17. Perfect Numbers, Proportions, and the Properties of Quantity

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Lecture Notes

Main Topics #

Perfect Numbers and Proportional Relationships #

• A perfect number equals the sum of its divisors (e.g., 6 = 1 + 2 + 3)
• Second perfect number is 28
• When numbers form a proportion, the product of outer terms equals the product of inner terms
• Example: If 2:4 = 7:? then 2 × ? = 4 × 7, establishing proportionality
• Proportional relationships in mathematics are clearer than in other disciplines
• Understanding proportions with numbers is foundational for understanding proportions in other realms

Divisions of Quantity by Two Contradictories #

• When crossing two divisions by contradictories, results vary
• Substance divides into universal/particular and substance/accident = four parts
• Not all two-fold divisions by contradictories yield four parts
• Aristotle’s division of plot (before/not before AND after/not after) does not yield four logically coherent parts
• Some combinations are impossible (e.g., something neither before nor after anything else has no place in a plot)

The Two Divisions of Quantity #

• First division: Discrete vs. continuous (based on common boundary of parts)
  • Line, surface, body, place have parts with common limit AND position toward each other
  • Number and speech have parts with neither common limit nor position
  • Time has parts with common limit BUT NOT position toward each other
• Second division: Based on position of parts
  • Quantities either have position toward each other or do not
  • No quantity has position without common limit
• Continuous quantities are divisible forever (natural philosopher’s definition)
• Parts are like matter (divisible); whole is like form (unifying boundary)

The Unique Status of Time #

• Although continuous, time’s parts lack position because past and future do not coexist
• Time is also like number in being discrete (a numbered number, not abstract number)
• Time numbers the before and after in motion
• Parts of time have order but not position—this explains why only one category (not two) addresses time

Quantity’s Individuation and Proximity to Substance #

• Most accidents are individualized by their subject or substance
• Quantities with position are individualized by themselves—they can be individuated from their own nature
• This shows quantity is closer to substance than other accidents
• Example: The dimensions of bread in the Eucharist are individual but not universalized; they are individuated by their own positional nature
• Color and taste are individualized by being in the quantity of bread
• Quantity acts somewhat like substance in its capacity for self-individuation

Individuation in Material Substances #

• Multiple men can exist in one room because there is enough flesh, blood, and bones for each
• Similarly, multiple chairs can exist because there is enough wood
• Matter subject to quantity is the beginning of individuation in material things
• Angels lack this material composition; their distinction is wholly formal (no two angels are of the same species)
• This explains why there can be many particular substances of the same kind (multiple instances of “man” or “chair”)

Quantity vs. Descartes’ Confusion #

• Descartes confused body (species of quantity) with body (species of substance)
• Similar confusion in Plato and Pythagorean thought: treating numbers as substances rather than quantities
• Thomas Aquinas notes quantity has special affinity with substance
• This confusion is understandable but represents a philosophical error

Thomas Aquinas on Quantity’s Properties #

• Quantum per se: That which is quantum in itself
  • Line and surface—things whose definitions include quantity
  • Number—which we speak of as existing by themselves
  • Body and place
• Quantum per accidens: That which is quantum through another (accidental quantification)
  • Things quantified through their subject’s quantity
  • Example: White is quantified per accidens when the white surface is large
• Quantity alone among accidents has division in itself (after substance)
• Whiteness cannot be divided; thus it is individuated only through its subject
• Only in the genus of quantity are some things signified as subject (substance-like) and others as modification (accident-like)

The Properties of Quantity #

• Does not have a contrary: Equal and unequal are not true contraries; they are relative
  • Large and small belong to relation (toward something), not quantity itself
  • The same quantity can be called large relative to one thing and small relative to another
  • If they were contraries, something would be contrary to itself
• Not said more or less: One three is not “more three” than another three
  • This parallels substance: one man is not “more man” than another
  • Contrasts with quality: one sweet thing can be sweeter than another
• Equal or unequal: A distinctive property showing quantity’s internal structure

Why Two Divisions for Quantity (but only one for Quality) #

• The second division emphasizes quantity’s unique property: position (positio)
• Positio (order of parts in a whole) appears in quantity’s definition
• Position shows quantity’s special connection and likeness to substance
• Descartes’ error stems partly from not recognizing this distinction

Key Arguments #

Why Quantity Has Special Connection to Substance #

1. Accidents are normally individualized by their subject
2. Quantities with position are individualized by themselves (through their own positional nature)
3. Therefore, quantity approximates substance in capacity for self-individuation
4. This explains Thomas’ statement that quantity is propinquior (nearest) to substance among accidents

Why Multiple Instances of the Same Kind Can Exist #

1. Individuation in material things is caused by matter subject to quantity
2. Different portions of matter (flesh, wood, dough) are individuated by their quantity
3. Therefore, there can be many men, many chairs, many cookies of the same kind in one place
4. This is impossible without divisible matter—why we cannot have five men sharing the same flesh and blood

