87. Order, Symmetry, and the Problem of First in Motion

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Lecture Notes

Main Topics #

Order vs. Symmetry #

• Order (strict sense): Before and after; a relationship of temporal or causal sequence
• Symmetry: A relational property without before-and-after; balance and correspondence
• Berquist illustrates the difference through multiple examples:
  • A Midsummer Night’s Dream: Symmetrical structure (Athens → forest → Athens) without temporal priority
  • Mozart Masses: The Kyrie and Agnus Dei correspond to each other; Gloria and Sanctus correspond; Credo stands alone
  • Platonic dialogues: The Euthyphro and Crito are short and correspond to each other; the Meno and Phaedo are long and correspond
  • McNeill’s division of world history: Eastern dominance, Eurasian balance, Rise of the West (symmetrical, not ordered)
• The Trinity exemplifies relations (πρός τι) that are symmetrical without before-and-after; distinction is based on relations, not on temporal or causal priority

The Paradox of Becoming (Hegel’s Paradox) #

• The problem: Something changes from not-sphere to sphere. If a last instant exists when it is not-sphere and a first instant when it is sphere:
  • If these are the same instant: the thing both is and is not a sphere (contradiction)
  • If these are different instants: there is time between them, but nothing exists between contradictories
• Heraclitus and Parmenides: Heraclitus spoke as if opposites become each other (day becomes night); Parmenides concluded change is impossible

Aristotle’s Solution: The Asymmetry of Limit #

• There is a first instant when something is in the state to which it has changed (e.g., first now when it is a sphere)
• There is no last instant when something is in the state from which it has changed (e.g., no last now when it is not-sphere)
• This asymmetry follows from the nature of becoming itself: to change from A to B is to leave A behind
• When the change is completed (a now), the thing is in B, not in A; A has been left behind
• The completion of motion is indivisible (a now); the period of non-being is continuous and lacks a last part

Infinite Divisibility and the Absence of First in Motion #

• Key principle: Time and motion are continuous; continuous things are divisible forever and have no first part
• For any claimed first moment of motion at time T, motion must have already begun at T/2, T/4, etc.
• Therefore: There is no first time at which motion begins
• Similarly, for any claimed first distance traveled, half that distance was traveled earlier
• This does not mean motion never begins; rather, there is no indivisible first instant of beginning
• This differs from the completion of motion, which does have a first indivisible instant

Change and Habit: The Paradox of Acquiring Virtue #

• The apparent contradiction: One acquires virtue by performing virtuous acts, but if one is performing virtuous acts, one must already possess virtue
• Solution: Before and after acquiring a habit (ἕξις), one performs the same acts, but differently:
  • Before: with difficulty, instability, effort, and mistakes
  • After: with ease, stability, and consistency
• The acts are not exactly the same in mode before and after acquisition
• This parallels motion: as one continues moving, one “builds up” the completion of the motion; before moving any distance, one has already moved some distance

Key Arguments #

The Infinite Divisibility Argument #

1. Time and motion are continuous
2. Continuous things are divisible without limit
3. For any moment T claimed to be first in motion, an earlier moment T/2 must exist where motion had already begun
4. Therefore, no first instant of motion exists
5. But completion of motion is indivisible (a now), so a first instant of completion does exist

The Argument from Contradiction #

1. Something changes from not-sphere to sphere
2. At some point it ceases being not-sphere and becomes sphere
3. If the last instant of not-being and first instant of being are identical: both is and is not sphere (contradiction)
4. If they are distinct: time exists between contradictories (impossible)
5. Therefore: the last instant of not-being does not exist; only the first instant of being exists
6. This resolves the paradox without denying change

The Argument from the Nature of Change #

1. To change from A to B is to leave A and be in B
2. When change is completed, A has been left behind
3. The completion of change is indivisible (a now)
4. Therefore: in that now, the thing is in B, not in A
5. The now that marks completion belongs to the achieved state, not to the previous state

Important Definitions #

Order (ὅρος, in strict sense): A relation involving before and after; temporal or causal sequence

Symmetry: A balanced relation without before-and-after; mutual correspondence without priority

The Now (νῦν): The indivisible limit of time; itself not a time but a boundary between past and future

Continuous: Divisible forever without terminating in indivisible parts; having no first or last part in the sense of actual division

First (πρῶτος): In motion, two senses exist:

• No first instant of motion’s beginning (due to infinite divisibility)
• A first instant of motion’s completion (indivisible)

Change (μεταβολή): The imperfect act of that which is in potency insofar as it is in potency

Examples & Illustrations #

The Doorway Example #

When passing through a doorway from outside to inside: at the completion of the passage (a single indivisible instant), one is fully inside. There is no last instant of being outside; there is a first instant of being inside.

