“We are beginning an inquiry into civilization,” writes Collingwood, “and the revolt against it which is the most conspicuous thing going on at the present time.” The time is the early 1940s.
It could be the late 2010s. Civilization is being attacked in the old way, by autocrats driven by their passions. Local habitat destruction—the burning of crops, the salting of the earth—may always have been with us; general climate degradation is a new form of attack.
Some persons who attack us may do so in the name of civilization. They may qualify it as white civilization, or they may deny that this is what they mean.
In the conclusion reached by Collingwood in the thirty-sixth of the forty-five chapters of The New Leviathan of 1942, the personal is political:
36. 51. Civility as between man and man [sic], members of the same community, is not only what constitutes the civilization of that community relatively to the human world; it is also what makes possible that community’s civilization relatively to the natural world.
As Collingwood explained in the previous chapter, “Behaving ‘civilly’ to a [person] means respecting [their] feelings: abstaining from shocking [the person], annoying [them], frightening [them], or (briefly) arousing in [them] any passion or desire which might diminish [their] self-respect” (35. 41).
I understand respect for a person’s feelings to include, for example, not dismissing those feelings as irrational. Fear of immigrants, or of white genocide, is neither rational or irrational; our response will be one of those, whether the fear be in ourselves or in others. No automatic response (such as giving in) will be necessarily correct.
In taking up The New Leviathan chapter by chapter, starting in January, 2014, I wrote of what came to mind. This may have been a lot, even too much. Sometimes I have gone back to preface an article with a summary. Returning now to Chapter I, I supplement my original analysis with the following, where I indicate implicit sections of the chapter with ranges of paragraph numbers.
Being under attack is a current condition of civilization (1. 12). Civilization is a condition of communities (1. 13); community, of persons (1. 14). In the other order, these are the parts of the book, listed in the subtitle: “man [sic], society, civilization and barbarism.”
We shall apply the “principle of the limited objective,” though Collingwood will name it as such only in Chapter XXXI, “Classical Physics and Classical Politics.” An aspect of the principle is, “Take time seriously” (31. 68). This is what “Christendom” does, as we are told in Chapter XXVI, “Democracy and Aristocracy.”
We ask therefore what problem we need solve now.
1. 16. About each subject we want to understand only so much as we need in order to understand what is to be said about the next.
More precisely, “We of the twentieth century hold ourselves bound to the tradition in these matters laid down by Bacon and Descartes in the seventeenth: to speak not merely ‘to the subject’ but ‘to the point’ (1. 19).
We are starting out with a study of humans quâ beings that form communities (1. 22). This means we shall study the mind (1. 21). Collingwood is careful to distinguish this study from physiology and other sciences of the body. Collingwood is careful, presumably because others are not; and this is a reason why I am returning to Collingwood’s chapter now.
For some of us, the greatest certainty is the world out there, studied by physics, physiology, chemistry, and all of the other natural sciences. No deity can be as certain as that external world. For Collingwood, the starting point is what we do in the world. Some of us may not think we can do much. This will make reading Collingwood difficult.
A natural science gets to define for itself what it is studying: matter or life, as the case may be.
There is no knowing ahead of time what that object of study is; neither is the job of knowing it ever done.
To assert the opposite is to initiate scientific persecution, “the persecution of scientists for daring to be scientists” (1. 57).
Writing in Prague at Logic Colloquium 2019, I propose examples from mathematics.
In the session of the colloquium called “Foundations of Mathematics,” one talk (pdf) concerned the use of diagrams to prove the Intermediate Value Theorem. I cannot locate now a quotation from an analytic philosopher, to the effect that real proofs do not use diagrams; but such a dogmatic assertion would be scientific persecution. The real proof does not sit on a page, but lives in our minds; we also call what puts it there a proof, and this can be words or pictures.
Mikhail Katz recently alerted me to criticism of so-called non-standard analysis. Such criticism is scientific persecution when, as apparently in the case of the late Errett Bishop, it is based on a dogmatic preconception of what mathematics ought to be.
Teachers of calculus ought to take the approach they prefer, while being prepared to suggest another one to the interested student. I was such a student, in an undergraduate analysis course, before I started graduate courses in the fall of 1989. Though the teacher belittled the non-standard approach, he directed me, when I asked, to Abraham Robinson’s book. This was not actually the best reference. It was too abstract, and Keisler’s book might have been better. However, analysis is going to be difficult, however you do it. Thus the great Donald Knuth, creator of our beloved TeX program, was foolish to write in 1988, as quoted in the blog of Alexandre Borovik,
If I were responsible for teaching calculus to college undergraduates and advanced high school students today, and if I had the opportunity to deviate from the existing textbooks, I would certainly make major changes by emphasizing several notational improvements that advanced mathematicians have been using for more than a hundred years … I’m sure it would be a pleasure for both students and teacher if calculus were taught in this way.
Well sir, teach a calculus course that way, and tell us how it went! Put your money where your mouth is (as I was told in toothpaste ads in childhood). The proof of the pudding is in the eating.
Our “mind is made of thought” (1. 61); but we must not misunderstand this by reading theory for thought:
1. 68. It would be a more disastrous mistake in the science of mind to forget that thought is always practical than to forget that it is sometimes theoretical.
The first three propositions of Euclid’s Elements are practical, each one ending with the formula ὅπερ ἔδει ποιῆσαι, “which was to be done.” What was to be done in this case was to cut a given shorter length from a longer. The fourth proposition is theoretical, ending ὅπερ ἔδει δεῖξαι, “which was to be shown”: this being that if two sides and the included angle of one triangle are respectively equal to the same in another, then so are the third side, the triangle itself, and the remaining angles.
The remainder of the chapter is a warning against the misuse of expertise.
1. 83. Man as body is whatever the sciences of body say that he is. Without their help nothing can be known on that subject: their authority, therefore, is absolute.
1. 84. Man as mind is whatever he is conscious of being.
We have different ways to approach the universe. If you feel the need to say, “My way or the highway,” then you are denying the unity of the universe. If all is one, then there can indeed be no other way than yours; but if somebody seems to deny this anyway, then maybe you are the one who is confused.
There is foolish resistance to expert opinion on vaccines and climate change.
There is also foolish acceptance that experts get to judge the correctness of our language. On the contrary, we get to decide whether we have said what we wanted to say. Language may be a skill that must be learned; and yet I note my grandfather’s recollection of a teacher at Beloit College, “who tried to teach writing, which nobody can.” Success at journalism was up to my grandfather; however, his teacher, Roscoe Ellard, “was instrumental in getting me my first job—with the United Press in Chicago, pay $25 a week” (in 1924).
Not everybody can put pen to paper, or finger to keyboard, and come up with something satisfying. At least some persons do not think they can. When I worked at a farm after college, and the season was winding down, a comrade asked whether I had been good in English class: he wanted help in writing a job application.
Collingwood may have little sympathy for your sense of inadequacy. He expects you to get over it. In Chapter XXXII, “Civilization as Education,” he will tell you not to send your children to professionals, but to educate them yourself, mainly just by being with them, letting them run around on their own, and not allowing anybody else to interfere.
In my original essay on Collingwood’s opening chapter, I wrote in several parts, which I have only recently distinguished with headings.
Photos by me or Ayşe Berkman, Prague, August 10–15, 2019
]]>After Descartes gave geometry the power of algebra in 1637, a purely geometrical theorem of Apollonius that is both useful and beautiful was forgotten. This is what I conclude from looking at texts from the seventeenth century on.
Naturally there are versions of the theorem for the parabola and hyperbola; but when a point P moves along an ellipse, as in the animation above (which you can probably enlarge by clicking; or if there is a problem, as one person experienced, try the still version below), the theorem is that the blue triangle (which has rising rules from left to right) is always equal either to the red trapezoid (with falling rules), when it is a trapezoid, or to the red triangle less the green triangle (cross-hatched). In letters, the triangle XPY is always equal to the trapezoid VXY*E*, whose sides may cross one another, but which can always be understood as the triangle VKE* less the triangle KY*X. In brief,
XPY = VXY*E*,
though our sign of equality dates only from Robert Recorde’s Whetstone of Witte of 1557. Descartes uses a different symbol for equality; Apollonius, none.
Apollonius applies the foregoing theorem as follows. By adding quadrilateral XYX*Y* to both sides of the equation, and using the equality of triangles VFE and E*FD, we obtain
Y*PX* = EYX*D,
as suggested in the animation below. In algebraic or “analytic” terms, we have effected a change of coordinates and shown that, with respect to the new coordinates, the ellipse has the same equation as before. The original coordinate axes were VK and VE*; the new, DK and DE.
Descartes’s 1637 Géométrie
In De Sectionibus Conicis, Novo Methodo Expositis, Tractatus of 1655, John Wallis used Cartesian methods to establish the change-of-basis theorem that we have just proved geometrically. According to Morris Kline in Mathematical Thought from Ancient to Modern Times (1972), Wallis “was probably the first to use equations to prove properties of the conics.” In my judgment, Wallis’s argument concerning change of coordinates is tedious and opaque. It is easier to follow the argument of Jan de Witt in Elementa Curvarum Linearum, Liber Primus of 1659, though it helps that an English translation, by Albert W. Grootendorst, was published in 2000. I have looked at Ian Bruce’s translation of Euler’s Introductio in Analysin Infinitorum of 1748; at the 1773 English translation (Geometrical Treatise of the Conic Sections) of Hugh Hamilton’s De Sectionibus Conicis, Tractatus Geometricus of 1758; and at more recent books. Nobody uses geometry—areas—as Apollonius does.
The best algebraic proof of change of coordinates in an ellipse is by means of the affine transformation called an elliptical rotation. In his commentary on Apollonius, Boris Rosenfeld suggests that this rotation (he calls it a “turn”) is what Apollonius’s theorem amounts to. In a sense this is true, but it ignores how radically Descartes has changed our way of thinking about mathematics.
We can translate the first equation above into the modern form
y^{2} = ℓx(2d − x)/2d
for some ℓ (the latus rectum or upright side), where d is the length of VK. Then we can verify that, if satisfied by (a, b), the equation is fixed by the transformation
x′ = cx/d − 2by/ℓ + a,
y′ = −bx/d − cy/d + b,
where c = d − a. However, these computations have nothing obvious to do with our earlier computations with areas.
I return to Apollonian modes of thinking, to fill in the details of why the earlier geometric equations hold. Our ellipse has center K, and VW is a diameter. As a diameter, VW bisects all chords that are parallel to the tangent VE*. In particular, PX is half of such a chord and is thus an ordinate with respect to the given diameter. This ordinate cuts off from the diameter two abscissas, VX and XW. Also DM is an ordinate, and KV is extended to E so that KE is a third proportional to KM and KV (that is, as KM is to KV, so is the latter to KE). Then PY is drawn parallel to DE, which will turn out to be tangent to the ellipse.
