16. Quantity, Measure, and the Nature of Number

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Lecture Notes

Main Topics #

The Definition and Nature of Quantity #

• Quantity is understood primarily through examination of its parts
• Distinguished by whether parts have a common boundary (meeting place)
• Every quantity consists in a multiplication of parts
• Matter and composition are essential to understanding quantity

Primary Division: Discrete vs. Continuous #

• Discrete Quantity: Parts do NOT have a common boundary
  • Includes: number, speech
  • Number is the first and simplest species
• Continuous Quantity: Parts meet at a common boundary
  • Includes: line, surface, body, place, time
  • Line = one-dimensional; surface = two-dimensional; body = three-dimensional
  • Place and time are extrinsic measures, others are intrinsic

The Concept of Measure #

• Measure is that by which quantity is made known to reason
• In number: measured by the indivisible one (simplest and most absolute measure)
• In continuous quantity: measured by external arbitrary units (foot, meter, etc.)
• The simplicity of the measure of number explains why discrete precedes continuous in Aristotle’s order
• One has no position; the point is closer to the one than other things, but points have position

One, Being, and Multitude #

• One convertible with being: Indivisible being; transcends all genera; is quasi-genus, not properly a genus
• One as measure of number: Determined to genus of quantity; has ratio of measure
• Plurality/Multitude: Opposed to one convertible with being; also quasi-genus
• Number: Plurality measured by one; a species of discrete quantity
• These are opposed as relatives (towards something), not as contraries

Time: A Unique Case #

• Defined as “the number of before and after in motion” (ἀριθμὸς κινήσεως)
• Despite being a number (discrete), it is continuous and divisible forever
• Parts of time do not exist together (past and future do not coexist), unlike parts of line or surface
• Divisibility forever follows from divisibility of motion itself
• Aristotle demonstrates this: faster runner covers more distance in same time; both distance and time divisible forever

Position of Parts (Secondary Division) #

• Some quantities have parts with position relative to each other: line, surface, body, place
• Others lack such position: number, speech, time
• This provides another way to distinguish quantities beyond discrete/continuous distinction

Key Arguments #

Why Quantity Must Be Understood Through Parts #

• Discrete vs. continuous distinction is based on the fundamental nature of how parts relate
• The relationship of parts to the whole reveals the nature of the quantity itself
• This is more fundamental than other modes of distinction

Why Discrete Precedes Continuous #

• The measure in discrete quantity (the one) is simpler and more absolute
• The one is indivisible and has no position, making it the most perfect measure
• Although continuous has affirmative definition (parts meet) and discrete has negative (parts don’t meet), simplicity of measure takes precedence

Why Time is Both Discrete and Continuous #

• Time is a number (ἀριθμὸς), hence discrete in species
• But it measures motion, which is divisible forever, so time is divisible forever
• Parts of time do not coexist (unlike parts of line), yet time itself is infinitely divisible

On Composition and Simplicity in God #

• Thomas distinguishes six kinds of composition in creatures
• God is not composed in any of these ways
• In every composition, one part is potency and another act; God is pure act
• Even individual creatures have composition (essence + individual characteristics)
• Angels escape matter-form composition but not the composition of individual essence

Important Definitions #

• Quantity (ποσόν/quantum): That which is divisible into parts that are actually in it
• Discrete Quantity (ποσὸν διωρισμένον): Quantity whose parts do not have a common boundary
• Continuous Quantity (ποσὸν συνεχές): Quantity whose parts meet at a common boundary
• Number (ἀριθμός/numerus): Plurality (πλῆθος/multitudo) measured by one (ἓν/unum)
• Measure (μέτρον/mensura): That by which quantity is made known to reason
• Position (θέσις/situs): The spatial relationship of parts to each other in a quantity
• One (convertible with being) (ἓν/unum convertibile cum ente): Indivisible being; transcends all genera; quasi-genus
• One (as measure) (ἓν ἀρχὴ ἀριθμοῦ): The indivisible unit by which number is measured; determined to genus of quantity
• Time (χρόνος/tempus): The number of before and after in motion (ἀριθμὸς τοῦ πρότερον καὶ ὕστερον ἐν κινήσει)

Examples & Illustrations #

On Number and Measure #

• Two is the only even prime number
• Three measures six (odd can measure even); but no even number measures an odd number
• Five and seven are prime—no other number can measure them
• Odd and even is one division of number; prime and composite is another
• A number can be divided into odd/even or into prime/composite—multiple divisions possible

On Position #

• Different parts of a room wall are laid out in position relative to each other
• Parts of a line have position; parts of a surface have position; but parts of time do not
• Past and future do not exist together, so time lacks the position property of spatial quantities

On Motion and Divisibility #

• If one runner is faster than another:
  • Faster runner covers more distance in same time
  • Both distance and time are thereby shown to be divisible forever
  • Slower runner covers lesser distance in same time
  • The time it takes faster runner to cover that lesser distance is proportionally less
  • This process continues infinitely

On Individual Substance #

• Berquist’s own example: “I am a two-footed animal with reason born of a Swedish father and an Irish mother”
• This composition (genus + species + individual characteristics) exists in all creatures
• Does not exist in God or angels
• “I’m an odd number” (referencing his attachment to five)

Notable Quotes #

“Every quantity consists in some multiplication of parts.”

“Number is a plurality measured by one” (Pluralitas mensurata per unum)

“The one, according as it is simply said to be an indivisible being, is convertible with being.”

“Time, although a continuous quantity, does not have parts that are laid out toward each other, for the past and future do not exist together.”

“In every composition, one part is to the other part as act is to potency, or both parts are to the whole as potency to act. But God is pure act.”

“For the beauty of order” (Pro pulchritudine ordinis)—God makes creatures unequal not because of sin or defect in nature, but for the beauty of order.

“Plurality, or multitude, according as it signifies divided beings, is not determined to some genus, but is quasi-genus.”

Questions Addressed #

Why does Aristotle distinguish discrete and continuous quantity? #

• To understand the fundamental nature of quantity through the relationship of parts to wholes
• Discrete quantities have parts without common boundaries; continuous have parts meeting at boundaries
• This distinction reveals how reason apprehends quantity itself

Why is the one in number simpler than any measure of continuous quantity? #

• The one is indivisible and has no position
• Continuous quantities require external, arbitrary measures (foot, meter, etc.)
• Therefore discrete (number) precedes continuous in Aristotle’s order despite continuous having an affirmative definition

How can time be both a number (discrete) and divisible forever (continuous property)? #

• Time is defined as the number of before and after in motion
• Because motion itself is divisible forever, time—being the measure of motion—is also divisible forever
• Yet its parts (past, future) do not exist together, distinguishing it from spatial quantities

Are the one convertible with being and the one that measures number the same? #

• No. The one convertible with being is quasi-genus transcending all genera (like being itself)
• The one as measure of number is determined to the genus of quantity
• They have different ratios and different determinations

In the Trinity, is three persons more than one person? #

• No. The multitude of persons in the Trinity is not numerical multitude
• “Three is not more than one, and the Father and Son is not more than the Father”
• This is a different kind of multitude, not measured by one in the way number is

Can even number measure odd number? #

• No. If an even number measured an odd, then two (which measures all evens) would measure an odd
• This would make the odd divisible into two equal parts, which is impossible
• But odd can sometimes measure even (three measures six)

What composition exists in individual creatures? #

• Six kinds: matter-form, body-accident, substance-accident, essence-existence, substance-essential parts, universal-individual
• Each individual creature has: essence (what it is) + individual characteristics (who it is: parents, accidents, etc.)
• Angels escape matter-form composition but not essence-existence distinction or universal-individual composition