41. Aristotle's Arguments Against Anaxagoras and the Second Difference in Quantity

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Lecture Notes

Main Topics #

Anaxagoras’s Position and Its Difficulties #

• Core claim: Everything is mixed in everything; all things come from all things
• Motivation: Cannot get something from nothing; observes that grass becomes cow, cow becomes man, etc.
• Consequence: Must posit infinite, infinitely small pieces of everything in everything
• Central problem: How to fit infinity of pieces into finite bodies? Answer: make them infinitely small—leading to contradictions

The First Difference Between Mathematical and Natural Quantity #

• Already established: Natural philosophy considers quantity as it pertains to natural bodies; mathematics considers quantity in abstraction from natural bodies and their sensible qualities
• Example: A geometrical sphere has no matter, sensible qualities, color, weight, or taste—purely mathematical abstraction
• Implication: This recognition opened Aristotle to discovering a second difference

The Second Difference Between Mathematical and Natural Quantity #

• Core principle: Natural quantities have limits in the direction of large or small due to the natures of things
• Mathematical contrast: Pure mathematics considers quantity in separation from the natures of things; therefore cannot foresee these natural limits
• Key distinction: A mathematical line can be divided infinitely (same ratio); a natural body cannot be divided infinitely (must take out same amount each time)
• Significance: This second difference distinguishes modern 20th-century physics from classical (Newtonian) physics
• Historical markers:
  • Classical physics (17th-19th centuries): Galileo, Kepler, Newton—based on principle of fewness
  • Modern physics (20th century): Quantum theory and relativity—based on discovery of limits in natural quantities

Key Arguments Against Anaxagoras #

Argument 2: The Limit Argument (Foundation for Arguments 3 & 4) #

• Structure: If-then refutation: if parts can fall below any size, then wholes can fall below any size
• Logical principle: Part-to-whole relationship is necessarily correlated
• Empirical observation: Different kinds of animals and plants have definite size limits; no animal exists at any arbitrary size
• Conclusion: Parts cannot fall below any size; there must be a smallest piece of flesh, bone, etc.
• Status: Uses this result to refute Anaxagoras’s claim that parts are infinitely small

Argument 3: The Exhaustion Argument #

• Builds on: Result from Argument 2—smallest pieces of flesh, bone, etc. exist
• Logic: If you must take out at least the smallest amount each time, you cannot extract infinite amounts from a finite body
• Consequence: Generation must eventually stop (contradicting Anaxagoras’s eternal generation)
• Key contrast with mathematics: Mathematical line can be divided infinitely because there is no smallest division; natural body cannot because the smallest piece exists
• Illustration: Perkowitz selling mathematical lines (never runs out) vs. Perkowitz giving water from canteen (eventually runs out due to molecular minimum)

Argument 4: The Composition Argument #

• Premise: Uses result from Argument 2—smallest pieces exist
• Logic: In the smallest piece of flesh, there cannot be any bone; if part of it were bone, then only part would be flesh, making something smaller than the smallest—contradiction
• Application: In smallest piece of bone, there cannot be any flesh
• Conclusion: Not everything is inside everything; directly contradicts Anaxagoras’s core claim

Argument 5: The Infinite Infinities #

• Problem: If every elementary particle is composed of all the rest, then you have infinities composed of infinities of infinities
• Contradiction: Each elementary particle getting smaller and smaller—contrary to laboratory experience that all electrons have same mass/size
• Modern physics parallel: Quantum physicists recognize that every elementary particle is composed of all the rest only potentially, not actually

Arguments 1, 5, and 8: Grouped by Common Principle #

• Common thread: Based on principle of fewness (simplicity)
• Argument 1 (unknowability): If principles are infinite, they cannot be known; but nature gives no impossible desires
• Argument 8 (principle of simplicity): Empedocles explains same phenomena with four elements plus love/hate; Anaxagoras requires infinity of principles; fewer principles are better

Arguments 6 & 7: Particular Difficulties #

• Argument 6: Anaxagoras confuses substance and accident—tries to separate bone from its color, when only substances can be separated from each other, not accidents from substances
• Argument 7: Assumes all becoming is addition of like to like; in fact, unlike things combine (wood and cement)

Important Definitions #

Act and Potentiality (Ability) #

• Distinction central to solution: Anaxagoras makes actually present what is only in potentiality
• Potentiality: Not knowable by itself; known only through acts it produces
• Act: What is present and actual
• Application: Everything that can come to be from matter is in matter only in ability, not in act

Smallest Piece (Minimally Divisible Unit) #

• Flesh: The smallest piece of flesh that still counts as flesh; nothing smaller counts as flesh
• Bone: The smallest piece of bone that still counts as bone
• Natural principle: Due to the nature of the substance itself, there exists a minimum threshold

Substance vs. Accident #

• Substance: What exists in itself (man, bone, flesh)
• Accident: What exists in something else (color, shape, quality)
• Confusion in Anaxagoras: Treats accidents as if they were substances; tries to separate color from bone

Examples & Illustrations #

Perkowitz’s Retail Business Example #

• Selling mathematical lines: Customer comes in, Perkowitz sells half the line; next customer gets half of remainder; can continue forever—never runs out because mathematical lines have no smallest division
• Giving water in desert: Perkowitz gives half his canteen to first person; half of remainder to second person; eventually runs out because water molecules have a definite minimum size
• Lesson: Illustrates the second difference—mathematical quantity vs. natural quantity with limits

