The Three Roads of Knowledge: Natural, Common and Private Ways of Proceeding

Orientation #

The recording opens with the close of the previous session (“it’s time for lunch… we’ll finish in our next class… finish talking a little bit about this distinction of the three roads, and then we’ll go into this consideration of the private roads and how they should be determined”), then the prayer and this class. He refers back to “the other day” for the before-and-after in the acts of reason and to Thomas’s proemium to logic (prev. lecture), and to “another day” for the order of determination. He points forward to “this last reading” on certitude and precision. The recording cuts off as he turns to the text (“Let’s look now at the…”). Verified source: the passage of Thomas distinguishing ordo demonstrandi from ordo determinandi matches Aquinas, In Physics I, lect. 1 (handout “Note for Book One of Natural Hearing, Reading I”, p. 11). The previous class introduced the “road” or way of proceeding, and the next takes up the final reading of Book II on custom and the certitude proper to each subject.

The class, in order #

1. Recap: the natural road (“what is more known to us”) #

Point: restate the first of the three roads before turning to the two made by reason.

• The natural road runs from the senses into reason. Before and after along it was distinguished (prev. lecture) in two ways: order among kinds of knowing, and order in things known; the latter again in two: order in which different things are known, and order in which one and the same thing is known.
• Governing rule for both: what is more known to us but less known by nature (simply) comes before what is more known by nature but less known to us.

2. The common road of reason (“reason is used in all reasoned-out knowledge”) #

Point: the second road is the road of reason as reason, taken up by logic, because reason is used in every reasoned-out knowledge.

• Following Thomas’s proemium to logic: three acts of reason — understanding what a thing is; understanding true or false (composing in affirmation, dividing in negation); reasoning.
• Euclid’s Elements shows the common road most clearly: definitions (first act), postulates and axioms (second act), theorems (third act). In every reasoned-out knowledge one must understand what before affirming or denying, and statements come before reasoning since reasoning puts two statements together to reach a third.
• Further before-and-afters in the acts of reason: we guess before we know; we think about something before understanding it; reason discurses before it sees. Unfolded: guessing before knowing → grasping before judging → dialectical before demonstrative reasoning (“except perhaps in geometry”; arithmetic easier) → part before whole, since reasonable guesses hit part of the truth.
• Thinking before understanding = going from confused to distinct (thinking out a definition or a division). Here the natural road is presupposed: from examples to definition, as in nearly all Plato’s dialogues — Meno “what is virtue?”, Euthyphro “what is piety?”, Theaetetus “what is science?”; the answerer gives examples, Socrates: “I asked you for one thing, you gave me many.” Reason: a thing is singular when sensed, universal when understood, so singular examples come before the universal definition.
• Same in reasoning: induction before syllogism (induction from many singulars to universal; syllogism begins from universal); in rhetoric, example before enthymeme (example goes singular to singular; enthymeme starts from what is true for the most part — “boys will be boys” — likelihoods or signs).
• Student question: does the natural road explain why there is no fourth figure? Answer: no; that belongs to the treatise on the syllogism. People posit a fourth figure “because they’re resolving to imagination rather than to reason”: the middle term is either between the two extremes, above them, or below them — a “fourth figure” just renames one of these. “I used to call that playing checkers.”

3. The private road: way of defining and making beginnings known (“the way of defining”) #

Point: the private road of each reasoned-out knowledge fits its matter (and, as Boethius adds in De Trinitate, how our reason stands to that matter); limiting himself to the three parts of looking philosophy, he begins gathering its elements.

