88. Motion, Time, and the Now: Resolving Zeno's Paradoxes

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Lecture Notes

Main Topics #

Change and the Indivisible Now #

• When something changes from state A to state B, it completes the change in an indivisible moment
• At the moment of completion, the thing is no longer in state A but is in state B
• The change itself is indivisible, yet it involves leaving behind the prior state
• When ceasing to be (dying), there is a first moment when one is not, which is indivisible
• Scripture references the damned descending “in puncto” (in a point)—an indivisible descent not occurring in time

The Now and the Sharing of Time #

• The fundamental problem: There cannot be the same now shared between being and non-being
• If one admits a last now in which something exists in the state from which it is proceeding, contradiction arises
• Either that now is the same as the first now of the new state (contradiction) or there is time between them (but time cannot be between two nows)
• Therefore, there is no time between being alive and being dead

Motion and Rest in the Now #

• Neither motion nor rest exists in a single indivisible now
• A body in motion and a body at rest, when considered in the same now at the same location, have no difference in that now
• Motion and rest are properties of bodies in time, not in an indivisible instant
• Difference between moving and resting bodies lies in their relation to what follows in time, not in the now itself

Key Arguments #

The Problem of Infinite Divisibility #

• The modern response to Zeno (polygon approaching circle): You can pass through an infinity of divisions because the divisions are only potentially infinite, not actually infinite
• A straight line approaching a circle will never equal it; if it did, a straight line would be a curved line—a contradiction
• The acceptance of infinite divisibility does not create a logical problem if we distinguish potential from actual division

Science and Idealizations #

• Einstein, Heisenberg, and de Broglie all recognize that science is based on idealizations that depart from reality
• Example: The law of inertia (a body in the absence of external forces remains at rest or in uniform motion) has no actual examples in experience
• As friction is reduced and conditions simplified, bodies travel further, suggesting the ideal case; but this is imaginative, not real
• The margin of error in laboratories may stem from theory falseness rather than instrument inaccuracy
• Logical positivists make science impossible by demanding theories perfectly fit facts

Imagination vs. Intellection #

• Imagination tends to make potential things actual
• Heisenberg observes that concepts of natural language (ordinary language) have greater stability in knowledge growth than precise scientific terms, because scientific terms are based on idealizations departing from reality
• When imagining the now, the mind tends to expand it into a small duration of time, making it more actual than it is
• Intellection can understand indivisibles (like the point) negatively—through successive negation rather than positive imagination

Act and Potency in Understanding Motion #

• Act is more known to us than potency
• To manifest that a body has moved, Aristotle compares an equally fast body that stops (and thus clearly has moved) to a body that continues moving
• The body that stops makes the “has moved” more actual and evident than the continuing body
• This is why one object at rest and one in motion, at the same position in the same now, show a difference: the resting body manifests through its actuality what the moving body only has potentially

Important Definitions #

The Now (Nunc) #

• An indivisible limit of time
• Does not occupy time itself
• Possesses neither duration nor divisibility
• Cannot contain motion or rest
• Analogous to the point in a line

Motion (Kinesis) #

• The act of transition from one state to another
• Characterized as an “imperfect act”—the act of something insofar as it remains in potency
• Defined fundamentally in terms of act and potency, not in terms of before and after
• The least actual of all acts, yet most known to us through sensation

Indivisible #

• A limit point or moment that cannot be divided into parts
• The now is indivisible; time is not
• Change is completed in an indivisible instant

Before and After (in Aristotle’s treatment) #

• Distinguished from the definition of motion itself
• Motion is defined by act and potency; before and after characterize the order in time
• Time is related to the before and after of motion, not to the definition of motion
• Both before-and-after and opposition (the basis of distinction) pertain to the one and the many

Examples & Illustrations #

The Two Bodies in the Doorway #

• One body standing in the door; another walking through it
• At a single moment when the moving body passes the stationary one, the moving body is in an “imperfect act” (a state that will progress further)
• The standing body remains in that state; the walking body will not
• Yet in that single now, there is no difference in their actual state

Two Bodies at Different Speeds #

• Aristotle’s argument: Two equally fast bodies, one stops, one continues
• The body that stops clearly manifests that it has traversed a distance
• The body moving at equal speed must have traversed the same distance, though this is less obvious since it continues
• Act manifests potency: the stopped body’s obvious actuality makes the continuing body’s achievement manifest

The Cone and Circle Problem #

• Cutting a cone parallel to its base produces two circles
• If all possible cuts are made actual, the result would be a cylinder, not a cone
• This shows that lines between any two lines exist only potentially until actually made
• Making all potential divisions actual fundamentally changes the thing itself

