The Continuous and Its Basic Role: Beginning, End, and the Point Without a Limit

Orientation #

He refers back to “the first reading” of the present book, which they have been in, and to “the second definition” of the continuous given “back in the earlier books here of the book three, I think it was”; also to things “we saw” earlier (the four causes; eternity as negation of time; the Anaxagoras fragment; the senses of before “they’ve had the text here from Aristotle”). The previous class treated Aristotle’s argument against motion being composed of indivisibles and the senses of before; the next continues with why points cannot be continuous with or touch one another. A text-matching system found in the audio the handout “A Note on Practical Philosophy”, p. 9, and DHB Vol. III, The School of Aristotle, p. 1078.

The class, in order #

1. How basic is the philosophy of the continuous? (“the philosophy of the continuous, how fundamental is this”) #

Point: after the opening prayer he asks how fundamental the continuous is for philosophy and for knowledge generally; he cannot think of a modern philosopher’s work on it, only a book by the physicist Louis de Broglie (the continuous and the discrete in modern science).

• Division recalled: two chief kinds of philosophy, looking [ed.: speculative] and practical; looking philosophy has three kinds, mathematical, natural, and wisdom or first philosophy.
• Natural philosophy: motion, place or distance, and time, seen at the beginning of Book III, are all continuous; so understanding the continuous is “absolutely fundamental” for it.
• Geometry (mathematical philosophy): it is about continuous quantity and assumes what is shown here. Between any two points you can draw a straight line, but if two points could touch without coinciding and remain two, could you draw a straight line between them? No; and the first reading shows two points cannot touch without coinciding. Geometry also always assumes you can keep bisecting a line, keep inscribing square in circle in square getting smaller and smaller, always bisect an angle.
• Arithmetic: important not because numbers are continuous but because number arises from the division of the continuous (the Anaxagoras fragment, “no smallest of the small… no greatest of the great”): divide one line, you have the first number, two; divide again, three; divide forever, and numbers increase forever. The potential infinity of numbers corresponds to the infinite divisibility of the continuous.

2. The continuous and the knowledge of immaterial things (“by the negation of the continuous”) #

Point: even for the soul, the angels and God, none of which is continuous, the philosophy of the continuous is basic, because in this life we know them more by what they are not.

1. If we saw God face to face the philosophy of the continuous would not be needed to consider Him (though in seeing God we would understand the continuous).
2. But in this life we understand more what God and the angels are not: e.g. in the Summa Contra Gentiles Thomas shows there are understanding creatures with will, then shows they are not a body, not continuous. So immaterial substances and God are understood in part by the negation of the continuous; so too the immateriality of our own reason and hence of our soul.
3. In the philosophy of the soul Aristotle says the proper object of human reason is what it is to be something you can sense or imagine, hence something continuous or tied to the continuous, so reason never understands in this life without imagining. Hence in On Sense and the Sensible (with Thomas’s commentary) “we understand nothing without the continuous and time” (time itself continuous; the images involve continuous and time).
4. Boethius’ De Trinitate with Thomas’s articles: objection, we understand nothing without the continuous and time, but God is not continuous or in time, therefore we don’t understand God. Reply: we understand God by the negation of the continuous, as we saw eternity is the negation of time (of the things found in time’s definition). Conclusion: that we understand nothing without the continuous shows how basic it is to all our thinking.

3. The most basic words take their first meaning from the continuous (“every cause is a beginning, but not every beginning is a cause”) #

Point: the words Aristotle studies in Metaphysics V, used everywhere, especially in the axioms known by themselves by all men, and most of all in wisdom, begin from the continuous.

