31. God's Infinity and Its Distinctions

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Lecture Notes

Main Topics #

God’s Infinity vs. Other Types of Infinity #

• God’s simple infinity: God alone is infinite in His essence/nature because His very nature is His existence (ipsum esse subsistens)
• Infinity in magnitude: No created body can be infinite in size because any actual body must possess a determined substantial form, which entails determined quantity
• Infinity in multitude: No infinite multitude can exist in act; infinite multitude is possible only in potency (successively)
• Qualified infinity (secundum quid): Created intellectual substances (like angels) are infinite in a qualified way—not limited by matter, yet their existence is contracted to a particular nature

The Distinction Between Matter and Form in Relation to Infinity #

• Infinity pertaining to matter: Infinity as potential, unformed, unlimited—this is imperfect infinity
• Infinity pertaining to form: When form is not received in matter (as in angels), it has a kind of perfection but remains contracted by reception in a particular nature
• Pure form (God): Unlimited because not received in anything; infinity as pure actuality is perfection
• Quantity falls upon matter; quality falls upon form. Therefore infinite pertains to division (matter) not addition (form)

Natural Bodies Cannot Be Infinite in Magnitude #

• Every natural body has a substantial form that is determined
• To a determined form, there necessarily follow determined accidents, including quantity
• Each living thing has natural limits to its size (ant vs. elephant)
• An infinite body would have no natural motion, since natural motion occurs when a body is outside its proper place (which an infinite body could never be)

Mathematical Bodies Cannot Be Infinite in Act #

• Any body that exists in act must exist under some form
• Form necessarily brings with it figure (shape)
• A figure is defined as that which is comprehended by limit(s)
• Therefore, any actual mathematical body must be finite
• When geometers say “let this line be infinite,” they mean it can be extended as much as needed—not actually infinite in act

Infinity in Division vs. Infinity in Addition #

• Infinite divisibility of continuous quantity pertains to matter and potency
• One cannot go in the opposite direction (infinite addition) because addition approaches the whole/form
• Motion and time can be infinite in potency but not in act (they exist successively, not all at once)
• Magnitude as a whole exists in act and therefore cannot be infinite

The Problem of God’s Power #

• Objection: If God has infinite power, why can’t He make an infinite thing?
• Resolution: It is against the definition of the made (created) that its very essence be its own existence. Since a made thing must have existence received from another, it cannot be simply infinite. God’s omnipotence does not extend to contradictories (e.g., making something both made and infinite in the way God is)

The Understanding and Infinity #

• The human mind seems to have an infinite power because understanding grasps universals, which extend to an infinity of singulars
• Resolution: The mind’s power extends to infinity only secundum quid (in a qualified way). Understanding is not in matter, which gives it a kind of unlimitedness, but the mind itself is not infinite
• The grace of Mary is described as infinite in some sense, but not in the fullest sense as Christ is infinite

Matter and Infinity #

• Prime matter (first matter) does not exist in the real world except through form
• First matter is in potency only, not in act
• Therefore, even if it could be called infinite, it is not infinite in act but only in potency

Key Arguments #

For the Possibility of Infinite Magnitude (Objections) #

1. Mathematicians use infinite lines: Geometry demonstrates with infinite lines; therefore infinite magnitude must be possible

   • Response: Geometers do not take lines to be actually infinite, but rather finite lines from which they can subtract as much as needed. They call this “infinite” as a manner of speaking.

2. Infinite not against definition of magnitude: Finite and infinite are properties of quantity, so infinite should apply to magnitude

   • Response: Though infinite is not against the definition of magnitude in general, it is against the definition of any particular species of magnitude (e.g., a cubit magnitude, a triangular magnitude). Since nothing can exist in a genus without existing in some species, no infinite magnitude can exist.

3. Divisibility goes both ways: Magnitude is divisible forever (in the direction of diminution); therefore it should be able to increase forever (in the direction of addition)

   • Response: Infinite pertains to division/matter, not addition/form. Through division one approaches matter; through addition one approaches the whole (form). Infinity belongs to the side of matter only.