The properties of the ellipse are derived initially from the cone from which it is cut, as in the animation above (or the still version below), with apex A and base the circle with diameter BC. The chord DD′ is at right angles to, and is therefore bisected by, the diameter. Hence the square on the ordinate DM is equal to the rectangle bounded by the abscissas BM and MC. The same is true in a section of the cone parallel to the base; thus the square on PX is equal to the rectangle bounded by QX and XR. By the similarity of triangle QXV to BMV and XRW to MCW, the ratio of the squares on PX and DM is the product of the ratios of VX to VM and of XW to MW.
Now we are in the plane of the ellipse. The square on PX varies jointly as the abscissas VX and XW. But the square also varies as the triangle XPY, and the trapezoid VXY*E* varies jointly as the abscissas VX and MX. In case P coincides with D, the triangle is equal to the trapezoid. Therefore it is always equal.
The algebraic equation involving the latus rectum is Proposition 13 of Book I of the Conics of Apollonius of Perga; the consequence concerning ratios, Proposition 21. The equation of triangle and trapezoid is Proposition 43; the consequence concerning change of basis, Proposition 50. The equation of triangles used to derive this consequence is later isolated as Proposition 1 of Book III.
For the record, I made the animations as follows. I drew the still diagrams in LaTeX with the pstricks, pst-3dplot, and pst-eucl packages. I included a parameter for an angle. Then, with the \multido command, I could at once make a pdf file of 360 diagrams, one for each angle, each diagram on a new page. In The Gimp program, I imported the pdf file as layers, then exported as a gif file. (The procedure seems to have changed a bit since “Self-similarity.”)
When I first posted this article, VE* in the 2-d animations was accidentally equal to what is labelled as KL in the 3-d animation. Since one should not assume that this will always be so, I have made it not so.
Still versions of the animations added July 31, 2019.
]]>This is a preliminary report on two recent films:
The report is preliminary, not because there is going to be another, but because I have seen each film only once, and I may see one of them again. I remember that François Truffaut liked to see films at least twice. I would guess that I read this in The Washington Post, in an appreciation published when Truffaut died; however, he died on October 21, 1984, during the first semester of my sophomore year in Santa Fe, when I would not have been reading the Post. While in college, I did enjoy seeing some films twice, or a second time; Truffaut’s own 400 Coups was an example, a French teacher having shown it to us in high school.
The two films that I am reviewing concern young adults trying to find their own way in the world, in defiance of their elders. We all have to do this. In every generation, some will do it more defiantly than others. Heraclitus can be defiant, he of Ephesus and thus one of the Ionian philosophers, whose spirit I imagine to haunt the Nesin Mathematics Village. A further reason to bring up Heraclitus will be—gold.
The Wild Pear Tree is Ahlat Ağacı in Turkish; we watched it on what was probably an “unauthorized copy,” though Ayşe had bought this from one in a publisher’s chain of bookshops in Istanbul (there are outlets also in Ankara and Izmir). We had missed Ceylan’s film in the cinema; however, it was three hours long, and I would have had a hard time without English subtitles. We watched the DVD on three consecutive nights, pausing between sessions after long conversations in the film. In the first of these, young Sinan, who fancies himself a writer, has cornered an author in a bookstore and tagged along with him as he tries to walk home. In the second conversation, Sinan has met up with two young village imams, one of whom he knows from childhood. The imams take conservative and liberal approaches, respectively, to interpreting the Quran. The conservative imam is an authoritarian whose response to a challenge is to make excuses or drop the subject.
We the Coyotes we saw in a cinema during the Istanbul Film Festival, though probably again what we watched was an electronic version. The image was not very bright, perhaps because the projection required more than a strong lamp. Old-fashioned long-playing records are being marketed again, apparently because audiophiles have found that analogue reproductions are better than digital; surely then cinephiles are finding the same thing.
Whether We the Coyotes was seen in North American cinemas on actual film or not, apparently the movie has now been “retitled Anywhere With You for its digital release in the United States and Canada.” The “anywhere” of the new title is Los Angeles, to which two young lovers make a road-trip, hoping to start a new life.
I appreciate seeing what life is like for persons my students’ age. In Coyotes, Amanda’s parents think Jake is a loser. He is hours late to meet Amanda after the LA job she has been counting on falls through. He has good news though: his rapper homeboy has given him some primo medical marijuana. Unfortunately Amanda and Jake cannot stay with the friend, because he is living out of his car.
Back in Illinois, Amanda’s helicopter parents may have driven Amanda to seek freedom with Jake in the first place. We don’t actually see the parents, though we hear the mother over the phone, as Amanda watches Jake frolic in the Pacific Ocean. LA is still a shithole; but the youngsters see potential.
In the other film, there is a wild pear tree growing in a field, above a shithole of a provincial town, near ancient Troy along the Hellespont. Sinan comes home there from university, where he has studied to be a teacher like his father. He has no enthusiasm for teaching though. His friend who could not be a teacher went to the police academy instead; now he beats students when they are detained at demonstrations.
Sinan’s mother has had to take babysitting jobs. Her husband plays the horses and owes gold coins to his friends. Ayşe and I have given gold coins to brides in Turkey, but I had not been aware that people made loans in this form. It does make sense in a country where inflation is unpredictable and religion forbids usury.
Gold coins were invented here, more than a millenium before the religion. The relevant part of the country then was Lydia, where Croesus would reign; and by the account of Herodotus (I.94),
The customs of the Lydians are like those of the Greeks, except that they make prostitutes of their female children. They were the first men whom we know who coined and used gold and silver currency; and they were the first to sell by retail [that is, without having made the product]. And, according to what they themselves say, the games now in use among them and the Greeks were invented by the Lydians.
Does Herodotus make that opening remark with irony? He may well not, but just tell things as he sees them. He says the Lydians invented games—dice, knuckle-bones, and ball, but not draughts—as a way to pass the time during a famine.
Three-hour films are another way to pass the time. When the film is by Nuri Bilge Ceylan, the viewer must be given to quiet contemplation. She must be free of the corruption bemoaned by Pascal when he wrote,
all the unhappiness of men arises from one single fact, that they cannot stay quietly in their own chamber.
One of Ceylan’s first films is called The Clouds of May in English, but in Turkish, Mayıs Sıkıntısı: “The Boredom of May.”
Passing the time is the subject of an essay called “Intentional Obscurity in Ancient Writings,” given as part of the Introduction to Volume IV of the ten Loeb volumes of Hippocrates. According to the words of W. H. S. Jones from 1931, which impressed me when I chanced on them in Santa Fe,
The modern man has perhaps too much to think about, but before books and other forms of mental recreation became common men were led into all sorts of abnormalities and extravagances. The unoccupied mind broods, often becoming fanciful, bizarre, or morbid. To quote but two instances out of many, the “tradition” condemned by Jesus in the Gospels, and the elaborate dogmas expounded at tedious length by the early Fathers, were to some extent at least caused by active brains being deprived of suitable material. It is a tribute to the genius of the Greeks that they found so much healthy occupation in applying thought to everyday things, thus escaping to a great extent the dangers that come when the mind is insufficiently fed.
The Gospel reference is presumably to such passages as in Mark 7, where the Pharisees, “except they wash their hands oft, eat not, holding the tradition (παράδοσις) of the elders.” Jesus tells them,
8 For the laying aside the commandment of God, ye hold the tradition of men, as the washing of pots and cups: and many other such like things ye do.
9 And he said unto them, Full well ye reject the commandment of God, that ye may keep your own tradition.
The commandment is to honor your father and mother; but Jesus accuses the Pharisees of denying material aid to their parents. The Pharisees keep their wealth to themselves, as long as they declare it consecrated to the deity. The Pharisees have missed the point of the Law, as Jesus tells it to those who have ears to hear, in the Antitheses. On the other hand, according to Mark 10,
29 And Jesus answered and said, Verily I say unto you, There is no man that hath left house, or brethren, or sisters, or father, or mother, or wife, or children, or lands, for my sake, and the gospel’s,
30 But he shall receive an hundredfold now in this time, houses, and brethren, and sisters, and mothers, and children, and lands, with persecutions; and in the world to come eternal life.
There is plenty here for young divinity students to debate. Meanwhile, Sinan has to debate with himself whether to share with his importunate father the money that his mother has given him for travelling to the teaching certification exam.
Is it true then, what W. H. S. Jones suggests, that we moderns have too much to think about? In my last post, “Piety,” I questioned a suggestion about modernity by another Loeb translator, Fowler: “Instruction in methods of thinking may perhaps seem needless to modern readers; … however … in Plato’s times it was undoubtedly necessary.” Whether born in 500 BCE or 2000 CE, we all need instruction in thinking, if we will take it. However, nothing in the world will make us think, unless we care to do it.
The Loeb volume of Hippocrates includes the fragments of Heraclitus. According to Jones,
The religious teaching of Heracleitus appears to have been directed against customs and ritual rather than against the immoral legends of Homer and Hesiod.
Such legends include the eating of his children by Cronus, discussed also in “Piety.” Jones extrapolates from words of Heraclitus as in Diels’s Fragment 5, or CXXX in the enumeration of Bywater, which Jones follows:
When defiled they purify themselves with blood, just as if one who had stepped in mud were to wash himself in mud.
Nuri Bilge Ceylan does not happen to show us any blood sacrifices, though they surely occur in the town and village that he depicts. He does show us how, when going to a wedding, the conservative young imam borrows a gold coin from the senile retired imam, who happens to be Sinan’s grandfather.
The Lydian invention of coinage inspired Heraclitus. In his own words, quoted by Plutarch, presented as Fragment 90 of Diels (and XXII of Bywater),
πυρὸς τε ἀνταμοιβὴ τὰ πάντα καὶ πῦρ ἁπάντων ὅκωσπερ χρυσοῦ χρήματα καὶ χρημάτων χρυσός.
For Fire everything is an exchange and Fire for everything, just as for gold, money and for money, gold.
The translation here is by Eva Brann, venerable tutor of St John’s College, who points out that χρήματα is commonly translated as goods, rather than money. When minted coins were a new idea, one might well remark on the possibility of exchanging lumps of gold or silver for a standardized little disc, produced by means of fire, and containing those precious metals in a fixed λόγος or ratio, as the alloy called electrum. This is what I understand from Brann’s meditation called The Logos of Heraclitus (Paul Dry Books, 2011).
In Ceylan’s movie, Sinan has written a meditation on life in his region, though we are given no evidence that anybody else has advised him or read his work. He still tries to get somebody to fund publication. After he begs, borrows and steals the money to publish the book on his own, it is no surprise that he ends up like Thoreau, who said, referring to A Week on the Concord and Merrimack Rivers, his own self-published first book, “I have now a library of nearly nine hundred volumes, over seven hundred of which I wrote myself.”
The Wild Pear Tree portrays the sort of people I have been living among for almost twenty years. When I see Sinan’s younger sister, doing schoolwork as she sits with her mother watching TV, I think this is why some of my own students may not get much done. Other students will follow their own muse, like Sinan.
