47. Definitions, Numbers, and the Nature of Form

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Lecture Notes

Main Topics #

Plato’s Theory of Forms and Its Problems #

• Plato believed that truth requires the way we know to be the way things are (ἡ γνῶσις = ἡ οὐσία)
• Because we know mathematical objects (triangles, spheres, cubes) in separation from sensible matter, he concluded these universals must exist separately in reality
• Aristotle’s response: the mind can consider things in separation that don’t actually exist in separation
• Plato’s forms would be indefinable because definitions require genus and difference (analogous to matter and form), yet pure forms lack matter
• Therefore, Plato’s separated forms cannot adequately explain the possibility of definition itself

The Structure of Definition: Genus and Difference #

• A definition is composed of genus (what is more material) and difference (what is more formal)
• Example: “man” is defined as “rational animal” where “animal” is genus and “rational” is difference
• The genus-difference structure parallels the matter-form structure
• Not every genus has a genus above it; there must be a highest genus beyond which division stops
• If every genus had a higher genus, we could never complete a definition and would know nothing

The Four Comparisons Between Definitions and Numbers #

First Comparison: Neither is infinitely divisible

• Continuous quantities (lines) are infinitely divisible: any line can be bisected into two smaller lines indefinitely
• Numbers have a limit: the unit (one) is indivisible and marks the end of division
• Similarly, definitions reach a highest genus that cannot be further divided
• This is why we can achieve knowledge by definition, unlike with continuous quantities

Second Comparison: Adding or subtracting changes the whole

• Removing one from a number produces a different number
• Adding or removing any element from a definition produces a different nature
• Example: body + life = plant; plant + sensation = animal; animal + reason = man
• Remove reason and you no longer have man; remove sensation and you no longer have animal
• This is analogous to how one is not still five; it becomes six

Third Comparison: Unity through act and potentiality

• A number is not merely a collection of units but achieves unity through actualization
• Six is able to be seven; the addition of one makes it actually seven
• The one actualizes the potential of six to be seven
• Similarly, a definition achieves unity when the difference actualizes what the genus contains in potentiality
• Example: triangle is able to be equilateral; adding “equilateral” actualizes this potential
• This mirrors matter and form: form actualizes matter’s potential without requiring an external unifying agent
• The clay doesn’t need to be “screwed together” with its shape; the shape is the actualization of the clay’s potential

Fourth Comparison: Neither admits of more or less (degree)

• No number is “more” a number than another; five is not more five than four is four
• No substance is “more” a substance than another by virtue of its form
• All men are equally men; differences in health, strength, beauty, or intelligence are accidental variations, not substantial differences
• Just as seven is completely seven and five is completely five, so too one man is completely man and another man is completely man

The Problem of Locke’s General Idea #

• Locke struggled to understand how we can think of “triangle in general” when every imagined triangle is equilateral, isosceles, or scalene
• He confused images (which must be determinate) with thoughts (which can be indeterminate)
• Solution: equilateral, isosceles, and scalene are contained in the definition of triangle in potentiality, none in actuality
• The definition “a plane figure contained by three straight lines” is indifferent to which type of triangle it is
• When we add “equilateral” to triangle, we are not bringing in something entirely new but actualizing what was already potential in the definition

The Unity of Matter and Form #

• Form does not need an external third thing to unite it with matter, unlike composite things (which need glue, nails, screws)
• This is because form is the actuality of matter’s potential—they are intrinsically one
• The difference in a definition actualizes the genus in the same way that form actualizes matter
• Just as we don’t “screw together” the shape and the clay, we don’t need external unity for matter and form

Key Arguments #

The Argument from the Possibility of Definition #

• Premise 1: We have genuine knowledge by definition (e.g., what a square is)
• Premise 2: If every genus had a genus above it, we would need to know infinitely many genera before knowing anything
• Conclusion: There must be a highest genus that is not itself a species of something higher
• Significance: This parallels how numbers have a limit in the unit, proving definitions are more like numbers than like continuous quantities

The Argument from the Structure of Definition #

• Premise 1: Genus relates to matter and difference relates to form (as the more material and more formal)
• Premise 2: A definition requires both genus and difference to be complete
• Premise 3: Pure forms (as Plato conceived them) have no matter and therefore no genus
• Conclusion: Platonic forms would be indefinable, contradicting Plato’s own project of defining things like piety and courage

The Argument from Actualization #

• Premise 1: Six is able to be seven only through the addition of one
• Premise 2: One actualizes what was potential in six
• Premise 3: This actualization does not require an external unifying principle
• Conclusion: Just as the last unit actualizes the previous number, the difference in a definition actualizes the genus; form is the actualization of matter’s potential

Important Definitions #

Definition (ὁρισμός) #

An expression of what a thing is through genus and difference. Definitions are determinate and indivisible in the sense that they have a finite, necessary structure. They cannot be infinitely divided because there is a highest genus that cannot be further determined.

