78. Continuous Magnitude, Motion, and Time: Infinite Divisibility

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Lecture Notes

Main Topics #

The Three Foundations of Philosophy and Modern Rejection #

Berquist emphasizes that all philosophy depends upon three truths:

1. Natural Understanding: Some statements are naturally known without reasoning
2. Natural Desire to Know: We naturally desire to understand what we don’t naturally know
3. Knowledge Through the Natural: We come to know unknowns through knowns

Modern philosophy has systematically rejected these, particularly the first—the idea that we naturally know anything with certainty. This rejection has caused philosophy to collapse into a “zigzag” pattern where each thinker diverges without building on predecessors. Modern philosophers (Descartes, Hobbes, Marx) replaced the natural desire to know with a desire for power and control.

The Problem of Indivisibles #

Aristotle argues that magnitude, motion, and time cannot be composed of indivisibles. If they were:

• Something would simultaneously move and have moved (contradiction)
• Something would rest in each indivisible part while moving the whole (contradiction)
• Motion would not be from motions but from “having moved” (self-contradiction)

The Argument from Faster and Slower Bodies #

Aristotle demonstrates infinite divisibility through two naturally known truths about relative motion:

Premise 1: In equal time, the faster body covers more distance than the slower body

Premise 2: The faster body covers the same distance in less time than the slower body

Method: By alternating these truths, one can show both distance and time must be divisible forever:

• Faster body goes distance AB in time T
• Slower body goes lesser distance AC in time T (divides distance)
• Faster body must cover distance AC in less time than slower body (divides time)
• This process continues indefinitely

The Definition of Continuous #

That which is divisible into parts that are themselves divisible forever. The continuous is distinct from:

• Touching (contiguous): Things with a common limit but not one being
• Next (adjacent): Touching things that are not continuous

Natural Philosophy vs. Mathematics #

The natural philosopher is wiser than the mathematician regarding continuous quantities:

• The natural philosopher proves that magnitude, motion, and time are divisible forever
• The mathematician assumes infinite divisibility as a presupposition
• Geometry must assume points can be placed anywhere between any two points, which presupposes infinite divisibility

The Hierarchy of Knowledge #

Higher knowledge distinguishes itself from lower knowledge and determines its proper use:

• Reason distinguishes itself from sensation and imagination
• Natural philosophy distinguishes itself from mathematics and determines mathematics’ utility in physics
• Theology distinguishes itself from both philosophy and mathematics

The modern problem: Modern philosophers rejected revealed theology and thus lost the wisdom needed to order philosophy properly. They attempt to accomplish in philosophy what should be done in theology.

Key Arguments #

The Self-Refuting Nature of Denying Natural Knowledge #

If one denies that any statements are naturally known:

• The denial itself is a statement
• One cannot deny that statements exist without making a statement
• Therefore, “statements exist” is naturally known
• The very act of denying natural knowledge presupposes natural knowledge

The Alternating Argument for Infinite Divisibility #

Given two bodies where A is faster than B:

1. A covers distance X in time T
2. B covers lesser distance Y in time T (distance is divided)
3. A must cover distance Y in time less than T (time is divided)
4. This alternation continues infinitely
5. Therefore, both distance and time are infinitely divisible

The Impossibility of Indivisible Motion #

If motion were composed of indivisible motions D, E, F corresponding to indivisible distances A, B, C:

• When motion D is present, the mobile has covered distance A but hasn’t “gone through” A (contradiction with divisibility requirement)
• The mobile must either move and have moved simultaneously (impossible) or rest in each part while moving the whole (contradiction)

Important Definitions #

Continuous (συνεχές) #

That which is divisible into divisible parts forever; cannot be composed of indivisibles.

Indivisible (ἀδιαίρετον) #

That which has no parts and cannot be divided—such as a point or instant. A point cannot have an end or limit, making it impossible for two points to touch and form a continuous line.

Natural Understanding (intellectus) #

The natural knowledge of truths grasped by the mind without reasoning them out. Distinguished from episteme (reasoned-out understanding that must be demonstrated).

The Natural Desire to Know #

The intrinsic human inclination to understand things beyond what is naturally known—the basis for all philosophical inquiry.

Examples & Illustrations #

The Faster and Slower Bodies (Aristotle’s Clearest Example) #

Two bodies with equal velocity take the same time to cover equal distance. A slower body covers less distance in equal time. A faster body covers the same distance in less time. By alternating these facts:

• Faster body: distance AB in time FG
• Slower body: distance AC (less than AB) in time FG
• Faster body covers AC in time FI (less than FG)
• Each step divides distance and time further
• Process continues indefinitely

The Bisection Problem #

When one cuts a straight line, a point appears. If every cut produces a point, must not the line be composed of points? Berquist uses Nixon’s joke: “Any way you slice it, it’s still baloney.” If every slice produces baloney, what is it made of? The fallacy: confusing actual composition with potential divisibility—the point exists in potency until actualized by the cut.

Euclid’s Fifth Theorem (Book II) #

Illustrates that infinite divisibility allows unexpected relationships:

• Square 5×5: area 25, perimeter 20
• Rectangle 4×6: area 24, perimeter 20 (same perimeter, less area)
• Rectangle 2×10: area 20, perimeter 24 (less perimeter, still less area)
• The difference in area always equals the square of the difference between sides
• This shows how infinite divisibility permits geometric relationships that naive imagination would deny

The Inscribed Square and Circle #

Euclid proves: inside any square, inscribe a circle; inside any circle, inscribe a square. Since this is always true:

• Inside square 1, inscribe circle 1
• Inside circle 1, inscribe square 2
• Inside square 2, inscribe circle 2
• And so on infinitely
• There is no smallest square or circle (no minimum), just as there is no maximum

Notable Quotes #

“The complete dependence of philosophy upon the natural.”

“If you revolt from the natural right, that would lead to the complete collapse of philosophy.”

“It’s one thing to never have had anything, another thing to have had something and given it up.”

“Wisdom is to speak the truth and to act in accord with nature.” (Heraclitus, as cited)

“You should spend more time trying to judge whether what somebody says is true or false than what he means.”

Questions Addressed #

Can We Know Anything Naturally? #

Modern skepticism denies natural knowledge, but this is self-refuting. The very statement “we know nothing” is itself a statement claimed to be known. Therefore, at minimum, we naturally know that statements exist.

How Can We Prove Infinite Divisibility? #

Through two simple, naturally known facts about faster and slower bodies. By alternating these truths, we demonstrate that both distance and time must be divisible forever. This is a natural proof more fundamental than mathematical assumptions.

Why Do Modern Thinkers Imagine Lines Are Made of Points? #

The human mind struggles to understand potency (potential divisibility). When we cut a line and get a point, we imagine the point was actually there, composed within the line. However, the point exists only in potency until actualized by the cut. This confusion leads to the false doctrine of composition from indivisibles.

What Is the Relationship Between Natural Philosophy and Mathematics? #

Natural philosophy proves what mathematics must assume. The natural philosopher demonstrates infinite divisibility; the geometer presupposes it. Therefore, natural philosophy is wiser and can judge mathematics’ proper role in physical inquiry.

How Should Reason Relate to Nature? #

Kant’s approach—commanding nature like a judge commands a witness—is only part of the answer. Reason must first listen to nature like a student, then investigate it like a judge. Modern philosophy’s error was in dismissing the listening phase entirely.