Summa I, Q7, Articles 2–3: Can anything besides God be infinite, in essence or in size? (Simply vs. in some way, first matter, and useful fictions in physics)

Orientation #

He opens: “you’re up to the second article in question seven,” the first article (whether God is infinite) having been done; he recalls what “we saw earlier” about being and good simply / secundum quid, and about showing the mathematical line divisible forever (prev. lecture). He closes with “a little break now before we go on to the multitude” (a. 4). The previous class covered q. 6 and the opening of q. 7; the next takes up infinite multitude and God’s presence in things.

The class, in order #

1. The plan of q. 7 and “coerced by the truth” (“as if coerced by the truth”) #

Point: q. 7 runs from God’s infinity (a. 1) to whether anything else is infinite in essence (a. 2), then to continuous and discrete quantity (aa. 3-4).

• Aristotle, first book of “Natural Hearing” [ed.: Physics]: all Greeks who thought about change explained it by contraries without giving a reason, “as if coerced by the truth.” Thomas (Summa Contra Gentiles) uses the phrase of the ancients who all took the beginning to be infinite, quasi a veritate coacti; once they saw there could be no infinite quantity they had to find the beginning infinite some other way, and eventually reach God’s infinity.

2. Article 2: the objections and sed contra (“proportionate to its what nature”) #

Point: three reasons something besides God seems infinite in essence.

1. Power is proportionate to nature; God’s nature is infinite, so His power is, so He can produce an infinite effect (the quantity of a power is known through its effect). Virtus here means power, excellence of power, not virtue.
2. Whatever has infinite power has infinite nature; the created understanding grasps the universal, which extends to an infinity of singulars (“no odd number is even” covers a potential infinity of numbers). Anaxagoras’s first thing about mind is that it is unlimited; Heraclitus anticipated it: you never come to the end of the soul, so deep is its reason. Thomas will admit the intellect is infinite in some way. Aside: popes speak of Mary’s grace as in some way infinite, though not as Christ’s; “mediatrix of all grace.”
3. First matter (he prefers “first” to “prime”) is other than God and is infinite. Sed contra: the infinite cannot be from a beginning (Physics III); everything apart from God is from a first beginning.

3. Respondeo: infinite secundum quid, on the side of matter and of form (“secundum quid and not simpliciter”) #

Point: something apart from God can be infinite secundum quid, not simpliciter, the same distinction seen earlier with being and good (what is being simply is good only secundum quid, and conversely).

• The distinction corresponds to the second kind of mistake outside of words; understanding the one helps understand the other. In logic he leaves it until the four kinds of mistakes.
• Matter side: everything in act has a form, so its matter is limited by form (wood could be chair, table, door, an infinity of things, but by its form it is this table). Yet under one substantial form matter remains in potency to many accidental forms, so it is infinite secundum quid, as the wood is an ability for an infinity of shapes. Translating secundum quid: “in some way,” “in some qualified/imperfect way.”
• Form side: forms in matter are simply finite, contracted by the matter. Created forms not received in matter but subsisting by themselves (the angels, as Thomas holds) are infinite secundum quid, not limited by matter; but since such a form has existence and is not its existence, its existence is received and contracted to a determinate nature, so it is not infinite simply as God is, who is His existence (“I am who am”).
• Aside: Kierkegaard, a positive relationship with a woman makes you very limited (he broke his engagement); his mother’s neighbourhood news putting his father to sleep, “attracts to the here and now.”

4. Digression: the distinction in the first divisions of being (“being simply, and accidental being”) #

Point: simpliciter / secundum quid is already at work in the two main divisions of being.

• Substance vs. accident: substantial being is being simply, accidental being is being secundum quid. When he came to be a geometer he did not come to be; when generated he did. Coming into this room is not coming to be; leaving it he ceases to be in this room, the added qualification is the sign that it is not so simply.
• Diminutives as qualification: “Annette” (little Anne, dropped when she got older), “daddy,” “goody” (a candy is a goody; you would not call God or wisdom a goody).
• Act vs. ability: chairs in the room, yes; chairs in the trees, only “chairs in ability,” something able to be a chair. Being in act is being simpliciter, being in ability secundum quid.
• The same distinction was used of the good earlier; now of the infinite, something more particular.

