15. Continuous and Discrete Quantity: Definitions and Measures

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Lecture Notes

Main Topics #

Division of Quantity #

• Quantity is divisible into parts that are actually in it
• Two primary species: discrete quantity (number) and continuous quantity (magnitude)
• Discrete: parts do not meet at a common boundary
• Continuous: parts meet at a common boundary

Two Definitions of Continuous Quantity #

In Logic (Categories):

• Definition: “that whose parts meet at a common boundary” (affirmative definition)
• Focuses on the form and wholeness of the continuous
• Appropriate for logical consideration since logic deals with things as they are in reason

In Natural Philosophy (Physics):

• Definition: “that which is divisible forever” (negative definition)
• Emphasizes material divisibility and process
• Appropriate because natural philosophy considers matter and brings matter into its definitions
• More concrete, process-oriented approach

Why the Definitions Differ #

• Logic considers things as they are in reason/mind
• Natural philosophy considers things as they are in being
• The logician defines by form (the whole)
• The natural philosopher defines by matter (the parts)
• Both approaches are valid but serve different purposes

The Point vs. One Distinction #

• A point is indivisible but has position
• The one (unit in number) is indivisible but has no position
• Platonists would say: a point is one with position
• This distinction matters for understanding why lines cannot be composed of points

Measure and Species of Quantity #

• Measure is “that by which quantity is made known to reason”
• Different species of quantity may have different ratios of measure
• Examples: number measured by one; length measured by foot or meter; time measured by duration; speech measured by long/short syllables
• Things with different measures may still be the same kind of quantity (e.g., wall length and sofa length are both continuous quantity measured in the same units)

Why Discrete Comes Before Continuous #

• Despite continuous quantity having an affirmative definition, Aristotle prioritizes discrete
• The one is “simpler, absolute, and indivisible”
• Numerical measure is more perfect and exact than physical measures
• Physical measures (foot, meter) are arbitrary and subject to variation
• The one is universal and applies to all numbers

Key Arguments #

The Infinite Divisibility Argument #

• A line can always be divided into two smaller lines
• Division ceases only when you reach a point (indivisible)
• A point has no parts; two points cannot touch except by coinciding
• If two points coincide, they are one point, not two
• Therefore: a line cannot be composed of points
• Conclusion: continuous quantity is infinitely divisible

The Motion/Speed Argument #

• If one body is faster than another, they cover different distances in the same time
• The faster body covers distance D in time T
• The slower body covers distance d (less than D) in the same time T
• When we divide distance, we must proportionally divide time
• This demonstrates both distance and time are infinitely divisible
• Shows how Aristotle proves divisibility using natural observation

Affirmation Before Negation #

• Affirmation must come before negation in definition
• One must know what is being negated before understanding negation
• Yet discrete quantity (negatively defined: parts without boundary) comes before continuous quantity (affirmatively defined: parts meeting at boundary)
• This apparent contradiction is resolved by the priority of measure: numerical measure is more perfect

Important Definitions #

• Quantity (ποσότης/posas): That which is divisible into parts actually present within it
• Discrete Quantity: Multitude or plurality; parts without common boundary (number is the paradigm)
• Continuous Quantity: Magnitude; parts with common boundary (length, surface, body, time, place)
• Measure: That by which the quantity or size of something is made known to reason
• Whole: Related to parts as form is to matter; comes to be from its parts which exist in it
• The One (ἕν): Indivisible unit; simpler and absolute; basis of numerical measure
• Point (σημεῖον): Indivisible but with position; cannot compose a line

Examples & Illustrations #

Mathematical/Logical Examples #

• Seven can be divided: 7 = 2 + 5 or 3 + 4, but parts remain distinct
• A line bisected produces two lines, which can be bisected again indefinitely
• In Euclid: two is a part of six (6 = 3 × 2), but four is not a part of six
• Prime numbers can measure some numbers but not all; the one measures all numbers

Physical Examples #

• Wine bottle: 1.5-liter bottle contains more wine than 0.75-liter bottle (different extrinsic measurements of the same kind of quantity)
• Room measurement: measuring room length in feet vs. measuring wall length for a sofa fit (different measurements, same kind of quantity)
• Worms can be divided with both parts remaining alive; plants can be divided into multiple substances

Sensory/Practical Examples #

• Poetry meter: Latin/Greek poetry measured by long and short syllables; English poetry by accented/unaccented syllables
• Thomas’ prayer verse: “Visus toctus gustus in te faldi tur” (visual example of metrical patterns)
• Container volume: measured by gallons or cubic feet, different from linear measurement

Questions Addressed #

Why does Aristotle prioritize discrete quantity over continuous despite its negative definition? #

• Because measure is more perfect in number (the one) than in continuous magnitude
• The one is absolute, indivisible, and universal
• Physical measures are arbitrary, variable, and imperfect
• The accuracy and simplicity of numerical measure take precedence over the form of the definition

How can continuous quantity be infinitely divisible if it cannot be composed of points? #

• As long as division produces two lines (or surfaces, or bodies), further division is possible
• The infinite divisibility is a potential property, not an actual composition
• Division never actually completes; there is always more to divide

Why do logic and natural philosophy define continuous quantity differently? #

• Logic considers things as they exist in the mind/reason
• Natural philosophy considers things as they exist in being
• Logician emphasizes form and unity (parts meeting at boundary)
• Natural philosopher emphasizes matter and process (divisibility)
• The difference reflects different approaches to the same reality

Can different species of quantity be compared for equality? #

• Only if they share a common measure
• A line and surface cannot be equal or unequal (different types of continuous quantity)
• Time and length can be compared in practical contexts (e.g., distance takes 24 hours), but this is somewhat playing with language