Genus (γένος) #

The more material aspect of a definition; what is common to a group of things. Genus contains the potential for different specific forms.

Difference (διαφορά) #

The more formal aspect of a definition; what distinguishes one species from others within the same genus. The difference actualizes what the genus contains in potentiality.

Actuality (ἐνέργεια) and Potentiality (δυνάμις) #

Fundamental principles of being. Potentiality is the ability to be something; actuality is the realization of that ability. Form is the actuality of matter’s potentiality.

Number (ἀριθμός) #

A collection of units actualized into a unity. Unlike continuous quantities, numbers are determinate and not infinitely divisible. Each number is completely itself; there is no “more” or “less” in being a number.

Examples & Illustrations #

The Hierarchy of Living Things #

Body (matter) + life form = plant → plant + sensation form = animal → animal + reason form = man. Each addition creates a new type of being, just as adding one to a number creates a new number. Removing any element returns you to the previous level or to something that is no longer what it was.

The Triangle and Its Determinations #

The definition “a plane figure contained by three straight lines” allows for equilateral, isosceles, or scalene triangles. These three possibilities are in the definition in potentiality. When we specify “equilateral,” we are actualizing one of these potentials, not adding something entirely external. Similarly, “equal or unequal” three lines are able to be either, but neither in actuality until determined.

The Cat (Tabitha) as an Example of Non-Degree in Substance #

One cat may be more beautiful than another (a matter of accidental differences), but no cat is “more a cat” than another cat. Tabitha was not more a cat than other cats despite being superior in other qualities. What makes something a cat admits of no degree.

Fingers and the Thumb #

In common speech, people say “four fingers and a thumb” rather than “five fingers,” treating the opposable thumb as standing out due to its distinction. This linguistic convention reflects how something that adds distinction may receive a new name while others retain the original name.

The Continuous Divisibility Argument #

A faster body covers more distance in the same time as a slower body, or the same distance in less time. This proves both distance and time must be divisible forever, because if they stopped being divisible, the faster and slower bodies would cover identical distances in identical times, contradicting the definition of faster and slower.

Notable Quotes #

“Our mind can consider in separation sometimes things that don’t exist in separation, right? When one can be understood or known without the other.” — Berquist, articulating Aristotle’s response to Plato on the nature of abstraction

“If you add or subtract anything, you have a different kind of thing, just like you add or subtract one from a number and you have a different number.” — Berquist, explaining the second comparison between definitions and numbers

“The falsity comes in when you say that what is separated in our knowledge, truly, our true knowledge, is separated in things.” — Berquist, identifying Plato’s fundamental error regarding universals

“To get the nature of the thing is to get its number, right?” — Berquist, summarizing the deep analogy between definitions and numbers

“Is one man more a man than another?” — Berquist, illustrating why substance admits of no degree through form

Questions Addressed #

Why did Plato’s theory of forms lead to the problem of indefinability? #

Answer: If forms are pure form without matter, they lack the genus-difference structure that enables definition. But Plato needed to define things (piety, courage, etc.). Therefore, his separated forms undermine his own philosophical project of definition through the dialectical method.

How can we think of a general triangle when every triangle we imagine is determinate? #

Answer: The difference between images and thoughts. Images must be determinate (equilateral or isosceles or scalene), but thoughts can grasp what is indeterminate by understanding the potential. The definition of triangle includes equilateral, isosceles, and scalene in potentiality, none in actuality.

Why is a number not just a collection of ones? #

Answer: If a number were merely a collection of ones, it would be like a heap or pile with no unity. But a number is one because the last unit actualizes the potential of the previous number to be that number. Unity comes through actualization, not aggregation.

How does the comparison between definitions and numbers clarify the nature of form? #

Answer: Just as a number achieves unity when the last unit actualizes the potential of the previous number, a definition achieves unity when the difference actualizes the potential contained in the genus. By analogy, form is the actualization of matter’s potential. Neither requires an external unifying principle.

Why must there be a highest genus that is not itself a species? #

Answer: If every genus had a genus above it, we would need to know infinitely many genera before completing any definition. But we do achieve knowledge by definition (e.g., knowing what a square is). Therefore, there must be a highest genus (like “being” or “substance”) that terminates the series of determination.