The Point in a Line #

• When you bisect a line, you make a point that was there only potentially now actual
• The point becomes more actual through division
• Just as the now is to time, the point is to the line—a limit and division, not a composition
• Motion is to the “has moved” as time is to the now: a continuous magnitude, not composed of its limits

Lincoln’s Rhetorical Riposte #

• Douglas called Lincoln “two-faced”
• Lincoln replied: “If I had another face, would I wear this one?”
• After this jest made with seriousness, Douglas could not resume the attack on two-facedness
• Illustrates how the serious aspect of life can be brought into sharp focus such that the comic aspect recedes

The Navy Radio Antenna #

• A sailor drilled a hole in the ship to get antenna reception through the metal
• Did not realize the ship was not fully loaded, so water rose to the level of the hole
• Illustrates the many comic aspects of military life that coexist with its serious tragedy

Notable Quotes #

“The cause of difficulty is not in us but in the thing itself.” — Boethius (cited by Berquist, referring to the difficulty of understanding motion)

“Men do not understand the things they meet every day, although they think they do.” — Heraclitus (cited by Berquist, applied to motion and time)

“When we imagine something, we make it actual in our imagination.” — Heisenberg’s student Sacha (cited by Berquist)

“Nothing is before or after itself.” — Aristotelian axiom (fundamental to understanding that multiplicity is necessary for order)

“Nothing is the beginning of itself.” — Aristotelian axiom (cited by Berquist, showing that multiplicity is necessary for causation)

“The least actual of all acts is motion, but it is most known to us.” — Aristotle (cited by Berquist, illustrating the gap between what is knowable by nature and what is knowable to us)

“Nothing is more misleading than a clear and distinct idea.” — Louis de Broglie (cited by Berquist, contrary to Descartes)

Questions Addressed #

Can there be a difference between two bodies in the same now if one is moving and one is resting? #

• Initial difficulty: In an indivisible now, neither motion nor rest exists; both are properties in time. So how can there be a difference?
• Resolution: While there is no difference in the now itself, there is a difference in their relation to what follows. The resting body will remain; the moving body will continue. Through the principle that act is more known than potency, the resting body’s actual completion of its state manifests the moving body’s achievement more clearly.

How can one complete an infinite task if infinite divisions must be traversed? #

• Zeno’s assumption: All the divisions must be actually traversed.
• Resolution: The divisions are only potentially infinite. A continuous magnitude is infinitely divisible in potency but not composed of infinite actual parts. One traverses a finite distance.

Is there time between the last moment of one state and the first moment of another? #

• Apparent problem: If there is a last now of A, is it also the first now of not-A (contradiction)? If not, what time lies between them?
• Resolution: There is no last now of being alive; there is a first now of being dead. The change is completed in an indivisible now. There is no time between them because there is no “between” in the indivisible.

How can imagination and intellection differ in understanding the indivisible? #

• Difficulty: Imagination makes everything continuous and tends to expand the indivisible into small durations.
• Resolution: Intellection understands indivisibles negatively—by successive negation (e.g., the point has no width, no depth, no length, nothing at all). Imagination cannot grasp this purely negative knowledge.

What is the relationship between the definition of motion and the before-and-after of motion? #

• Question: Motion is defined by act and potency, but time is related to before and after. Are these the same reality differently conceived?
• Answer: Yes—they are the same reality but conceived differently. Motion (as imperfect act) and the before-and-after in motion (as order in time) are two ways of grasping the same phenomenon. The definition emphasizes the metaphysical nature (act and potency); the before-and-after emphasizes the temporal order.

How do tragedy and comedy relate to complementarity? #

• Problem: When one aspect of life is brought into sharp focus, the other recedes. War is hell (tragic), yet many comic things happen in war.
• Resolution: Like complementarity in physics, tragedy and comedy are complementary aspects of life. To be in one mood, one must leave the other. Yet both are real aspects of the whole.

Key Structural Insights #

The One and the Many in Metaphysics #

• Distinction presupposes multiplicity (distinction is when “this is not that”)
• Order and before-and-after presuppose multiplicity (nothing is before or after itself)
• Both opposition/distinction and before-and-after are listed among names pertaining to the one and the many
• This shows why resolving the problem of motion requires understanding the metaphysical foundation of distinction and order

The Imperfection of Our Knowledge #

• What is most actual by nature (God as pure act) is least knowable to us
• What is least actual (motion) is most known to us through sensation
• This gap reveals the imperfection of the human mind, not of reality
• It parallels how we love private goods before common goods, and candy before wisdom—signs of our weakness, not reality’s