A hundred generations earlier in these lands (by Eva Brann’s reckoning in the Postscript of her book), Heraclitus followed his own muse.
Ἐδιζησάμην ἐμεωυτόν
I searched myself
—thus reads Diels’s Fragment 101 (Bywater LXXX). The verb here, δίζημαι, is one that Herodotus uses (at VII.142) to describe how the Athenians sought the meaning of an oracle from Delphi concerning the impending Persian invasion. Earlier (at VII.103), not long after his army has crossed the Hellespont, around where Ceylan made his film, Xerxes uses the same verb to tell the exiled Spartan king Demaratus that, if an ordinary Spartan soldier would face ten Persians, he (Xerxes) would look for the king to face twenty.
Even man for man, Xerxes thinks, the Persian army must be stronger than the Spartan; for the Persians fight under compulsion, while the Spartans, being free, can flee.
Not so, says Demaratus. “Free they are,” he says of his compatriots,
yet not wholly free; for law (νόμος) is their master, whom they fear much more than your men fear you. This is my proof—what their law bids them, that they do; and its bidding is ever the same, that they must never flee from the battle before whatsoever odds, but abide at their post and there conquer or die.
Here is the paradox that confounds persons who confuse freedom with unpredictability. The greater the freedom with which the Spartans embrace their law, the greater the certainty that they will obey it.
Mathematics can be done only in freedom: freedom to pursue new thoughts. The thoughts may not be new to the world; they must fit the world, and they can never rest in contradiction with others’ thoughts. This is a discovery of Heraclitus, by Brann’s account:
I think Heraclitus is first, or if not first—for who can ever prove primacy in thought?—at least all on his own, in going within; indeed, derivativeness in this matter would be self-contravening. Moreover, he knows just why this introspection is not an invitation to that inward idiosyncrasy (literally “private temperament”) for which we have the euphemism “subjectivity.” The reason is that there is a two-fold commonality to our thinking: It is both a common capacity of humankind, and it is concerned with what is in its very nature common.
Brann does have to grant that, “in spite of the universality of the encounter and the common capacity, humankind does not listen often.” Amanda’s aunt does not listen: this is why Amanda decides, on the morning of her job interview, that she and Jake can stay no longer with the aunt in LA. As for Sinan, in the end only his father listens to what he has to say in his book.
What Heraclitus has to say is,
Thought is common (ξυνόν) to all. Men must speak with understanding (ξὺν νόῳ) and hold fast to that which is common (τῷ ξυνῷ) to all, as a city holds fast to its law, and much more strongly still. For all human laws are nourished by the one divine law. For it prevails as far as it wills, suffices for all, and there is something to spare.
This is Bywater’s XCI, as translated by Jones, or Diels’s 113 and 114 (from Stobaeus). I insert Greek as Brann does, to highlight the puns that she thinks are intended. In each of its three instances (one plural), “law” here is νόμος, as in the words that Herodotus attributes to Demaratus.
]]>The post below is a way to record a passage in the Euthyphro where Socrates say something true and important about mathematics. The passage is on a list of Platonic passages that I recently found, having written it in a notebook on May 23, 2018. The other passages are in the Republic; here they are, for the record, with some indication of why they are worth noting (translations are Shorey’s, originally from 1930 and 1935 in the old Loeb edition):
Now I turn to the Euthyphro, which is assigned the alternative title On Holiness. The dialogue’s Wikipedia article may reflect its reception in contemporary philosophy departments; my concerns may differ.
Politics may be different, but you cannot come to blows over mathematics. Ayşe points this out in her courtroom statement:
… matematikte kavga edemezseniz, varsa hatayı söylersiniz, karşınızdaki düzeltebilirse düzeltir, düzeltemezse iddiasını geri çeker, ya da sizin hata dediğiniz şey hata değildir, bu sefer de siz hatalı olduğunuzu kabul edersiniz.
… you cannot fight in mathematics: if you see an error, you say so; if the other person can correct it, they do so; if they cannot, they withdraw their claim; or perhaps what you called a mistake was not, and then you accept that you were in error.
Socrates makes a similar point, in the dialogue of Plato called the Euthyphro (7B):
ἆρ᾽ ἂν εἰ διαφεροίμεθα ἐγώ τε καὶ σὺ περὶ ἀριθμοῦ ὁπότερα πλείω, ἡ περὶ τούτων διαφορὰ ἐχθροὺς ἂν ἡμᾶς ποιοῖ καὶ ὀργίζεσθαι ἀλλήλοις, ἢ ἐπὶ λογισμὸν ἐλθόντες περί γε τῶν τοιούτων ταχὺ ἂν ἀπαλλαγεῖμεν;
If you and I were to disagree about number, for instance, which of two numbers were the greater, would the disagreement about these matters make us enemies and make us angry with each other, or should we not quickly settle it by resorting to arithmetic?
“Of course we should,” says Euthyphro. Of course.
The translator here is Harold North Fowler, in the Loeb edition of Plato, Volume I, first published in 1914. (My printing is from 1982; a new edition, with new translators, was published in 2017.) Jowett’s earlier translation (reprinted in the Plato volume of the Great Books of the Western World, 1952) is similar:
Suppose for example that you and I, my good friend, differ about a number; do differences of this sort make us enemies and set us at variance with one another? Do we not go at once to arithmetic, and put an end to them by a sum?
The word sum seems to be Jowett’s interpolation. What Jowett and Fowler translate as arithmetic is in Greek λογισμός, for which the first meaning given by Liddell and Scott is counting; their first quotation is from Thucydides (4.122), about counting days to determine whether a city revolted against the Athenians before or after they concluded an armistice with the Spartans.
The word λογισμός is evidently connected to λόγος and thus to our logic as well as words like epistemology. Further back, there is a connection to the Latin origin of the stem of collecting. This last word itself describes the supposed meaning of the Indo-European root *leg-. A meaning that developed from collecting is speaking, which is part of the meaning of recounting and telling. Another meaning of telling is counting. As William Strunk says, in a passage emphasized by E. B. White (and which nicely uses the subjunctive mood):
Vigorous writing is concise … This requires not that the writer make all sentences short or avoid detail and treat subjects only in outline, but that every word tell.
Every word should tell; every word should count.
We never fight over which of two numbers is greater; we just count them up and see. So says Socrates, and Euthyphro agrees. One may however wonder whether Socrates himself, or Plato, has a counterargument in mind.
Some of us do argue over which of two numbers is greater. They could be numbers of persons attending different American Presidential inaugurations. Arguing may be appropriate here. At least, refusal to agree would be appropriate, when a President asserts, contrary to the evidence, that his inauguration was the biggest ever. Unfortunately his people have parroted the lie, as if accepting the dictum that 2 + 2 = 5.
The rhetorical point of Socrates remains sound. Everybody will agree that there are things that cannot be argued about, because there is a clear method for resolving disputes. Socrates gives also the example of comparing objects by weighing them (7C).
There are however matters over which people contend bitterly (7C–D):
περὶ τίνος δὲ δὴ διενεχθέντες καὶ ἐπὶ τίνα κρίσιν οὐ δυνάμενοι ἀφικέσθαι ἐχθροί γε ἂν ἀλλήλοις εἶμεν καὶ ὀργιζοίμεθα; ἴσως οὐ πρόχειρόν σοί ἐστιν, ἀλλ᾽ ἐμοῦ λέγοντος σκόπει εἰ τάδε ἐστὶ τό τε δίκαιον καὶ τὸ ἄδικον καὶ καλὸν καὶ αἰσχρὸν καὶ ἀγαθὸν καὶ κακόν.
But about what would a disagreement be, which we could not settle and which would cause us to be enemies and be angry with one another? Perhaps you cannot give an answer offhand; but let me suggest it. Is it not about right and wrong, and noble and disgraceful, and good and bad?
Euthyphro agrees with Socrates here. Socrates takes him to have agreed with more: that the topics mentioned are the only ones that cause persons to become enemies. Since the gods are sometimes at odds with one another, as we see in Homer and Hesiod, they must therefore disagree about what is good and bad. Since Euthyphro has said that the holy is dear to the gods, and the unholy not, the same thing can be both holy and unholy.
For Socrates, this is a problem, because Euthyphro has given an affirmative answer to the question (5D),
ἢ οὐ ταὐτόν ἐστιν ἐν πάσῃ πράξει τὸ ὅσιον αὐτὸ αὑτῷ, καὶ τὸ ἀνόσιον αὖ τοῦ μὲν ὁσίου παντὸς ἐναντίον, αὐτὸ δὲ αὑτῷ ὅμοιον καὶ ἔχον μίαν τινὰ ἰδέαν κατὰ τὴν ἀνοσιότητα πᾶν ὅτιπερ ἂν μέλλῃ ἀνόσιον εἶναι;
Is not holiness always the same with itself in every action, and, on the other hand, is not unholiness the opposite of all holiness, always the same with itself and whatever is to be unholy possessing some one characteristic quality?
We should not be so ready to agree. According to Fowler’s Introduction,
The purpose of the [Euthyphro] is in part to inculcate correct methods of thinking, more especially the dialectical method …
Instruction in methods of thinking may perhaps seem needless to modern readers; even they, however, may find it interesting, and in Plato’s times it was undoubtedly necessary.
When recognized as having been written on the eve of what would turn out to be the Great War, such progressive words become naïve or foolish, unless intended ironically. Every child born on this planet will need lessons in thinking, though many of the lessons may not be explicitly delivered. When a young couple spend their time staring silently at their mobiles, like the pair next to me in the food court of the local shopping mall the other day, perhaps their baby is not getting a good lesson.
Does Fowler agree with Socrates that holiness must be some one thing, and unholiness some one other thing? If we are Moderns, we do not agree with Socrates, at least by Collingwood’s account in The New Leviathan (1942); for we have adopted the principle of the limited objective.
31. 62. Ancient sciences aimed at an unlimited objective. They defined their aims by asking questions like: ‘What is Nature?’ ‘What is Man?’ ‘What is Justice?’ ‘What is Virtue?’ A question of this sort was to be answered by a definition of the thing. From this definition, which had to state the ‘essence’ of the thing defined, implications could be derived, each implication being the statement of some ‘property’.
To seek an “essence” was to try to answer a question of the form, “What is x?”
31. 67. To a question in this form, for example: ‘What is Nature?’ modern science answers: ‘I do not know. What the essence of nature is nobody knows, and nobody need care. When they asked that question the Greeks were asking a question too vague to be precisely answered.’
31. 68. Limit your objective. Take time seriously. Aim at interpreting not, as the Greeks did, any and every fact in the natural world, but only those which you think need be interpreted, or can be interpreted (the two things are not, after all, so very different); now, choose where to begin your attack. Select the problems that call for immediate attention. Resolve to let the rest wait.