Against Infinite Magnitude #

1. Every body has a surface, which is a limit
2. Every natural body has a determined substantial form
3. To a determined form there follows determined quantity
4. Therefore, nothing natural can be infinite in magnitude

Natural Motion Argument #

• Every natural body has some natural motion
• An infinite body cannot have natural motion in a straight line (it would never be outside its place, which is required for straight-line motion)
• An infinite body cannot have circular motion around a center (an infinite body has no center)
• Therefore, an infinite natural body is impossible

Figure and Limitation #

• Anything in act must exist under some form
• The form of quantified being is figure
• A figure is that which is comprehended by limit(s) (as Euclid says)
• Therefore, any actual magnitude must be finite

Important Definitions #

Infinitus (Infinity) #

• In essence/nature (secundum essentiam): Only God is infinite in this way; His essence is His existence, unlimited by form or matter
• In magnitude (secundum magnitudinem): Impossible for created things
• In multitude (secundum multitudinem): Impossible in act; possible only in potency (successively)
• Secundum quid (in a qualified way): Applied to created intellectual substances and potential infinities; fundamentally different from God’s simple infinity

Virtus (Power/Excellence) #

• In Latin, virtus can mean virtue but also power and the excellence of power
• The quantity of a power is known through its effects

Forma (Form) and Materia (Matter) #

• Substantial form: Determines species and nature; entails determined accidents including quantity
• Accidental form: Properties and qualities added to substance
• Parts relate to the whole as matter relates to form

Continuous Quantity vs. Discrete Quantity #

• Continuous quantity: Parts meet at a common boundary/limit (e.g., parts of a line meet at a point)
• Discrete quantity: Parts do not meet at any common boundary (e.g., the number 7 has parts 2 and 5 that don’t meet)
• Continuous is divisible forever; discrete (numbers) is not

Examples & Illustrations #

On Natural Bodies and Determined Quantity #

• Ants vs. elephants: Each species has natural limits; an ant cannot be as large as an elephant because their substantial forms determine their sizes
• Wood can become many different things (chair, table, door) but is limited by its actual form to being one thing at a time
• Fire’s substantial form determines its accidental properties: hotness, lightness, dryness

On Divisibility #

• A line can be divided forever into smaller lines, never reaching points (which would be nothingness) or parts that touch without coinciding
• The faster body analogy: If a faster body covers the same distance in less time, and the slower body covers less distance in more time, this alternating divisibility shows time and distance are divisible infinitely
• The hexagon inscribed and circumscribed in a circle: These operations can continue indefinitely in both directions (getting bigger or smaller)

On the Understanding #

• Knowing what an odd number is covers infinite numbers; saying “no odd number is even” concerns an infinity of things
• The understanding can grasp the universal, which extends to an infinity of particulars

On Modern Language and Diminution #

• Annette means “little Ann”—the diminutive suffix indicates a qualified or lesser version
• “A goodie” is something that has goodness but not the greatest goodness (unlike God)
• Coming into a room: I cannot say “I came to be,” but must qualify and say “I came to be in this room”

On Useful Fictions in Mathematics #

• Astronomers can consider the earth as a point when calculating celestial distances (a useful fiction, though the earth is not a point)
• Parallel postulate: We imagine two infinite lines, but what we actually imagine are finite lines extended as needed

Questions Addressed #

Is Something Other Than God Able to Be Infinite in Essence? #

• Objection 1: God’s infinite power should enable Him to produce an infinite effect

  • Response: Against the definition of “made” is that its essence be its own existence. Therefore, God cannot make something infinite in the way He is infinite, just as He cannot make something both made and not made (contradictory)

• Objection 2: The created understanding has infinite power (grasps universals covering infinite singulars); therefore intellectual substance is infinite

  • Response: This power extends to infinity only secundum quid, because the understanding is a form not in matter (anima intellectiva). But the created form, even when not in matter, has existence that is received and contracted to a determinate nature. Only God, whose existence is not limited to some nature it is the existence of, is infinite simply

• Objection 3: Prime matter is infinite; therefore something besides God can be infinite

  • Response: Prime matter does not exist in vera natura (the real world); it exists only in potency, not in act. Even as a potential principle, its potency does not extend beyond natural forms.

Can There Be an Infinite Magnitude in Act? #

• Resolution: No. Any actual body must have some form; form brings figure; figure means comprehended by limits; therefore no infinite magnitude in act is possible

Can Motion and Time Be Infinite? #

• Resolution: Motion and time can be infinite in potency/successively (since each indivisible moment is both a beginning and an end), but not in act as a whole. They exist differently from magnitude.

Notable Quotes #

“All the ancient philosophers attributed the infinite to the first beginning, as if coerced by the truth, even without giving a reason.”

• Demonstrates how human reason naturally inclines toward recognizing infinity in the ultimate principle

“The infinite that belongs to quantity holds itself on the side of matter, for through division of the whole one approaches to matter.”

• Explains why infinite divisibility pertains to matter (and division), not form (and addition)

“When geometers say ’let this line be infinite,’ they mean let it be as long as we need it to be, not actually infinite in act.”

• Clarifies the meaning of mathematical infinity as potential, not actual