14. Heraclitus, Change, and the Axiom of Contradiction

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Lecture Notes

Main Topics #

Heraclitus’s Emphasis on Change and Flux #

• Change (metabole) is the most evident feature of the natural world and the proper starting point for philosophy
• All things flow; nothing remains identical (the river fragment: we step and do not step into the same rivers)
• Change occurs between contraries: what becomes warm was cold; what dries was wet; what hardens was soft, etc.
• Understanding the world requires understanding change itself; if you don’t understand change, you understand nothing

The Apparent Contradiction in Change #

• Heraclitus points out that change seems to involve contradiction: things both are and are not
• Examples: “We step and do not step into the same rivers”; “We are and we are not”; “The sun is new every day”
• The inductive argument (DK-126): All instances of change are between opposites, suggesting change is fundamentally between contraries
• The apparent contradictions are expressed in language: “One opposite becomes the other” (e.g., the sick becomes healthy)
• If “becomes” means “comes to be,” then one opposite would be the other—a real contradiction

Parmenides’s Response: The Axiom of Contradiction #

• Parmenides denies the reality of change because it seems to violate the axiom of contradiction
• The axiom of contradiction: Something cannot both be and not be at the same time in the same way
• Parmenides establishes this axiom explicitly: “It is impossible both to be and not to be”
• Consequence: If change involves real contradiction, change must be an illusion
• Parmenides calls those who deny the axiom “two-headed mortals”—monsters because it is so unnatural

The Central Dichotomy #

• Heraclitus affirms the reality of change (seemingly admitting contradictions)
• Parmenides denies change to preserve the axiom of non-contradiction
• Most thinkers historically follow Heraclitus because change is evident to the senses
• However, Parmenides is “a man to be feared and reverenced” because the axiom of contradiction is more certain
• The two seem to have nothing in common: one affirms what the other denies

Finding Common Ground #

• Despite their disagreement, both Heraclitus and Parmenides accept something in common: that you must choose between affirming change or affirming the axiom of contradiction
• The very fact that change seems to contradict the axiom proves the axiom is at work in their thinking
• To resolve disagreement, one must find common ground that both parties share
• Those who deny the axiom of contradiction use reasoning (which presupposes the axiom) to justify their denial—a self-refuting position

Aristotle’s Resolution #

• Aristotle accepts that the axiom of contradiction is more certain than the appearance of change
• Therefore, apparent contradictions in change must not be real contradictions
• The key: Different things or aspects are involved in what appears to be a contradiction
• Example: “Changing at rest” is not a real contradiction because the body is in motion while emotions come to rest (different things)
• Example: The sick becoming healthy is not a contradiction because the sick person (substrate) persists while the property (sickness) changes

Hidden Harmony and Apparent Harmony #

• “The hidden harmony is better than the apparent”
• People often walk in apparent harmony, thinking their beliefs cohere, when actually hidden contradictions lie beneath
• The Socratic method leads people from apparent harmony into contradiction, revealing the need for deeper understanding
• Beneath the revealed contradiction lies a true, hidden harmony—the deeper truth
• Slave boy example: The boy thinks a square with twice the side length is twice as big (apparent harmony), but calculation reveals contradiction (16 ≠ 2×4, though 4 = 2×2)

The Usefulness of Opposites #

• “The opposite is useful”: Opposition and contradiction serve the development of truth and knowledge
• Opposition is useful in law (prosecutor vs. defendant), politics (opposing parties), economics (competition), aesthetics (contrast), and character development (hardship)
• Opposition is also useful epistemologically: contradiction reveals the need to examine one’s beliefs more carefully
• Beauty arises from contrast: stars against black sky, light and shadow in paintings, candlelight on diamonds
• From opposites comes the most beautiful harmony

The Question of Fire as First Matter #

• Heraclitus proposes fire: “All things are exchanged for fire and fire for all things”
• Fire uses an economic metaphor: just as all goods can be exchanged for gold and vice versa, all things are exchanged for fire
• Comparison to previous guesses: Water is finer than earth; air is finer than water
• In terms of subtlety and penetration, fire is finer than air and water
• However, in terms of formlessness, fire is worse (more definite qualities: hot and dry)
• The puzzle: Why would Heraclitus choose fire, which seems to have more qualities and less formlessness?