Socrates’s problem is that Meletus has indicted him for impiety (5C). Euthyphro is indicting his own father for murder: the murder of Euthyphro’s hired man, who got drunk and murdered one of the enslaved men of the family. Euthyphro’s father tied up the killer and threw him in a ditch, then sent to Athens for advice on what to do next. Meanwhile the man died of exposure.
Euthyphro is progressive, like Fowler perhaps. Most persons would not charge kin with murder of non-kin. Socrates is like most persons here, but Euthyphro is not (4B).
It is ridiculous, Socrates, that you think it matters whether the man who was killed was a stranger or relative, and do not see that the only thing to consider is whether the action of the slayer was justified or not, and that if it was justified one ought to let him alone, and if not, one ought to proceed against him, even if he share one’s hearth and eat at one’s table.
Euthyphro recognizes the principle of equality before the law, and he is going to stick with it. On September 24, 2016, I was applauded for quoting the Universal Declaration of Human Rights on this principle, during my talk about Thales in the ruined Roman theater of Miletus. Persons or things can be equal without being the same, and this is important for ancient Greek mathematics, as in some theorems that are attributed to Thales:
For Euthyphro, respecting the principle of equality is piety. It does not matter that his father was himself trying to do the pious thing by sending to Athens for an ἐξηγητής, an exegete, somebody whom Liddell and Scott define as an “expounder, interpreter … at Athens, of sacred rites or customs, modes of burial, expiation, etc., spiritual director.” Cited passages illustrating this meaning are from the Euthyphro itself.
In going after his father, Euthyphro is a man of principle. He has also tradition on his side: the mythical tradition whereby Cronus castrated Uranus (thus giving birth to Aphrodite, as recalled incidentally in “Antitheses”), and Zeus imprisoned Cronus.
The tradition would seem to continue, here in the land from which the Greek gods were “stolen”—by the perverse account of a young tour guide in Antalya, whose ideas may have had some tenuous connection to the Blue Anatolia Thesis (Mavi Anadolu Tezi). Not finding a good source for the Thesis in English, I quote and translate words of Murat Belge (Birikim, October 2006):
Önce “tez”den başlayalım: Halikarnas Balıkçısı’nın (Cevat Şakir) işlediği kadarıyla bu tez büyük ölçüde Batı Anadolu’yla, İonia ile sınırlıdır. Cevat Şakir, İonia ile bugünkü Yunanistan arasında büyük kültürel-entellektüel farklar olduğuna inanmıştır. Ona göre, Tales’ten Demokritos’a maddeci Yunan felsefesi İonia’nın ürünüdür, Anadolu’da yaşayan filozoflar tarafından geliştirilmiştir. Ama bu demokratik felsefe, Yunan karasında Sokrates ve Platon’un elinde idealist ve totaliter bir felsefeye dönüştürülmüştür.
First let’s begin with the “thesis”: as far as the Fisherman of Halicarnassus (Cevat Şakir) worked it out, it is mainly limited to Western Anatolia, to Ionia. Cevat Şakir believed there were great cultural-intellectual differences between Ionia and what is today Greece. According to him, materialist Greek philosophy, from Thales to Democritus, is a product of Ionia, developed by philosophers living in Anatolia. But this democratic philosophy was turned into an idealist, totalitarian philosophy on Greek soil, in the hand of Socrates and Plato.
Freya Stark addresses the same geographical distinction in Ionia: A Quest (1954; Tauris Parke, 2010), which I quoted at greater length towards the end of the Thales article:
Curiosity ought to increase as one gets older … Whatever it was, the Ionian curiosity gave a twist for ever to the rudder of time. It was the attribute of happiness and virtue. To look for the causes of it is a hopeless quest in Greece itself; the miracle appears there, perfect, finished and inexplicable. But in Asia Minor there may be a chance, where Thales of Miletus, ‘having learnt geometry from the Egyptians, was the first to inscribe a right-angled triangle in a circle, whereupon he sacrificed an ox.’
Eleven years after founding the Turkish Republic in Anatolia and Thrace, Mustafa Kemal took the name Atatürk, “Father Turk.” The current ruler of the people called Turks would seem to be working out a damnatio memoriae, as by razing the Atatürk Culture Center on Taksim Square, as well as planning the demolition of Atatürk Airport (if the new Istanbul Airport can be finished).
In the Republic, Socrates rejects the tradition that has come to be called Oedipal (377D–8A):
Hesiod and Homer … methinks, composed false stories which they told and still tell to mankind … There is, first of all, the greatest lie about the things of greatest concernment, which was no pretty invention of him who told how Uranus did what Hesiod says he did to Cronos, and how Cronos in turn took his revenge; and then there are the doings and sufferings at the hands of his son. Even if they were true I should not think that they ought to be thus lightly told to thoughtless young persons.
Euthyphro is a thoughtless young person, who thinks he knows everything worth knowing. Socrates brings him to confusion on this point, just as he (Socrates) is getting ready to be tried for doing this kind of thing to others. It may be safer to teach young persons mathematics.
]]>At the end of Collingwood’s New Leviathan (1942), we reach a chapter whose theme is that of my more recent articles on grammar.
As history, Collingwood’s last chapter is difficult, for the reasons that trouble Herbert Read at the beginning of his Concise History of Modern Painting (revised 1968, augmented 1974). Read opens his first chapter with a passage from Collingwood’s Speculum Mentis (1924):
To the historian accustomed to studying the growth of scientific or philosophical knowledge, the history of art presents a painful and disquieting spectacle, for it seems normally to proceed not forwards but backwards. In science and philosophy successive workers in the same field produce, if they work ordinarily well, an advance; and a retrograde movement always implies some breach of continuity. But in art, a school once established normally deteriorates as it goes on … Whether in large or in little, the equilibrium of the aesthetic life is permanently unstable.
Collingwood’s observation suggests a theme of “Antitheses”: once we identify a rule for living, then we may obey it only in letter, not in spirit.
Herbert Read goes on to observe, in his own voice and then Collingwood’s again, what is also to our present purpose:
So wrote one of the greatest modern philosophers of history and one of the greatest philosophers of art. The same philosopher observed that contemporary history is unwritable because we know so much about it. ‘Contemporary history embarrasses a writer not only because he knows too much, but also because what he knows is too undigested, too unconnected, too atomic. It is only after close and prolonged reflection that we begin to see what was essential and what was important, to see why things happened as they did, and to write history instead of newspapers.’
The people who are the subject of the last chapter of The New Leviathan are occupying the newspapers. Collingwood refers specifically to the writings of German émigrés in Britain that we looked ahead to earlier:
45. 76. There are some wines which, they say, do not travel. The same is true of these samples of modern German politics. On reflection, it seems only natural that an author who has taken part in a long and nerve-racking political battle in one country should arrive in another with his nerves shot to pieces and a determination (very likely unconscious) that his new audience should fully realize how invincible is the man or party or machine that defeated him.
Collingwood does not say that the German émigrés will damp the British war effort; but he suggests it. In his chapter on duty, he said that C, Christ, might discharge the responsibility undertaken by B, the Believer, for the sin of A, Adam (17. 33); but now he observes that some duties are not so transferable:
45. 71. This is because the obligation here in question is not only an obligation that the act should be done, it is an obligation that it should be done by B; if A puts in his oar and takes it upon himself to do it, the result may be (must be, if the, obligation is one of those which admit of being discharged only by one irreplaceable agent) that A does what he is under no obligation to do, and that what B is under obligation to do is not done.
We could take B here to be the British, who need to fight the war, undiscouraged by any of the A, the Alemanni. However, Collingwood does not now exhibit such cleverness with the letters A and B; and anyway, their roles can be reversed. The people B, ruled from Berlin, have to civilize themselves, and aliens A cannot do it for them.
Collingwood recalls an observation made in “Decline of the Classical Politics,” that this classical politics “was unadvisedly and unsuccessfully transplanted into the soil of Prussia by Frederick the Great” (45. 85). Now he suggests that it was not until the time of Bismarck that German barbarism began to form (45. 49). He declines to consider this an inconsistency (45. 87). We are doing history now, which is about things that become, not things that are (45. 22).
The barbarists of the past may seem always to have been so (45. 12); but now, since Germans that we know personally seem not to be innately barbarist (45. 13), the same may be true of everybody whom we have considered barbarist (45. 14). Or say the Germans were always “bad neighbours,” at least latently so (45. 15); we still cannot name conditions that are bound to bring out this barbarism (45. 17).
By Collingwood’s metaphorical account, the Germans were schoolboys, trained in grammar and logic by strict schoolmasters (45. 36). Now they have revolted (45. 37). Says an imaginary Nazi psychologist,
45. 28. ‘And why not? What Nazis call thinking with your blood is a much quicker way of thinking than the old-fashioned way of doing it with your brains.’
45. 29. Certainly, provided that you sometimes have someone with brains at your elbow to check your results and see that they are right; or, failing that, do not care whether they are right or wrong.
45. 3. I am not sure that Nazis understand what logic is for; at any rate they talk as if they were proud of believing a lot of nonsense about it.
45. 31. Everyone who has digested Locke’s Essay knows that it is a great mark of folly to over-estimate the value of logic, or to think that anything can be done with it that cannot be done just as well without it.
45. 32. But the Nazis advocate ‘thinking with your blood’ as if it were a new and revolutionary idea; which it could only be for a generation slavishly taught, in sheer defiance of Locke to think exclusively with their brains.
45. 33. Exclusively, I say; for therein lies the whole difference between thinking like a sane man and thinking like a Nazi.
I have not read Locke’s Essay Concerning Human Understanding (at St John’s College we read only from his Two Treatises of Government). Nothing indeed requires logic, if we understand this—as Collingwood does elsewhere—as a criteriological science, giving an account, or logos, of the standards, or criteria, that we are already wont to apply in our endeavors. I tried to work out the idea of a criteriological science in “A New Kind of Science.”
In “Writing Rules,” I took issue with some academics’ ideas about teaching grammar. Collingwood himself takes issue with Matthew Arnold, who was a school-inspector as well as a writer. We have seen Collingwood’s idea that children ought to be free-ranging and home-schooled. Apparently Arnold would instead send British children to schools like those in Germany, which ended up spawning Nazis (45. 82).
If we want to see for ourselves, Collingwood suggests Arnold’s Friendship’s Garland (1871). The title continues: Being the conversations, letters, and opinions of the late Arminius Baron von Thunder-ten-Tronckh. Arminius would seem to be an imaginary Westphalian nobleman, and Arnold seems to respect his advice for Britain, though Arnold’s true opinion could possibly be hidden in irony. In one reported conversation, Arminius tells Arnold (pp. 52–3),
you were talking of compulsory education, and your common people’s want of it. Now, my dear friend, I want you to understand what this principle of compulsory education really means. It means that to ensure, as far as you can, every man’s being fit for his business in life, you put education as a bar, or condition, between him and what he aims at. The principle is just as good for one class as another, and it is only by applying it impartially that you save its application from being insolent and invidious. Our Prussian peasant stands our compelling him to instruct himself before he may go about his calling, because he sees we believe in instruction, and compel our own class, too, in a way to make it really feel the pressure, to instruct itself before it may go about its calling.
