Which Definition Comes First? Circularity in Aristotle's Reasoning

Orientation #

He says he has been “away from Book Six a while” (teaching, then an advanced course that did not fill) and returns to the chapter on the continuous; what “always comes to my mind” is the faster/slower argument “we’ll meet later on in the third reading.” The verified sources place the class on the handout “Distinction of the Roads in Our Knowledge” (p. 1) and the printed text of Physics VI (DHB Vol. III, p. 923). The previous class treated the axiom that nothing is its own limit and why a line cannot be composed of points; the next continues Book VI on continuity, contact and succession.

The class, in order #

1. The faster and slower body divides time and distance together (“time and distance are divisible forever”) #

The point: there is an argument, coming in the third reading, that shows time and distance divisible forever independently of the definition of the continuous.

1. There are faster and slower bodies; the faster covers the same distance in less time — so the time is divided.
2. In that lesser time the slower body goes a lesser distance (else it would not be slower) — so the distance is divided.
3. The faster body must cover that lesser distance in less time again (else it would not be faster) — so the time is divided again; and so on. Conclusion: time and distance together must be divisible forever. Seeing that “to some extent independently,” one can then “argue from the first to the second” — which sets up the whole class.

2. The two Summas order the attributes of God differently (“five attributes of the substance of God”) #

The point: the same five attributes can be reasoned in different orders, and something is learned from each.

• Summa Theologiae: simple, then perfect, then infinite, then unchanging, then altogether one; here Thomas reasons from simplicity to unchangeableness.
• Summa Contra Gentiles: unchanging first (then simple, then perfect), so he cannot reason from simple to unchanging; he reasons the other way “to some extent.”
• You can show God unchanging before showing him simple, and simple before showing him “fully” unchangeable; he once assigned a paper on the difference. Aside: in theology “it gets a little complicated” — stick to the two Summas.

3. The question put: is Aristotle’s chapter circular? (“reasons from the definition of continuous”) #

The point: Aristotle first reasons from the (first) definition of continuous to the second (divisible forever), but toward the end of the chapter seems to reason from the second to the first — and “that’s the way I usually do it.” Reply in outline: no vicious circle if (a) the first is taken as a probable statement — “it seems to you, you’re imagining that way,” as his students do — or (b) there is some reason for the first other than this one; Aristotle has not yet given that other reason here but does so later (segment 1). Insofar as the first is known independently, reasoning back from it is not circular.

4. Worked parallel: the first road in our knowledge (“the natural road from the senses into reason”) #

The point: the same “going around” happens with the first road in our knowledge, and both directions are legitimate. Syllogism he gives students: the first road in our knowledge is the natural road; the natural road is the road from the senses into reason; therefore the first road is from the senses into reason.

• Manifesting the minor: “nature” here in its last sense, what a thing is; a thing must be what it is before it can be anything else; hence the natural road is first.
• Manifesting the major: the nature of man is two-footed animal with reason; as animal he has senses; in the generation and development of a thing the generic comes before the particular; so he goes naturally from his animal nature (senses) into reason. Biology as further manifestation: in the fertilized egg one first sees cell division and growth (what we share with plants), then the senses much later, finally reason.
• Is it first because natural, or natural because first? The middle term goes “first because natural”; but one could also start from experience that the child’s first road is from his senses (he thinks only about what he senses) and reason from first to natural, without yet thinking it a natural road. “They’re kind of convertible in a way.”

5. Wonder and philosophy as knowledge of causes (“philosophy begins in wonder”) #

The point: a second convertible pair, from the “private road” of a science and the proemium of wisdom.

• The private way of proceeding in a science is above all its way of defining and demonstrating, because defining answers what, demonstrating answers why; these are the philosopher’s questions because (Theaetetus, Metaphysics) philosophy begins in wonder, a desire to know what and why. So: from wonder as beginning to philosophy as knowledge of causes.
• But in the proemium Aristotle shows wisdom is a knowledge of causes, even first causes, before he mentions wonder — comparing the man of experience and the man of art. The man of experience may succeed better in doing because he is close to the singular: one cures not man but this man. Example: his wife and Demerol after childbirth — the standard dose gave her violent headaches, so next time she asked for less and relieved her discomfort better than nurse or doctor with their universal art. Example: the psychologist he knew as a freshman admitting that, with too few psychologists to know patients, they classified you and gave the type’s treatment, which “often made them worse.”
• Still we think the man of art more knowing and wiser because he knows why — the doctor over the pharmacist; wisdom consists in knowing, and knowing why, more than in doing; so it is knowledge for its own sake. All shown independently of wonder. Then in the third reading he reasons from wonder to wisdom being a “looking” knowledge: had it begun in hunger, thirst, wealth or warmth it would be practical. Could one reason the reverse — from wisdom being for its own sake to its beginning being wonder rather than hunger? Yes, since wonder is the desire to know for its own sake; “both ways.” Aside: De Koninck had taught the Physics since the 1930s (he studied under him 1958–60 and 1963–64) and said he always saw something new; at Columbia they read the Physics “on a weekend” — the perennial question of how much time to spend on a text.