Unity and Reason #

• “It is wise, listening not to me, but to reason, to agree that all things are one”
• Reason naturally seeks unity and simplicity; order is based on something one
• “Those who speak with understanding must be strong in what is common to all”
• What does it mean to listen to reason? Listen to what is common to your reason and my reason—what reason naturally knows
• What is common to all rational minds is what is natural to reason itself
• To understand natural things is to understand what is natural to reason

Key Arguments #

The Inductive Argument for Change Between Contraries (DK-126) #

• Premise: Multiple instances of change exhibit a pattern:
  • What becomes warm was cold; what cools was hot
  • What dries was wet; what moistens was dry
  • What hardens was soft; what softens was hard
  • What becomes sick was healthy; what becomes healthy was sick
• Conclusion: Change universally occurs between contraries

The Problem of Contradiction in Change #

• Premise 1: We say “one opposite becomes the other” (e.g., sick becomes healthy)
• Premise 2: “Becomes” means “comes to be”
• Conclusion 1: If one opposite comes to be the other, then one opposite is the other
• Conclusion 2: This violates the axiom of contradiction (something both is and is not)
• Consequence: Either change is real (and contradictions are real) or the axiom of contradiction holds (and change is impossible)

Why the Axiom of Contradiction Cannot Be Rationally Denied #

• Premise 1: Those who deny the axiom use reasoning to justify the denial
• Premise 2: Reasoning presupposes the axiom of contradiction
• Conclusion: Denying the axiom while using reasoning is self-refuting
• Analogy: Like trying to prove the fifth postulate by assuming the fifth postulate
• Implication: Even those who deny the axiom implicitly accept it in their thinking

Aristotle’s Resolution: Apparent vs. Real Contradiction #

• Premise 1: The axiom of contradiction is more certain than the appearance of change
• Premise 2: Change appears to involve contradiction
• Conclusion: Apparent contradictions in change are not real contradictions
• Method: Distinguish the things or aspects involved in the apparent contradiction
• Example: Motion and rest are not a real contradiction when different things are at rest and in motion

The Method of Finding Common Ground #

• Premise 1: When two parties disagree, they cannot resolve the disagreement by appealing to what they disagree about
• Premise 2: They must find something common that both parties accept
• Conclusion: Use what is common to both parties to adjudicate between them
• Application to Heraclitus and Parmenides: Both accept that one must choose between affirming change or affirming the axiom; this shared ground proves the axiom is at work

Important Definitions #

Change (Metabole) #

• The process by which something comes to be different from what it was
• Always occurs between contraries (opposites)
• Requires a substrate that persists through the change (the same person who was sick and is now healthy)
• Apparent contradictions in change dissolve when one distinguishes the substrate from the property

The Axiom of Contradiction #

• Something cannot both be and not be at the same time in the same way
• The most certain principle of thought and being
• Foundation of all reasoning and intelligibility
• So fundamental that even denying it presupposes its truth
• Explicitly established by Parmenides but implicit in all rational thought

Hidden Harmony (Harmonia Krypte) #

• The true order underlying apparent contradictions
• Revealed through dialectical examination (the Socratic method)
• More valuable than apparent harmony because it reflects reality
• Emerges when one recognizes the inadequacy of apparent harmony and seeks deeper understanding

Cosmos #

• A beautiful, ordered whole
• Contrasts with chaos (disorder)
• Implies intelligibility and purposeful arrangement
• The universe as an ordered system, not a random heap

Examples & Illustrations #

The River Fragment #

• Heraclitus: “It is not possible to step twice into the same river”
• Berquist’s experience crossing the Mississippi River many times and the Hudson with his son at West Point
• The water is always flowing and changing, yet we call it the same river
• Illustrates the paradox: We do step into the same river (by name) and we do not (the water is different)

Apparent Contradictions in Daily Life #

• Travel and Rest: People say “I find travel restful,” yet travel is motion. Resolution: The body is in motion while emotions come to rest
• The Restless Child: A child forced to sit still becomes restless. Resolution: The body is at rest while emotions are agitated
• Day and Night: Day becomes night and vice versa, yet they seem opposed. Resolution: They are one in the sense that one becomes the other through time

The Slave Boy and the Square (Plato’s Meno) #

• The slave boy thinks a square with twice the side length is twice as big
• Calculation reveals this is false: side 2 gives area 4; side 4 gives area 16 (not twice 4)
• The boy is led from apparent harmony (thinking his beliefs cohere) into contradiction (recognizing they don’t)
• Hidden harmony is then revealed: The diagonal of the original square gives the side of a square twice as large

The Bow and the Lyre #

• The bow: Pulling back with one hand and pushing forward with the other seem to work at cross-purposes
• Yet this tension creates the force to propel the arrow forward with great power
• Apparent opposition (different hands working in opposite directions) produces unity (the arrow flying forward)
• Illustrates how apparent contradictions can work together toward a unified purpose

The Diamond and Candlelight #

• A diamond looks more beautiful under candlelight than under bright electric light
• Candlelight creates contrast: darkness interspersed with light, making the diamond glitter more
• Under uniform bright light, the diamond appears garish, less beautiful
• Beauty requires contrast between opposites

The Painting: Rembrandt’s Man with Helmet #

• If flooded entirely with light, a painting loses beauty
• If entirely in shadow and darkness, a painting loses beauty
• The contrast between light and shadow creates beauty and depth
• Sunrise and sunset are more beautiful than midday because of the interplay of light and dark