This all sounds sensible. Should not people be qualified for what they do? In that case though, somebody has to assess qualifications, and it is hard to say who this should be. I see the problem in Turkey, where there are good-faith efforts to make admissions and promotions unbiassed and objective; but then, as suggested earlier, once rules for these things are in place, then people can figure out ways around them.
When I was a graduate student, my office-mate from Taiwan was studying for the U.S. naturalization test, but she did not understand trial by jury. Should not experts be doing the judging? I tried to explain that a jury trial was a right, which one could waive if one wanted.
What shall we do? For Collingwood, the arch-barbarists are the Ottoman Turks, when they are at the height of their power (45. 93); but even they “are no exception to the rule which elsewhere, to the best of my knowledge, is unbroken: the rule that barbarists in the end have always been beaten; a rule which I state here merely as the conclusion arrived at by the inductive study of cases” (45. 94). Such an induction is not a proof; to draw the universal conclusion that barbarism always fails, one would have to give a reason—as I suggested in the post “On Gödel’s Incompleteness Theorem.”
All Collingwood is prepared to do is close his book by saying (45. 96),
I think it not wholly without interest to read once more how the professed champions of barbarism, embattling themselves time and again to make an end once for all of the thing we call civilization, have not so much perished at the stroke of lightning from heaven as withered away in the very hour of their victory, or even after it, until those who once feared their rage come first to despise, and then utterly to forget, those who once set themselves up as champions of that which needs no champion, and would not even tolerate a champion if it was the sheer force it pretends to be.
So Collingwood ends his race with death, and so I end this reading of the last book of his that he saw to press. It may be worth while to recall from An Autobiography (1939) that Collingwood learned from his artist-parents
]]>what some critics and aestheticians never know to the end of their lives, that no ‘work of art’ is ever finished, so that in that sense of the phrase there is no such thing as a ‘work of art’ at all. Work ceases upon the picture or manuscript, not because it is finished, but because sending-in day is at hand, or because the printer is clamorous for copy, or because ‘I am sick of working at this thing’ or ‘I can’t see what more I can do to it’.
The last part of Collingwood’s New Leviathan (Oxford, 1942) is “Barbarism.” The first chapter of the part is “What Barbarism Is”; the remaining chapters describe examples of barbarism in turn. The fourth and last example is the one that Britain is fighting as Collingwood writes.
In reviewing Collingwood’s book, we have reached only the third example, in Chapter XLIV. This chapter is called “The Turks.” Such a title needs an explanation or a disclaimer. The Turks in question are those that:
This list of three achievements is from ¶44. 83, which then continues: “at Constantinople they lost their last chance and committed themselves to a career as the sick man of Europe.”
Perhaps instead of Constantinople, Collingwood meant to name Vienna, which the Ottomans went on to besiege, but could not take. On the other hand, in each of the three listed events, the Turks were not exactly victorious over Europe, but only “in a good way to be victorious” (emphasis mine). In none of the events was European civilization destroyed, and at Constantinople the victors lost their momentum: “the Turks were never able, even at the height of their power, to register upon the body of Europe the knock-out blow that they needed for a decisive victory” (44. 84).
By Collingwood’s account, the decisive victory would have been for barbarism. Another kind of victory was won instead, as described in an important passage. Since the Turks “fail[ed] to keep the ice from packing around them,” in the metaphor from ¶41. 63,
44. 85. The result is visible in the Turkey of to-day, a country no longer to be tempted by a recollection of her ancestors’ thievery, but leading an honest and upright life.
44. 86. Those who remember the operations of 1915 and 1916 in the Dardanelles and in Mesopotamia may be glad that the Turks, who were then against us, are now for us.
44. 87. What is the cause of this change? It was because, during the same years in which the Germans turned to thievery, the Turks turned to honest ways.
This seems like a non-answer. It is not. To explain what the people called Turks were and are to Europe, it is no good to refer to material causes like “race,” or weather, or natural resources. The Turks simply were one thing, but now they have changed.
One might name Mustafa Kemal as the one who led Turkey to what Collingwood calls “an honest and upright life.” A general theme of Collingwood is that such changes are never complete. We may note today:
Collingwood’s line of thinking about how people change is found in his first published work, “The Devil” (1916), which we looked at in the context of Chapter XLI. However, the subject in “The Devil” is rather a negative change than the the positive one that we have been considering. The so-called Devil is needless as an hypothesis to explain evil, because
evil neither requires nor admits any explanation whatever. To the question, “Why do people do wrong?” the only answer is, “Because they choose to.” To a mind obsessed by the idea of causation, the idea that everything must be explained by something else, this answer seems inadequate. But action is precisely that which is not caused; the will of a person acting determines itself and is not determined by anything outside itself.
Nonetheless, when some evil purpose requires collective effort, it is likely to fail, because an evil will as such cannot cooperate with another. This is a theme of Collingwood now.
Calling a chapter about barbarism “The Turks” needs a disclaimer, and I have tried to suggest one. Titles like “The Turks” or “The Germans” should be understood to refer, not to an eternal race or nation, but to some people acting more or less together for a particular time. At the beginning of Chapter XLIV, Collingwood himself issues a disclaimer for his whole list of exemplary barbarisms.
“Accidental resemblances” is my term. Collingwood uses the metaphor of “mimicry” in the natural world. A caterpillar may “imitate” a twig (44. 17). I do not know whether Collingwood means to allude to insects’ evolution of camouflage. If birds do not eat twigs, then butterflies may evolve so that their larvae look like twigs. One might attempt a similar materialistic explanation of facts in the human world; however, such an explanation would not be historical. Natural evolution is by inheritance of features that originate by random mutation in gametes. Inherited features may be “selected” for, but the selection is not considered to be an act of will. However, history is about what is willed.
Another disclaimer may be needed. Collingwood is not quite writing history—not in the sense that he described in “Democracy and Aristocracy,” when objecting to revolution as a pseudo-scientific term in history:
26. 79. To stop being surprised when the course of history waggles, and to think of it as waggling all the time; to stop taking sides, and to think of ‘heroes’ and ‘villains’ alike as human beings, partly good and partly bad, whose actions it is your business to understand; this is to be an historian.
If only for rhetorical purposes, Collingwood is now taking sides and considering the Turks of the past as villians.
For Turkish history, Collingwood’s sources are
I don’t suppose the bare story has changed much with further research, especially in the summary form that Collingwood gives. The moralizing tone may have changed; but Collingwood is writing a polemic, in a literal or etymological sense. He aims to contribute to the success of a real war—in Greek, πόλεμος.
Collingwood distinguishes between the Seljuks and the Ottomans, the latter of whom “in the later Middle Ages succeeded, with no very clear title, to the name which the Seljuks had made glorious or nefarious in the earlier
Middle Ages” (44. 2).
I considered the glory of the Seljuk name in “Turks of 1071 and Today.” In the year referred to, the Seljuk leader was chivalrous to the Greek emperor whom the Seljuks had just defeated at Manzikert. However, the Seljuks went on to conquer so much of Asia Minor that, according to Collingwood, the conversion of the inhabitants to Islam must have been forcible (44. 27). When the Seljuks took Jerusalem in 1076, they desecrated the holy places, as far as Christians were concerned (44. 28).
So much for the barbarity of the Seljuks, whose empire was by then “in a state of decay, and the history of the Crusades need not command our attention”—says Collingwood; but I would draw our attention to the Fourth Crusade of 1204, in which Latin Christians turned on their Greek co-religionists, capturing Constantinople and desecrating its holy place (St Sophia).
By Gibbon’s account at the end of his Chapter LVII, what the Seljuks did in Jerusalem was not so bad, compared to what Fatamid caliph al-Hakim had done:
A spirit of native barbarism, or recent zeal, prompted the Turkmans to insult the clergy of every sect; the patriarch was dragged by the hair along the pavement and cast into a dungeon, to extort a ransom from the sympathy of his flock; and the divine worship in the church of the Resurrection was often disturbed by the savage rudeness of its masters. The pathetic tale excited the millions of the West to march under the standard of the Cross to the relief of the Holy Land; and yet how trifling is the sum of these accumulated evils, if compared with the single act of the sacrilege of Hakem, which had been so patiently endured by the Latin Christians!
What al-Hakim had done was to obliterate the holy places, by Gibbon’s account a few pages earlier. Did the Europeans mount a crusade then? No, they only persecuted the Jews:
The temple of the Christian world, the church of the Resurrection, was demolished to its foundations; the luminous prodigy of Easter was interrupted, and much profane labour was exhausted to destroy the cave in the rock, which properly constitutes the holy sepulchre. At the report of this sacrilege, the nations of Europe were astonished and afflicted; but, instead of arming in the defence of the Holy Land, they contented themselves with burning or banishing the Jews, as the secret advisers of the impious Barbarian.
Evidently such details do not fit Collingwood’s rhetorical purpose.
In Anatolia, Christians of one kind or other re-established control over some part of what the Seljuks had earlier taken. Collingwood’s concern is with how the Ottoman Turks brought this land again under Muslim rule. Gibbon apparently blames “the political errors of the Greek emperor” (44. 33). Collingwood finds this analysis flawed, but proposes an alternative, which would seem to be materialistic. The Seljuks had induced (Collingwood does not say how) a decline in population (44. 34; by a misprint, the text reads 43. 34). This decline led to:
“The Turks from the beginning of their history established themselves in a parasitic position relatively to their Christian neighbours” (44. 4). Collingwood’s immediate example is the Janissaries, consisting of Christian captives. They might correspond in general purpose, as well as name, to England’s later New Model Army. Lacking an explicit correlate to “model,” the Turkish term yeniçeri has the direct translation “new army.” However, Collingwood gives the unexplained translation “bright faces.” In 1826 these bright faces “were abolished, characteristically, by massacre” (44. 41).
Collingwood reviews the examples of barbarism so far given.
If the sequence of examples is meant to exhibit a decline or a worsening, then “barbarism as we know it today” would seem to involve utter lack of scruple or trustworthiness.
Collingwood elaborates on the example of Orhan Gazi, son of the founder of the Ottoman (Osmanlı) dynasty:
44. 44. In 1346 Orchan married the Greek princess Irene and became an ally of the Emperor her father; but the alliance was precarious from the start; Orchan looked upon the Greek world merely as so much prospective plunder, and never intended that any treaty he had made with its members should bind him for a moment after it had outlived its utility to himself.
44. 45. If anyone doubted this, he need not wait long for evidence. Before the negotiations for the marriage of Irene were complete, Orchan had been in treaty for the hand of Anne of Savoy, whom he threw over when a richer alliance offered; but in the meantime he used the earlier negotiations as an opportunity for obtaining permission to hold a slave· market at Gallipoli.
