Aquinas Summa I q.86 a.2–3: Can Our Mind Know Infinitely Many Things? Potential vs. Actual Infinity, Meno's Paradox, and Knowing Contingent Things

Orientation #

He opens: “we’ve got two, three articles today,” and hopes to do “the next three next time.” He says “we’re up to article two, I guess, question eighty-six.” He refers back to things “we’ve gone through before”: the either-or syllogism on points touching, matter as principle of individuation, and the logic distinction “simply / not simply.” The beatific vision “we’ll be talking about in the later questions,” on how we know things above us. Verified sources: q. 86 a. 2 and a. 3 matched against the text; the a. 1 match is not reflected in the transcript, which begins at a. 2.

The class, in order #

1. Article 2: the four objections (“able to know an infinity of things”) #

Point: he reads the four objections that our understanding can know an infinity of things.

1. God exceeds all infinities, and we can know God; therefore much more all other infinities.
2. Our understanding is apt to know genera and species; some genera have infinite species — numbers, ratios, figures (three-sided, four-sided, five-sided…).
3. If one body did not impede another from being in one place, an infinity of bodies could be there; one understandable form does not prevent another in the same understanding (many things are known “in habit”); so the mind could have in habit a knowledge of infinite things.
4. Since our understanding is not the power of a bodily organ, it seems in some way an infinite power, and infinite power is capable over infinite things.

Asides: Anaxagoras’s “great fragment” — the first thing he says of mind is that it is infinite; Heraclitus said it before him. There are signs of something infinite about the mind: it is always learning something new, always inventing, and when it knows a universal (“no odd number is even”) it makes a statement in a way about an infinity of things — “but you have to specify in what way it’s knowing that infinity.” Feuerbach (“between Hegel and Karl Marx”): the infinite is God (quoting some theologian); man’s mind is infinite; therefore man’s mind is God. Aristotle in Physics I criticizes an early thinker [name garbled in recording; ed.: probably Melissus] for mixing two senses of infinite — no beginning or end in time vs. no limit in size. Feuerbach in the 19th century makes the same mistake: “fallacy of equivocation, mixing up different senses of the same word.”

Sed contra: Physics I, “the infinite, insofar as it is infinite, is unknown.”

2. The body: infinite in potency, one after another (“proportional to its object”) #

Point: because a power is proportioned to its object, the understanding stands to the infinite as its object does.

1. Our object is the what-it-is of a material thing.
2. In material things there is no infinite in act, only in potency, “according as one succeeds to another” (Physics III).
3. Therefore in our understanding the infinite is found in potency, taking one thing after another: we never understand so many but that we could understand more. Example: counting — “Daddy, can you count to a hundred?… to a thousand?” — until one realises one can always count to something larger.

Not in act: the mind cannot at once know more than it knows through one species. Comparison: the mind is to the form it understands as wax or clay to its shape — clay shaped as a sphere cannot at the same time be a cube; put it in the cube shape and it ceases to be a sphere. So there are not two forms of the same kind “in complete act” in the mind. And the infinite has no one species, “otherwise it would have the definition of a whole and perfect.”

3. The divisibility of the line, and why points make no line (“a straight line is divisible forever”) #

Point: Thomas leans on Aristotle’s account of the infinite in the Physics; the continuous is “divisible forever,” but the division is never completed.

• Bisection (a theorem of geometry): bisect a line and you get two shorter lines, which, being lengths, can be bisected too. Do you ever cut a line without getting two shorter lines? No. The only alternatives: (a) two nothings — but something cannot be made of nothing; (b) two points — but that assumes a line can be put together from points.
• The either-or syllogism (recalled): two things touch either part to part, part to whole, or whole to whole (i.e. coincide); one could also speak of edge touching edge, the edge being not a part but the limit. A point has no parts and no distance inside it to distinguish point from limit; so the only way points touch is to coincide; if they coincide they have no more length than one point — for ten, a hundred, a million or “an infinity of points, like the mathematicians say.” So points cannot compose a line.
• Alexander [a student, or commentator; unclear in recording] pointed out: as you divide, you never reach a shortest line, but the number of lines keeps increasing (two, three, four…). So if the straight line is divisible forever, number can increase forever — but you never complete the dividing; it is “always more reduced to act, but never completely made actual.” Text continued: the infinite is not understood except part after part, as is clear from its definition in Physics III — that of which, taking its quantity, there is always something further to take; likewise a greater number is always able to be taken. So the infinite is not known in act unless all its parts were numbered — impossible.