5. Replies to article 2 (“this would be for contradictories”) #

• Ad 1: it is against the definition of the made that its essence be its existence; so it is against the notion of the made to be simply infinite; God, with infinite power, cannot make something made and infinite simply, for that would be contradictories at once. Grade-school challenge: can God make a stone so big He can’t lift it? Omnipotence does not extend to contradictories: a square circle is nothing; so too something equal to Himself, something infinite, and (a later question) something not dependent on Him for being.
• Ad 2: the understanding extends “in some way” (quodammodo, which he links to quidam quid) to the infinite because it is a form not in matter, either wholly separate (angels) or a power not the act of any organ (On the Soul III), as in the understanding soul joined to a body (“understanding soul” = anima intellectiva; Shakespeare’s “my understanding soul”). Feuerbach’s “the human mind is infinite, the infinite is God” comes from never distinguishing infinite simply from infinite secundum quid. Aside: his wife’s plants and the “souls” around us; Fido among “the faithful departed.”
• Ad 3: first matter does not exist by itself in rerum natura (in the real world) since it is being in ability only; even as potency it is infinite only secundum quid, its potency reaching only natural forms, not the forms that are angels; easier still, it always exists through some form and is thereby contracted. Democratic man thinking he has all possibilities ends up doing something very limited, “not a big secundum quid”; you can’t walk three directions at once.

6. Two kinds of quantity and two definitions of the continuous (“the parts meet at a common”) #

Point: before a. 3, the categories and natural philosophy give different accounts of continuous quantity.

• Categories: continuous quantity’s parts meet at a common limit (line at a point, surface at a line, body at a surface); discrete quantity’s parts do not (seven as two and five, three and four).
• Natural philosophy: the continuous is divisible forever. Cutting a line cannot end in two nothings (the line would be made of nothing) nor two points (points touching coincide and have no length), so always two shorter lines. Aristotle shows it for time and distance together: the faster body covers the same distance in less time, the slower a lesser distance in that time, and so on alternating.
• Numbers are not divisible forever: the one is simpler than the point; you can’t have half a point, so no “one and a half points.” Hence Metaphysics VIII compares definitions to numbers: not every part of a definition can or need be defined.
• Thomas explains why “divisible forever” suits natural philosophy (going toward parts, which are like matter) and “parts meeting at a common boundary” suits logic (toward the whole, like form). Aristotle recalls the logical definition in natural philosophy and adds others there. “Very subtle things … how subtle Aristotle is.” “Only a guy from Laval would point that out”; Thomas said it first.
• Ways of defining differ by science: theology’s man as imago Dei, natural philosophy’s two-footed animal with reason. Aside: the plucked chicken thrown over the wall, “It’s a man for you.”

7. Article 3: the objections and sed contra (“Let this line be what infinite”) #

Point: four reasons an actually infinite magnitude seems possible.

1. No falsity in the mathematical sciences (Physics II, “no lie in those abstracting”: against Plato, who held the way we know must be the way things are, so spheres exist in separation), yet geometers use infinite magnitude: “let this line be infinite.” He suggests this is just a way of talking, “as long as we need it.”
   • Aside on ways of speaking: he read SCG on the Holy Spirit this morning; the Macedonians on “do not sadden the Holy Spirit” (understood metaphorically); “God is the cause of all things” (not of Himself, though He is a thing); “no one knows the Father except the Son” (what belongs to the divine nature is understood of all, Thomas says); “now I know that you love me” to Abraham means “now I have made known.”
2. What is not against a thing’s definition is not impossible for it; finite and infinite seem to be passions or properties of quantity.
3. Magnitude is divisible forever (Physics III mentions it; shown in Book VI); contraries come about the same thing; addition is opposed to division, so magnitude can increase forever. Euclid Book IV: inscribing and circumscribing squares and circles, bigger and bigger, smaller and smaller.
4. Motion and time have continuity from the magnitude (Physics IV; VI treats all three); it is not against their definition to be infinite, since each indivisible in time and circular motion is both beginning and end. Aside: time was the Greeks’ example of what always is (no time when time was not); Augustine raises it, Heisenberg quotes Augustine; the children’s book, “a time before there was any time.” Sed contra: every body has a surface; a surface is the limit of a limited body; so no infinite magnitude, and likewise surface and line.

8. Respondeo: no natural body infinite (“infinite in one’s nature … in one’s magnitude”) #

Point: infinite in nature and infinite in magnitude differ; a. 2 settled the first, a. 3 asks the second.

