19. Anaxagoras on Matter, Mind, and Infinite Divisibility

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Lecture Notes

Main Topics #

Anaxagoras’s Inductive Argument for Everything in Everything #

• Begins with observable induction: grass becomes cow, cow becomes man, man becomes food for lions, and so forth
• Conclusion: everything eventually comes from everything else
• Preservation of the principle “you cannot get something from nothing” (οὐκ ἐκ μή ὄντος): if cow comes from grass, cow must already exist in grass in some form
• Because new things come into being forever (each spring), there must be infinite pieces of everything inside everything
• To fit infinite pieces into finite bodies, these pieces must be infinitely small

The Principle of Infinite Divisibility #

• Anaxagoras: “Nor is there a smallest of the small, but there is always a smaller. For what is cannot cease to be by being cut.”
• The continuous (τὸ συνεχές) is that which is divisible forever
• Cannot be composed of points: points have no parts and cannot touch except by coinciding (which produces no length)
• When dividing a line, you never reach points or nothing—only shorter lines
• This principle is mathematically sound but creates problems when applied to natural material bodies

Anaxagoras and Modern Elementary Particle Physics #

• Twentieth-century physicists discovered any elementary particle could eventually produce all other elementary particles
• Scientists concluded every elementary particle is composed of all the rest—exactly Anaxagoras’s logic
• Modern physics uses language strikingly similar to Anaxagoras about infinitely small pieces
• However, modern physics had to reintroduce Aristotle’s concept of potency (δύναμις/ἀκτουαλότης) to escape the problems Anaxagoras faced
• Anaxagoras gropes toward the idea of ability/potency but cannot quite express it

Mixing and Separation vs. Generation and Corruption #

• Anaxagoras denies true generation and corruption: “The Greeks do not rightly take coming into being and perishing. Nothing comes to be or perishes.”
• What appears as birth is mixing; what appears as death is separation
• A newborn child is not truly a new being but a new mixture of pre-existing pieces
• A corpse is not truly destroyed but separated into different arrangements
• This preserves the axiom that nothing comes from nothing and nothing becomes nothing
• Anaxagoras, like Empedocles, cannot understand how inward/qualitative change is possible without violating fundamental principles

The Principle of Conservation: “All Things Are Forever Equal” #

• From the fragments: “These things having been thus separated, it is necessary to know that all things are neither more nor less.”
• Because something would come into existence out of nothing if there were more, or cease to exist if there were less
• Modern mathematical science is fundamentally based on this principle
• Algebraic equations express this: in 2x + y = 2(x + y), you cannot lose a y or gain a y—nothing comes from nothing, nothing goes into nothing
• Conservation laws in physics (energy, momentum, mass) are expressions of this principle
• This framework assumes that only change of place occurs in the world; all other apparent changes are rearrangements

The Mind (Nous) as Separate and Unlimited #

• Mind is fundamentally different from all other material things
• “Other things have a part of everything, but mind is unlimited”
• Three characteristics: unlimited (in knowing ability), self-ruling (αὐτοκρατές), and self-knowing
• Mind is the thinnest and most subtle of all things, yet most penetrating
• Mind causes the circular motion that separates the heavy (settling at center) from the light (flying outward)
• The good arrangement (τάξις) of parts in animals and plants is due to mind, not chance
• Circular motion as a cosmic principle: humans use circular motion to separate things and generate force (boxing, cyclotrons, amusement park rides)

Relative Size and Quantity #

• Large and small are relative terms (πρός τι) not absolute quantities
• A hundred people is a big dinner in your house but a small crowd at a baseball game
• A seven-foot man is tall compared to ordinary humans but small compared to a redwood tree
• Examples: Betelgeuse contains the entire solar system; molecules in a glass of water; proportions (as the ape is to man, so man is to God)

Key Arguments #

The Infinite Divisibility of the Continuous #

1. What is cannot cease to be by being cut (you cannot cut something into nothing)
2. Therefore, when you divide a line, you get shorter lines, not indivisible points
3. Since you always get something divisible, you can always divide further
4. Conclusion: There is no smallest of the small; the continuous is divisible forever
5. Mathematical proof by contradiction: Points have no parts and cannot touch except by coinciding; coinciding produces no length; therefore a line cannot be composed of points

The Argument Against True Generation and Corruption #

1. If something truly came into being, it would come from nothing (before it existed, it was not)
2. But you cannot get something from nothing (you cannot get blood from a turnip)
3. Therefore, nothing truly comes into being—only mixing of pre-existing pieces
4. Similarly, nothing truly perishes—only separation of existing pieces
5. Implication: All apparent qualitative changes are actually rearrangements of matter through change of place

Why Everything Must Be in Everything #

1. Things perpetually come into being in the natural world (each spring brings new growth)
2. If grass becomes cow, and nothing comes from nothing, cow must already exist in grass
3. If things keep coming into being forever, there must be infinite pieces of everything in grass
4. To fit infinite pieces into a finite blade of grass, pieces must be infinitely small
5. Conclusion: “All things were together, unlimited in number and in smallness”

Important Definitions #

τὸ συνεχές (The Continuous) #

• That which is divisible forever
• Distinguished from discrete quantity (number) by the fact that parts meet: two parts of a line meet at a point; two parts of a circle meet at a line
• Cannot consist of indivisible units
• Examples: lines, surfaces, bodies, time, place