• Why start here: looking philosophy arises from wonder, and the philosopher wonders what and why. Posterior Analytics II compares the question what-is-it with why-is-it, definition with demonstration, and their connection. Hence two headings — way of defining, way of demonstrating — plus a third, the order of determination (Thomas) or consideration (his own term).
• Element 1, way of defining: Natural Hearing II distinguishes natural philosophy (defines with matter and motion) from mathematics (without); Wisdom VI adds first philosophy — mathematics keeps “intelligible” (or “imaginable”) matter, extension; wisdom defines without even that.
• Element 2, more generally, the way beginnings are made known: natural philosophy by the senses (as it defines with sensible matter); mathematics in the imagination (to understand a right angle you must imagine a line meeting a line making equal angles); logic and wisdom cannot use sense or imagination, since they deal with immaterial things — easier to see in logic, which comes before the sciences in the order of learning.
  • Parmenides: Socrates imagines the universal as a sail covering things, resolving to imagination, and it fails.
  • Modern mathematicians and logicians substitute classes (a multitude of individuals, imaginable) for the universal (one said of many, the same in each).
  • Locke trying to imagine a triangle in general, neither scalene nor isosceles nor equilateral — “all confused.”

4. Element 3: the way of judging (“go back to imagination to judge”) #

Point: judgment (separating true from false by some beginning in our knowledge) goes back to whatever made the beginnings known, so it differs by science.

• Thomas on Boethius: can judgments about divine things resolve to imagination? No. Natural philosophy’s judgment goes back to the senses (Aristotle “in many places, like in the book on the Universe” [ed.: De Caelo]); geometry’s to the imagination; wisdom and logic to reason; theology “maybe back to what the Church teaches.”
• Geometrical examples of judging in the imagination:
  • The fifth “parallel postulate” (his brother Mark): it is really about lines meeting, because one cannot imagine lines of infinite extent never meeting, but can imagine lines meeting.
  • The fourth theorem (Mark: not really a theorem in the full sense): triangles with an equal angle contained by equal sides — place one on the other and see in imagination that they coincide.
  • Diameter bisecting the circle (a postulate not in the list): students say “by definition,” but the definition of diameter is a line from the circumference through the centre to the opposite side; that it halves the circle is shown (Thales) by flipping one half over the other — if the circumferences do not coincide, the radii are unequal, contradiction.
  • Fourth postulate, all right angles equal: not just the definition, since two lines each bisected do not have equal parts; superpose the lines — if the uprights do not coincide, one angle is a part of the other and the whole is bigger than the part, contradicting the hypothesis of equal division.
• But God, angel, soul, the universal cannot be judged by imagination: to imagine is to make singular (Locke again).
• Student objections and replies: (a) no perfectly straight line in sense or imagination? — one can have a flat surface in imagination without finding one in sense; “I don’t know if the circles in my imagination are perfectly circular” — “that’s a defect in your imagination, then, Father.” (b) Aren’t you also going back to reason there? — yes, but you must be able to imagine these things; solid geometry is harder because harder to imagine (cardboard models over the lamp, paper as “a crutch for imagination”). (c) You need only imagine part and reason the rest — yes, but you still go back to imagination. Euclid begins with plane geometry rather than arithmetic because geometry is more imaginable.

5. Elements 4–6: the way of demonstrating (“from the cause to the effect”) #

Point: three differences in demonstration, all traced to subject matter.

• Element 4: from cause to effect or effect to cause. Geometry mostly cause to effect (its properties are relations following on quantity, treated abstractly); natural philosophy and wisdom, at first, effect to cause (senses know only outward accidents; immaterial things known only through effects); later one may see why. Latin: demonstratio quia and propter quid (the second name is garbled in the recording).
• Element 5: which causes. Geometry perhaps only form (“why are these angles equal? because the lines are straight”); natural philosophy all four; wisdom chiefly end and form, mover to some extent, matter not at all. Aside: Warren Murray at Minnesota, where “Allen” the so-called Aristotelian said Aristotle looks for four causes everywhere; Warren would ask “what are the four causes of a triangle?” Aristotle is not offering a system: in each science one asks which of the four kinds is relevant, even for each phenomenon. God is a cause “in two and a half senses”: mover/maker, end, and exemplar (half), not intrinsic form, not matter.
• Element 6: rigor/certitude. Geometry universal and rigorous; natural philosophy, and a fortiori practical philosophy, true for the most part, because of matter or free will (“even that great rule of two or three might have a few exceptions”). Nicomachean Ethics I: not the same certitude in every science, as not the same precision in every art — his wife cutting a dress in cloth vs. chipping wood or marble vs. dynamiting the presidents’ busts in the Dakotas. Equally absurd to demand that the rhetorician demonstrate (who will be next president? “I thought Bush was going to win” — but wait for election night) and to let the geometer persuade (students wanting to measure the angles instead of proving). One demands more certitude than possible, the other accepts less than is so; both are mistakes. “Boys will be boys, but not always.”
• Student query confirming the count: six elements so far; order of consideration not yet reached.