• Even if fire or air went on forever, its nature would be limited to a species by form and to an individual by matter.
• Body is “complete magnitude” (length, width, depth), taken mathematically (quantity only) or naturally (matter and form).
• Argument 1 (form): every natural body has a determined substantial form; determined accidents, among them quantity, follow a determined form; so every natural body has a determined quantity, both toward the large and the small. Physics I, ninth reading, against Anaxagoras: if flesh and bone could get smaller forever so could the animal, but each kind has definite limits; the ant is not as big as the elephant (thank God), the adult elephant not as small as the ant.
• Argument 2 (motion, more Aristotle’s own way): every natural body has a natural motion; an infinite body can have none. Not straight, since straight-line motion is natural only outside one’s place (fire and earth go to their places and rest), and an infinite body occupies all places, each indifferently its own. Not circular, since one part must be carried to where another was, but lines from the centre grow more distant as they lengthen, infinitely so in an infinite body, so one would never arrive at the place of another. “Kind of subtle.”
• The second argument depends on the older physics, the first does not. Aside on modern physics: uniform straight-line motion treated as rest, needing a mover only to disturb it; but to consider a motion as not a motion is a fiction, perhaps a useful one, like the astronomer taking the earth as a point (Thomas mentions this), bodies reduced to points, or McNeill’s “fictitious sharpness of historical dates.”
• Aristotle argues against an infinite body in Physics III from the bodies his contemporaries admitted, then repeats it in the book of the universe [ed.: De Caelo] once he knows there is something more. Thomas notes Aristotle uses examples acceptable to predecessors though false (Nicomachean Ethics: end of household science is wealth, corrected in Politics I); as he himself uses H2O or proton-electron to illustrate, not prove; Milo’s mean is not ours.

9. No mathematical body infinite (“nothing is an act except through its”) #

Point: an imagined mathematical body must be imagined under some form; the form of the quantified as such is figure, and figure (Euclid) is what is comprehended by a limit or limits (circle one, square four); so it is finite.

• Aside (from his brother Marcus): the “parallel postulate” is not about lines extended forever never meeting, which cannot be imagined; Euclid says that lines on which a straight line makes angles less than right angles will meet, which can be imagined. Whatever you imagine is finite.

10. Replies to article 3 (“the parts had themselves in the notion of what”) #

• Ad 1: the geometer needs not a line infinite in act but a finite line from which he can subtract as much as he needs, and this he calls infinite, a way of speaking.
• Ad 2: infinite is not against the notion of magnitude in general but against every species (bicubit, tricubit, circular, triangular); nothing is in a genus that is not in some species. Student: isn’t this inductive, from particulars to the general? He restates: no particular magnitude is infinite, therefore none in the genus.
• Ad 3: the infinite of quantity is on the side of matter: division goes toward the parts, which have the notion of matter; addition goes toward the whole, which has the notion of form. Aristotle, after the four causes and three corollaries, generalises matter so that all parts from which a thing is put together are its matter (wood of the table, bronze of the statue, and then soldiers the matter of the army, parents and children of the family, the baptized of the church). Hence infinity in division only, not in addition; this goes back to quantity falling upon matter and quality upon form (Categories); note the understanding’s infinity secundum quid falls in quality.
• Motion and time are not wholes in act but successive, ability mixed with act; magnitude is a whole in act; so the infinite of quantity is repugnant to the totality of magnitude, not of time or motion, since being in potency belongs to matter (“my walking home is not entirely there or here or anywhere”). He briefly loses his place: “I’m getting confused here.”

11. Why articles 3 and 4 at all: infinite as privation and as negation (“privation or lack … merely a negation”) #

Point (answering a student): Thomas wants to show nothing else is infinite as God is, and also the senses in which God is not infinite.

• Infinite said of a line is a privation, lacking limits it should have; infinite said of God is merely a negation: negating the contraction of form by matter and of existence by a nature, things God cannot have at all.
• Cf. q. 3 a. 1: “deeper than the hill, longer than the earth” is metaphorical; taken properly, God’s infinity would be spatial, as some picture Him spread out everywhere; Newton calls infinite space the “sensorium of God.”
• In the SCG infinite comes last: God not a body and altogether simple (no body going on forever like Newton’s space), and one (no infinite multitude), then the way He is infinite. Here a. 1 comes first and aa. 3-4 secondarily eliminate the quantitative senses. God has no length, width, depth, nor even position like a point.