6. Back to the text: the third-from-bottom and bottom paragraphs (“it’s clear that everything continuous is divisible”) #

The point: the layout of the chapter itself shows the reverse direction, and Thomas does not remark on it.

• Thomas’s commentary: Aristotle reasons first from the first definition of continuous, then from the second.
• Last paragraph: everything continuous is divisible into things always divisible; for if into indivisibles, indivisible would touch indivisible, which is impossible — arguing from the absurdity to “divisible forever.”
• Third paragraph from the bottom: “if a thing is divided into those things in which it is, it would be divided into indivisibles; but nothing continuous is divided into that which is without parts” — put affirmatively, everything continuous is divided into what has parts, i.e. is divisible forever; and because of that it cannot be divided into indivisibles. So the reverse order again.
• He himself reasons from the second to the first; the first can be taken as probable (already seen “back in Book Three”) or supported by other reasons — “a little bit of that in our thinking.”

7. Why Thomas can go from simple to unchanging: the third thing in change (“what changes is necessarily composed”) #

The point: the reasoning from simplicity to unchangeableness rests on Physics I, reading 11.

1. If the hard becomes soft, hardness itself cannot become soft; there must be a subject in the hard besides hardness that loses it and acquires the contrary — butter. Without a real distinction between butter and its hardness, butter could never soften.
2. Health cannot become sickness; if the healthy in no way can become sick, the healthy is always healthy and the sick always sick — “tough luck.” So a third thing, e.g. the body, able to be either but not both at once; while actually one, still able to be the other.
3. The single word “healthy” does not distinguish health from that in which health is; it is the latter that becomes sick — moving from confused knowledge to the truth that what changes is composed. Syllogism, second figure: whatever changes is composed; God is not composed; therefore God does not change. Plato sees the connection in the Phaedo: what can change is the composed, what seems unchangeable the simple (things are destroyed by taking apart); he hints the soul is simple and indestructible without proving it simple.

• ST: q. 2 existence, q. 3 simplicity; he reasons from simple, from infinite and from universally perfect to unchanging, all shown before it. SCG: unchanging comes first because the argument from motion is far more developed — the first and second of its five proofs are from motion; where ST has one middle term for a premise, SCG has three; it syllogizes not only from the definition of motion but from Book Six (motion depending on a mover). Thomas is then “very quick,” almost taking unchangeableness for granted and moving to eternity.

8. Convertible theorems in Euclid (“don’t be begging the question”) #

The point: geometry shows the same pattern — A proves B, but B might be provable otherwise and then used to prove A (or C proved by X rather than B).

• I.5 and I.6: equal sides give equal angles; equal angles give equal sides — the reverse. His brother Mark used to say the converse shows the thing is a property in the strict sense.
• I.47 and I.48: the Pythagorean theorem turned around — if the square on one side equals the squares on the others, the angle is right.
• Convertible in logic: every A is B and every B is A, as a strict property: every two is half of four and every half of four is two; but two is less than ten and not everything less than ten is two.
• Student exchange on whether 48 is proven from 47: he “can’t remember”; a student recalls it constructs a right angle on the given triangle. Left loose: “a little bit of food for thought.”

9. Which direction gives more the why? (“more why it is so”) #

The point: one in fact reasons more from the impossibility of composing the continuous out of indivisibles to its being divisible forever than the reverse; the reverse works from the probable, or from the faster/slower argument, which shows that but perhaps not as fully why.

• Student: is one then arguing from the division in logic to prove the division in natural philosophy? “I don’t know that.” One reasons from what the indivisible is: the point has no parts and no edge or limit, so the continuous cannot be composed of it — and the same fact shows it cannot be continuous on the first definition: how can what has no limit have a common limit? Two semicircles share the diameter; two parts of a straight line share the point that ends the left part and begins the right, because a limit must be other than that of which it is the limit — “the axiom we’re talking about.” Like Euclid I.6 resolving to “the whole is greater than the part,” it goes back to the axioms.
• Both continuous and contiguous (touching) suppose things with a limit other than themselves, either one limit or limits together; the point “can’t have a limit at all. It’s a limit that has no limit.”

10. “Everything has a cause” — and a limit with no limit (“who made God”) #

The point: the CCD child’s “everything has a cause” is not even true of limits. Every limit having a limit would mean an infinity of limits to limit anything.

11. The principle of simplicity (“the genuine physicist believes obstinately”) #

The point: the mind naturally rejects an infinity of causes or limits; modern science is built on this yet fears the uncaused cause.