44. 46. This permission gave him a formal claim, which he retained in spite of renouncing the intended marriage, to occupy positions on the European side of the Dardanelles. This is how the Turk obtained his foothold in Europe.
Collingwood has made at least one mistake here that seems not to be due to Gibbon: Irene was the wife of John Cantacuzene, who would later be emperor; and Orhan married the couple’s daughter, Theodora. By the account of John Julius Norwich (A Short History of Byzantium, p. 344), the marriage was for love, at least on one side:
Although John deplored the Turks as much as did the rest of his countrymen, on the personal level he had always got along with them remarkably well. With Orhan he quickly established a close friendship, which became yet closer when the Emir fell besottedly in love with Theodora, the second of his three daughters and, in 1346, married her.
The marriage was supposed to be useful for John, who could use the military support of the Turks in order to return to Constantinople as emperor. He was opposed by—Anna of Savoy, widow of the deceased Emperor Andronicus and thus mother of the heir presumptive, John. I find no suggestion (in any of my sources) that Orhan negotiated to marry Anna, except in this sentence of Gibbon’s Chapter LXIV:
By the prospect of a more advantageous treaty, the Turkish prince of Bithynia was detached from his engagements with Anne of Savoy; and the pride of Orchan dictated the most solemn protestations that, if he could obtain the daughter of Cantacuzene, he would invariably fulfil the duties of a subject and a son.
Engagements need not be for somebody’s hand in marriage: this meaning of the term is only labelled 2d in the definition of “Engagement” in the Oxford English Dictionary. There also seems to be no reason for Orhan not to ask of Cantacuzene something else that his rival Anna offered. Here is what Gibbon says presently about that:
In the treaty with the empress Anne, the Ottoman prince had inserted a singular condition, that it should be lawful for him to sell his prisoners at Constantinople or transport them into Asia. A naked crowd of Christians of both sexes and every age, of priests and monks, of matrons and virgins, was exposed in the public market; the whip was frequently used to quicken the charity of redemption; and the indigent Greeks deplored the fate of their brethren, who were led away to the worst evils of temporal and spiritual bondage. Cantacuzene was reduced to subscribe the same terms …
As for the Turkish foothold in Europe, Lord Kinross (The Ottoman Centuries, p. 38) blames it on the Christian mercenaries called the Catalan Grand Army, originally “under the command of a lawless soldier of fortune, Roger de Flor.” The aforementioned Andronicus’s grandfather Andronicus had hired the mercenaries for unspecified purposes; but they became troublesome, and “Roger de Flor was ill-advisedly murdered in the Emperor’s palace,” so the Catalans called in the Turks, “their former enemies,” for help in wreaking revenge.
Collingwood acknowledges the Christians’ share in their own downfall. He blames further success of the Ottomans on:
Collingwood elaborates on the lesson learned too late by Radak:
44. 67. The rules of the game, as understood by the Turks, are that there are no sides; you play, as children call it, all against all. In such a state of things, one player may have a kind of ascendancy over another, such that this other obeys the orders he gives him, strictly speaking they are not orders but what I have called (20. 5) force, and the giving of them I call not the giving of orders but the bringing of force to bear on someone.
We considered the danger of the Albigensian refusal to swear oaths. But perhaps at least the Albigensian heretics would insist that, as Christ enjoined, their every word had the strength of an oath.
I drafted this article while being diagnosed for the sciatic pain that struck on February 11, 2018, and turned out to involve partial foot drop. Printouts from MRI are the background for the book photos. I had surgery with Yunus Aydın on February 19, 2018.
Here is the defense (savunma) of Ayşe Berkman before the 36th Heavy Penalty Court (Ağır Ceza Mahkemesi) of Istanbul, January 10, 2019, against the charge of making propaganda for a terrorist organization (terör örgütü propagandası yapmak).
The crowd from the courtroom when the session was over.
From a tweet of the Peace Academics
The defense is given in two forms:
The English translation is by the defendant and David Pierce, the latter being the owner of this blog. Relevant articles of the blog include
In the courtroom in Çağlayan (conveniently located within a half-hour walk of our flat), Ayşe was one of five persons defending themselves against the same charges. The prosecutor read out the same boilerplate response to each defense. The five defendants will return to court on April 2.
Bianet have Ayşe’s statement, as was announced in a tweet; they have also, together with photographs, beyanlar of many of the Barış Akademisyenleri who have been brought to trial. Statements such as Ayşe’s were also live-tweeted by the Peace Academics.
After the defense.
From Bianet
Sayın Başkan ve Sayın Üyeler,
Your honors,
Barış istediğim için, daha fazla insan ölmesin dediğim için karşınızda bulunuyorum. Aralık 2015 ve Ocak 2016’da acı haberler peş peşe geliyordu. Hiç durmayan sivil ölümleri, suya, gıdaya ve sağlığa erişimin engellenmesi, ölülerin buzluklarda saklanması, Miray bebeğin halasının kucağında bir keskin nişancı tarafından öldürülmesi, Taybet Ana’nın cenazesinin yedi gün boyunca açıkta kalması, cenazeye ulaşmaya çalışan yakınlarının vurulması, okuduğum haberlerden bazılarıydı. Ben de her gün aldığım bu haberlerden dolayı derin bir üzüntü duyuyordum ve ne yapacağımı bilemez perişan bir haldeydim. Bu metni internette görünce belki bir faydası olur umuduyla hemen imzaladım. Bunu yaparken kafamda tek bir amaç vardı: daha fazla insanın ölmesini engellemek.
I stand before you, because I asked for peace and said no more people should die. In December of 2015 and January of 2016, the bad news was unceasing. Civilian deaths, one after the other; the blocking of access to water, food, and medical treatment; the storing of dead bodies in home freezers; a sniper’s killing of Baby Miray in the lap of her aunt; the lying in the open for seven days of Mother Taybet’s corpse; the shooting of relations who tried to reach the body: these are some of the stories that I read. Hearing such news daily, I felt deep sadness; I was in a wretched state, not knowing what I could do. Seeing the text of a petition on the internet, I signed it right way, in hopes that it might do something. I had one idea in mind: to keep more people from dying.
Daha önce hem sizin mahkemenizde, hem de diğer ağır ceza mahkemelerinde aynı iddianame ile yargılanan meslektaşlarım 2015’te yaşananları ayrıntılı olarak anlattıkları için, ben tekrar etmeyeceğim. Ancak Birleşmiş Milletler veya Avrupa Konseyi gibi bağımsız kuruluşlar tarafından yayımlanan raporlardan da maalesef o günlerde aldığımız haberlerin doğru olduğunu öğrendik.
In both your courtroom and others, colleagues charged under the same indictment have explained the details of the events of 2015; I am not going to repeat them. However, from reports published by such independent bodies as the United Nations and the Council of Europe, we have learned that the news we heard in those days was unfortunately correct.
Olayların üzerinden 3 yıl geçmiş olmasına rağmen, bu raporları yanlışlayan hiçbir çalışma yapılmadı. Oysa benim Türkiye Cumhuriyeti devletinden bir vatandaş olarak beklentim, bu raporların doğruluğunu veya yanlışlığını ortaya koyması ve doğruysa da hak ihlallerine neden olan kişileri bulup cezalandırmasıdır.
Even though three years have passed, no attempt has been made to contradict the reports. Nonetheless, as a citizen of the Republic of Turkey, I expect the truth of those reports to be confirmed or refuted; if confirmed, I expect the persons who caused these violations of rights to be found and punished.
Belki matematikçi olduğum için, belki de matematikte de mantığa yakın bir alanda çalıştığım için iddianameyi okurken en çok dikkatimi çeken, iddianamenin izlediği mantıksal yöntem oldu.
Perhaps because I am a mathematician, or because I work in a part of mathematics that is close to logic, as I was reading the indictment, what most drew my attention was the logical pattern that it followed.
Liseden mezun olduktan sonra, Matematik Bölümü’ne girdim ve o gün bugündür yani 30 yıldır hiçbir zaman matematikten kopmadım. Lisans, yüksek lisans ve doktora derecelerimin hepsini matematikte yaptım. Bence matematiğin en güzel yanı, dayanaksız ifadelere yer olmamasıdır, matematikte her söylediğinizi kanıtlamanız gerekir.
After graduating from high school, I entered a mathematics department, and from that day since, for thirty years, I have never been separated from mathematics. My undergraduate, master’s, and doctoral degrees are all in mathematics. For me the most beautiful aspect of mathematics is that unsupported claims have no place. What you say in mathematics, you have to prove.
Ben bu davalar başlayana kadar, hukukta suçlamaların da aynı matematikteki gibi sağlam kanıtlara veya delillere dayandırılması gerektiğini sanıyordum, çünkü eğer bir suçlama yapmak için delil şart değilse, herkes herkes hakkında istediği suçlamayı yapabilir.
Until I started reading the claims against us, I thought that, just as in mathematics, any legal accusations had to be supported by sound evidence and proof; for if evidence were not a condition for making an accusation, then anybody could accuse anyone of anything.
Bize gelen iddianamede delil yerine dayanağı açıklanmamış birçok kişisel yorum yazılmış. İddianameden birkaç örnek vermek istiyorum. “Bildirinin esas amacının … olduğu anlaşılmış”, “dikkatlice incelendiğinde açıkça görülecektir”, “görünürde yasalara uygun ancak esasta yasalara aykırı bir şekilde”, “görünürde A ama özünde B”, gibi yargılarla sonuca giden bir iddianame ile suçlanıyorum.
In the indictment against us, in place of evidence, there are some unexplained personal interpretations. I want to give a few examples from the indictment. “It is understood that the petition’s real aim was …”; “on careful examination it is clearly seen that …”; “in appearance it is legal, but in reality it is illegal”; “in appearance it is A, but in essence it is B”—with such judgments concludes the indictment whereby I am accused.
İkinci dikkatimi çeken nokta, iddianamenin bir varsayım üzerine kurulmuş olmasıdır. Savcıya göre, devleti eleştirmek terör propagandası yapmakla eşdeğerdir, savcı bunu doğru kabul edip, suçlamalar yapmıştır. Oysa, vatandaşlar, devleti eleştirebilir, yanlış gördüğünde daha iyisi yapılsın diye bu yanlışa dikkat çekebilir. Hatta bunların yapılması gereklidir, ancak bu şekilde daha iyiye daha güzele gidebiliriz. Savcı “eleştiri = terör” varsayımı ile hareket ettiği için bizim terör propagandası yaptığımız sonucuna varmıştır. Oysa, yanlış bir varsayımla başlandığında ve mantık kuralları çerçevesinde ilerlendiğinde, doğru veya yanlış hiç fark etmez, bütün cümlelerin kanıtlanabileceği, matematikçiler ve mantıkçılar tarafından bilinen en temel gerçeklerden birisidir. Hukukçuların da bu kuralı bildiklerini düşünüyorum. Bu nedenle, biz matematikte aksiyom adını verdiğimiz varsayımlarımızı seçerken oldukça dikkatli davranırız, aksi takdirde kurduğumuz aksiyom sistemi çöker, çünkü çelişkiler dahil, her cümlenin kanıtlanabildiği hiçbir işe yaramayan bir sistem haline gelir.