Animal and Plant Size Variations #

• Observed fact: Different kinds of animals have definite maximum and minimum sizes
• Cats: Variations among cats, but within definite limits; no cat at arbitrary size
• Elephants: Different limits than cats
• Plants: Trees in local region don’t grow as tall as California redwoods; grass doesn’t grow as tall as trees
• Conclusion: Refutes the consequence that wholes can be any size

Smallest Piece of Flesh Example #

• Logical principle: In the smallest piece of flesh, there is no bone; if there were, part would be bone, part would be flesh, making something smaller than smallest—contradiction
• Same for bone: In smallest piece of bone, there is no flesh
• Application: Refutes claim that everything is in everything

Notable Quotes #

“You can’t get something from nothing”

• Fundamental principle shared by all Greek natural philosophers; led Anaxagoras to his position

“Nature allows to hide” (Heraclitus)

• Discovery of limits in nature is a sign we’re getting closer to the natures of things

“The well-known formula” (Anaxagoras)

• Describing how every elementary particle is composed of all the rest
• Easier to speak this way, but leads to contradictions

“The same way with the modern physicist”

• If physicist wants to put actually in any elementary particle all things extractable from it, contradictions follow (same as Anaxagoras)

Questions Addressed #

Why does Aristotle group arguments 2, 3, and 4 together? #

• Answer: They are linked logically—argument 2 establishes the smallest pieces exist; arguments 3 and 4 use this result to refute other positions of Anaxagoras
• Domino theory: Overthrowing one position knocks over the others in sequence

How can infinite pieces fit in a finite body? #

• Anaxagoras’s answer: Make them infinitely small
• Aristotle’s response: This contradicts observable limits in actual things; parts cannot fall below any size

Why can a mathematical line be divided infinitely but not a natural body? #

• Answer: Mathematical line has no smallest division (pure abstraction); natural body has smallest piece due to its nature
• Consequence: Cannot take infinite amounts of same size from finite natural body

What is the connection between ancient Anaxagoras and modern quantum physics? #

• Similarity: Both struggle with fitting infinite divisibility into finite wholes
• Modern resolution: Quantum theory recognizes limits (Planck constant) and potentiality (not actuality) of what can be extracted from particles
• Aristotelian principle: Modern physics discovering that natural quantities have limits due to their natures

How does this relate to the distinction between classical and modern physics? #

• Classical physics (17th-19th centuries): Based on principle of fewness/simplicity; assumes what’s true of quantity in math is true in nature
• Modern physics (20th century): Discovers limits in natural quantities—quantum theory (minimum in direction of small), relativity (maximum in direction of large)
• Significance: Modern physics returns to Aristotelian recognition of second difference

Why does Anaxagoras confuse substance and accident? #

• Root error: Attempts to separate bone from its color, treating accident as if it were substance
• Logical point: One substance can be separated from another substance; accident cannot be separated from substance
• Broader confusion: Shows lack of clear distinction between two fundamental divisions of being

Connections to Modern Science #

Chemistry #

• Atom: Smallest piece of a chemical element
• Molecule: Smallest piece of a chemical compound
• Principle: Different atoms have different sizes; every atom of hydrogen has same size
• Aristotelian foundation: Based on principle that smallest pieces exist

Quantum Theory #

• Planck’s quantum hypothesis (December 1900): Energy cannot be given or received in any amount; there is a smallest amount (quantum)
• Key discoveries:
  • Einstein (1905): Need quantum to understand light
  • Bohr (1913): Understand atomic structure through quantum
  • Late 1920s: Quantum theory perfected
• Characteristic: Recognition of limit in quantity of actual things in direction of small
• Minimum length hypothesis: Heisenberg and others theorized minimum length (10^-13 cm) in study of elementary particles

Special Relativity (Einstein, 1905) #

• Limit in direction of large: Speed of light as maximum speed
• Classical contrast: Newtonian physics assumed no limit to speed; something could always go faster
• Modern principle: Speed of light cannot be exceeded

General Relativity (Einstein, 1915) #

• Cosmological implications: Led to discovery that universe may be finite (rather than infinite as early Greeks thought)
• Historical reversal: From medieval/Renaissance infinite universe back to Aristotelian finite cosmos
• Cosmic limits: Perhaps limits even in time (Big Bang theory)

Elementary Particles and Modern Physics #

• Quantum physicists’ approach: Every elementary particle potentially composed of all the rest, but not actually
• Heisenberg’s recognition: Difficult to speak this way; easier to say actually composed, but leads to contradictions
• Observed fact: All electrons have same mass/size despite theoretical complications

Two Frontiers of Modern Physics #

• Quantum physics: Limits in direction of small (minimum energy, minimum length)
• Cosmology: Limits in direction of large (finite universe, possibly limited in time)
• Common feature: Both discover limits foreseen from pure math but discovered empirically

Logical Structure of the Arguments #

If-Then Refutation Method #

• Form used: If A is so, then B is so; but B is not so; therefore A is not so
• Application: If parts can fall below any size, then wholes can fall below any size; but wholes cannot fall below any size in nature; therefore parts cannot fall below any size
• Common in refutation: Standard way to overthrow a position by showing its consequences are false

The Role of the Smallest Piece #

• Established in Argument 2: Smallest pieces of flesh, bone, etc. must exist
• Used in Argument 3: Cannot take infinite amounts of same size from finite body
• Used in Argument 4: Smallest pieces cannot contain other substances
• Cascading effect: One established point overthrows multiple subsequent positions