• The words are divided into three groups; the first group begins with beginning, and its first meaning is the beginning of this table, the beginning in the continuous. Every cause is a beginning, not every beginning a cause, so beginning is more general than cause; other meanings are seen by likeness to the beginning in the continuous. Example: “the curb is the beginning of the campus”, the curb on Salisbury Street.
• In the third group, end or limit: first meaning, the end of the table or of a line; sometimes end is used for both ends (“the two ends of the table”, two endpoints), so end is in a way even more universal than beginning. Then end of a motion (also continuous); then purpose, that for the sake of which; last, end or limit in the sense of a definition.
• One: the continuous is one of its fundamental, almost first, meanings. Socrates’ joke: asked for a definition, someone gives many examples, as if you asked for one plate and, dropped and broken, it is no longer continuous, so now many instead of one. Everything is in some way one: the simple very much so; the composed exists only if its parts are united.
• Being in the sense of quantity: so close to substance that Descartes confused the continuous, extension, with substance.

4. Why this belongs to natural philosophy and not geometry: proportional causes (“assigning the property to what it’s proportional to”) #

Point: geometry is the science of continuous quantity, yet it is natural philosophy that determines what the continuous is (divisible forever, not composed of or divided into indivisibles), and the geometer assumes it. Two reasons.

1. Continuous is broader than what geometry studies: time and motion are continuous, and it is not geometry’s to say what time or motion is.
2. From logic (Prior and Posterior Analytics): the highest demonstration is propter quid, giving the cause, and causes and effects must be proportional; you must find exactly why something belongs to something.
   • The Meno: Socrates’ demonstration to the slave boy is a particular case of the Pythagorean theorem. Square twice as big: the boy says side twice as long, which gives four times as big. Socrates cannot assume the Pythagorean theorem, so he shows something more particular: four equal squares fill a bigger square four times as big; draw the diagonals; each diagonal cuts its square in two equal triangles (formally from Euclid’s fourth theorem, two triangles with equal sides containing an equal angle, shown by superposition, since only one straight line joins two points); the base angles are each half a right angle, so two make a right angle; the inner figure is a square made of four halves of four squares, hence exactly twice the original, and it stands on the diagonal. “That’s a good argument, nothing wrong with that argument at all.”
   • But what is shown is a case of the isosceles right triangle, whereas the property belongs to every right triangle: Euclid I.47, the culminating theorem of Book I, shows it for any right triangle, e.g. the 3-4-5 triangle, which is not isosceles. So the property belongs to it insofar as it is a right triangle, not insofar as isosceles, “kind of subtle thing there”. I.47 is a more perfect demonstration because it assigns the property to what it is proportional to; though the slave-boy version is easier to follow, and historically men may have seen the particular first.
   • “Super example”: Euclid I.5, the base angles of an isosceles triangle are equal. Is that a property of the isosceles as divided against the equilateral (only two sides equal)? No: an equivocal use of isosceles. Common pattern: one of two species keeps the common name, the other gets a new name. Animal divided into man and beast, but the beast keeps the name animal, so “animal and man”; in one sense man is an animal, in another not. The beast keeps the name because it adds nothing noteworthy to animal; man adds reason and will. So isosceles broadly means two sides equal, saying nothing of the third; the triangle with only two equal keeps the name, the one with the third also equal gets a new name. Greek: skelos, leg; isos, equal; his legs spread and a line between his feet form an equal-leg triangle whether the feet are close or a leg’s length apart. Student: is Euclid taking isosceles as including the equilateral or as divided against it? Answer: the general sense. From I.5 you can show all angles of the equilateral equal; and from I.6 (angles equal, sides equal) that no two angles of a scalene are equal. So I.5 is not understood as private to the isosceles as distinguished from the equilateral.
   • Application: the reason a line, the motion over it, and the time of that motion are divisible forever and not composed of indivisibles “is really the same reason for all three, proportionally speaking, at least”. So, as in I.47, it is appropriate to bring them together and show it for all; shown in geometry it would hold only for the line. Building it up as Socrates does would not be bad, but not the most universal.