The second point that I notice is that the indictment is built on an assumption. According to the prosecutor, criticizing the state is equivalent to making terror propaganda. Accepting this as correct, the prosecutor makes his accusations. However, citizens can criticize the state, and when they see something wrong, they can draw attention to the wrong, in order to correct it. Doing this is even necessary, if we want to be a better country. By setting out with the assumption that criticism equals terror, the prosecutor arrives at the conclusion that we made propaganda for terror. However, if we start with a false assumption and follow the rules of logic, we can prove any statement, whether true or false: this is one of the basic facts known to mathematicians and logicians. I thought jurists would know this rule too. In mathematics, while choosing our assumptions, which we call axioms, we are very careful, since otherwise our axiomatic system will collapse; it will be useless, if we can prove all statements, including contradictions.
Şimdi terör propagandasına temel oluşturan en büyük suçlamaya gelmek istiyorum: Ben Bese Hozat isimli kişiden talimat almışım ve bu talimat doğrultusunda barış bildirisini imzalamışım. Bu suçlamayı önce bir mantıkçı gözü ile irdelemek istiyorum. Bu suçlama bir varlık cümlesidir, yani bir şeyin (bu örnekte talimat almış olduğumun) varlığını iddia etmektedir ancak ispat verilmemektedir. Zaten ispat verilemez, çünkü ben kimseden talimat almadım, ama bunu bir tarafa bırakalım. İddianameyi hazırlayan savcı burada zaten herhangi bir delil göstermeye çalışmıyor, tek dayanağı bu kişi tarafından açıklama yapıldı, ondan sonra bildiri yayınlandı. Yani A, B’den sonra oldu; demek ki A’yı yapanlar B’yi yapandan talimat aldı.
Now I should like to talk about the strongest basis for the terror propaganda accusation. Supposedly I received an order from someone called Bese Hozat, then behaved accordingly and signed the petition. First of all, I should like to analyze this sentence from the perspective of a logician. This is an existential statement; that is, it asserts that something exists (in this example, the receiving of an order); however, no proof is given. Of course it cannot be proven, because I did not take any orders; but let us leave this aside for the moment. The prosecutor who prepared the indictment does not try to show any evidence here; the only basis for their claim is the observation that this person made a declaration, and our petition followed. In other words, A happened after B, and hence those who did A received their orders from the person who did B.
Sayın heyet, mutlaka sizin de bildiğiniz ünlü bir örnek vardır, hatırlatmama izin verin. “Dondurma satışları arttıktan sonra denizde boğulmalar da arttı” biçiminde. Burdan çıkarılacak mantıksal sonuç, elbette dondurma yemenin denizde boğulmaya neden olduğu değildir. Açıkça yaz gelmiş, havalar ısınmış ve insanlar serinlemek için farklı yöntemler uygulamaktadır. Burda en azından iki olay arasında gerektirme olmasa da yine de bir bağ bulabildik. Savcının kurduğu mantığa göre bu bağ “Dondurma yemek gerektirir boğulmalar” olmalıydı ancak bu örnekte doğru analiz “Yazın gelmesi gerektirir dondurma yemek ve boğulma” ’dır.
Your honors, there is a famous example that I am sure you are aware of, but let me recall it here: “After the ice cream sales increased, the number of drownings in the sea also increased.” The logical conclusion that one can derive from this sentence is of course not that eating ice cream leads to drowning. Obviously, the summer came and people use different methods to cool off. The relation between the two events here is not necessitation. According to the logic used by the prosecutor, the relation should have been “eating ice cream entails drowning”; however, in this example, the correct conclusion is, “summer entails eating ice cream and drowning”.
Çeşitli verilerin taranması ile aralarında herhangi bir gerektirme ilişkisi olmayan olaylar bile birbirini gerektiriyormuş gibi görülebilir. Bu konuda İngilizce’de istatistikçilerin kullandığı güzel bir söz vardır “correlation is not causation” derler, yani “korelasyon gerektirme değildir” ya da daha serbest çeviri ile her ilişki nedensellik ilişkisi değildir.
In fact, by studying various data, events that have nothing in common may seem to entail one another. Statisticians have a nice expression for this, they say “Correlation is not causation.” We can restate it as “not every relation is a relation of necessitation.”
Dolayısı ile bana yöneltimiş olan “talimat alma” suçlamasının dayanağı bir mantıkçının veya bir istatistikçinin kabul edebileceği bir dayanak değildir. Kanıtsızdır, kanıtlanması da mümkün değildir. Herhangi bir delile dayanmayan suçlamaların hukukta bir değerinin olmasına çok şaşırdığımı bir kere daha ifade etmek isterim.
As a result, the basis of the accusation of taking orders is not the kind that a logician or a statistician can accept. It has no proof, and it is impossible to prove anyway. I should like to point out once more that I am surprised that an accusation with no basis has a value in law.
Bu konu ile ilgili olarak söylemek istediğim diğer husus, beni bir şeyin varlığı ile suçlayan iddianameye karşı elbette yapılabilecek en doğru savunma, suçlamanın yanlış olduğunu kanıtlamaktır, bu da suçlamanın değilini kanıtlamaya denktir. Yani benim beraat edebilmem için Bese Hozat isimli kişiden talimat almadığımı kanıtlamam gerekmektedir. Keşke bu matematiksel bir cümle olsaydı, çünkü matematikte bir şeyin olmadığını kanıtlamanın yolları vardır ancak yaşamla, dünyayla veya uzayla ilgili cümlelerde bir şeyin yokluğunu kanıtlamak genellikle mümkün değildir. Bu felsefede bilinen bir husustur ve bunu vurgulayan en güzel örneklerden birini ünlü mantıkçı ve aynı zamanda uluslararası savaş suçları mahkemesini kurmuş olan Bertrand Russell vermiştir. Russell yanlış olmasına rağmen yanlışlığı kanıtlanamayacak cümlelere çarpıcı bir örnek olarak “Dünya ve Mars arasında, eliptik bir yörüngede güneşin etrafında dönen porselen bir çaydanlık vardır.” cümlesini vermiştir. Bu cümle ile benim Bese Hozat’tan talimat almış olduğum cümlesi mantıksal olarak aynı kategoridedir. İşte iddianame beni yüzyıllardır felsefeciler, mantıkçılar ve matematikçiler tarafından çok iyi bilinen bir açmaza bu şekilde düşürmektedir.
Here is another point that I should like to make about this issue. The best defence against an indictment that makes an accusation against me is of course to show that this accusation is false, which is equivalent to proving the negation of the accusation. In other words, to be acquitted, I have to prove that I have not received an order from the person called Bese Hozat. I wish this were a mathematical statement, because in mathematics there are ways to prove non-existence; however, in general, it is impossible to prove a statement of non-existence, if it is about life, earth, or space. This is a well-known fact in philosophy, and one of the best examples was given by the famous logician Bertrand Russell, who happens to be the founder of the International War Crimes Tribunal as well. The striking example given by Russell of a false statement that is impossible to disprove is, “Between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit.” This sentence and the claim that I took an order from Bese Hozat fall into the same logical category. You see how the indictment puts me in an impasse that has been known to philosophers, logicians, and mathematicians for over a century.
İddianame ile ilgili olarak irdelemek istediğim son nokta, iddianamedeki “hiçbir ülkede terör propagandasına izin verilmeyeceği” iddiası. Savcının terör propagandası tanımını devleti eleştirmek olarak alırsak, bunu yanlışlayan birçok örnek vardır. Örneğin, aralarında Jean Paul Sartre ve André Breton’un da olduğu bir grup aydın tarafından, 1960 yılında Fransız ordusunun Cezayir’de yaptığı işkenceleri ve insan hakları ihlallerini kınayan, bu savaşı caniyane ve saçma olarak nitelendiren ve Fransızları orduya itaatsizliğe davet eden 121’ler Manifestosu yayımlanmıştır. Bildiri o zaman Fransa’da büyük tepki çekmiş, gazetelerde basılması yasaklanmış, birkaç imzacı öğretim üyesinin derslerine ara verilmiş ve 121 imzacıdan 29’u hakkında iddianame hazırlanmışsa da, iddianame işleme konmamış, dolayısı ile hiçbir imzacı ceza almamıştır.
The last point that I should like to examine in the indictment is the claim that no country will allow terror propaganda. If we use the prosecutor’s definition of terror propaganda as criticising the state, there are many counterexamples. For instance, in 1960 the “Manifesto of the 121” was announced in France by a group of intellectuals, including Jean Paul Sartre and André Breton, condemning the tortures and human-rights violations committed by the French army in Algeria. They called the war criminal and absurd, and they invited the French people to disobey the army. There was a big reaction to the manifesto in France; its printing in newspapers was banned, and the lectures of a few university professors who signed the petition were suspended. Even though indictments were prepared for 29 signers out of 121, in the end they were not issued; hence, none of the signers were prosecuted.
İkinci örnek, Amerika Birleşik Devletleri’nden. Vietnam Savaşı sırasında birçok Amerikan üniversitesinde savaşı protesto eden gösteriler düzenlenmiş ve üniversiteler savaş karşıtlığının merkezi haline gelmiştir. Gösterilerde şiddet kullananlar tutuklanmıştır ancak savaş karşıtı görüşlerini dile getirdiği için hakkında dava açılan bir akademisyen olmamıştır. Çarpıcı bir örnek olarak, Harvard Üniversitesi Senatosu yayımladığı bildiride “barışın tesisi için ABD askeri birliklerinin derhal Vietnam’ı terk ederek ülkeye dönmelerini” istemiştir.
My second example is from the United States. During the Vietnam War, American universities became the centers of anti-war movements, and many protests took place. Protestors who were violent in the events were arrested; however, not a single academic was prosecuted for making anti-war statements. One of the striking examples from those days is the declaration made by the Harvard University Senate saying, “the most reasonable plan for peace is the prompt, rapid, and complete withdrawal of all US forces. We support a united and sustained national effort to bring our troops home.”
Vereceğim son örnek İsrail’den. Altı ay kadar önce İsrail Ulus Devlet Yasası’nın kabulünün ardından, birçok aydın, yazar, sanatçı ve akademisyen yasayı protesto eden çeşitli bildiriler yayınlamıştır. Bunlardan bir tanesi, devletlerinin Filistinlilere karşı etnik temizlik yaptığını, köyleri yakıp yıktığını, İsrail mahkemelerinin ise bu yapılanları yasal bulduğunu söyleyerek, çok geç olmadan uluslararası kuruluşları yardıma çağırmıştır. İsrail’de de hiçbir imzacıya dava açılmamıştır.
