64. Time as the Number of Motion: Before and After

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Lecture Notes

Main Topics #

• The Three Senses of “Before”: (1) Before in time (one thing comes before another temporally); (2) Before ontologically (one can exist without the other, but not vice versa, as exemplified by numbers: one can exist without two, but two cannot exist without one); (3) Before in reasoning/discourse (the order in which we come to know things)
• The Relationship Between Magnitude, Motion, and Time: Before-and-after is first found in place/magnitude (as spatial division); because magnitude is continuous and divisible, motion over magnitude is proportionally continuous; consequently, time which measures motion is also continuous
• Time as Number, Not Motion: Time is the number of motion according to before and after, not motion itself. It is a “numbered number” (the number of something continuous), distinct from the abstract number by which we count
• The Paradox of the Now: The now presents a classical philosophical puzzle: if the now is always the same, then all events would be simultaneous (making time meaningless); if the now is always different, when does it cease to exist? It cannot cease when it is, and there is no “next” now (like there is no next point on a line)
• The Resolution Through Proportion: The now is to time as the thing in motion is to motion. Just as a moving object remains the same as to what it is (still a baseball) but is always other as to where it is (its position), the now is always the same as to what it is (always “now”) but always other in its temporal position (before and after)

Key Arguments #

Why Before-and-After is First in Place/Magnitude #

• One place is not moving; it has a firm position
• Division of magnitude into before-and-after parts is immediately evident
• Since magnitude is continuous, motion over magnitude is proportionally continuous
• Since motion is continuous, time measuring the motion is proportionally continuous

Why Time is the Central Sense of “Before” #

• Although before-and-after is first ontologically found in place/magnitude, Aristotle makes before-and-after in time the central sense
• This is because before-and-after becomes the very definition of time (unlike in the definition of motion, which does not include before-and-after)
• In daily usage, “before” and “after” instinctively refer to time (e.g., “I will do this before I do that”)
• The now divides past from future and is essential to our understanding of time

The Distinction Between Motion and Time #

• Motion is only in the thing moving or where it is moving
• Time seems to be everywhere and in all things
• Motion can be faster or slower; time itself is not faster or slower (we speak metaphorically of time “going fast”)
• The numbering of before-and-after in motion constitutes time, not the motion itself

The Problem of the Now’s Identity #

If the now is always the same:

• All events would occur at the same now
• The American Revolution and modern events would be simultaneous
• Time would be meaningless

If the now is always different:

• When does a now cease to be? Not when it is (it exists then)
• Does it cease in a later now? But there is no “next” now (unlike discrete numbers)
• Therefore, it would be simultaneous with all intermediate nows—an absurdity

The Resolution: The Proportion #

• The now is always the same as regards what it is (it is always the present moment)
• The now is always other in its temporal position (before and after)
• This parallels the thing in motion: always the same as to what it is, always other as to where it is
• The distinction is real and proportional, not arbitrary

Important Definitions #

Time (χρόνος): The number of motion according to before and after. Not motion itself, but the measure of motion derived from numbering the successive before-and-after divisions within motion.

Numbered Number vs. Numbering Number:

• Numbered number (ἀριθμὸς ἠριθμημένος) = the number of something (e.g., “three yards,” “ten days,” time)
• Numbering number (ἀριθμὸς ἀριθμῶν) = the abstract number by which we count (e.g., the number three in mathematics)
• Time is a numbered number because it is always the number of some motion; arithmetic deals with numbering number

The Now (τὸ νῦν):

• The division between past and future
• Not a part of time (as a point is not part of a line), but a limit of time
• Always the same as regards its nature, always other in its temporal position
• Known by perception of before-and-after in motion

Proportion (ἀναλογία): A likeness of ratios; not merely a single ratio. For example: as 2 is to 3, so 4 is to 6 (two different ratios that are alike). Essential for understanding how the now relates to the thing in motion.

Examples & Illustrations #

The Sleeping Man #

• When a man falls asleep and wakes up without perceiving change, he does not perceive time’s passage
• He takes the earlier now and later now as one and the same now
• Only when he distinguishes two distinct nows (before and after) does he become aware that time has passed

The Baseball in Motion #

• A ball hit in the infield and caught in the outfield is the same ball as to what it is
• But it is always other as to where it is (different position)
• This demonstrates that real identity can coexist with real difference of position
• Applied proportionally to the now: it is the same now (same as to what it is) but always in different temporal position (before and after)

Measuring Distance and Time #

• Part of a road AB comes before part BC
• Motion over AB comes before motion over BC
• Time measuring motion over AB comes before time measuring motion over BC
• Proportion: As AB is to BC, so motion-over-AB is to motion-over-BC, so time-for-AB is to time-for-BC

Robinson Crusoe Counting Time #

• Stranded on an island, Robinson Crusoe marks a piece of wood
• Each sunset (before) is followed by another sunset (after)
• By counting these divisions, he numbers the before-and-after in the sun’s motion
• This is essentially what all timekeeping does: number the before-and-after in some motion (sun’s motion, watch circulation, etc.)

