74. The Continuous, Indivisibles, and the Axiom of Distinction

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Lecture Notes

Main Topics #

The Continuous and Its Definitions #

• First Definition (from Logic): “That whose parts meet at a common boundary”
  • More appropriate to logic because it emphasizes how parts unite to form a whole (form-like)
  • Example: parts of a line meet at a point; parts of a surface meet at a line
• Second Definition (from Natural Philosophy): “That which is divisible forever” or “always divisible into divisibles”
  • More appropriate to natural philosophy because it emphasizes infinite divisibility into parts (matter-like)
  • Both definitions are convertible and equivalent

The Argument Against Composition from Indivisibles #

Aristotle argues that a continuous thing (like a line) cannot be composed of indivisibles (points) because:

1. Points have no parts, so they cannot touch part-to-part or part-to-whole
2. Points have no edges or boundaries (a boundary would imply divisibility)
3. Therefore, points can only touch whole-to-whole, which means they must coincide
4. If points coincide, they produce no length at all
5. Thus, infinite points cannot compose a line

The Problem of Touching and Next #

• Touching: Things whose edges are together while remaining distinct
• Points cannot touch in this sense because they have no edges
• Next (ἐχόμενα): Things between which there is nothing of the same kind
• Two points cannot be “next” to each other because between any two points there is always a line
• Therefore, the continuous has no atomic units

Fundamental Axioms #

The Axiom of Distinction (Limite and that which it limits):

• “The edge and that which it is an edge of are other” (ἐτέρα)
• There must always be a distinction between a limit and that which it limits
• A point cannot be the limit of itself
• This is self-evident once understood and requires no proof

The Axiom of Before and After:

• “Nothing is before or after itself”
• There must always be a distinction between what is before and what comes after
• Applied proportionally to magnitude, time, and motion
• Related to the axiom that “nothing is the beginning of itself”

The Hierarchy of Limits #

• A body is limited by surfaces
• A surface is limited by lines
• A line is limited by points
• A point has no limit—it is the ultimate limit
• Some limits have limits; the point (ultimate limit) does not
• This parallels the First Cause in metaphysics

Privation vs. Negation #

• Privation (στέρησις): Non-being of something a thing is able to have and should have
  • Example: A straight line extending infinitely lacks endpoints (it should have them but doesn’t)
  • Example: Blindness in a man (he should have sight but lacks it)
• Negation: Simple non-being, not a lack
  • Example: A point lacks a limit (but cannot have one by nature)
  • Example: A chair is not blind (it is not the sort of thing able to have sight)
• When we say God is “infinite,” this is negation (no limit to perfection), not privation (no lack)

Key Arguments #

Why Points Cannot Touch or Be “Next” #

• Points are indivisible and have no parts
• All four ways things can touch require either parts touching parts, parts touching wholes, or edges touching edges
• Points have no parts and no edges
• If two points touch whole-to-whole, they coincide and become one point
• Coinciding points produce no length
• Therefore, points cannot compose a line while remaining distinct

Why Infinite Divisibility Follows from the First Definition #

• If parts meet at a common boundary (first definition), that boundary is itself something
• The boundary is distinct from what it bounds (axiom of distinction)
• Therefore, on either side of the boundary are parts that can be further divided
• This divisibility proceeds forever (second definition)
• Thus, the two definitions are equivalent

The Relationship Between Definitions #

• Reasoning from first to second: If parts meet at boundaries, and boundaries are distinct from what they bound, then there must always be further division
• Reasoning from second to first: If always divisible into divisibles, division never terminates in indivisibles, so parts must meet at boundaries
• Both directions are valid because the definitions are convertible

Important Definitions #

• Continuous (συνεχές): That whose parts meet at a common boundary; that which is divisible forever into divisibles
• Touching (ἁπτόμενα): Things whose edges are together but remain distinct
• Next (ἐχόμενα): Things between which there is nothing of the same kind
• Indivisible (ἀδιαίρετον): Without parts; a point or moment
• Limit/Edge/Boundary (πέρας): That which marks the end of something and is necessarily distinct from what it limits
• Discrete Quantity: Things whose parts do not meet at a common boundary (e.g., numbers: 3 and 4 in the number 7 do not meet at a point)

Examples & Illustrations #

The Restaurant Sink #

Berquist uses an anecdote about a complaining restaurant patron whose complaint about an unusually high sink illustrates weak vs. strong reasoning:

• Weak reason: “I had this experience once, therefore it is better”
• Strong reason: Understanding why the alternative is worse (it would fail if you didn’t breathe; breathing is before philosophizing in the order of being)
• Applied to breathing vs. philosophizing: Stop breathing for an hour vs. stop philosophizing for an hour—which is worse? The worse is to stop breathing. Therefore, breathing must be better.

The Chaucer and Shakespeare Comparison #

• Chaucer is before Shakespeare in time (14th century vs. 17th century)
• Does temporal priority mean Chaucer is a better poet? No.
• This illustrates the fallacy of equivocation: being before in one sense (time) does not imply being before in another sense (goodness)

Faith, Hope, and Charity #

• Charity is greater than faith (a greater good)
• Yet the loss of faith is worse than the loss of charity
• Why? Because faith is before charity in the order of being: charity cannot exist without faith, but faith can exist without charity
• Therefore, losing the lesser good (faith) entails losing the greater good (charity)
• The loss of the greater good (charity) can leave you with the lesser good (faith)

Just to Live vs. to Live Well #

• Which is better: to live or to live well? Clearly, to live well.
• But to live is before living well in the order of being
• One cannot live well without living
• Therefore, the argument that what is worse to lose must be better does not apply here

Notable Quotes #

“An opinion without a reason for it is an ugly thing.”

Attributed to Socrates; used to establish the importance of rational justification for claims

“The edge and that which it is an edge of are other.”

Aristotle’s foundational axiom establishing necessary distinction between a limit and what it limits

“Nothing is before or after itself.”

Fundamental axiom about the nature of priority and posteriority

“If two points coincide, how much length do you have? Well, as much as one point. Just no length at all.”

Berquist’s pedagogical clarification of why points cannot compose a line

Questions Addressed #

Can Two Points Touch and Remain Distinct? #

• No. Points have no parts, so they cannot touch in any of the four ways things touch
• If they touch whole-to-whole, they must coincide
• If they coincide, they are no longer two points but one

Can There Be a “Next” Point? #

• No. Between any two points, there is always a line
• The definition of “next” requires that nothing of the same kind be between them
• But a line (which is of the kind “continuous magnitude”) always lies between points
• Therefore, the continuous has no atomic units

Why Does the Point Have No Boundary? #

• If a point had a boundary, that boundary would be something other than the point
• That “other” thing would constitute a part of the point
• But a point is indivisible, by definition, and cannot have parts
• Therefore, a point has no boundary

How Can Limits Themselves Have Limits? #

• A limit that has a limit is not an ultimate limit
• Example: A line is a limit of a surface, but the line itself is limited by points
• This hierarchy continues until reaching the point, which has no limit
• This parallels metaphysics: there are causes that have causes, but there must be a First Cause that has no cause

What is the Difference Between “Lacks a Limit” and “Has No Limit”? #

• A straight line extending infinitely lacks a limit (privation): it is the sort of thing able to have limits but does not have them
• A point has no limit (negation): it is not the sort of thing that can have a limit by nature
• When we say God is “infinite,” we mean negation, not privation: God has no limit to His perfection, but this is not a lack or deficiency
• The chair does not see, but we should not say the chair is blind (it is not the sort of thing that can see), whereas a blind man lacks sight (he is the sort of thing that should see)