69. The Continuous: Definitions, Divisibility, and Becoming

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Lecture Notes

Main Topics #

The Continuous and Its Fundamental Importance #

• The continuous is fundamental to understanding motion, place, and time
• Aristotle states: “We do not think without an image and therefore we do not think without the continuous in time” (Sense and Sensible, Memory and Reminiscence)
• The continuous is more basic than geometry because it presupposes continuous quantities
• Most people cannot transcend the continuous in their thinking (e.g., imagining eternity as endless time rather than timeless)

The Continuous in Language and Knowledge #

• Basic words like “beginning” (ἀρχή), “end” (τέλος), “before” (πρότερον), “after” (ὕστερον), and “in” (ἐν) are fundamentally tied to the continuous
• These words originate from the continuous and are then extended metaphorically to other domains:
  • “Beginning” applies to: foundation of a house, the prince in a city, principle (ἀρχή), axioms, definitions
  • “End” (τέλος) extends to purpose and definition (Greek: ὅρος; Latin: terminus, meaning limit or end)
  • “Road” (ὁδός) extends from physical path to the path of reasoning
• Words derived from the continuous (“in,” “beginning,” “end”) are more extendable than private sensibles (“red,” “green”)
• This is because the continuous is more understandable and less tied to sensible matter than color qualities

Three Related Concepts: Continuous, Touching, Next #

• Continuous (συνεχές): That whose parts have a common boundary
  • Example: two semicircles meeting at a diameter; the diameter is the end of one and beginning of the other
  • Alternative definition: that which is divisible forever
• Touching (ἁπτόμενα): Limits are together but not identical
  • Example: two rooms sharing a wall; the wall is limit of both but they remain distinct
• Next (ἐφεξῆς): Ordered but not touching
  • Example: neighboring houses without a shared wall
• These form an ordered hierarchy, each adding to the previous

The Common and Private Sensibles #

• Common sensibles (κοινὰ αἰσθητά): Sensed by more than one sense
  • Examples: surface, shape, motion, size
  • These are tied to the continuous
• Private sensibles (ἴδια αἰσθητά): Proper to one sense
  • Examples: color (sight), sound (hearing), smell (smell), flavor (taste)
  • Less extendable in meaning; cannot be universally applied like words from the continuous

Key Arguments #

The Problem of Becoming (Parmenidean Paradox) #

• The Problem: When something becomes a circle (transitions from not-circle to circle), it appears to create a contradiction:
  • If the last instant of being non-circular is also the first instant of being circular, the thing both is and is not a circle (contradiction, as Hegel argues)
  • If they are distinct instants, there must be time between them, but in that interval the thing must either be or not be a circle—leading to infinite regress
• Aristotle’s Solution (from Physics Book VI): There is no last instant in which something is not a circle, but there IS a first instant in which it is a circle
  • This may seem arbitrary at first but makes sense upon analysis
  • Example: In death, there is a first instant of being dead but no last instant of being alive
  • Analogously: At the beginning of life, there is a first instant of being (alive) but no last instant of not being
• Why This Matters: Understanding becoming as non-contradictory requires understanding the asymmetry in the continuous structure of time

Zeno’s Paradoxes and the Infinite Divisibility of the Continuous #

• Zeno’s Dichotomy Paradox: Before reaching the door, one must traverse half the distance; before that, half of that; infinitely many halvings must be completed, making motion impossible
• Achilles and the Tortoise: The swift Achilles cannot overtake the slow tortoise because he must first reach where it was, but by then it has moved further; this repeats infinitely
• Zeno’s Purpose: To defend Parmenides’ denial of change by showing that motion involves logical impossibilities
• Aristotelian Resolution: The infinite divisibility of the continuous is potential, not actual. One can traverse an infinite number of potential divisions in finite time because the continuous nature of motion and time permits this.

Reason Knows the Continuous in a Non-Continuous Way #

• The Principle: The way of knowing does not have to be the way the thing is
• Examples of Non-Correspondence:
  • Remembering the past: The way I know the past (present memory) is not the way the past is (now gone)
  • Knowing shape through different senses: The eye knows shape through color; touch knows it through hardness. The way of knowing differs, but the object is the same
  • Definition of a continuous thing: The definition of an equilateral and right-angled quadrilateral is not itself continuous; the genus and differences do not meet at a common boundary
• Implication for Immateriality: If reason knows the continuous in a non-continuous way without falsifying the continuous thing, then reason itself is not continuous. Since bodies are continuous, reason is not a body.

Important Definitions #

Continuous (συνεχές) #

• Primary Definition: “That whose parts meet at a common boundary” or “that which has a common limit”
• Alternative Definition: “That which is divisible forever”
• Key Property: No last division; always divisible further
• Not Composed Of: Indivisibles (points, nows, moments). If composed of indivisibles, two indivisibles could be adjacent without a line between them, which is impossible in true continuity.

