75. The Continuous: Composition, Divisibility, and Circular Reasoning

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Lecture Notes

Main Topics #

The Four Ways Things Can Touch #

Berquist establishes a framework for how magnitudes can come together:

• Whole touching whole
• Part touching part
• Whole touching part
• Edge touching edge (potentially distinguishable from parts)

This taxonomy becomes crucial for examining whether points can compose a continuous magnitude.

The Impossibility of Points Composing a Line #

Since points have no parts, they cannot touch in any of the four ways listed above except whole-to-whole. When two points touch whole-to-whole, they coincide and produce only one point. Therefore:

• Ten points coinciding = no length
• Hundred points coinciding = no length
• Million or infinity of points coinciding = no length
• Conclusion: A line cannot be composed of points

The Implication for Divisibility #

If a line cannot be divided into points (indivisibles), then:

• Division must always yield shorter lines (divisibles)
• The line is therefore divisible forever
• The continuous is always divisible into divisibles

The Problem of Circular Reasoning #

Aristotle appears to reason in two directions:

1. From first definition to second: The continuous cannot be composed of indivisibles (first definition) → therefore it is divisible forever (second definition)
2. From second definition to first: The continuous is divisible forever (second definition) → therefore it cannot be composed of indivisibles (first definition)

Thomas Aquinas notes this apparent circularity but does not address it explicitly. Berquist raises the question: which direction represents the proper demonstration?

Berquist’s Resolution of the Circularity #

• The reasoning from impossibility of composition to divisibility forever appears more fundamental and explanatory
• The second definition might be accepted as “probable” or known from student experience (they readily accept infinite bisection)
• Independent arguments for the second definition may exist (e.g., the faster/slower body argument mentioned for Reading III)
• Therefore, both directions of reasoning can be valid without strict circularity
• Analogy: In Euclid’s geometry, convertible theorems (like V and VI on isosceles triangles) can be proven from each other yet also proven independently

The Axiom: “The End and That Which It Is an End Are Other” #

Berquist notes that a point cannot be the limit of itself—a limit must be other than that of which it is the limit. This prevents:

• A point from having a boundary (which would require it to be other than itself)
• Two points from sharing a common boundary
• Infinite regress in limits

Distinction Between Continuous, Touching, and Next #

• Continuous (Συνεχές): Parts have a common boundary
• Touching (Ἁπτόμενα): Boundaries are together; can occur without sharing one boundary
• Next (Ἐφεξῆς): Nothing of the same kind between them

Critically, nothing of the same kind can exist between points because a line always lies between any two points.

Key Arguments #

Argument from the Definition of Continuous #

• The continuous is that whose parts have a common boundary
• Points have no boundary (they are themselves limits)
• Therefore, two points cannot constitute continuous parts
• Therefore, the continuous cannot be composed of points

Argument from Impossibility of Points Touching #

• Points have no parts, so they cannot touch part-to-part, whole-to-part, or part-to-whole
• Points can only touch whole-to-whole, which means coinciding
• Coinciding points produce no additional length
• Therefore, infinite points still produce no length
• Therefore, a line cannot be made from points
• Therefore, division of a line never terminates in points
• Therefore, the line is always divisible into divisibles

Argument from Reductio ad Absurdum #

• Suppose the continuous could be divided into indivisibles
• Then indivisible would touch indivisible
• But indivisibles cannot touch (established above)
• Therefore, the continuous cannot be divided into indivisibles
• Therefore, the continuous is always divisible into divisibles

Important Definitions #

Continuous (Συνεχές): That whose parts have a common boundary; equivalently, that which is divisible forever into divisibles

Indivisible (Ἀδιαίρετον): That which has no parts; a point in magnitude, a now in time, a unit in number

Divisible: That which has parts; that which can be separated into smaller parts

Touching (Ἁπτόμενα): Having extremities or boundaries together

Next (Ἐφεξῆς): Having nothing of the same kind between them

Examples & Illustrations #

The Playground Analogy #

Berquist recalls lining up as students in grade school—each student next to the next, like houses in a row. But this requires “nothing of the same kind” between them (no other student). However, two points cannot be next to each other because there is always a line between any two points.

The Line Bisection #

When bisecting a line, you get two shorter lines, not two points. You can bisect again and again, always obtaining shorter lines. This demonstrates that division always yields divisibles, never terminating in indivisibles.

The Demerol Example #

After childbirth, a woman received the standard dose of Demerol but experienced a violent headache. She discovered through experience that a smaller dose relieved her discomfort better than the universal prescription. This illustrates that particular experiential knowledge can exceed universal scientific knowledge in effectiveness, yet science (knowing why) remains wiser than mere experience.

Faster and Slower Bodies #

Berquist mentions (referencing Reading III) that when a faster body covers the same distance as a slower body, it does so in less time, dividing time. The slower body then covers less distance in that same time, dividing distance. This reciprocal relationship demonstrates that both time and distance are divisible forever.

Notable Quotes #

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

“So you can’t make a line by putting points together, right? Do you see that?”

“Can you cut something up into nothing? … If you could cut something up into nothing, it would be made out of nothing. … So if you could cut something up into nothing, it would be made out of nothing, which is absurd.”

“So you always end up with shorter lines, and you cut or bisect a straight line, you end up with two points. And there was nothing. So if you always end up with shorter lines, you can cut again and cut again and cut again.”

Questions Addressed #

Can the continuous be composed of indivisibles? #

No. Since points have no parts, they cannot touch in any of the four modes except whole-to-whole. Whole-to-whole touching means coinciding, and coinciding points produce no length. Therefore, infinite coinciding points still produce no line.

What happens when you divide a line? #

You always get shorter lines, never points. If you always get shorter lines, you can always divide further. Therefore, the continuous is divisible forever.

How can Aristotle reason from both definitions without circularity? #

The reasoning from the impossibility of composition (first definition) to divisibility forever (second definition) appears more fundamental and explanatory. The second definition can be known from experience or from independent arguments. Therefore, one can reason from both without strict circularity, similar to convertible theorems in geometry.

Why is the axiom “the end and that which it is an end are other” important? #

It prevents a point from being its own limit or boundary, and prevents infinite regress in limitations. It shows why points cannot have boundaries and why the continuous cannot be composed of boundless points.

Can two points be “next” to each other? #

No. “Next” requires nothing of the same kind between them. But a line (of the same kind as a point in magnitude) always lies between any two points.