Metaphysics VIII, Reading 3 (Part 2): Definitions and Forms Are Like Numbers

Orientation #

He opens: “the second part of the third reading,” where Aristotle compares definitions — “especially the formal thing” — with numbers. He closes by noting they have now gone through the third chapter. He refers back to arguments “we used” about dividing the line and about genus in logic (prev. lecture / earlier logic course).

The class, in order #

1. Definitions are like numbers, not like the continuous (“a definition is a number in a qualified way”) #

Text read: “if substances are in some way numbers, they are so thus and not as some say as units. For a definition is a number in a qualified way. For it is divisible and indivisibles. For definitions are not unlimited, and number is also such.” Thomas’s commentary notes Aristotle likewise compares thoughts and definitions to numbers in On the Soul I.3 — rather than to the continuous.

2. Why the continuous is divisible forever (“you can always bisect the line”) #

Argument that bisection never ends:

1. Ending would require either dividing into two nothings (then the line is made of nothings — absurd), or a line two points long.
2. A line two points long presupposes a line can be put together out of points (“composed is really the last word for put together”).
3. Two points touching: part-to-part, part-to-whole, whole-to-whole, or (a possible fourth) at the edge. A point has no parts, so the first two fail; the “edge” of a point would make it a little circle, so that fails; only whole-to-whole, i.e. coinciding, remains.
4. Two coinciding points have no more length than one, which is no length at all — and the same for a hundred, a million, an infinity.
5. Therefore a line cannot be composed of points; division always yields two smaller lines, forever.

Aside argument from fast and slow: a faster body covers a given distance in less time; in that lesser time the slower covers a lesser distance; alternating these, time and distance are each shown divisible forever. If they were not, the faster body would cover the same distance in the same time as the slower — i.e. it would not be faster.

3. Definitions do stop: not every genus has a genus (“does that go on forever?”) #

Euclid’s square is divided into quadrilateral, then rectilinear plane figure, then plane figure, then figure. Two reasons the regress stops:

1. If it did not, one would have to know infinitely many genera before knowing anything, and so know nothing by definition — but we do come to know what a square or rhombus is by definition.
2. There would always be a more universal name than any name — but nothing is more universal than being, or than “something.”

Numbers likewise stop: seven divides into three and four, three into two and one, two into two ones, and there it stops — “unless you’re one of these crazy mixed-up modern mathematicians.” The one, beginning of number, is even simpler than the point, since the point has position and the unit has none. Hence definitions are like numbers; and since the form completes what a thing is and the definition signifies what the thing is, forms are like definitions, definitions like numbers. The same holds proportionately of statements: there must be something known without definition, and first statements naturally known, not known through other statements.

4. A next thought, as there is a next number (“is there a number in between?”) #

Between two unequal lines there is always a line in between in length, because of infinite divisibility; between six and seven there is no number (and between six and seven points, no point). Thoughts are like numbers here: there is a next thought. Examples: in the syllogism “what changes is composed; God is not composed; therefore …” there is no thought between premises and conclusion. In definition: quadrilateral → right-angled (rectangle) → equilateral (square), with nothing in between.

5. Second comparison: add or subtract one, and you have a different number (“you have a different number”) #

Text (third paragraph, p. 5): add or subtract one from a number and you have a different number; so with the definition and “the what was to be the what it is.”

Shakespeare on man: “What is a man, if his chief good and market of his time be but to sleep and feed? A beast, no more.” He imitates it for number: “What is a five, if it be half of eight? Four, no more” — half of eight is to four as the beast’s chief good is to the beast.

The ladder: body + life = plant; + sense = animal (the sense of “beast”); + reason = man. Take away reason, a beast; take away sense, a plant; take away life and growth, a body, a stone. So adding or subtracting gives a different kind of thing, as with numbers. “That’s very important to see.”

Aside: Thomas returns to this in the Trinity, in the article on the Holy Spirit proceeding from the Son as well as from the Father — there must be some order between them, since immaterial things are like numbers. “You come back to the text for that.”

6. Third comparison: what makes a number one (“by what it is one if it is one”) #

Text: “it is necessary that a number be one by something which these thinkers cannot state.” If a number were merely a collection of ones, it would be a heap or a pile, not one thing. Rather each added one makes actual what the previous number was able to be: six is able to be seven, and the added one actualizes that ability — a unity of act and ability, matter and form. So too the difference determines the ability of the genus.

Locke and Berkeley: Locke, confusing thought with image, asks whether the three lines of the general triangle are equal, two equal, or none, and concludes it is “all none of these.” Berkeley replies that is silly, it cannot be all none of them, therefore there are no general ideas. Berquist’s answer: all of them in ability, none of them in act — Locke cannot see the distinction, so states a contradiction; Berkeley, unable to escape the contradiction, denies general ideas (“like talking about a square circle”). Adding “equilateral” to triangle therefore determines what is already in the definition in ability; it does not bring in something entirely other.

Contrast case: my hand picking up the chalk is not actualizing any ability of the chalk — that is a heap (book, glass, chalk), one thing brought to another. In definition and number, one is the actuality of the other.

Clay is able to be sphere, cube or pyramid; one of these abilities is made actual by the form it receives from my hands.