Why Time is Both Discrete and Continuous #

1. Time is continuous because it is divisible forever (like motion)
2. Time is a numbered number—it numbers the before and after in motion (like discrete quantity)
3. Yet time’s parts lack position because past and future do not coexist together
4. Therefore, time fits both categories but has unique properties distinguishing it from both

Why Proportional Understanding Requires Numerical Clarity #

1. If one does not understand proportion with numbers, one cannot understand it in other domains
2. Numbers provide the clearest example of proportion (e.g., 2:4 = 7:14)
3. Without this foundation, students confuse proportional relationships in qualities, substances, and other categories
4. Mathematical abstraction is foundational to understanding proportions generally

Important Definitions #

• Perfect number: A number equal to the sum of its proper divisors (those that measure it)
• Proportion: A relationship where the first is to the second as the third is to the fourth; the product of outer terms equals the product of inner terms
• Positio (position): The order of parts in a whole; distinguishes certain quantities by whether parts have spatial relationship to each other
• Divisible forever: A property of continuous quantity—a line can always be divided into smaller lines without reaching a final, indivisible part
• Parts with common limit: Characteristic of continuous quantities like line, surface, body, and place where parts meet at a boundary
• Parts without common limit: Characteristic of discrete quantities like number and speech where parts are separate
• Quantum per se: That which is a quantity in itself (like line, surface, number)
• Quantum per accidens: That which is quantified through another (like white when the surface is large)
• Numbered number: A multitude applied to actual things (like “three men”)
• Abstract/numbering number: Number in itself (like “three” considered absolutely)

Examples & Illustrations #

Perfect Numbers #

• 6 is the first perfect number (divisors: 1, 2, 3; sum: 1+2+3=6)
• 28 is the second perfect number (divisors include 1, 2, 4, 7, 14)
• Finding next perfect number becomes increasingly difficult mathematically

Proportional Relationships from 28 #

• From 28: 2 and 14 multiply to 28; 4 and 7 multiply to 28
• Therefore: 2:4 = 7:14 (proportional)
• From 100: Any two factors form a proportion when arranged by increasing values

Student Misconceptions #

• When asked “2 is to 3 as 4 is to ?”, students frequently answer “5” instead of “6”
• This shows lack of understanding of proportional relationship
• Without this understanding, students cannot grasp proportions in other philosophical domains

The Grandmother’s Cookies #

• A grandmother makes dozens of cookies at Christmas using enough dough for each
• This illustrates why many objects of the same kind can exist: sufficient divisible matter
• Contrast with impossibility of five men sharing one body: insufficient flesh for division

Multiple Chairs in a Room #

• Several chairs of the same kind can exist in one room
• Why? There is enough wood to make multiple instances
• This exemplifies individuation through divisible matter subject to quantity

The Eucharist’s Dimensions #

• Dimensions of bread in Eucharist are individual (not universal)
• Yet they are not individualized through a subject (as color would be)
• Quantity individualizes itself through position
• This shows quantity’s unique capacity among accidents

Notable Quotes #

“If you don’t know what a proportion is with numbers, how are you going to know what a proportion is with these other things?” — Berquist, emphasizing the foundational importance of mathematical proportion

“It’s beautiful they’re still helping us. I think I was knowing it. It shows you how quantity… is close to substance in some way.” — Berquist, noting how properties of quantity reveal its proximity to substance

“Quantity is acting a bit like a substance.” — Berquist, explaining why quantity can individuate itself unlike other accidents

Questions Addressed #

Why does Aristotle divide quantity twice (discrete/continuous AND by position)? #

The first division captures the nature of parts’ boundaries; the second captures their spatial relationship. Together they fully characterize quantity. Time is exceptional: continuous yet lacking position because its parts (past, future) do not coexist.

How can matter subject to quantity explain individuation in material things? #

Divisible matter allows multiple instances of the same form (multiple men, chairs). Each instance occupies different matter, and this matter—subject to quantity—is what grounds their individuation. Angels lack this material composition, so their individuation is purely formal.

Why is quantity closer to substance than other accidents? #

Quantity alone can individuate itself through position (the order of its parts). Other accidents (color, taste) are individualized only through their subject. This self-individuation approximates substance’s mode of existence.

What does it mean that large and small are relations, not quantities? #

The same quantity can be called large (relative to something smaller) and small (relative to something larger). If they were contrary qualities, something would be contrary to itself—impossible. Therefore they belong to relation, not quantity itself.

Why must mathematical proportion be understood first? #

Proportion is clearest in numbers because abstract relationships are not obscured by material accidents. Understanding “2:4 = 7:14” provides the model for understanding proportions in any domain—qualities, substances, virtues, etc.