86. Motion, Continuity, and the Paradox of First

Subscribe in Podcast App | Download Transcript

Lecture Notes

Main Topics #

The Paradox of First in Motion #

• First in completion: When something changes from state A to state B, there is an indivisible first moment (a νῦν/now) at which it first is in state B
• No first in beginning: There is no first distance traveled, no first time at which motion begins, because continuity is divisible forever
• The asymmetry: Motion has a definite end but an indefinite beginning

The Nature of Continuity #

• Infinite divisibility: Everything continuous is divisible forever and is not composed of ἀτόμα (indivisibles)
• Nows are not adjacent: Two νῦν cannot be contiguous; there is always time between any two moments
• Limits are indivisible: While the continuous thing itself is divisible, its limit (σημεῖον/point in space, νῦν in time) is indivisible

The Problem of Beginning vs. Completion #

• Completion: When I arrive in Boston, there is a first νῦν in which I am in Boston—this is indivisible
• Beginning: When I leave Worcester, there is no first distance I travel; before traveling any distance d, I must travel d/2, and before that d/4, etc., infinitely
• Application to change: When becoming a sphere, there is a first νῦν in which it is a sphere, but no last νῦν in which it is not a sphere

The Mutual Dependence of Changing and Having Changed #

• Neither has priority: Before any motion is completed, some motion has already occurred; before any motion occurs, some motion has already been completed
• Continuous entailment: At any division of time, the thing must have changed in each part if it has changed in the whole
• Distinction from discrete things: Numbers have a genuine first (3 comes after 2); continuous quantities do not

The Order of Distinction and Order #

• Distinction precedes order: Things can be distinct without having order, but cannot have order without distinction
• In being: Distinction is prior to order in ontological status
• In knowledge: We must perceive distinction before we can perceive order

Key Arguments #

Argument for No First Distance Traveled #

1. If I travel from Worcester to Boston, do I first travel one mile?
2. No—I must travel half a mile before the whole mile
3. Do I first travel half a mile? No—I must travel a quarter mile first
4. Since distance is continuous and infinitely divisible, no point of termination exists
5. Conclusion: There is no first distance I have traveled

Argument for No First Time of Motion #

1. Suppose T is the first time I am moving from Worcester to Boston
2. If I am moving during the whole of T, then I am moving during the first half of T
3. If I am moving during the first half of T, then T cannot be the first time I was moving
4. Any time can be similarly divided
5. Conclusion: There is no first time at which motion begins

Argument for Having Changed Before Changing (Reading 8) #

1. Everything that changes does so ἐν χρόνῳ (in time)
2. Time is divisible by any νῦν into two parts
3. If something has changed in the whole time, it has changed in each part
4. In each νῦν, no motion occurs; therefore, it must have already ἀλλοιώθη (changed)
5. Conclusion: Before completing any motion, something has already moved some distance

Argument Against Contradiction in Becoming #

1. Hegel’s objection: When becoming a sphere, at some moment it is both sphere and not-sphere (contradiction)
2. Aristotle’s solution: The νῦν at which it first is a sphere is not a time but a limit
3. There is no last νῦν in which it is not a sphere, but a first νῦν in which it is a sphere
4. Since the νῦν is indivisible and not time itself, no contradiction occurs

Important Definitions #

Νῦν (Now) #

• The indivisible limit between times
• Not itself a duration but the boundary of time
• Characterizes the moment at which a change is completed

Ἀτόμα (Indivisible) #

• Without parts; cannot be divided
• The νῦν is ἀτόμα
• The σημεῖον (point) is ἀτόμα as the limit of a line

Continuous (Συνεχές) #

• Infinitely divisible; composed of parts sharing a common boundary
• Characterized by the axiom that there is no first part
• Examples: magnitude, time, motion

Discrete (Disjunctive) #

• Has a definite number of parts
• Has a genuine first element
• Examples: the natural numbers (3 comes first after 2)

Proton (First) #

• In motion: that before which nothing is before and after which nothing is first
• Can mean first in time, first in causality, first in knowledge
• Applied equivocally depending on context

Distinction (Distinctio) vs. Order (Ordo) #

• Distinction: Two things are distinct when one is not the other
• Order: One thing is before or after another; order presupposes distinction
• Axiom of order: Nothing is before or after itself

Examples & Illustrations #

The Worcester to Boston Journey #

• When I have arrived in Boston, there is a first νῦν at which I am in Boston (indivisible)
• But there is no first distance I traveled—any distance d can be divided into d/2
• Illustrates the asymmetry between completion (definite) and beginning (indefinite)

The Becoming a Man Example #

• A person becomes a man; let us say born January 18th
• At the instant of birth, there is a first νῦν of being born
• But was he first born the whole day? No—if he was born in the morning, he wasn’t “first born” on that day
• Any period of time can be similarly divided, showing the νῦν must be ἀτόμα

The Piano Through the Doorway #

• How much of the piano comes through the door first?
• Do I first shove the whole foot through? No—half a foot gets through first
• Half a foot? No—a quarter foot gets through first
• Since the body is continuously divisible, there is no first amount

The Two Bodies at Equal Speed #

• Body A moves from X to Y in time T
• Body B starts at X and moves equally fast but stops at some point C (the midpoint)
• Body B clearly has moved to C
• Body A, while continuing, has also moved to C before completing its full journey to Y
• Therefore: before the full distance is traveled, some distance has already been traveled

Lines and Points Analogy #

• On a line: there is no first length
• Before every length, there is another length
• But at the boundary of each length is a σημεῖον (point), which is ἀτόμα
• Motion relates to distance as a νῦν relates to time: the end is indivisible, but the beginning is infinitely divisible

Questions Addressed #

Is there a first moment when motion begins? #

Answer: No. Because time is continuous and infinitely divisible, any moment one specifies as ‘first’ can be divided, and motion must have already occurred in the first half. Only by positing that time is composed of ἀτόμα (which Aristotle rejects) could there be a true first moment.

Is there a first distance traveled? #

Answer: No, for the same reason. Any distance d can be divided into d/2, and d/2 must be traveled before d. Since this divisibility is infinite, there is no first distance.

How can motion avoid the contradiction of both being and not being? #

Answer: The νῦν at which change is completed is not itself a time but the limit of time—it is ἀτόμα. Thus, while there is a first νῦν at which the thing is in state B, there is no last νῦν at which it is in state A. No moment exists at which it is both A and not-A.

What is the relationship between changing and having changed? #

Answer: They are mutually entailed in continuous motion. Before any motion is completed, some motion has already occurred; before any motion occurs, some motion has already been completed. Neither has temporal priority—they presuppose each other.

How does this apply to understanding substances and change in theology? #

Answer: This analysis resolves apparent contradictions, such as in the doctrine of transubstantiation: there is a first νῦν at which the body and blood of Christ are present, but no last νῦν at which the bread and wine are present. This avoids the heresy of coexistence.