Categories (“Is a man one thing with many parts”) #

1. Aristotle’s example of a substance is a man; a dog, cat, horse. If man is a substance there is change of substance, since men come to be and die.
2. Deny it and man becomes many substances arranged, like a word of letters — no more one than an army, family or city. The early Greeks and many modern scientists imagine a man as earth, air, fire, water or atoms and molecules “poured together.” He asks the class’s inward experience: are you one thing?
3. A change from man to dog needs a subject, as sphere to cube needs clay. But if that subject were an actual substance like clay, man and dog would be accidents, not substances — contrary to the starting point. So the subject is substance in ability: able to be man or dog, not both at once.
4. Known by proportion — first matter : man, dog :: clay : sphere, cube — and knowable only so: by itself it is neither actual substance nor actual accident, and ability is never known through itself but through its act (seeing, walking, thinking); pure ability most of all “easily escapes our understanding.”

4. Elements and elementary particles (“no elementary particle is eternal”) #

The point: the same argument from the lowest forms of matter, less sure than from man and dog.

• If earth, air, fire, water are the first forms of matter and they change (burn, dissolve), something underlying is substance in ability. Less sure, because we are not sure they are the first forms.
• Modern science: water is not first (hydrogen, oxygen); atoms are not first (proton, electron, neutron — “elementary particles”). High-energy experiments: no elementary particle is eternal; Heisenberg: “the complete mutability of matter.”
• Is there something smaller below? A new question: when one particle is shot at another, the target disappears, and sometimes what results has more mass than the original — unlike breaking water into lighter hydrogen and oxygen. So the elementary particles are not made of smaller things; yet they change into one another, so there is an underlying subject that is not an actual substance. Heisenberg: we know matter only through the forms of matter.
• Heisenberg and Pauli (Nobel Prize, quantum theory) then studied elementary particles; from any particle you can eventually get all the rest; hence the “well-known formula”: every elementary particle is composed of all the rest — exactly Anaxagoras’s way of speaking, and it runs into the same difficulties. Both reveal the mind’s difficulty in seeing that you can get out of something what is not actually in it.

5. Chairs out of the next room, chairs out of trees (“Can you get money out of my pocket”) #

• No money in the pocket, no money out of it; no chairs in the other room, no chairs out of it — that sense of “in” is in a place, and what is in a place is actually there.
• Chairs come out of trees: they were in the trees “potentially,” in ability — a much more hidden way of getting something out.
• So the most common fallacy, equivocation, is at work: one imagines being in matter as being in a place. Weizsäcker (Heisenberg’s student): when we imagine, we make it actual in imagination — “false imagination,” the fundamental cause of deception on the side of our powers.

6. Eight meanings of “in”, to the fifth (“part in a whole”) #

Physics IV (the “Natural Hearing”) gives eight meanings; Thomas orders them as Aristotle teaches in Metaphysics V (“fifth book of Wisdom”). He goes through five:

1. In place: we and the chairs in this room.
2. Part in a whole: teeth in the mouth. Very close to (1): a tooth loosened so it comes in and out is in the mouth as in a place, not as a part; a disc sawn from the tabletop and put back is no longer part of the table.
3. Genus in species, and generally part of a definition in the definition: square = equilateral right-angled quadrilateral; quadrilateral, equilateral, right-angled are actually parts of it. This is more in the mind, while (2) is what the senses reach — so (2) is more known to us and comes first.
4. Species in genus: man in animal (animal is in man in the third sense; man is in animal in the fourth). Like (2)/(3) it is part in a whole, but the genus is a universal whole (Greek katholou from kata + holon; English “particular” from “part”; later Greek merikon from meros). Here what is in is in only in ability, not actually.
5. Form in matter — very like species in genus; again in ability.

• Boundary: senses 1–3, actually in; 4 and 5, in ability.
• Later sense: “I’ve got you in my power” — a different, active ability (“out of my hands,” “in God’s hands”). He stops at five.

7. Locke’s triangle and Anaxagoras’s matter (“it’s all and none of these”) #

• Locke on the general idea of triangle (three-sided plane figure): are the sides equal, or two, or none? He answers “all and none of these” — trying to imagine triangle in general, which cannot be done (any imagined triangle is equilateral, isosceles or scalene). Berkeley then: this is nonsense, so there are no general ideas.
• The right answer: the sides are none of these actually, all of them in ability; Locke cannot understand the ability of the genus.
• The same confusion arises with matter: the things that come out of matter are all “in there” and none — Anaxagoras says all in there, as infinitely small, indistinguishable pieces. Aristotle (in the Metaphysics, “I guess”) says if he had listened to what Anaxagoras was trying to say he would have had the truth.