Touching (ἁπτόμενα) #

• Things whose limits are together but not identical
• The limits coincide in place but remain distinct

Next (ἐφεξῆς) #

• Things that are ordered in sequence but do not touch
• There exists space or some distinction between them

Indivisible (ἀδιαίρετον) #

• That which has no parts and cannot be divided
• Examples: points, nows, moments
• The continuous is NOT composed of indivisibles

Examples & Illustrations #

The Border Analogy #

• The border between the United States and Canada serves as the upper limit of the U.S. and the lower limit of Canada
• The same line is the boundary for both countries
• Application to Philosophy:
  • The proof of the unmoved mover is at the border between natural philosophy and metaphysics
  • The proof of the immortality of the soul is at the border between natural philosophy and first philosophy
  • Like geographic borders, these proofs have the character of a common boundary between two domains
• Related Concept: Man as microcosm on the horizon between the material and immaterial worlds, mentioned by Arab philosophers and Democritus

Continuous Proportion in Numbers #

• Continuous Proportion: “Four is to six as six is to nine”
  • The end of the first ratio (six) is the beginning of the second
  • Exhibits a likeness to continuity though numbers themselves are not continuous
• Discrete Proportion: “Two is to three as four is to six”
  • Three and four do not coincide; no common term
• Why Transferred to Numbers: Geometry is more sensible and continuous; we tend to transfer mathematical language from geometry to arithmetic

Continuous Arguments, Definitions, and Divisions #

• Continuous Definitions: Definition of “square” includes “quadrilateral,” which includes “plane figure”; each definition is a boundary for the next
• Continuous Divisions: Dividing “rectilineal plane figure” into “triangle” and “quadrilateral,” then “quadrilateral” into “square,” “oblong,” “rhombus,” “rhomboids”—each division is continuous with the next
• Continuous Syllogisms: The conclusion of one syllogism becomes a premise of the next (prosyllogism)
• Continuous Induction: An induction arrives at a general statement that is then used in a subsequent syllogism
• Chain Principle: As a chain is only as strong as its weakest link, reasoning is only as strong as its weakest definition or division

Childhood Memory #

• Berquist recalls kindergarten: each student had a drawer with a toy tied to it for identification
• He brought a toy soldier but could not play with it because it had to remain at school for educational purposes
• Sister Kindness (Καλοσύνη) was the teacher
• Later, attempting to open a coconut, he broke it and couldn’t retrieve it
• Philosophical Point: These memories illustrate how moments pass away and cannot be possessed; we can only know them through memory in the present

Questions Addressed #

How can the continuous be infinitely divisible without being composed of indivisibles? #

• The continuous is divisible potentially, not actually
• No matter how finely you divide the continuous, further division is always possible
• At any given moment, only finitely many actual divisions exist
• This resolution avoids the impossible position of composition from indivisibles

How can we think about non-continuous things (like God) if we cannot think without the continuous? #

• We must negate the continuous
• God is described as incorporeal (ἀσώματος), having neither birth nor death, lacking continuity
• Negation allows us to think beyond the continuous

Can the same thing be known in multiple different ways? #

• Yes. Aristotle shows this with sensible properties:
  • Shape can be known through the eye (via color) and through touch (via hardness)
  • If the way of knowing had to match exactly how the thing is, one object could not have two ways of knowing it
  • But it does, so the correspondence is not one-to-one
• Implication for Theology and Philosophy: Theologians and philosophers can have different but compatible ways of knowing the same truth

Is there a last moment of life or only a first moment of death? #

• Answer: There is a first instant in which one is dead, but no last instant in which one is still alive
• Parallel: At the beginning of becoming, there is a first instant of being, but no last instant of not-being
• This asymmetry resolves Parmenidean paradoxes about becoming

Notable Quotes #

“We do not think without an image and therefore we do not think without the continuous in time.” — Aristotle, from Sense and Sensible and Memory and Reminiscence (cited by Berquist)

“You don’t realize how fundamental the continuous is in our, what, in the very words that we use.” — Duane Berquist

“Most people can’t transcend [the continuous] at all. Whatever it is must be somewhere. They can’t transcend it and continue this.” — Duane Berquist, on the difficulty of thinking beyond spatial continuity

“If we never think without an image, and therefore without the continuous, how can we think about something that is not continuous? Well, we have to negate the continuous.” — Duane Berquist, on thinking about incorporeal realities

“You have to understand what the continuous is before we can see that reason is not continuous, right? And you’ll see it even in the Dialectic of the first book about the soul, that Aristotle is arguing against thinking, right, being something continuous.” — Duane Berquist, on the importance of understanding the continuous for proving immateriality of reason