The Universe as Ordered vs. Chaotic #

• The question: If fire (a mindless substance) is the origin of all things, why is the universe ordered and beautiful rather than chaotic?
• Analogy: Fire raging through buildings leaves “a heap piled up at random”
• If the universe originated from mindless matter or a mindless mover, it should be chaotic
• Yet the universe is beautiful and intelligible—suggesting the need for Mind (nous) as a cause

Notable Quotes #

“We step and do not step into the same rivers.” (Heraclitus, DK-49A)

• Expresses the apparent contradiction in change: the river is the same yet not the same

“The hidden harmony is better than the apparent.” (Heraclitus, DK-54)

• Hidden truth is more valuable than surface-level apparent agreement
• Points to the Socratic method of revealing deeper contradictions and understanding

“It is wise, listening not to me, but to reason, to agree that all things are one.” (Heraclitus, DK-50)

• Reason naturally seeks unity and simplicity
• To listen to reason is to listen to what is common to all rational minds

“Those who speak with understanding must be strong in what is common to all.” (Heraclitus, Fragment on Method)

• To resolve disagreement, one must appeal to what is common to both parties
• Understanding requires grasping what is universal and shared

“It is impossible both to be and not to be.” (Parmenides)

• The axiom of contradiction in its most explicit form
• The most certain principle of thought and being

“A two-headed mortal” (Parmenides, characterizing those who deny the axiom of contradiction)

• One who denies the axiom would be a monster, so unnatural is the denial
• Emphasizes the fundamental nature of the axiom

“The opposite is useful.” (Heraclitus, DK-67)

• Opposition serves truth, justice, beauty, and character development
• Principle applies to law, politics, economics, aesthetics, and epistemology

“From those differing comes the most beautiful harmony.” (Heraclitus, related to DK-8)

• Beauty and order arise from the interplay of contraries
• Contrast and opposition create harmony and intelligibility

Questions Addressed #

How can change be real if it involves contradiction? #

• The Problem: Change seems to require something to both be and not be (e.g., the sick person becomes healthy)
• Parmenides’s Answer: Change cannot be real; it is an illusion because it violates the axiom of contradiction
• Heraclitus’s Answer: Change is real; the apparent contradictions must be addressed through language and deeper analysis
• Aristotle’s Resolution: Apparent contradictions are not real when one distinguishes the substrate (the person) from the property (sickness)

What must both Heraclitus and Parmenides accept as common ground? #

• The Agreement: Both accept that one must choose between affirming change or affirming the axiom of contradiction
• The Implication: The very fact that change appears to contradict the axiom proves the axiom is already at work in their thinking
• Consequence: Even Heraclitus, in denying that one must choose, is using the axiom to make his case

How can one resolve disagreement between opposing views? #

• The Method: Find common ground that both parties accept
• Example: Scientists design experiments that both theories can make predictions about, then conduct the experiment to decide
• Principle: You cannot resolve disagreement by appealing to what you disagree about
• Application: To understand Heraclitus and Parmenides, find what they share despite their disagreement

What is the epistemological value of contradiction? #

• The Function: Contradiction reveals inadequacy in one’s understanding
• The Socratic Method: Leading someone from apparent harmony into contradiction forces them to examine their beliefs more carefully
• The Result: Hidden harmony (deeper truth) emerges when one recognizes and works through contradiction
• The Principle: Contradiction is useful for developing knowledge, not an obstacle to it

Why does fire seem to be a puzzling choice for first matter? #

• The Problem: Fire has definite qualities (hot, dry), making it a worse candidate for formless matter than air or water
• In fire’s favor: Fire is more subtle and penetrating than air or water
• The Deeper Issue: Fire may represent both matter (substrate) and mover (cause of change), suggesting the need to distinguish these two principles
• Anticipation: This distinction becomes explicit in later thinkers (Empedocles, Anaxagoras, Aristotle)

What does it mean to “listen to reason” rather than listen to me? #

• The Puzzle: There is my reason, your reason, and everyone else’s reason—which reason should we listen to?
• The Answer: Listen to what is common to all rational minds; listen to what reason naturally knows
• The Principle: What is common to your reason and my reason is what is natural to reason itself
• The Application: Understanding natural things means understanding what is natural to reason

Philosophical Significance #

The Centrality of Change #

Heraclitus emphasizes that change is the most evident feature of experience and therefore the proper starting point for philosophy. Understanding change is fundamental; if you don’t understand change, you understand nothing.

The Power and Limits of Reason #

The axiom of contradiction is so fundamental that denying it self-refutes—one must use reason (which presupposes the axiom) to justify the denial. This shows the power of reason to recognize its own foundational principles.

The Method of Common Ground #

To resolve genuine disagreement, one must find principles that both parties accept. This method is more productive than simply asserting one’s own position, and it reveals that apparent opposites often share deeper common ground.

Apparent vs. Real Contradiction #

Aristotle’s distinction between apparent and real contradictions resolves the tension between Heraclitus and Parmenides. This shows how philosophy can synthesize opposing views by making finer distinctions.