8. Paradox in ordering the fourth and fifth (“the clay or its shape”) #

• Part in whole is more sensible than genus in species, so it comes first; what is most like (3) is (4), species in genus. Yet (4) is mental, while form in matter (5) seems to be in the sensible world (shape in the chalk, in a body) — one might have expected (5) before (4). Thomas orders by nearness to the third meaning.
• Test for an “unprepared person”: clay sphere moulded into a cube — what changed, the clay or its shape? The average person says the shape: speaking as if the genus (shape) were the subject underlying the two species, sphere and cube. A sign the genus is more known to us than the matter. Locke makes the genus contain the differences actually and lands in contradiction.

9. Aristotle recounts Anaxagoras’s reasoning (“nothing comes to be from what is not”) #

He reads the first three paragraphs of the handout. Two premises:

1. By induction: from anything in nature you eventually get everything else — grass → cow → “Berquist has a steak” → Berquist fed to the lions → worms → bird → cat → “pushes up daisies.” The same appears in elementary-particle experiments.
2. Common to all the Greek naturalists: you cannot get something from nothing. Conclusion: everything is inside everything — imagined, falsely, as actually in there, hence everything composed of everything (if hydrogen and oxygen come out of water, water is composed of them; so any particle is “composed of all the rest”). “Clear as mud.”

• Added premise: coming to be goes on forever, so an infinity of pieces of everything in everything — fitted into a finite blade of grass only by being infinitely small, like mathematicians’ line “composed of an infinity of points” (false; they imagine what is able to be on the line as actually there). Anaxagoras’s position: an unlimited multitude of everything, unlimitedly small, in everything.
• Objection: then why call this a dog and that a tree? Answer: by what a thing has most of — the “Anaxagorean way of naming,” which we follow: the Psalms classed as prayers/sapiential though they contain history and prophecy; the Comedy of Errors called a comedy though it has serious scenes.

10. Numbering the eight arguments; the principle of fewness (“fewer beginnings, fewer causes, are better”) #

Numbering from the bottom of page 4: (1) “if the unlimited as unlimited is unknown”; (2)–(4) “in addition”; (5) “further”; (6) “it is not knowingly said”; (7) “nor does he rightly take the coming to be”; (8) “better to take fewer limited beginnings,” as Empedocles.

• Thomas’s division: the first seven against the position in itself vs. the eighth by comparison with Empedocles; within the seven, the first five against what is essential to the position vs. six and seven on more particular difficulties a more careful thinker could avoid.
• Berquist’s grouping: 8, 5 and 1 together, because they share with modern physical science (Galileo, Kepler, Newton, to Einstein) the principle of fewness or simplicity: fewer beginnings are better if they are enough. Newton’s rules: nature is pleased with simplicity and affects not pompous and superfluous causes; Galileo looked for the simplest way a falling stone might increase speed; Kepler spent ten years seeking the simplest figure fitting the planets and settled on the ellipse (not as simple as a circle, but Ptolemaic circles on circles on imaginary centres got too complicated; the ellipse has two centres and the sum of distances is always the same).
• The eighth argument: Anaxagoras needs an infinity of beginnings to explain coming-to-be going on forever; Empedocles explains the same with four elements and love and hate. Anaxagoras’s coming-to-be is a straight line (keep taking cow, lion, man, worm out of the grass forever — needs an infinity inside); Empedocles’s is a circle (love gathers, hate separates, forever with finite materials).
  • A bag of blocks vs. a finished toy at Christmas (“the Empedoclean principle”): build, knock down, build again; a finished toy breaks and needs an infinity of toys.
  • The rich man who wants to run forever: a straight road must be infinite; a circular track is finite.
  • Recycling vs. straight-line production: resources are not infinite.
  • Assume the same teacher each Tuesday rather than hundreds of identically trained look-alikes.
  • Ancients and moderns both explain day and night with one sun (whether the sun circles the earth or the earth turns) rather than a new sun shot across and extinguished each day (as in poetry) — which would need an infinity of suns.

