The Art of Division: Integral and Universal Wholes, Distinction, and the Rule of Two or Three

Orientation #

He refers back to earlier classes: “we saw Thomas dividing order into four. Remember that”, distinguishing “the senses of a word, like we did … a long ago”, and the five predicables “we went … from names”. He closes by announcing he will be away from Thursday until the 31st. Per the verified sources, the class works from his handouts (“Note for Book One, Reading Eleven”; “The Matter and Order of Wisdom”; “Second Road in our Knowledge”; “Foreword to Logic”) and matches Aquinas’s Physics I, lectio 11, though the transcript itself names only “the first chapter of the Physics”. The previous class treated the third distinction in the logic of definition and how one arrives at a definition; the next moves from definition to the logic of statement, opening with a review of this material on division and distinction.

The class, in order #

1. Two kinds of whole (“the distinction of the parts of some whole”) #

Division (here meaning divisions that are not definitions) is defined as the distinction of the parts of some whole; since there are two kinds of whole, logic distinguishes two kinds of division.

• Integral or composed whole vs. universal whole: he defines each by an affirmative and a negative part, and the two definitions are “just the reverse” of one another.
  • Composed whole: put together from its parts but not said of them. Chair: put together from seat, legs, back, but “the leg is not a chair”. Word cat: put together from C, A, T, but C is not the word. Definition: put together from genus and differences, but “the genus is not a definition”.
  • Universal whole: said of each of its parts individually but not put together from them. Quadrilateral is said of square, oblong, rhombus, rhomboid, trapezium but not composed of them; animal is said of man, dog, cat, horse. Otherwise “man is an animal” would mean man is composed of cat, dog, elephant: “absurd”.
• The composed whole is the first meaning of “whole”; the Greek word for genus/the general comes from the Greek word for whole, and what comes under it, the particular, “is like a part”.

2. Every science divides both ways (“every one of them divides in both of these ways”) #

He claims, and shows inductively, that every reasoned-out knowledge divides in both ways.

• Grammar (subject: the sentence): parts of speech, noun, verb, adjective, adverb, are parts of a composed whole (a noun is not a sentence, but “man walks” is); declarative, question, command, prayer are parts of a universal whole (each is a sentence, sentence is not composed of them).
• Geometry: Euclid divides a circle into two semicircles and a parallelogram into two triangles (composed whole); parallelogram into square, oblong, rhombus, rhomboid (universal whole); “sometimes the same thing in both ways”.
• Chemistry: atom into proton, electron, neutron (composed); the periodic table, hydrogen atom, nitrogen atom (universal).
• Biology (a student supplies “trunk”): tree into leaves, roots, trunk, branches (composed); broad-leaf tree vs. needle tree (universal).
• Statement: a noun is not a statement, but affirmation and negation are statements, and statement is not composed of them.
• Locke’s error: saying the triangle in general is “all or none” of its species mixes up the two kinds of division, as if animal were dog, cat, horse “run all together and mixed up”. No: “you separate out what they have in common” and leave aside the differences.

3. Distinction, division, definition (“every division is a distinction, but not every distinction”) #

Three “D” words, ordered from more to less universal.

1. A definition has parts and breaks something into parts, so a definition is in a way a division; but not every division is a definition, so division is more universal. Aristotle in the first chapter of the Physics speaks of definition as diairesis, dividing into parts.
2. Division and distinction are used loosely as the same, but strictly division requires a whole already; distinction is most universal. Every division is a distinction, not every distinction a division.

• Trinity: Thomas is careful here. One distinguishes Father, Son and Holy Spirit (a real distinction, Son from Father, Spirit from both), but he denies this is a division; God is not divided into them, which would make them parts and God a composed whole, “a very serious mistake about the Trinity”.
• Distinctions that are not divisions: the senses of a word (as with “whole” just now), per se vs. accidental, simply vs. in some respect. Important in thinking, but no whole and part in the strict sense. He usually leaves such distinctions to the treatment of famous fallacies, which rest on overlooking them.
• Student: “Define a distinction.” Answer: the basis of distinction is opposition, so to explain distinction one must study the four kinds of opposites; division adds that the things distinguished are parts of some whole. Distinction comes before order (as said before); Aristotle treats opposition in the Categories and “the fifth book of wisdom” [ed.: Metaphysics V].
• Student: the “distinctions” of the Commentary on the Sentences? A somewhat different sense: investigating a question to come to more distinct knowledge, like “speech” and “rhetoric” being different senses.

4. Division should exhaust (“there’s still something in the basket”) #

A division must exhaust all the parts; some derive “divide” from a word meaning “empty”: you empty the whole out, nothing left. Dividing triangles into equilateral and isosceles leaves scalene “still in the basket”.

5. My invention: the rule of two or three (“divide into two or three”) #

Is there a number into which one should divide? No rule true universally, he thinks, but one true for the most part, reached dialectically from the two famous thinkers who always divided into a fixed number.