His words #

• “coerced by the truth” / quasi a veritate coacti: Aristotle’s phrase for the Greeks’ explaining change by contraries; Thomas applies it to the ancients on the infinite beginning.
• “Natural Hearing” [ed.: Physics].
• virtus: in Thomas sometimes power, excellence of power, not virtue.
• “ability” [ed.: potency]; “being in ability” = being secundum quid.
• “first matter” [ed.: prime matter], his preferred rendering.
• secundum quid vs. simpliciter: “in some way / qualified / imperfect way” vs. “simply, without qualification.”
• “the second kind of mistake outside of words”: the fallacy the simpliciter/secundum quid distinction answers.
• quodammodo: “in some way”; he links it to quidam quid.
• “understanding soul” = anima intellectiva, against “sensing soul.”
• in rerum natura: “in the real world”; first matter does not exist there by itself.
• “book of wisdom” [ed.: Metaphysics], eighth book: definitions compared to numbers.
• imago Dei: theology’s definition of man.
• “complete magnitude”: body, having length, width, depth.
• “reading” [ed.: lectio]: ninth reading of Physics I.
• “the book of the universe” [ed.: De Caelo].
• “useful fiction”: uniform motion as rest, earth as a point, sharp historical dates.
• “privation or lack” vs. “negation”: infinite of a line vs. infinite of God.
• “sensorium of God”: Newton’s name for infinite space.

Texts #

Read in class

• Thomas Aquinas, Summa Theologiae I, q. 7, a. 2 (objections, sed contra, body, replies).
• Thomas Aquinas, Summa Theologiae I, q. 7, a. 3 (objections, sed contra, body, replies).
• Euclid, definition of figure (“comprehended by limit or limits”) and the “parallel postulate,” quoted. Mentioned
• Aristotle, Physics I (change by contraries; ninth reading against Anaxagoras), II (no lie in abstracting), III (infinite not from a beginning; divisible forever; no infinite body), IV (motion and time from magnitude), VI (the continuous, time and distance).
• Aristotle, Categories, chapter on quantity.
• Aristotle, On the Soul III (understanding not the act of an organ).
• Aristotle, Metaphysics VIII (definitions like numbers); the four causes and three corollaries (Physics or Metaphysics, unspecified).
• Aristotle, Nicomachean Ethics (ends of arts; Milo), Politics I (household science).
• Aristotle, “book of the universe” [ed.: De Caelo].
• Euclid, Book IV (inscribing and circumscribing).
• Thomas Aquinas, Summa Contra Gentiles (on the infinite; on the Holy Spirit and the Macedonians).
• Thomas Aquinas, Summa Theologiae I, q. 3, a. 1 (God not a body; Job’s “deeper than the hill”).
• Scripture: “I am who am”; “do not sadden the Holy Spirit”; “no one knows the Father except the Son”; “now I know that you love me” (Abraham); “setting off in every direction.”
• Anaxagoras, Heraclitus, Plato, Kierkegaard, Feuerbach, Augustine, Heisenberg, Newton, McNeill, Shakespeare.

His questions #

• Whether something other than God can be infinite in its essence or nature? Only secundum quid (matter under a form, subsisting forms, the understanding), never simpliciter, since a created form is not its existence.
• Can God, with infinite power, make an infinite thing? No: it would be contradictories at once, like a square circle, a stone He cannot lift, or something equal to Himself.
• What is wrong with Feuerbach’s “the human mind is infinite, the infinite is God”? He never distinguished infinite simply from infinite secundum quid.
• Why is the continuous defined as “divisible forever” in natural philosophy but by “parts meeting at a common boundary” in logic? Parts are like matter, the whole like form; natural philosophy deals with matter, logic with form.
• Could something limited in what it is be unlimited in size? No: a determined form carries a determined quantity, and an infinite body has no natural motion.
• Why can’t magnitude be added forever if it can be divided forever? Division goes to the parts (matter), where the infinite lies; addition goes to the whole (form).
• Isn’t ad 2 arguing inductively from particular magnitudes to the genus? No species of magnitude is infinite, and nothing is in a genus except in some species.
• Why does Thomas do articles 3 and 4 at all? To show nothing else is infinite as God is, and to remove from God the quantitative senses of infinite (privation), keeping only negation.
• Was there a time when time was not? Raised (Greeks, Augustine, Heisenberg, the children’s book), left open.