71. The Continuous and the Problem of Indivisibles

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Lecture Notes

Main Topics #

The Continuous and Related Concepts #

• Continuous (τὸ συνεχές): That whose parts have a common boundary; the end of one part is the beginning of another
• Touching (ἁπτόμενα): Things whose limits are together but not identical; no common boundary
• Next (ἐφεξῆς): Things between which nothing of the same kind exists; no touching required
  • Example: two houses can be “next” without touching; successive thoughts are “next” but not continuous

The Fundamental Role of the Continuous in Knowledge #

• We cannot think without images, and images are tied to the continuous in time
• Basic philosophical vocabulary (beginning, end, before, after, in, out) derives from understanding the continuous
• Understanding continuity is prerequisite for understanding what is NOT continuous (immaterial realities like God, angels, the human soul)

The Problem of Composition from Indivisibles #

• A line cannot be composed of points
• A surface cannot be composed of lines
• A body cannot be composed of surfaces
• Motion cannot be composed of indivisible motions
• Time cannot be composed of indivisible nows

Key Arguments #

Why Two Points Cannot Form a Line #

Berquist develops Aristotle’s argument in Physics Book VI:

1. The Problem of Common Boundaries: If two points could touch, they would need either:

   • A common boundary (but points have no parts, so no distinction exists between the point and any edge)
   • Whole-to-whole contact (which means they coincide, producing no additional length)
   • Part-to-part contact (impossible, as points have no parts)

2. The Coincidence Result: When two points touch whole-to-whole, they coincide and have no more length than one point, which has zero length. One hundred, one thousand, one million, or infinite points coinciding still produce zero length.

3. The Absence of “Next” Points: There is no “next” point on a line. Between any two points, infinitely many others exist. If you claim point A and point B are next to each other, there is always a line between them, and therefore always another point in between.

4. The Same Reasoning Applies to All Indivisibles: The argument holds for surfaces and lines, and by extension for motion and time.

The Dichotomy Problem (Zeno’s Paradox) #

• To traverse a distance, one must first traverse half the distance
• Before that, half of the half, and so on infinitely
• Resolution: The continuous is infinitely divisible but not composed of indivisibles. An object traverses the entire distance through continuous motion without needing a “last” instant of not-being-at-destination.

Important Definitions #

Point (σημεῖον) #

• An indivisible limit of a line
• Has no magnitude whatsoever (no length, width, or depth)
• Can be derived logically:
  • Bodies exist
  • Bodies come to an end → surfaces exist (boundary with no depth)
  • Surfaces come to an end → lines exist (boundary with no width)
  • Lines come to an end → points exist (boundary with no length, width, or depth)
• A limit that has no limit (unlike surfaces and lines, which themselves have limits)
• Cannot be distinguished from its own edge (giving it an edge would assign it magnitude)

Limit (πέρας) and Beginning (ἀρχή) #

• Every cause is a beginning, but not every beginning is a cause
• Beginning is more general than cause
• End or limit is more general than beginning
• A limit can have a limit: surface is a limit of a body (and has limits); line is a limit of a surface (and has limits); point is a limit that has no limit
• God is a cause that has no cause; a point is a limit that has no limit

Examples & Illustrations #

Deriving the Existence of Points #

Berquist leads students through a Socratic sequence:

1. Bodies exist (undeniable)
2. Bodies come to an end → surfaces exist (with length and width, but no depth)
3. Surfaces come to an end → lines exist (with length, but no width)
4. Lines come to an end → points exist (with neither length, width, nor depth) Conclusion: Points have no magnitude at all, yet they are real.

The Cone Paradox (Democritus) #

• If you bisect a cone parallel to the base, the two circular cross-sections are equal
• Yet if all such sections are equal, the cone becomes a cylinder
• If sections are unequal, the cone has jagged edges
• Resolution: There is no “next” section in a continuous cone; the continuous admits no such discrete layers

Two Semicircles and Common Boundaries #

• Two semicircles share a diameter as a common boundary
• The diameter is the end of one semicircle and the beginning of the other
• This illustrates the continuous properly
• But points cannot have edges in common because they have no parts to distinguish edge from interior

Forms of Touching Between Geometric Objects #

• Two circles can touch whole-to-whole (externally tangent)
• One circle can touch part of another (or vice versa)
• For points (which have no parts), only whole-to-whole contact is possible
• When two points touch whole-to-whole, they coincide

Notable Quotes #

“The continuous is that whose parts have a common boundary.”

• Aristotle’s definition, foundational to the entire analysis

“If a magnitude is composed of indivisibles, the motion over it will be from indivisible motions.”

• Sets up the reductio ad absurdum

“Something will have moved without moving… for it will have gone through A without going through it.”

• Demonstrates the contradiction when motion is composed of indivisibles

“There is not an edge and some other part of the indivisible… It is a limit.”

• Explains why points cannot have boundaries: the distinction between limit and limited thing requires the limited thing to have parts

“Since indivisible [has] without parts, it is necessary that the whole touch the whole… [but if] the whole touches the whole, it will not be continuous. For the continuous has one part other than another.”

• Aristotle’s argument showing that points coinciding cannot form a continuous line

Questions Addressed #

Can a line be composed of points? #

No. Points have zero magnitude. Whether points touch whole-to-whole (coinciding) or part-to-part (impossible) or whole-to-part (impossible), no configuration of points can produce a magnitude. Infinite zeros do not sum to a positive quantity.

Is there a “next” point on a line? #

No. Between any two points on a line lie infinitely many other points. The continuous is infinitely divisible; no “next” element exists in a continuous magnitude. This is essential to what makes something continuous.

How does this resolve Zeno’s dichotomy paradox? #

Resolution: While space is infinitely divisible, an object need not traverse infinitely many “final steps.” There is no “last” moment of not-being-at-the-destination; rather, the object continuously transitions until the instant it arrives. The infinite divisibility of the continuous is compatible with traversal precisely because the continuous is not composed of indivisible parts.

Why is understanding the continuous important for metaphysics and theology? #

Answer: To understand immaterial substances (God, angels, the human soul), we must first understand continuity and then negate it. Students often confusedly imagine immaterial realities using continuous imagery (e.g., picturing a thought as if it has shape). Understanding the continuous precisely—and understanding that immaterial things lack these properties—prevents false imagination and enables genuine understanding of what is truly non-corporeal.