Asides: the Oxford translation makes Aristotle speak of an “infinite number”; his old teacher (name unclear in recording, “Kasurik”) who made him learn Greek said Aristotle would not say that — a contradiction in terms — “look it up in the Greek”; the word “number” is not there. If something can be numbered it is not infinite; hence Shakespeare couples “countless and infinite.” He recalls an objection in the treatise on the Trinity: the Athanasian Creed says God is immensus, not measurable, yet He would be limited to three persons — “interesting objection,” which needs what number means in the Trinity.

4. Not in habit either; the two infinities compared (“neither in act nor in habit”) #

Point: we cannot know the infinite in habit, for the same reason.

1. In us habitual knowledge is caused by actual consideration: “by understanding something we become knowing” — e.g. by understanding Euclid’s demonstrations one acquires the habit of geometry.
2. If I cannot actually understand one demonstration, I cannot have its theorem in habit; so if I cannot actually understand the infinite, I cannot have it in habit — unless all the infinites were considered by numbering in succession, impossible. Conclusion: neither in act nor in habit, only in potency. God’s mind is not limited at all in what it knows; ours is “unlimited” only in that it can always know more — its infinity corresponds to its knowledge being always incomplete. So Feuerbach grossly confuses two kinds of infinity.

Asides: Feuerbach’s “little perverse work” The Essence of Christianity — God did not become man, man is God; Marx and Engels read it young and were convinced. In college, when the French positivists’ “religion of humanity” with days set aside for worshipping humanity was described, he broke out laughing. Marx in the preface to his doctoral thesis says explicitly the human mind is the highest divinity.

5. Reply 1: God’s infinity is of form, matter’s of privation (“a form which is not limited”) #

Point: God is infinite as a form not limited by matter; a material thing is infinite by privation of formal termination.

• A line is limited by points; with no endpoints it would be infinite, and would lack something. God’s infinity is a mere negation of the imperfection that would come from being in matter.
• Metaphysics V (“fifth book of wisdom”) on “perfect”: Homer is perfect in his kind — plots, characters, diction; Mozart a perfect musician — perfect in some genus; God is perfect as lacking nothing and not limited to any kind — “that’s the difference.”
• Form is known through itself; matter without form is unknown. So the material infinite is unknown in itself, but the formal infinite (God) is known in itself and unknown to us only by the defect of our understanding, which in this life is apt to know material things and knows God only through material effects — the first proof, from motion: motion depends on a mover, moved movers on an unmoved mover. In glory the defect is removed and we see God in His essence, without comprehending Him (never as much as He is knowable); 1 John 3:2, “we shall see Him as He is.”
• Second clay example: clay can take perhaps an infinity of shapes, but only one at a time, never all at once; when it gets a new one it loses the one it had. That infinity is tied to matter, based on lack of actuality; God’s is based on being perfect form.

6. Reply 2: species not imagined (“except perhaps in general”) #

Point: since we know forms by abstraction from images, species of numbers and figures one has not imagined are not known in act or habit, except in general. Example: I know what an odd number is — that covers an infinity — but I cannot know all odd numbers in particular; I could think three, five, seven, nine all day and never think all of them. To know them in general and in universal principles is to know them in potency, confusedly and indistinctly, “not distinctly one from the other.”

7. Reply 3: forms enter successively (“successively enter into the place”) #

Point: bodies in one place need not enter successively so as to be numbered; but understandable forms enter our understanding successively, since we cannot understand many at once, so they are numbered and not infinite. Example: understanding triangle, then parallelogram, then in Euclid Book IV pentagon and hexagon — the forms come in successively; you never have an infinity of figures in the mind.

8. Reply 4: infinite in power, knowing “in some way” (“extends itself to infinite individuals”) #

Point: the understanding is infinite in its power as not limited by a bodily organ, knowing the universal abstracted from individual matter, so of itself extending to infinite individuals — but its ability is never fully actualised, at least in this life.

• Contrast with sense: eye limited to colour, ear to sound; reason knows colour, sound, smell.
• “Everything is something; besides everything there’s nothing; knowing what something is, I know everything” — in a very general, indistinct way. His students: “I’m going to tell my dad I know everything now.” The distinction from logic: simply vs. in some limited way (secundum quid) — the “second kind of mistake outside of language.”
• Meno’s objection: you cannot investigate what you don’t know. Replies by examples: people are paid to find the cause of cancer — do they know what they look for? Secundum quid. Police at a murder don’t know who did it but know they seek the murderer of so-and-so, which guides them (enemies? threats? who profits?). A student’s question: if he knows what he wants to know, why ask; if not, how can I help? — to ask an intelligent question you must in some way know what you don’t know. His standard example: “How many students are in class today?” — I don’t know the number, but knowing I seek the number of students tells me to count; I reach twenty-eight “with the greatest of ease,” having known it only generally before.
• So in knowing what an odd number is, man knows an infinity of things in potency and confusedly. Aristotle at the beginning of wisdom [ed.: Metaphysics I]: the wise man knows all things “so far as is possible for man” — not in particular, but maybe an infinity in some way; God knows all things simply. These are the two things Feuerbach [transcript: “Alfarabi,” an ASR error] confuses.

