The Role of Contradiction in Discovery: Heraclitus, Planck, Bohr, Heisenberg, Einstein and Theology

Orientation #

He refers back to last class, where the first reading of Aristotle’s Metaphysics III was read, and says he will not expound that text again but will refer back to it. Today’s whole class is a walk through his handout of parallel texts, under the title he gives these readings: “the role of contradiction in coming to know.” (Verified sources indicate handout pages, incl. “Texts on Necessity and Difficulty of Beginning Well”.) The neighbouring classes treat Book III’s knots and then its second reading.

The class, in order #

1. The plan of the handout (“the role of contradiction in coming to know”) #

Page 1: Heraclitus and the Aristotle text read last time. Page 2 onward: the great physicists of the 20th century, beginning with Max Planck, “the father of modern physics,” ending with Einstein, “who’s the authority.” Pages 5–6: theology. The purpose: despite the differences in how philosophy, science and theology proceed, some of the greatest minds in all three see something of the role of contradiction in discovery.

2. Heraclitus, “the father of the progress of the human mind” #

The fragments, each applied to knowledge:

• “War is the father of all things, the king of all things” — perhaps an exaggeration as to universality, but in particular war is the father of new knowledge.
• “The hidden harmony is better than the apparent harmony” — harmony is the opposite of contradiction. Most men, as the dialogues show, are in apparent harmony: their thoughts contradict each other or things, and they are unaware of it. Socrates finds them in contradiction. (Aside, from an earlier conversation: the courts — kicking a pregnant woman and killing the baby is homicide, abortion is not; so it is and is not.)
• Meno example: the slave boy says you double a square by doubling the side. Take a 2×2: it gives 4; 4×4 gives 16; 16 is not double 4. So one claim must be rejected — the one that goes is “the square whose side is twice as long is twice as big.” Socrates tells the master they have done the boy a good turn: (1) freed from error; (2) he knows he doesn’t know; (3) he will want to know. Then the second conversation: three more equal squares fill out a square four times as big; the diagonals cut it so that the square on the diagonal is half of four, i.e. twice the original. That is the hidden harmony — the true harmony, really free of contradiction.
• “Nature loves to hide” — note he says nature, not triangle or square; harmony about nature is hidden at the start. (Aside: a book in the bookshop’s science section titled Nature Loves to Hide, with no acknowledgement of Heraclitus.)
• “If you do not expect the unexpected, you will not find it, for it is hard to be found and difficult.” Seemingly a contradiction — but when your thoughts contradict each other you realise something is hidden from you, and then you are expecting something.
• His last advice: dispute. Hence Thomas’s Quaestiones disputatae, and the Summa as an abbreviation of the disputed-question method: Monday-Wednesday-Friday lecture; Tuesday-Thursday one side argues the affirmative, one the negative, and the magister determines, answering the arguments of whichever side missed the truth — or both, if the truth lies between.

Titles he gives Heraclitus: “the father of the progress of the human mind”; “the central thinker of human thought.”

3. Planck: the fact that contradicts the accepted theory (“physics is now complete”) #

Planck is called father of modern physics — “modern” being 20th-century physics, “classical” that of Galileo, Kepler, Newton (17th–19th centuries). December 1900: the quantum hypothesis, from black-body radiation, where existing theories implied infinite energy — absurd. Energy is given or received only in discrete amounts. Five years later Einstein used the quantum for the photoelectric effect; 1913 Bohr for the atom; de Broglie: nothing in the physical world is made without the quantum. Heisenberg’s story: Planck walking in the park with his son — “I think I discovered something as great as Newton.”

The point: Planck had been told at university that physics was complete, principles all in place, go into biology if you want fundamental discoveries — and he made the most fundamental discovery. So he must know something about how discoveries are made. His text: the first impulse to revision of a theory is nearly always the discovery of facts that cannot be fitted into the existing theory; nothing is more interesting to the true theorist than a fact that directly contradicts the generally accepted theory, for that is his particular work.

4. Gamow (“staggering contradictions”) #

Gamow, Russian, studied under Bohr, Nobel Prize for contributions to quantum theory; his history of the thirty years (1900–1930) of quantum theory. His statement, which Berquist calls very strong: staggering contradictions between theoretical expectations on one side and observational facts, even common sense, on the other, are the main factors in the development of science.

5. Bohr: dialectical, heuristic logic (“just by emphasizing this conflict”) #

Bohr is in some ways the most influential — his institute in Copenhagen drew Oppenheimer, Heisenberg, Gamow, Pauli; Einstein by contrast was a loner. He saw the atom as a miniature solar system would have no stability, and that radiation from the atom is not continuous but jump-like: the quantum in the atom.