• Aristotle’s eight arguments against Anaxagoras: he groups the eighth, fifth and first as based on fewness — Empedocles explains the same with six principles, Anaxagoras with an infinity, and “everything inside everything” with infinitely small pieces each containing everything is as complicated as can be.
• Einstein calls simplicity the underlying principle of all natural science “from the Greeks … beyond”; Newton’s rules one, two and three are variations of it.
• Yet moderns fear an unmoved mover or a cause without a cause; the alternative, every cause has a cause, posits an infinity of causes — on its face contrary to simplicity.
• Max Born (explained the wave function; worked with Einstein in Berlin; sided with Heisenberg and Bohr; pointed Heisenberg to the Göttingen matrix mathematics), The Restless Universe: the genuine physicist believes obstinately in the unity and simplicity of nature despite appearances — said of the 92 (later about 100) elements, which seemed too many; the clue that the others were near multiples of hydrogen suggested something simpler repeated. If 92 is too many, infinity is too many.
• Day and night: Earth turning or Sun circling, but both assume one Sun; nobody proposes 365 Suns, though that would explain 365 days. (Aristotle was right that the Sun is not ordinary fire, which would burn out crossing the sky.)

12. Weizsäcker on the infinite universe and the loss of theology (“all these limits”) #

The point: the belief in an infinite universe is a substitute for the infinite God.

• Aristotle and the scholastics held a finite universe; the Renaissance returned to infinite extent; with general relativity a finite universe seemed more plausible, and 20th-century physics discovered limits everywhere (a maximum speed, a smallest energy). An older physicist grew angry at Weizsäcker’s lecture on limits with no objection to give; Weizsäcker concluded (The World View of Physics) that the infinite universe came in when theology was given up: the mind cannot rest in the limited, so with no God the universe had to be infinite, and its finitude frustrates them.
• Thomas on the Greeks up to Aristotle: they first took the material world as infinite, then found reasons it was finite, then saw that something immaterial must be infinite in a different way — already Anaxagoras with the unlimited mind. The moderns reverse the movement, attributing to nature what belonged to God. Closing: a science based on simplicity that denies the first cause forces itself into an infinity of causes; the mind naturally rejects that “before you get a reason why it is.” (Recording ends as a student arrives.)

His words #

• third reading; the section on faster and slower bodies (Physics VI).
• five attributes of the substance of God; simple, perfect, infinite, unchanging, altogether one.
• first road in our knowledge; the natural road from the senses into reason.
• nature (last sense); what a thing is.
• two-footed animal with reason; definition of man.
• private road of a science; its way of defining and demonstrating [ed.: proper method].
• proemium to wisdom; opening of Metaphysics I.
• looking knowledge; knowledge desired for its own sake [ed.: speculative].
• third thing in change; the subject besides the two contraries.
• second figure; the syllogism “whatever changes is composed; God is not composed.”
• convertible; every A is B and every B is A; sign of a property in the strict sense.
• contiguous; “the touching there,” things whose limits are together.
• a limit that has no limit; the point.
• principle of fewness / simplicity; explaining with fewer principles.

Texts #

Read in class

• Aristotle, Physics VI, first chapter on the continuous (DHB Vol. III p. 923): the third-from-bottom and last paragraphs.
• Handout, Distinction of the Roads in Our Knowledge, p. 1: the first road syllogism. Mentioned
• Thomas Aquinas, Commentary on Physics VI (same chapter); Physics I reading 11; Physics III (continuous as divisible forever); Physics I, eight arguments against Anaxagoras.
• Aristotle, Metaphysics I proemium (experience vs. art; wonder, “third reading”); Plato, Theaetetus; Plato, Phaedo.
• Thomas, Summa Theologiae I qq. 2–3 and the following on perfection, infinity, immutability; Summa Contra Gentiles I (five proofs, two from motion; unchanging first).
• Euclid, Elements I.5, I.6, I.47, I.48; axiom “the whole is greater than the part.”
• Einstein on simplicity; Newton’s rules; Born–Einstein correspondence; Max Born, The Restless Universe; Weizsäcker, The World View of Physics.

His questions #

• Is Aristotle reasoning in a circle from the first definition to the second and back? Not viciously: the first can be held as probable or known by another reason, and then reasoned from.
• Is the road from the senses into reason first because natural, or natural because first? The middle term runs “first because natural,” but experience of the child lets one reason the other way; convertible.
• Could you know that philosophy is a knowledge of causes without considering that wonder is its beginning? Yes — the proemium shows it from experience vs. art before mentioning wonder; and one can reason from “for its own sake” back to wonder.
• Why does the Summa Contra Gentiles show God unchanging first? Because its argument for the unmoved mover is much more developed (two proofs from motion, three middle terms, Book VI premises).
• Could you prove Euclid I.5 from I.6 or the reverse without begging the question? If one is proven independently it can prove the other; they are convertible — left as “food for thought.”
• Is one arguing from the division in logic to prove the division in natural philosophy? “I don’t know that” — one reasons from what the indivisible is: no parts, no limit.
• How can two points share the same limit? They cannot; a limit is other than what it limits, and a point has no limit at all.
• Who made God — doesn’t everything have a cause? No: not even every limit has a limit, else an infinity of limits to limit anything.
• Why not 365 Suns to explain 365 days and nights? If one Sun explains it, the mind naturally uses no more — the principle of simplicity.