7. No glue is needed (“what makes a soul and body one?”) #

From On the Soul: once you see matter and form as ability and act, it is stupid to ask what unites them — one is the actuality of the other’s ability, so no glue, nail or rope is needed. Contrast the parts of a chair, which must be screwed, nailed or glued, because one is not the act of the other’s ability. “I don’t usually screw together the shape of the clay and the clay.” Likewise you do not bundle together the ones and the number; the last one actualizes the previous number, as the difference actualizes the genus. Hence the genus must be understood first, since the differences determine an ability intrinsic to it. Aside on the phrase “I got his number”: to get a thing’s nature is like getting its number.

He recaps the three comparisons: (1) not divisible forever; (2) add or subtract and you have a different number/definition/nature — stone, plant, animal, man as one, two, three, four; (3) the parts are as ability and act, needed if the number is to be one thing and not a collection of units.

8. Aside: equivocal naming where the one that adds something gets the new name (“four fingers and a thumb”) #

Five fingers, or “four fingers and a thumb” — the opposable thumb stands out, so the four keep the name and it gets a new one. So too “animal” is kept for the beast and man is set against the animals. And five plus one keeps not the name five but a new name, six. His mother disliked his calling man an animal; he answered, “I don’t mean he’s just an animal” — calling man an animal is a bit like calling six a five, when it is a five plus one.

9. Fourth comparison: substance according to form has no more and less (“all men are created equal”) #

Text: “as number does not have the more and the less, neither does substance according to the form. But if any does, it is one with matter.”

Test case: proclaiming in slave-owning Athens or Rome that all men are created equal — some free, some noble, some slaves; some born healthier, stronger, handsomer, more intelligent, more courageous, temperate, mild or irascible. Answer: those are added virtues or dispositions; no man is more what a man is than another. Tabitha, “Queen of Bumblebee Circle,” was a more beautiful cat (the trash men admired her), but not more a cat than the other cats. That is what a thing is: add or subtract anything and it would be a different kind of thing.

10. Epilogue to chapter three, and Thomas’s use of it (“if ten really knew itself”) #

Aristotle closes on the generation and corruption of the said substances, how it can happen and how it is impossible — already touched in the first group of comparisons, where form as such is not generated or corrupted but the composite is — and on reducing things to number, “let it be determined as far as they say.”

Thomas keeps returning to this, e.g. explaining how God by knowing Himself knows everything else: if ten really knew itself it would know nine, eight, seven, and so on, since they are contained in the perfection of ten though something less. We know animals somewhat because they are like one unit less than us — without reason, but having what we have; plants less, being less like us.

Closing asides: “I feel closer to the plants and to the stones myself”; the experiments where plants grew straight with Mozart and twisted with rock and roll — “I don’t know how scientific these experiments were, but the results at least kind of pleased me.”

His words #

• “a number in a qualified way” — Aristotle’s phrase for the definition.
• “the what was to be the what it is” [ed.: quod quid erat esse / essence].
• “ability” and “act” [ed.: potency and act].
• “composed is really the last word for put together” — on “a line composed of an infinity of points.”
• “these crazy mixed-up modern mathematicians” — those who would divide the unit.
• “all of them in ability, none of them in act” — his solution to Locke’s general triangle.
• “I got his number” — getting the nature is like getting the number.

Texts #

Read in class

• Aristotle, Metaphysics VIII, third reading, second part — the comparisons of substance/definition to number; handout p. 5, third paragraph; the closing epilogue to ch. 3.

Mentioned

• Thomas Aquinas, Commentary on the Metaphysics (on the comparison to numbers rather than the continuous).
• Aristotle, On the Soul I.3 (thoughts and definitions compared to numbers); also On the Soul on what unites soul and body.
• Euclid’s definitions of square, quadrilateral, rectilinear plane figure (inferred: Elements I, definitions).
• Thomas Aquinas, on the Holy Spirit’s procession from the Son (order among immaterial things); on God knowing all else in knowing Himself.
• Shakespeare, “What is a man, if his chief good and market of his time be but to sleep and feed? A beast, no more.”
• Locke and Berkeley on the general idea of triangle.
• The Declaration of Independence: “all men are created equal.”

His questions #

• Does the bisection of a line ever come to an end? No — a line cannot be composed of points, so dividing always yields two smaller lines.
• Does the division of a definition into genus and difference go on forever? No — otherwise you would need to know infinitely many genera before knowing anything, and there would be something more universal than being.
• Is there something more universal than being, or than “something”? No.
• Is there a next thought, as there is a next number? Yes — nothing stands between the premises and the conclusion, or between rectangle and square.
• What makes a number to be one, and not a heap of units? The last unit makes actual what the previous number was able to be — act and ability, as form to matter.
• Are the three lines of the general triangle equal, two equal, or none? All of them in ability, none of them in act.
• What makes soul and body one? Nothing third — form is the actuality of matter’s ability; only things like chair parts need glue or nails.
• In what sense are all men created equal, given unequal health, strength, looks, intelligence, courage? No man is more what a man is than another; the inequalities are added dispositions, and more and less belong to substance only with matter.
• Was Tabitha more a cat than the other cats? No — more beautiful, but not more a cat.