The Sphere Example #

Something becomes a sphere (changing from not-sphere to sphere):

• There is a first now in which it is a sphere
• There is no last now in which it is not a sphere
• The time of not-being is continuous and lacks a final part

The Eucharist (Transubstantiation) #

Before and after the priestly consecration:

• Before: bread and wine exist
• After: the body and blood of Christ exist
• The problem: If both states exist, at what point does the change occur?
• Aristotle’s solution applied: There is a first instant when the body and blood are present; there is no last instant when bread and wine are present
• This avoids the heresy of consubstantiation (both existing in the same instant)
• Thomas Aquinas uses this Aristotelian framework to resolve the theological difficulty

The Polygon and Circle #

As one inscribes polygons in a circle with increasingly many sides:

• Each polygon approaches the circle but never equals it
• The circle is the limit, never actually reached
• Similarly, as one approaches an instant from before it, one approaches the first instant of motion/completion without ever reaching a “last instant” of the prior state

The Piano Habit #

One acquires the ability to play piano by playing piano:

• Before: playing with difficulty, errors, and great concentration
• After: playing with ease and consistency
• The acts are materially similar but formally different in mode
• This parallels how one continuously “builds up” completion of motion

Examples from Literature and Art #

• A Midsummer Night’s Dream: Symmetrical spatial movement (court → forest → court) without temporal priority among the movements
• Plato’s dialogues surrounding the Apology: Euthyphro and Crito (shorter) correspond; Meno and Phaedo (longer) correspond; the Apology stands uniquely in the center
• Mozart’s compositions: Careful knowledge of when to stop; artists like Titian know precisely when to cease
• Shakespeare: Exact sense of when to conclude, avoiding excess

Questions Addressed #

Q: Is there a first instant when motion begins? A: No. Because time is infinitely divisible, for any instant one identifies as first, an earlier moment can be found where motion had already begun. There is no indivisible first moment of beginning.

Q: Then how does change actually begin? A: Change does begin (it is real), but not by starting at a first indivisible instant. Rather, before any moment of the change, earlier moments of the change exist. The asymptotic approach of before and after continues indefinitely. Yet the completion of the change does occur at an indivisible first instant.

Q: How do we avoid the paradox that something both is and is not in the same instant? A: By rejecting the assumption that there must be a last instant of the prior state. The resolution is asymmetrical: there is a first instant of the achieved state, but no last instant of the prior state. This avoids saying both states coexist in the same now.

Q: Is the now a time or not a time? A: The now is not itself a time. It is the limit or boundary of time. There is no motion or change in the now itself, only in the time that it bounds.

Q: How does this resolve the theological problem of transubstantiation? A: By accepting that substance can change (from bread to body of Christ) at a first indivisible instant, but without requiring a last instant of the prior substance. The instant of consecration marks the first now of the achieved reality without needing to identify the final moment of the prior substance.

Q: Is the distinction between order and symmetry merely linguistic? A: No. Order involves before-and-after (real priority); symmetry involves balanced correspondence without priority. This distinction clarifies misunderstandings about the Trinity (which exhibits symmetry of relations, not order of causation) and about historical or literary structure (which may be symmetrical without being ordered).

Notable Quotes #

“The changing necessarily is in that which it has changed when it first has changed.” — Aristotle, Physics (Berquist’s reading)

“There is no first distance you have moved, and there’s no first what? Distance you’ll be moving.” — Berquist, explaining infinite divisibility

“It’s a strange reality, isn’t it?” — Berquist, on the paradox of motion

“He lacked the supreme gift of the artist.” — Conan Doyle, The Norwood Builder, cited by Berquist on knowing when to stop

“You both are and are not a sphere. So, it is a contradiction in what? Becoming, right?” — Berquist, formulating Hegel’s paradox

“The circle is a limit here of these polygons… you’re never going to reach… you’re never going to get a polygon that’s actually equal to the circle.” — Berquist, on limits and continuous divisibility