Four passages, each with his gloss:

1. Bohr’s 1913 lecture to the Danish Physical Society, a lecture he counted among his best: he hopes his hearers see the extent to which these considerations conflict with the classical theory of electrodynamics — “there is the war that Heraclitus spoke about” — and that just by emphasizing this conflict it may in time be possible to discover a coherence in the new ideas. Coherence = another word for harmony.
2. “Bohr’s turn of mind was essentially dialectical rather than reflective” — he needed dialogue to start his thinking; he would never outline a finished picture but go patiently through all phases of a problem, “starting from some apparent paradox” — i.e. a contradiction — and gradually leading to its elucidation, the untying of it. Those present said they were reminded of Socrates. (Dialectical, with the Greeks, means reasoning from probable opinions to contradictory conclusions, as Aristotle says — as in the third part of the Meno, where Socrates argues both that virtue can be taught and that it cannot.)
3. Logical analysis was for Bohr not a mere verification of consistency but “a powerful constructive tool, orienting the groping mind in the right direction”; he saw the heuristic power of logic in its dialectical character. He glosses heuristic from Greek heurein, to find or discover: logic’s power of discovery lies in its bringing out contradictions.
4. Bohr on Rutherford’s model: it made clear that atomic stability could in no way be accounted for by the classical (Newtonian) laws, and thereby pointed to the quantum postulate as the only possible escape from the acute dilemma; the sharpness of the contradiction made Bohr confident of the truth of the postulate. These are the two things Aristotle has in the second paragraph: the contradiction shows you where you are going, and tells you when you have arrived (the contradiction disappearing is the sign of progress).
5. Late remark: no progress is being made on transmutations of matter at very high energy because no process has been found exhibiting a sufficiently violent contradiction with what could be expected — note “expect the unexpected.”
6. Last: difficulties were for Bohr “merely the external appearance of new knowledge” — like the rind of an orange — and in an apparently hopeless contradiction he conceived the germ of a wider and more comprehensive order and harmony. The better harmony of Heraclitus.

6. Heisenberg: asking the right question is more than halfway (“the physicists learned to ask the right questions”) #

Story: Bohr in the 1920s knew more about the atom than anyone; at the “Bohr Festival” in Germany the student Heisenberg raised an objection in the question period; Bohr afterwards invited him for a walk and they talked atomic theory for hours — it influenced all Heisenberg’s later thinking. Heisenberg first proposed protons and neutrons in the nucleus, and formulated the principle of indeterminism, the greatest change in physics since Galileo, Kepler and Newton.

From the Gifford Lectures (Physics and Philosophy): after Bohr’s theory, spectroscopic and chemical material became available — “it was from this time on that the physicists learned to ask the right questions,” and asking the right question is frequently more than halfway to the solution. He draws it on the board: a straight line for the time and difficulty from ignorance to knowledge; the average man puts the question near the beginning, but Heisenberg puts it past the midpoint. His own witness: in philosophy he spends more time getting to the right question than answering it; in his doctoral thesis, after ten years on and off, he put together something Thomas said and something Dionne said and had the answer at once — and when he brought the question to Monsignor Dionne (“the best mind I’ve ever known”), the answer came immediately. [The exchange as transcribed is partly garbled.] Same in the Meno: Socrates only asks the right questions and the boy seems to teach himself.

Heisenberg’s next line: what were these questions? Practically all had to do with the strange apparent contradiction between the results of different experiments. Also the Heisenberg-Bohr episode in Copenhagen: Bohr goes skiing, Heisenberg asks the right question one night, is too excited and miscalculates, then calms down and all the numbers come out.

Back-reference: the fragment of Anaxagoras on mind. On one hand the mind is self-ruling (logic is the sign); on the other Anaxagoras says the ruler must be separated from the ruled, and the mind is not separated from itself. The contradiction let us ask the right question, and the answer: not two minds, not two parts, but some things known to the mind and some not — the mind is ruled in what it does not know by what it does know, in what is less known by what is more known. Until we saw the contradiction we did not really understand how the mind goes forward.

7. Einstein: from the horse’s mouth (“the relativity theory arose from necessity”) #

Aside on the proverb “get it from the horse’s mouth”: the horses down at the track are close enough to know who’s going to win — so don’t take another’s word on how relativity was discovered.