5. Is continuous equivocal? Beginning is: Melissus (“committing the fallacy of equivocation”) #

• Student: is continuous equivocal said of time, motion, magnitude? “Hard to say”; something perhaps, but not equivocal by chance, equivocal by reason.
• Beginning is clearly equivocal in Thomas and Aristotle, of the beginning of magnitude, of motion, of time. Physics I (“Natural Hearing”): Melissus argues being could not have come to be, has no beginning, hence no end, hence goes on forever; Aristotle says he passes from no beginning (and end) in time to no beginning or end in magnitude, the fallacy of equivocation.

6. Digression: the Apology, breathing vs. philosophizing, and the senses of before (“don’t pawn your uglies off on me”) #

Point: an example from his Introduction to Philosophy class of reasoning that trades on senses of before, tied to the Melissus fallacy.

• Apology: first part, how Socrates came to examine people; second part, he examines the Athenians for preferring goods of the body and exterior goods to goods of the soul (the reason why the soul’s goods are better is touched in other dialogues). Who is right?
• Argument: already shown that something is not good because you want it; syllogize: so not better because you want it more. So you cannot say the soul’s goods are better for Socrates because he wants them more. Example: is something sweet because it is white? No; then not sweeter because whiter. “The ball game is over”; you must give a reason. Colleague’s quotation from the Republic: “an opinion without a reason for it is an ugly thing”, “don’t pawn your uglies off on me”.
• Which is better, philosophizing or breathing? All raise hands for breathing. Reason offered: if you’re not breathing you won’t do anything else. That shows breathing is before philosophizing in the second sense of before (you can breathe without philosophizing, not the reverse), not the fourth sense (better). Chaucer before Shakespeare in time does not make him the better poet.
• He strengthens their argument: it is worse to stop breathing for an hour than to stop philosophizing; and the opposite of the worse is better (worse to kill a man than a mosquito, so a man is better; worse to kill a man than rob him, so his life is better than his money).
• “But there’s an exception to the rule”: when the lesser good is before the greater in the second sense, the lesser can be without the greater but not vice versa, the loss of the lesser is worse because it entails loss of the greater. Theological case: charity is greater than faith, yet Thomas speaks as if loss of faith is worse, since losing faith loses hope and charity. So breathing being before in being, its loss can be worse without its being the better. Simplest case: to live or to live well? Live well, though living is before in the second sense. “So that argument breaks down.”
• Back to Melissus: the senses of beginning are nevertheless closer to being the same, since one kind follows on the other; but having a beginning in time does not entail one in size or vice versa: Aristotle held the universe limited in size but not in time.

7. Reasoning starts from definition: the two definitions of the continuous (“that whose parts meet at a common boundary”) #

Point: as the Meno teaches (Meno wants to know whether virtue can be taught; Socrates says he can’t reason well about it without knowing what virtue is, and only looks at what seems reasonable on both sides), and as logic teaches more explicitly, reasoned-out knowledge rests on definition; so Aristotle begins with the definition of the continuous, and there are two.

• Both defined by parts, as Euclid defines number, a multitude composed of ones/units.
• A. Logic’s definition: that whose parts meet at a common boundary / have a common limit. It separates line, surface, body from number: the left and right parts of a line have a common point, the end of one as the beginning of the other; in a circle the diameter; the boundary between us and Canada. In seven, do the three and the four meet at a point or line? No: discrete quantity.
• B. Physics III: that which is divisible forever, appropriate to natural philosophy (logic’s is taken over here because logic is common to all sciences).
• Thomas’s reason why A fits logic and B natural philosophy, from the four causes: parts that compose something are like matter, whole is to parts as form to matter. Natural philosophy is about matter and goes toward parts and parts of parts, usque ad elementa (the Latin of the first reading), which is the direction of “divisible forever”; logic, like geometry, is about form (some make all logic formal logic), and A looks at how the parts are united to form the whole, as two semicircles brought together form the circle.
• He says he finds something puzzling here and will explain it in a moment [not reached in this recording]. Which definition does Aristotle reason from first? The first one.

8. The first paragraph re-read (“continuous whose edges are one”) #

Point: the thesis of the first reading, read again from the text.