• Plato (Socrates in the dialogues, e.g. the Sophist) always divides into two; Aristotle in the biological works criticizes the Platonists for it. Yet there is a reason, a syllogism: we should divide by opposites; opposites are two; therefore divide into two. Examples: numbers into odd/even (divisible into two equal parts or not), prime/composite (measured by another number or not); biologists must admit animal into vertebrate/invertebrate makes sense. A division not by opposites is bad: humans into “male and good” (one can be both), or “male and white”; male/female, good/bad are opposites that “can’t belong to the same thing”.
• But sometimes three is more natural: triangle into equilateral, isosceles, scalene; the Greeks’ government into monarchy, oligarchy, democracy (one, few, many); forcing these into two feels unnatural.
• Hegel (1770-1831) always divides into three; harder to give a reason, but Aristotle gives a sign in the book of the Heavens: three is the first number of which we say “all” (of two we say “both”), and a division wants all the parts; also beginning, middle, end; “reflection of the Trinity in a sense”. Marx, under Hegel’s influence at the universities, always divided into three, then came to suspect something artificial and forced in it.
• Combining the two: “don’t always divide into two … don’t always divide into three, but divide into two or three” is more probable than either. Still not “always”: for the most part.
• Divisions into more than three are, inductively, either two divisions crisscrossed or a division with subdivision:
  • Aristotle’s six governments (monarchy, tyranny, oligarchy, aristocracy, democracy, republic) crisscross good/bad (for the good of the whole vs. of the part) with one/few/many.
  • Thomas’s four orders (prev. lecture): order not made by reason vs. made by reason, the latter subdivided by that in which reason makes it, the act of reason (logic), the act of the will, an exterior act: two, one of them into three.
  • The seven sacraments: Thomas does not divide sacrament into seven but uses five distinctions (he first says “three, no, six”, then settles on five). Sacraments are sensible and like natural life. (1) Those who must be fed and directed (his grandchildren) vs. those who propagate and direct them: matrimony and orders against the other five, ordered to the individual himself, a bit like family vs. city; (2) matrimony vs. orders; (3) baptism, confirmation, Eucharist vs. penance and last anointing (the three necessities of life vs. the case of defect, sickness); (4) the three by analogy with bodily life, generated, grow, nourished, compared to the plant soul’s reproductive, growing, feeding powers; (5) penance vs. anointing. One could bypass this and say “seven”, but then you do not understand their distinction and relations.
  • The five predicables (prev. lecture): three divided against two, then proper vs. accident, and so on: three divisions to get five. Thomas’s division of the categories, said of individual substances by what they are, by something in them, by something outside them, then subdividing, is always into two or three.
• Exceptions he admits: Aristotle divides quality into four; he himself spent years dividing Shakespeare’s twenty-seven plays (setting aside the ten histories, which have special classification problems) into four kinds. “I am not a fanatic.”
• Why it fits: it fits what is divided and also our mind, which has “a very hard time understanding division into more than three”; that becomes an enumeration and not a division.

6. Hegel’s reasoning: a perverse imitation of the Trinity (“there is a perverse imitation of the Trinity”) #

A student asks whether he has a sense of Hegel’s reasoning for dividing into three.

• In the Trinity the Son proceeds from the Father and the Spirit from both; Hegel’s thesis, antithesis, negation of negation, reiterated, is a bizarre, imaginative imitation of that. Then he “suddenly jumps to matter”: a perverse imitation of “the Word was made flesh”, but in the heretics’ sense that the Word itself became flesh, thought becoming matter; then union of thought and matter rising in spirit, the Hegelian system as culmination of history. “More theology there than philosophy.”
• Feuerbach, the bridge from Hegel to Marx: The Essence of Christianity says the real meaning of Christianity is that man is God.
• Diagnosis: modern philosophers have given up the faith and hence theology, and therefore cannot distinguish theology from philosophy. This leads to the principle of the next segment.

7. It belongs to the higher knowledge to distinguish itself from the lower (“always belongs to the higher knowledge”) #

Principle: it always belongs to the higher knowledge, the one with more the character of wisdom, to distinguish itself from the lower and to show the order between them (since distinction is before order). He shows it inductively first.