9. Article 3 begun: contingent things (“does not know contingent things”) #

Point: he reads the objections, sed contra, and starts the body.

• Aside: the same questions are asked of God’s mind (infinity, contingents, future), with different answers.
• Contingent: what can be and not be; necessary: cannot not be; impossible: cannot be.
• Obj. 1: Ethics VI — the three virtues understanding, wisdom, science are about necessary, not contingent things; e.g. in Euclid, when straight lines intersect the opposite angles “must be equal.”
• Obj. 2: Physics IV — what sometimes is and sometimes is not is measured by time; understanding abstracts from time as from other conditions of matter.
• Sed contra: every science is in the understanding, and some sciences are of contingents — the moral sciences (human acts subject to free judgment) and the part of natural science on things generated and corrupted.
• Body: contingents can be considered as contingent, or as something necessary is found in them: “nothing is so contingent but that in it there is something necessary.” Socrates running is contingent, but the relation of running to motion is necessary. Contingency pertains to matter (what can be and not be); necessity follows form — hence complete necessity in geometry, where there is no matter (interior angles equal two right angles, exterior angle equal to the interior opposite). Matter is the principle of individuation (recalled): many chairs of one kind because there is enough metal; many identical window panes because enough glass; the dough rolled out makes many cookies because as extended it is divisible — “material cookies.”
• Class ends here, mid-body. Aside: his grandmother’s homemade sugar and Scotch-bread cookies, never equalled; her motto “I aim to please.”

His words #

• understanding — his rendering for intellectus; “our understanding is able to know…”
• ability — “the infinite in ability” [ed.: potency]
• in act / in habit / in potency — three ways of knowing; the infinite only in the last
• Natural Hearing — “the so-called Physics” [ed.: Physics]
• wisdom (fifth book of, beginning of) — [ed.: Metaphysics]
• divisible forever — his definition of the continuous
• the infinite — “that of which, taking its quantity, there is always something further to take”
• fallacy of equivocation — mixing different senses of the same word (Feuerbach, “infinite”)
• immensus — Athanasian Creed, “not measurable”
• simply vs. in some limited way / secundum quid — the logic distinction used against Meno
• contingent / necessary / impossible — can be and not be / cannot not be / cannot be
• principle of individuation — matter, illustrated by metal, glass, dough

Texts #

Read in class

• Thomas Aquinas, Summa Theologiae I, q. 86, a. 2 (objections, sed contra, body, four replies)
• Thomas Aquinas, Summa Theologiae I, q. 86, a. 3 (objections, sed contra, body begun)

Mentioned

• Aristotle, Physics I (infinite as infinite unknown; criticism of confusing two senses of infinite)
• Aristotle, Physics III (infinite in potency; definition of the infinite)
• Aristotle, Physics IV (time measures what sometimes is and is not)
• Aristotle, Nicomachean Ethics VI (understanding, wisdom, science about necessary things)
• Aristotle, Metaphysics V (senses of “perfect”); Metaphysics I (the wise man knows all things) (inferred, he says “beginning of wisdom”)
• Anaxagoras, fragment on mind; Heraclitus
• Feuerbach, The Essence of Christianity; Marx, preface to doctoral thesis
• Athanasian Creed (immensus); Thomas’s treatise on the Trinity, objection on three persons
• Euclid: bisection theorem, Book IV figures, vertical angles, angle sum of triangle
• Plato, Meno
• 1 John 3:2
• Shakespeare, “countless and infinite”

His questions #

• Can our understanding know an infinity of things? Only in potency, one after another — neither in act nor in habit.
• Do you ever bisect a line and not get two shorter lines? No; the alternatives (two nothings, two points) are both impossible.
• How can two points touch? Only by coinciding — so they never make a length.
• Do you ever have an infinity of lines from dividing? No; the division is always further reducible to act but never completed.
• In what sense is God infinite, and in what sense matter? God as form unlimited by matter; matter by lacking formal termination.
• Do the police know who they are looking for? Not simply, but secundum quid — the murderer of so-and-so.
• How did I direct myself to twenty-eight without knowing it? I knew it in general as “the number of students in class,” which told me to count.
• Does the student know what he is asking? In some way he must know what he doesn’t know, or he couldn’t ask intelligently.
• Why can you have many chairs, window panes, cookies of exactly the same kind? Because matter (metal, glass, dough) is subject to quantity and divisible.