• “If the continuous and the touching and the next have been determined before, continuous whose edges are one, touching whose edges are together, next of which there is nothing of the same kind between, it is impossible for anything continuous to be from indivisibles, as a line from points”, if line means something continuous and point something indivisible.
• What he will prove: something continuous cannot be put together from indivisibles; nor by their touching; nor even by their being next to one another (a row of houses: each house has a house next to it, on one side or both). Two points cannot be continuous, cannot touch and remain distinct, cannot even be next; “a very thorough way” of showing the continuous is in no way from indivisibles.

His words #

• looking philosophy: the speculative kind, divided against practical [ed.: speculative philosophy]
• wisdom / first philosophy: metaphysics
• Natural Hearing: [ed.: Physics]
• beginning: first meaning, the beginning of the continuous (of a table); every cause a beginning, not every beginning a cause
• end / limit: first meaning the end of the continuous; then end of motion; then purpose; then definition
• demonstration propter quid: giving the reason in the sense of the cause; requires proportional cause and effect
• isosceles: isos equal, skelos leg; “equal leg triangle”; used equivocally, broadly (two sides equal) or as divided against equilateral (only two)
• second sense of before / fourth sense of before: before in being (one can be without the other, not conversely) vs. before in goodness
• equivocal by chance vs. equivocal by reason
• fallacy of equivocation: Melissus mixing beginning in time with beginning in magnitude
• discrete quantity: number, whose parts have no common boundary
• usque ad elementa: “as far as the elements”, the direction natural philosophy goes, toward matter
• continuous / touching / next: edges one; edges together; nothing of the same kind between

Texts #

Read in class

• Aristotle, Physics VI, first reading, opening paragraph (definitions of continuous, touching, next; a line not from points)
• Euclid, Elements I, props. 4, 5, 6, 47 (argued at the board, not read verbatim); the Meno’s slave-boy construction
• Verified sources found by text-matching: handout “A Note on Practical Philosophy”, p. 9; DHB Vol. III, The School of Aristotle, p. 1078

Mentioned

• Aristotle, Physics III (beginning: motion, place, time; second definition of the continuous)
• Aristotle, Physics I (Melissus)
• Aristotle, Metaphysics V (beginning, end/limit, one)
• Aristotle, On the Soul (proper object of reason), On Sense and the Sensible with Thomas’s commentary
• Thomas, Summa Contra Gentiles (understanding creatures are not bodies)
• Boethius, De Trinitate, with Thomas’s commentary/articles
• Aristotle, Prior and Posterior Analytics
• Plato, Meno, Apology, Republic (quoted line)
• Anaxagoras, fragment; Louis de Broglie, book on the continuous and the discrete; Thomas/St Paul on charity greater than faith

His questions #

• Between any two points you can draw a straight line; if two points touched without coinciding and remained two, could you? No; the first reading shows two points cannot touch without coinciding.
• Is the philosophy of the continuous important for knowing angels and God? Yes in this life, since we know them by negation of the continuous (and eternity by negation of time).
• Why does determining what the continuous is belong to natural philosophy rather than geometry? Continuous is broader than geometry’s object (time, motion), and the reason is proportionally the same for line, motion and time, so it is shown for all together.
• Is the Pythagorean property a property of the isosceles right triangle or of every right triangle? Of every right triangle, as such (Euclid I.47).
• Is Euclid taking isosceles as including the equilateral or as divided against it? The general sense; from I.5 one shows the equilateral’s angles equal.
• Is continuous equivocal said of time, motion, magnitude? “Hard to say”; if so, equivocal by reason, not by chance; left partly open.
• Which is better, philosophizing or breathing? Breathing is before in being (second sense), not in goodness; the “opposite of the worse is better” rule has an exception when the lesser good is before the greater in the second sense.
• Which is better, just to live or to live well? To live well, though living is before in the second sense.
• Which definition of the continuous does Aristotle reason from first? The first, logic’s: parts meeting at a common boundary.