6. Element 7: the order of determination (“the order of determination or consideration”) #

Point: what is considered first and second differs in each part of looking philosophy, and wisdom runs almost the reverse of natural philosophy.

• Natural philosophy: from general to particular, and towards matter — nearly but not exactly the same direction. From change in general (eight books of Natural Hearing) to kinds of change; and among these, place before quality before substance before growth: bodies can move in place without change of quality but not vice versa (bodies must be brought together for a chemical reaction; clearer for Aristotle since heavenly bodies change place only); chemical change is more general than growth. Towards matter: Natural Hearing → the Universe → Generation and Corruption; again in the study of life, where Aristotle goes more into matter as he proceeds (Thomas at the beginning of the next book), yet begins with the soul because life as life is most known from inward experience.
• Geometry: simple before composed (circle → equilateral triangle in theorem 1; plane before solid; triangle before parallelogram) and, for the most part, equal before unequal: right angle defined before acute and obtuse, equilateral before isosceles before scalene, axioms of equality first and “whole greater than part” last; Book I ends with the Pythagorean theorem, and Book II’s theorems 12 and 13 (obtuse: square greater; acute: square less) are proved by it. So the equal is used to prove the unequal, and acute/obtuse are defined through the right angle (not as “the greater and lesser of unequal angles” — “wouldn’t be so useful”).
• Wisdom: in a way from less universal to more universal — Book IX rises from act and ability in reference to motion to their complete universal consideration; Books VII–VIII from material substance to a general understanding usable for immaterial substances; the one, from the one that begins number (a measure) to the one convertible with being. More basically, towards the immaterial: being, one, many (Books V–X) can be in matter but need not; then immaterial substances, which cannot be. Hence the more one goes on in natural philosophy the harder wisdom becomes “because of the way you’re accustomed to think.”
• Thomas in the Physics distinguishes ordo demonstrandi (e.g. effects to causes) from ordo determinandi (e.g. general to particular); elsewhere he notes the different ways of judging. Aristotle and Thomas treat these elements in various places; “this was my attempt to bring them together.”

7. Beginnings and judgment, again (“definition is a beginning”) #

Point: why the way of judging is filed with the way of making beginnings known, and how ethics differs.

• Ethics makes its beginnings known by experience: Thomas’s example that concupiscence diminishes by abstinence — the full Lenten fast, hard at first, easier towards the end. Without experience of being better disposed by repeated acts, one lacks the starting point of ethics.
• Thomas: the most authoritative judgment is by the definition. “No odd number is even” — judged with great certitude by definition; “no perfect number is prime” (a prime is measured only by one, whose sum cannot add up to it). Phaedo: pressed by Simmias, Socrates goes back to definition — one opposite cannot be the other; if one opposite is in the definition of a third thing, that thing cannot admit the other (good in the definition of virtue; life not in the definition of body, so a body can die, but life is in the definition of soul). “I don’t want to say the argument is entirely good,” but it shows definition’s role in judgment. Even the first proof of God’s existence rests on the definition of motion.
• So judgment, separating true from false by a beginning, corresponds to the way beginnings are made known: in geometry back to axioms, postulates, definitions; ultimately to the senses.

8. Student question: arithmetic and imagination (“how arithmetic resolves to imagination”) #

Point: arithmetic is more abstract than geometry but not wholly free of the continuous.

• Hard to see. A text of Thomas (he cannot now locate it) almost says the one that begins number abstracts from continuous quantity/intelligible matter — but then it would be the one convertible with being; shown to Warren Murray as striking evidence of how hard the two ones are to separate (“count the angels”). A point here and there is not the one convertible with being, yet the one beginning number is not here or there.
• Number arises from division of the continuous; numbers go on forever because the continuous is divisible forever.
• Thomas: even within one science one part is more immersed in matter than another; geometry more material (intelligible matter) than arithmetic. Hence the difficulty of the number three in the treatise on the Trinity: three persons are not more than two, two not more than one — separating the two senses of one.