From The Evolution of Physics, “the nearest thing to a history of the main ideas of physics”:

• Relativity “arose from necessity, from serious and deep contradictions in the old theory, from which there seemed no escape.” Why necessity? Not determinism — because the mind cannot accept a contradiction. And “the strength of the new theory lies in the consistency and simplicity with which it solves all these difficulties”: it arose from contradiction, and arrival is shown by the contradictions disappearing — Aristotle’s two points again.
• On quantum theory: reconciling the two contradictory views, matter and wave, is a fundamental difficulty which “once formulated must lead in the long run to scientific progress.” Context: Einstein would not go along with indeterminism; at the congresses (1927 or 1930) he posed objections to the Copenhagen interpretation, Bohr staying up nights to refute them; the photograph of the two coming out after Einstein’s last objection, Einstein looking a bit cocky, Bohr wanting to talk more — and Bohr found the flaw, which was that Einstein had not applied his own theory to the question. Einstein never publicly objected again, though he would not accept it.
• General statement: their aim is “to break down the wall of contradictions which frequently blocks the way of scientific progress,” and all the essential ideas of science — a universal affirmative, no exception — “were born in a dramatic conflict between reality and our attempts at understanding.” “War is the father of all.”

Two metaphors compared: Aristotle’s feet tied together by rope, where you look for the loose end and unravel; Einstein’s wall that blocks the way. Both say contradiction stops the mind, at least temporarily.

He lingers on dramatic: in the Poetics a plot consists in tying the knots and untying them; hence the philomythos, the lover of plots, is in a way a philosopher, since philosophy too sees knots and unties them. Twelfth Night: “O time, thou must untangle this. Not I. It is too hard a knot for me to untie” — and Aristotle says time is a good helper to the philosopher. Aristotle also observes that the younger or lesser poets are better at tying knots than untying them — “a poor story about the philosopher”: most philosophers can tie themselves and others in knots but cannot untie them; you get engrossed in a movie and the ending disappoints. Practice: with the Summa, read the objections and the sed contra, let the knot settle, try to untie it yourself, then see how Thomas does it — he sometimes gives students only the objections and sed contra, handing out the body and replies afterward.

8. Theology I: Christ and the Pharisees (Matthew 22) #

He follows the order of the decree on priestly training: scripture first, then the Fathers, then Thomas. Matthew 22 belongs to the chapters giving the reasons they wanted to put him to death; this one is where he refutes them. “Whose son is the Christ? David’s,” they say; then how does David, moved by the Spirit, call him Lord — “The Lord said to my Lord, sit at my right hand”? If David calls him Lord, how is he his son?

Our Lord points out an apparent contradiction between two passages of Scripture, and the Pharisees, who claim to know the Scriptures, cannot answer — as Socrates does with the slave boy, showing a man ignorant of what he claims to know. Hidden under the apparent contradiction is the unexpected truth: Christ is both God and man; as man descended from David, as God the Lord of David. Same with “the Father is greater than I” against his equality with the Father: as God equal, as man less — indeed as man less than himself as God. Comparison: the quantum physicist sees an apparent contradiction between experiments; Christ between one part of Scripture and another; in both the untying is a discovery. And Socrates and Christ share that particular reason for being put to death (only one of the reasons in Christ’s case).

9. Theology II: Augustine on heresy (“there must also be heresies”) #

Augustine, City of God: even the heretics yield an advantage to those who make proficiency, per the Apostle, “there must also be heresies, that they which are approved may be made manifest among you”; “the son who receives instruction will be wise, and he uses the foolish as a servant.” The heretic contradicts an article of faith — sometimes from misunderstood scripture, sometimes from philosophy — and the Fathers were forced to defend it, untying the apparent contradiction, and so came to a deeper understanding. “While the hot restlessness of heretics stirs up questions about many articles of the Catholic faith, the necessity of defending them forces us both to investigate them more accurately, to understand them more clearly, and to proclaim them more earnestly; and the question mooted by an adversary becomes the occasion of instruction.” Note necessity and forces — Einstein’s own words. God brings good out of evil.

Student question: what does “mooted” mean? — Simply raised by an adversary; not a real question in some sense. Otherwise you go on believing without penetrating what you believe, until someone says it makes no sense and gives a reason, and then you are forced to defend it and so understand it better. This is the role of contradiction in the patristic development of theology.

10. The decree on priestly training: individual truths, then their interconnection #

Three steps in the decree, glossed:

1. Biblical themes first — hence he began with Matthew.
2. What the Fathers of Eastern and Western Church contributed to the transmission and development “of the individual truths of revelation.” He notes the ad hoc character of much of Augustine (always answering some heretic on one article), against Thomas — though Thomas has ad hoc works too, the Contras.
3. The further history of dogma in relation to the history of the Church — the ecumenical councils, likewise mostly responses to heresy: early ones on the divinity of Christ or Mary as Mother of God; Trent on the sacraments, a development of sacramental theology. So the councils too are ad hoc, not treating the whole of the faith.
4. Finally, “in order that they may illumine the mysteries of salvation as completely as possible” — a very strong phrase — students should learn to penetrate them more deeply with the help of speculation, under the guidance of Thomas. And in Thomas every one of the hundreds of articles of the Summa begins from a contradiction developed in the objections: he is always arguing against himself, objecting to the truth. So under Thomas’s guidance the role of contradiction is central to everything you do. And the decree adds “to perceive their interconnections” — contrast the earlier “individual truths of revelation”: Thomas looks at the whole of the faith connectedly.