• Reason vs. senses: only reason can make the distinction; likewise imagination vs. reason, and so the order of senses or imagination to reason belongs to reason.
• Mathematics vs. natural philosophy: made in natural philosophy, which has more the character of wisdom (it was once thought to be wisdom); wisdom vs. both is made in wisdom.
• Theology vs. philosophy: the Greeks say God alone, or most of all, is wise; revealed theology, based on the word of God, shares God’s wisdom, so (Thomas, first question of the Summa) it has more the character of wisdom than philosophical wisdom, and the distinction and order belong to theology. The Greeks had no need of the distinction, having no revealed theology, so they did not mix them; the moderns, in an originally Christian civilization but having given up faith, gave up the very knowledge by which they could distinguish, and so mix them up, seeking in philosophy a substitute for theology. Comparison: the philomythos, lover of myth, who finds philosophy too high and seeks a substitute in literature. Hence the moderns’ chronic complaint against the senses (the Greeks knew they were imperfect) and their demand for a certitude in no way dependent on the senses, a more-than-human certitude proper to theology, ending in despair or skepticism.
• Rhetoric vs. political philosophy: Plato’s dialogues; the distinction belongs to political philosophy. Aristotle in the Rhetoric: some rhetoricians think themselves politically wise, partly from boastfulness, partly from ignorance.
• Speaking properly vs. metaphorically: you cannot define metaphor with metaphors, only exemplify it; metaphor “doesn’t know itself”, proper speech can define definition.
• Knowledge in words and statements vs. in mathematical symbols and equations: one can say in words what a word is, but “you can’t write an equation that says what an equation is”; the first thing for a wise man is to know himself. Heisenberg recognizes this: the mathematics is “wiser than we are”, and a scientist’s ability to put it in words is the criterion of how much he understands.
• Consequence: it belongs to the natural philosopher to say to what extent and how mathematics is useful in natural science; its limitations were seen by modern scientists and by Aristotle.
• Announcement: he leaves Thursday for “grandfatherly concerns”, back on the 31st, probably late that day.

His words #

• division (strict sense): the distinction of the parts of some whole; loosely, interchangeable with distinction.
• composed whole [ed.: integral whole]: put together from its parts, not said of them; the first meaning of “whole”.
• universal whole: said of each of its parts, not put together from them; the Greek word for genus comes from the word for whole; “the general”.
• the particular: what comes under the universal whole, “like a part”.
• diairesis: Aristotle’s word in Physics I for definition as dividing into parts.
• distinction: most universal of the three D’s; its basis is opposition (four kinds of opposites); comes before order.
• the rule of two or three: his own; for the most part divide into two or three; larger numbers are crisscrossed or subdivided divisions.
• enumeration: what a division into more than three becomes for our mind.
• the fifth book of wisdom [ed.: Metaphysics V].
• the book of the Heavens [ed.: De Caelo].
• last anointing [ed.: anointing of the sick / extreme unction].
• philomythos: lover of myth (Aristotle), something like a philosopher.
• perverse imitation: his characterization of Hegel’s triads (of the Trinity) and jump to matter (of the Incarnation).

Texts #

Read in class

• His handouts, per verified sources: “Note for Book One, Reading Eleven”; “The Matter and Order of Wisdom” p. 30; “Second Road in our Knowledge” p. 5; “Foreword to Logic” p. 8; matching Aquinas, commentary on Physics I, lectio 11.

Mentioned

• Aristotle, Physics I, first chapter (definition as diairesis).
• Aristotle, Categories and Metaphysics V (“fifth book of wisdom”), on opposition.
• Aristotle, biological works (criticism of Platonic dichotomy).
• Aristotle, On the Heavens (three the first number of which we say “all”).
• Aristotle, Rhetoric (rhetoricians who think themselves politically wise).
• Aristotle, six kinds of government (Politics, inferred); division of quality into four (Categories, inferred).
• Plato, Sophist and the dialogues (division into two; rhetoric vs. political philosophy).
• Euclid (circle into semicircles, parallelogram into triangles).
• Locke (the triangle in general).
• Aquinas, Summa Theologiae I, q. 1 (revealed theology more the character of wisdom); his division of the sacraments, of order (prev. lecture), of the categories; care about distinction vs. division in the Trinity.
• Aquinas, Commentary on the Sentences (student’s question on its “distinctions”).
• Hegel; Marx; Feuerbach, The Essence of Christianity.
• Heisenberg (mathematics “wiser than we are”).
• Shakespeare’s plays (his own four-fold classification).

His questions #

• Is the leg a chair? Is the seat a chair? No; but put together they make a chair: the composed whole.
• Is a semicircle a circle? Is a noun a sentence? No: divisions of composed wholes; but an affirmation is a statement: universal whole.
• Is distinguishing the senses of a word, or per se from accidental, a division or a distinction? A distinction, not a division; there is no whole and part in the strict sense.
• Define a distinction. Its basis is opposition; division adds that the things distinguished are parts of a whole.
• Is there any number into which you should divide, and can you give a rule? No universal rule; for the most part, two or three.
• Why should one divide into two? We divide by opposites, and opposites are two.
• How many distinctions does Thomas use to arrive at the seven sacraments? Five (after hesitating “three, no, six”), each into two or three.
• Do you have a sense of Hegel’s reasoning for dividing into three? A perverse imitation of the Trinity, then of the Incarnation.
• To whom does it belong to distinguish reason and senses, mathematics and natural philosophy, rhetoric and political philosophy, philosophy and theology? Always to the higher knowledge, the one with more the character of wisdom.
• Can you write an equation that says what an equation is? No; knowledge in symbols does not know itself, so knowledge in words has more the character of wisdom.