ἀπειρα (The Unlimited/Apeira) #

• Infinitely small pieces of all things present in all things
• Unlimited in multitude and in smallness
• Not the same as δύναμις (potency/ability), though Anaxagoras gropes toward that concept
• Too small to be perceived individually

Νοῦς (Mind/Nous) #

• The principle of order and cosmic motion
• Unlimited in its ability to know (unlike limited material things)
• Self-ruling: directs itself rather than being directed by external forces
• Self-knowing: knows itself, not merely other things
• The thinnest and most penetrating of all substances
• Separate from the mixture of all other things

Μῖξις and Διάκρισις (Mixing and Separation) #

• Mixing (μῖξις): the coming together of infinitely small pieces in new combinations (appears as generation)
• Separation (διάκρισις): the dispersal of pieces into different combinations (appears as corruption)
• These are the only real changes in the world according to Anaxagoras

Examples & Illustrations #

The Grass Chain #

Grass → cow (eaten by) → man (eaten by) → lion (dies, eaten by) → worm (eaten by) → bird (eaten by) → cat (dies, becomes) → daisies. This demonstrates how everything eventually comes from everything else, and how if we cannot get something from nothing, all these forms must have been present in grass initially.

The Baseball Hat at the Amusement Park #

As a child on a spinning ride, Berquist’s baseball hat flew off (light object moved outward) while he remained (heavier object stayed closer to center). This illustrates how circular motion separates light from heavy objects—the same principle Anaxagoras uses to explain cosmic order.

Circular Motion in Human Activity #

• Spinning a rock on a stick to throw with more force
• A boxer’s circular motion to generate power in a punch
• Cyclotrons accelerating particles then shooting them to split atoms
• All demonstrate that humans use circular motion to separate and divide things

Algebra and Conservation #

The equation 2x + y ≠ 2(x + y) because the second expression has an extra y. In algebra, nothing can come into existence (gain a y) or go out of existence (lose a y). This principle—that all things remain forever equal—underlies all mathematical science.

Relative Quantity #

• A hundred people: big dinner at your house, small crowd at a baseball game
• A hundred people: big class at university, negligible crowd at Yankee Stadium
• A seven-foot man: tall compared to ordinary humans, small compared to a redwood tree
• Betelgeuse: so large the entire solar system would fit inside it, yet relative to other cosmic objects

The Proportion of Knowledge #

As a child, a father seems to know everything. As a man, a father seems to know nothing compared to God. Like the proportion Heraclitus noted: “As the ape is to man, so is man to God.” Man is wise (Homo sapiens) only in comparison to the ape, not in comparison to God.

Notable Quotes #

“All things were together, unlimited in number and in smallness.” (DK1) Anaxagoras’s description of the primordial state and mixture

“Nor is there a smallest of the small, but there is always a smaller. For what is cannot cease to be by being cut.” (DK3) Expressing the principle of infinite divisibility of the continuous

“The Greeks do not rightly take coming into being and perishing. Nothing comes to be or perishes, but it is mixed and separated from existing things.” (DK17) Anaxagoras’s fundamental rejection of true generation and corruption

“All things are forever equal.” Berquist’s synthesis expressing the conservation principle underlying modern mathematics and science

“Other things have a part of everything, but mind is unlimited.” (DK12) Establishing mind’s unique separation from all material mixture

Questions Addressed #

How can everything come from everything without violating the axiom that you cannot get something from nothing? #

Resolution: Everything must already exist in everything else, though in infinitely small pieces imperceptible to us. When grass becomes cow, the cow was already present in the grass in infinitesimal form. This preserves both the principle of infinite divisibility and the axiom against creation from nothing.

How can infinite pieces of everything fit into a finite blade of grass? #

Resolution: The pieces must be infinitely small. Just as a mathematical line can be infinitely divided without becoming zero (you never reach indivisible points), material bodies can contain infinite pieces if those pieces are infinitely small. However, Berquist notes that Anaxagoras does not truly grasp the distinction between actual and potential (potency).

What is the connection between Anaxagoras and twentieth-century physics? #

Resolution: Elementary particle physicists, using purely empirical methods, discovered that any elementary particle could be transformed into all others. Following Anaxagoras’s logic—you cannot get something from nothing—they concluded every particle must contain all others. Modern physics then had to reintroduce Aristotle’s concept of potency to escape the absurdities Anaxagoras faced.

How is mind different from all other things in Anaxagoras’s system? #

Resolution: Mind (νοῦς) is unlimited in its knowing ability (whereas material things are limited), self-ruling (whereas other things are moved by external forces), and self-knowing (whereas other things only contain pieces of other things). Mind is separate from the unlimited mixture of all material things and causes cosmic order through circular motion.

Why does mind cause circular motion rather than other kinds of motion? #

Resolution: Circular motion is the most effective natural method for separating things. Heavy objects settle toward the center while light objects fly outward. Humans use this principle (spinning, cyclotrons, boxing) to generate force and separate materials. The cosmos, ordered by mind, uses the same principle to arrange all material things.

What is Berquist hinting will be important about mind in the next lecture? #

Resolution: Berquist mentions there is an apparent contradiction in the great fragment on mind (DK-12) that reveals something very important about Anaxagoras’s position. Only one student in many years of teaching has identified this contradiction. The resolution awaits the next lecture’s detailed discussion of the complete mind fragment.