11. Why this is less true of geometry (“nature loves to hide” applied again) #

He connects back to Metaphysics II on why natural philosophy and wisdom are more difficult than geometry. Discovery in wisdom and in natural philosophy is made by seeing contradictions and untying them; this he thinks is not so much true of discovery in geometry. Reason: before a theorem is proved in geometry, for the most part — there are exceptions, “very interesting exceptions” — men would not think opposite things about it; they might have no opinion at all about the Pythagorean theorem, but they would not hold opposite opinions.

Second, a great disposition for discovery is a good imagination — Sherlock Holmes always speaks of imagination, and the modern physicists almost put imagination above reason. Geometry is more accessible to the imagination: I can imagine what I talk about in geometry; I cannot imagine God, the angels, the soul, or the natures of things. He leaves the connection between these two points stated rather than worked out, and breaks for the class.

Aside: in the Augustine passage he recalls Watson calling himself “the whetstone” of Holmes — the whetstone of the wise; Shakespeare speaks that way too.

His words #

• The role of contradiction in coming to know — his title for this set of readings.
• Apparent harmony vs. hidden harmony — the state of most men’s unnoticed inconsistency vs. the true harmony really free of contradiction.
• Dialectical — with the Greeks, reasoning from probable opinions to contradictory conclusions, as Aristotle says.
• Heuristic — from Greek heurein, to find or discover; hence logic’s “heuristic power.”
• Paradox — he glosses Bohr’s “apparent paradox” simply as a contradiction.
• Quaestiones disputatae — the disputed question; the Summa an abbreviation of that method.
• Mooted — raised by an adversary.
• Philomythos — lover of plots, in a way a philosopher (Poetics).
• Classical / modern physics — 17th–19th centuries vs. the 20th; “classical” here not meaning Greek.
• Father of the progress of the human mind / central thinker of human thought — his titles for Heraclitus.

Texts #

Read in class (from the handout)

• Heraclitus, fragments: war the father of all; hidden harmony better than apparent; nature loves to hide; expect the unexpected; the concluding advice to dispute.
• Max Planck, on the revision of physical theory.
• George Gamow, on staggering contradictions in the development of science.
• Niels Bohr, close of the December 1913 lecture to the Danish Physical Society; four passages of memoir/commentary on Bohr, including his remarks on Rutherford’s model and on high-energy transmutations.
• Heisenberg, Physics and Philosophy (Gifford Lectures), on learning to ask the right questions.
• Einstein (The Evolution of Physics): relativity from necessity; matter and wave; the wall of contradictions and “all the essential ideas in science.”
• Matthew 22: “What is your opinion about the Christ? Whose son is he?”
• Augustine, City of God, on heretics, citing St. Paul.
• Decree on priestly training, on the ordering of dogmatic theology.

Mentioned

• Aristotle, Metaphysics III, first reading (read last class); Metaphysics II, on the difficulty of natural philosophy and wisdom vs. geometry; Poetics, on tying and untying the plot and on the lesser poets.
• Plato, Meno: the slave boy; the third part on whether virtue can be taught.
• Anaxagoras, fragment on mind.
• Thomas, Summa Theologiae and Quaestiones disputatae.
• Shakespeare, Twelfth Night; Conan Doyle’s Sherlock Holmes.
• de Broglie; Rutherford; Bohr–Einstein congress debates.

His questions #

• Harmony is the opposite of what? — Contradiction.
• How do you double a square? — Not by doubling the side (2×2=4, 4×4=16); by the square on the diagonal.
• How can you expect the unexpected? — When your thoughts contradict each other you know something is hidden, and then you are expecting something.
• What does heuristic mean? — From heurein, to find; the power of discovery, which logic has in its dialectical character.
• Why does Einstein say relativity arose from necessity? Is it determinism? — No: because the mind cannot accept a contradiction.
• What did Heisenberg’s “right questions” have to do with? — Practically all with the strange apparent contradiction between the results of different experiments.
• How can the mind rule itself, when the ruler must be separated from the ruled? — It is ruled in what it does not know by what it does know, in the less known by the more known.
• If David calls him Lord, how can he be David’s son? — As man he is David’s descendant, as God David’s Lord.
• What does “mooted” mean? — Raised by an adversary.
• Why is discovery in geometry not so much by contradiction? — Because men do not hold opposite opinions about theorems beforehand, and geometry is accessible to the imagination; the connection to imagination’s role he leaves stated, not worked out.