9. The order among the three roads (“the first road is presupposed the second”) #

Point: the roads are ordered — the natural road explains the common, the common explains the private.

• Natural → common: because we know the confused before the distinct, reason must define, divide and distinguish (angels and God do not); many heresies (Eutyches; Father and Son) come from failing to distinguish; equivocal words must be distinguished by reason, “as the moderns do ad nauseum” fail to. Induction before syllogism and examples before definition (Plato; a child asked “what’s a nose?” points: “that’s a nose”) go back to singular-when-sensed, universal-when-understood.
• Common → private: logic teaches that definition is the beginning of demonstration, hence the importance of the way of defining; Posterior Analytics teaches premises and conclusion are in the same genus, so (Thomas on ethics, “as we’re taught in the art of demonstrating”) if principles are true only for the most part, so are conclusions.
• One must see the distinction of the three roads before the order among them.

His words #

• looking knowledge / looking philosophy — theoretical, speculative knowledge [ed.: speculative]
• reasoned-out knowledge — science [ed.: scientia]
• road / way of going forward — modus procedendi, method
• Natural Hearing — [ed.: Physics]
• Wisdom — [ed.: Metaphysics]; “the book on the Universe” [ed.: De Caelo]
• ability — [ed.: potency], paired with act
• common road of reason — the way taken up by logic
• private road — the way proper to each science, fitted to its matter
• order of determination (Thomas) / order of consideration (his) — what is considered first and second; ordo determinandi vs ordo demonstrandi
• intelligible / imaginable matter — the extension mathematics defines with
• judgment — “separating the true from the false by some beginning in our knowledge”
• resolving — going back (of judgment) to sense, imagination, or reason
• demonstratio quia — from effect to cause; contrasted with demonstration from cause to effect [ed.: propter quid]
• the one that is the beginning of number vs the one convertible with being
• playing checkers — his name for the “fourth figure”

Texts #

Read in class

• Thomas Aquinas, In Physics I, lect. 1 — ordo demonstrandi vs ordo determinandi (verified source; handout p. 11)

Mentioned

• Thomas, proemium to logic/Posterior Analytics (three acts of reason) (inferred locus)
• Euclid, Elements I (defs., postulates 4 and 5, theorems 1 and 4, diameter bisecting the circle), II (theorems 12, 13)
• Plato, Meno, Euthyphro, Theaetetus, Parmenides, Phaedo
• Aristotle, Posterior Analytics II; Rhetoric (example and enthymeme); Prior Analytics, treatise on the syllogism (inferred title)
• Aristotle, Physics II (defining with/without matter and motion); Metaphysics VI, VII–VIII, IX, V–X; Nicomachean Ethics I
• Boethius, De Trinitate with Thomas’s commentary (resolving judgment to imagination)
• Thomas on ethics (concupiscence diminished by abstinence; principles and conclusions agree) — locus not stated
• Summa treatise on the Trinity (the number three)
• Locke on the general triangle

His questions #

• Does the natural road explain why there is no fourth figure? No; the middle term can only be between, above or below the extremes — a fourth figure is a renaming, from resolving to imagination.
• What are the four causes of a triangle? (Warren Murray’s playful question) In geometry perhaps only the form is relevant; each science must ask which causes apply.
• Is it obvious that the diameter bisects the circle, or is it by definition? Not by definition; it must be manifested in imagination by flipping one half over the other.
• Can you go back to imagination to judge the way God, an angel, your soul, or the universal is? No; to imagine is to make singular.
• In what senses is God a cause? Two and a half: mover/maker, end, and exemplar; not intrinsic form, not matter.
• How does arithmetic resolve to imagination? Hard to see; number arises from division of the continuous, so it is not wholly abstracted from it — left partly open.
• Why must our reason define, divide and distinguish, when angels and God need not? Because we know the confused before the distinct (the natural road).
• Why does induction come before syllogism, and examples before definition? Because a thing is singular when sensed and universal when understood.