11. The fifth argument: infinities of infinities (“could hardly be more complicated”) #

Inside everything an infinity of infinitely small pieces of everything; inside each of those, again an infinity; and so on forever — there must be something wrong. Max Born (The Restless Universe): the genuine physicist believes in the unity and simplicity of nature despite appearances — said of the periodic table’s roughly 92–100 “first matters” being too many; that is nothing beside infinities of infinities. Not among the four arguments against infinite beginnings in reading 11 (arguments 8 and 1 were used there); 5 is peculiar to Anaxagoras.

12. The first argument: the unlimited is unknowable (“nature does nothing in vain”) #

“More hidden,” so treated last. If the beginnings are infinite we cannot know them. What is wrong with that? Man has a natural desire to know the causes; nature does nothing in vain (Newton again), so the desire must be for something possible — as hunger and thirst do not guarantee food and water but show they are not impossible. A word with an infinity of letters: the national spelling champion could never know it.

• Student: what about pi — we know something about it and work with it but cannot read it out? He: yes, “irrational”; the continuous is divisible forever but cannot actually be divided completely; pi results from the Greek discovery that not every line has to every line the ratio a number has to a number (see Heath’s notes to Euclid V) — one can approximate. But does the mind have a natural desire to divide a line forever, or count forever? No — there is no goal there.
• He puts 8, 5, 1 together “to kill many birds with one stone” and to facilitate the comparison with modern science. Breaks off: “Now, the second, third, and fourth…”

His words #

• ability — [ed.: potency]; known only through its act.
• substance in ability only — the first matter, as opposed to an actual substance.
• first matter — “matter before other matters.”
• the Natural Hearing — [ed.: Physics].
• fifth book of Wisdom — [ed.: Metaphysics V].
• katholou / holon, merikon / meros — general from “whole,” particular from “part.”
• false imagination — making the potential actual in imagination (via Weizsäcker).
• the Anaxagorean way of naming — naming things by what they have most of.
• principle of fewness / simplicity — fewer beginnings are better if they are enough.
• Empedoclean principle (at Christmas time) — circle over straight line: blocks over a finished toy.

Texts #

Read in class:

• Aristotle, Physics I, on Anaxagoras (handout “Anaxagoras’ unlimited beginnings and their refutation,” first three paragraphs; arguments numbered from bottom of p. 4); Aquinas’s commentary, lectio 13, for the division of the eight arguments.
• Thomas Aquinas on Hebrews 1 (from the Catena Aurea volume), angels as spirits and flame of fire.

Mentioned:

• Aristotle, Categories (man as example of substance).
• Aristotle, Physics IV (eight meanings of “in”); Aquinas’s ordering per Metaphysics V.
• Aristotle, Metaphysics (Anaxagoras would have the truth if he listened to himself — “I guess”).
• Heisenberg, lectures on the unified field theory of elementary particles; Weizsäcker on imagination.
• Locke on the general idea of triangle; Berkeley’s reply.
• Newton, rules of reasoning; Galileo; Kepler; Einstein.
• Max Born, The Restless Universe (periodic table).
• Euclid, Elements V with Heath’s notes; reading 11 (four arguments against infinite beginnings).
• Psalms; Shakespeare, Comedy of Errors.

His questions #

• Is the first matter an actual substance or a substance in ability only? Substance in ability only; otherwise all change would be accidental.
• Is a man one thing with many parts, or many things arranged together? One thing — a man’s inward experience; otherwise no more one than an army.
• Why is the first matter knowable only by proportion? By itself it is neither actual substance nor accident, and ability is known only through act.
• Are the elementary particles eternal, or made of something smaller? Not eternal (Heisenberg); not made of smaller things, since collisions yield heavier particles.
• Can you get chairs out of the other room if there are no chairs there? Were the chairs in the trees? No; in the trees only potentially — a different sense of “in.”
• Are the three sides of triangle in general equal or unequal? None actually, all in ability — what Locke could not see.
• A clay sphere moulded into a cube: what changed, the clay or its shape? The average person says the shape — treating the genus as subject, a sign it is more known to us than matter.
• Why call this a dog and that a tree, if everything is in everything? Anaxagoras: by what it has most of.
• How long must the road be for a man who wants to run forever? Infinite — unless he runs a circular track, which is finite.
• What is wrong with the consequence that infinite beginnings are unknowable? Nature does nothing in vain; the natural desire to know causes must be for something possible.
• What about pi — do we have a natural desire to divide forever? Pi can be approximated but not read out; no, there is no goal in dividing or counting forever.