Hours, Days, and Continuous Quantity #

• One day is the time for one complete circulation of the sun
• Hours divide this circulation arbitrarily into 24 parts
• Despite arbitrary division, the before-and-after structure remains: motion from here to here comes before motion from here to here
• Because time is the number of something continuous, we speak of time as both “more/less” (discrete) and “long/short” (continuous)

The Sophist’s Distinction (Cariscus) #

• Cariscus in the Lyceum and Cariscus in the marketplace are the same person but in different places
• The sophists wrongly take these as different things
• Properly understood: the same thing (Cariscus) in different positions
• Analogy to the now: always the same now, always in different temporal position

Knowing Mother vs. Knowing Her as the One at the Door #

• You know your mother as your mother (what she is)
• But when she knocks at the door, you don’t immediately know her as the one knocking
• The same person, but grasped differently depending on context or position
• Shows how real identity can coexist with real difference in relation or position

Notable Quotes #

“Time is the number of motion according to before and after.” (Aristotle, Physics IV)

“Things in motion sooner catch the eye than what not stirs.” (Shakespeare, Troilus and Cressida, cited by Berquist)

“The now is to time as the point is to the line.” (Aristotle, Physics IV; explained throughout lecture)

“Since number is twofold, for we call number both the numbered and the numerable, and that by which we number. Time is the numbered [number], and not that by which we number. But arithmetic is about the number by which we number.” (John of St. Thomas, cited by Berquist)

Questions Addressed #

Why Does Aristotle Make Before-and-After in Time the Central Sense Rather Than Before-and-After in Place? #

• Although before-and-after is first found ontologically in place/magnitude, it becomes the central sense because before-and-after is constitutive of the very definition of time
• When we perceive or speak of time naturally, we think first of temporal before-and-after, not spatial
• The definition of time explicitly includes before-and-after; the definition of motion does not

How Can the Now Be Both the Same and Always Different? #

• The Same: As to what it is (its nature as the present moment, the boundary between past and future)
• The Different: In its temporal position (each now is before or after another now)
• This is possible through the proportion: the now to time is as the thing in motion to motion
• The thing in motion is similarly both identical (same baseball, same stone) and always in different position

How Do We Become Aware of Time’s Passage? #

• By perceiving change or motion
• By distinguishing before and after in that motion
• By numbering these distinctions (recognizing two nows as distinct)
• Without distinguishing before from after, we do not perceive time (the sleeping man example)

Why is Time a Number and Not the Motion Itself? #

• Motion is localized (only where the thing moves)
• Time is universal (same time for all things)
• We measure motion by numbering its before-and-after divisions
• Time is this numbering of before-and-after, proportional to the motion

What is the Relationship Between Continuous and Discrete Quantity in Time? #

• Time is the number (discrete aspect) of something continuous (the before-and-after in motion)
• Therefore, we can speak of time as both “more/less” (treating it as discrete number) and “long/short” (treating it as continuous magnitude)
• “Seven days is more than five days” (discrete); “This time is longer than that time” (continuous)
• This duality reflects time’s nature as the number of continuous motion

Methodological Observations #

On Choosing Examples #

• Berquist cites his teacher Cassuric: “You can tell how well a man understands by the examples he chooses”
• Aristotle’s examples are carefully chosen for philosophical significance
• Thomas Aquinas recognizes the significance: the father exemplifies a mover by nature; the advisor exemplifies a mover by reason
• God is called “Father” (not mother) because God is the mover/maker (cause), while matter is associated with “mater/materia”
• Modern philosophers often choose imaginatively striking but philosophically insignificant examples

On Language and Conceptual Development #

• Language often reflects the historical development of understanding
• Euclid’s terminology (“sides of six” rather than “factors”) reflects more concrete, original understanding
• Geometric terms (square, cube) retain concrete meaning even in arithmetic, showing continuity from sensible to abstract
• The word “road” (ὁδός) in knowledge reflects the order (before-and-after) in our thinking, not the spatial continuity of an actual road

On Definitions and Starting Points #

• Understanding often requires resolving imagination all the way back to first principles
• A point, by motion, generates a line; a line, by rotation, generates a circle; a circle, by rotation around its diameter, generates a sphere
• This imaginative resolution back to the point clarifies why lines have no width, circles are equidistant from center, etc.
• Without clear definitions and proper imagination, philosophy becomes arbitrary

Connections to Broader Philosophy #

To Natural Philosophy: Time is inseparable from motion and change; understanding time requires understanding magnitude, motion, and continuity

To Metaphysics: The paradox of the now relates to fundamental questions about substance, identity, and change; the distinction between what something is and where/when it is

To Logic and Epistemology: The three senses of “before” relate to how things are known and the order of reasoning; proportions are essential tools for understanding non-obvious relationships

To Future Lectures: The resolution of the now paradox through proportion prepares for understanding how continuous quantity (magnitude and time) differs from discrete quantity (number), and how division and composition work in continuous things