My last example is from Israel. After the acceptance of the Israeli Nation-State Law about six months ago, many intellectuals, writers, artists, and academics published various declarations protesting the law. In one of these declarations, signers said that their state is doing ethnic cleansing against Palestinians, and Israeli courts are legitimizing the destruction of entire villages; hence they called for external pressure before it is too late. No signer was prosecuted in Israel either.
Önemli olaylar karşısında, siyasetçilerle aydınların fikir ayrılığına düşmesi ve aydınların sorumluluk alarak, siyasetçileri uyarması tarihte sıklıkla rastlanan bir durumdur. Burada yalnızca üç örneğe değinebildim.
During important events, it frequently happens that politicians and intellectuals have disagreements, and intellectuals feel responsible and warn the politicians. Here, I give only three examples.
Sayın Heyet, 20 yıldır üniversite öğrencilerine matematik dersleri veriyorum. Sanırım bu süre içerisinde, toplam 3000 civarında öğrenci benden ders almıştır. Verdiğim derslerde ya da özel çalışmalarımızda, öğrencilerime elbette dersin ya da çalışmanın konusu ile ilgili teoremleri, bunların kanıtlarını veya problem çözme yöntemleri gibi matematiğin teknik taraflarını anlattım, ama aynı zamanda onlara özelde matematik, genelde de bilim kültürünü aşılamaya çalıştım. Bir bilim insanının kavramlara nasıl yaklaştığını, problemleri nasıl analiz ettiğini, bazen sorunun zorluğu karşısında herkesin bocalayabileceğini ama yine de sorunun peşini bırakmaması gerektiğini hem bende gözlemlesinler hem de yaşayarak öğrensinler istedim.
Your honors, I have been teaching mathematics to university students for twenty years. In this period, I suppose about three thousand students have taken courses from me. In lectures and in office hours, of course I explain the technical side of mathematics, such as the theorems of our topic, their proofs, and the methods of solving problems; but at the same time, I try to engender the culture of mathematics in particular and science in general. I want students both to observe in me, and to experience for themselves, how a scientist approaches concepts, analyzes problems, and must not break off the pursuit of a problem, though its difficulty may leave everybody not knowing which way to go.
Ancak onlara vermeye çalıştığım, üstelik de tüm bunlardan daha fazla önem verdiğim bir konu daha vardı. O da kısaca eleştirel yaklaşım olarak nitelendirebileceğim alışkanlığı edinmelerini istedim. Bu elbette yalnızca bilim insanlarında olması gereken bir özellik değil, her bireyde olmalıdır. Eleştirel yaklaşım ile kastettiğim, öncelikle kişinin kendi aklına güvenmesi, aklını bir otoritenin emrine sunmak yerine, düşünmeye üşenmemesi ve gördüğü, duyduğu bilgileri fikirleri sorgulayarak kendi akıl süzgecinden geçirdikten sonra onları kabul etmesi veya uygulamaya koymasıdır. Bu tip bir yaklaşım, herhangi bir kişiye, otoriteye, hocaya veya ders kitabına sorgulamadan inanmamayı gerektirir. Kişiler yanılabilir, kitaplar hatalı bilgi içerebilir, belli koşullarda doğru olan bir bilgi, koşullar değişince yanlış olabilir. Bu farklılıkları anlayabilmek ve doğruyu bulabilmek için elimizde kendi aklımızdan başka kullanabileceğimiz bir ölçü yoktur.
But there is one more thing that I try to teach students, something to which I give more importance than all that I have said. I want them to make a habit of what I can call, briefly, the critical approach. This is something needed not only by scientists, but by everybody. What I mean by the critical approach is, first of all, trusting one’s own mind, instead of giving it over to the command of an authority; not giving up on thinking, and, by investigating the information and ideas that one sees and hears, accepting them or putting them to work only after passing them through the filter of one’s own mind. In this approach, one must believe nobody, no authority, no teacher, no textbook, without investigation. People can be mistaken, books can contain misinformation, something correct under certain conditions can be wrong otherwise. To understand the differences and find the truth, there is no measure we can use, but our own mind.
Ancak öğrenciye ya da herhangi bir kimseye, “her şeye eleştirel yaklaşın”, “aklınızı kullanın” gibi sözler söyleyerek bu alışkanlığı kazandırmamız mümkün değildir. Ne şanslıyız ki, matematik anlatırken sözünü ettiğim noktaları vurgulamak ve uygulamak için karşımıza birçok fırsat çıkar.
However, we cannot habituate our students, just by saying words like “Approach everything critically,” or “Use your mind.” Fortunately, in explaining mathematics, quite a few opportunities arise to apply and emphasize the points that I have made.
Matematikte yanlış bir cümleyi ya da kanıtı kimse kimseye zorla kabul ettiremez. “Talimatla teoremi kabul etmek” diye bir şey yoktur. Her iddianızı kanıtlamanız gerekir. Aslında kanıtlamak da yetmez, karşınızdakinin de ikna olması gerekir. İkna olmamışsa, karşınızdaki size anlamadığı yeri sorar, orayı açıklamanız gerekir. Bu sefer yeni bir soru gelebilir, onu da açıklamakla yükümlüsünüz. Bu nedenle, matematikte kavga edemezseniz, varsa hatayı söylersiniz, karşınızdaki düzeltebilirse düzletir, düzeltemezse iddiasını geri çeker, ya da sizin hata dediğiniz şey hata değildir, bu sefer de siz hatalı olduğunuzu kabul edersiniz. Bu tartışmalar elbette ancak ifade özgürlüğünün olduğu bir ortamda yapılabilir. İdeal bir üniversitede, birinci sınıf öğrencisi, bir profesör’ün hata yaptığını fark ettiğinde bunu rahatlıkla ifade edebilir; herkes herkesle tartışabilir. Üniversite herkesin doğru bilgiyi aradığı yerdir.
In mathematics, nobody can make anybody accept a false statement or proof. There is nothing like “command acceptance of a theorem.” We have to prove our claims. Actually, proving them is not enough, but other persons have to be convinced. If they are not convinced, they ask about the points they do not understand, and you have to explain them. This time new questions may arise, and you have the burden of answering them. For this reason, you cannot fight in mathematics: if you see an error, you say so; if the other person can correct it, they do so; if they cannot, they withdraw their claim; or perhaps what you called a mistake was not, and then you accept that you were in error. Of course, such a discussion can happen only in an environment where there is freedom of expression. In the ideal university, if even a first-year student notices that a professor is in error, they can comfortably say so; anybody can dispute with anybody. The university is where everybody is looking for the truth.
Akıl, bilimsel tutum, dürüstlük, düşünce ve ifade özgürlüğü yalnızca matematiğin değil akademinin olmazsa olmazlarıdır. Yeni fikirler ancak özgür beyinlerden, özgür ortamlarda doğar. Ülke olarak daha ileri gitmek istiyorsak, her alanda düşünce ve ifade özgürlüğünü ödünsüz savunmalıyız.
Reason, scientific conduct, honesty, freedom of thought and expression: they are the sine qua non, not only of mathematics, but of the whole of academia. New ideas are born only in free minds, in a free environment. If we want to advance as a country, we must defend freedom of thought and expression everywhere, without compromise.
Sonuç olarak, aklı ve akılcı düşünceyi yaşamının merkezine koymuş biri olarak, “talimatla imza attığım” suçlamasını şahsıma yapılmış bir aşağılama ve hakaret olarak görüyorum. Bu hakareti reddediyorum.
To sum up my words, I shall say that I consider this accusation of “signing on orders” as the biggest denigration and insult that can be made to a person who puts reason and rational thought at the center of her life. I reject this insult.
Yalnızca bir bilim insanına ya da bir akademisyene değil, reşit olan herhangi bir bireye talimat aldığını söylemek bence en büyük hakarettir. Elbette başkalarından fikir ve görüş alınabilir ama kişi son kararını kendi akıl ve vicdan süzgecinden geçirdikten sonra vermelidir. Özellikle de topluma karşı sorumluluğu olan meslek gruplarına mensup kişiler: örneğin adaletin tesisini sağlayan siz hakimler, savcılar, avukatlar; sağlığımızı emanet ettiğimiz doktorlar, gelecek kuşakları yetiştiren öğretmenler ve daha niceleri.
Not only to a scientist or an academic, but to any adult, claiming that they are taking orders is the biggest insult. Of course, people can ask others’ opinions and thoughts, but they should make their decisions after filtering through their own minds and consciences—especially people who are responsible to society through their professions, such as judges, prosecutors, and lawyers like you, who administer justice; doctors, in whose hands we put our lives; teachers, who raise the next generations; and many others.
Doğrudan öğrencisi olmasam da, öğrencilerinin ve meslektaşlarının öğrencisi olduğum, Türkiye’deki akademik dünyaya hem doğrudan hem de dolaylı olarak büyük katkılar yapmış Cahit Arf Hocamız’ın adını burada saygı ve sevgi ile anmak isterim. Cahit Arf, 70’li yılların sonunda ODTÜ’de Dekanlık yaparken Genelkurmay Başkanı’nın talimatlarını kabul etmemiş ve hem üniversiteyi hem de öğrencilerini baskılara karşı korumuş birisidir.
At this point, with love and respect, I should like to mention our teacher Cahit Arf, who contributed a lot to the academic world of Turkey both directly and indirectly. Even though I was not his student, I was a student of his students and colleagues. In the late 1970s, when Cahit Arf was a dean at METU, he did not agree to follow the orders of the Chief of the General Staff, and he protected the university and his students against pressures.
Sözlerime son vermeden önce, yeniden bildiriye dönmek istiyorum. Bildiride barış müzakerelerine dönme çağrısı yapılmaktadır. Terör propagandası yapan bir metin barış çağrısı yapar mı, bunun yanıtını sizin takdirinize bırakıyorum. Ayrıca bildiride müzakereler başladığında, görüşmelerde gönüllü gözlemci olarak yer almak istediğimiz yazılıdır. Dolayısı ile barışın yeniden tesisinde elimizden gelen katkıyı sunmak istediğimiz açıkça ortadayken terörizm propagandası ile suçlanmamız da ayrı bir ironidir. Bildiri eğer bir şeyin propagandasını yapıyorsa o da barışın propagandasıdır, asla ve asla terörün değil. Hakaret içermemektedir, hepimiz için gerekli olan ve hepimizin sahip çıkması gereken ifade özgürlüğü sınırları içindedir. Vatandaş sorumluluğu ile imzalanmıştır.
Before I finish my statement, I should like to say a few more words about our petition. In the petition we make a call to resume the peace negotiations. Does a text which makes terrorism propaganda also make a call for peace? I leave it to you to decide. Also we mentioned in the petition that if the peace negotiations start, then we will volunteer to be observers in the process. Hence it is another irony that while we are openly declaring that we should like to give as much support as we can for achieving peace again, we are being accused of making propaganda for terrorism. If the petition is making propaganda of something, then it is for peace; never ever for terrorism. It contains no insults, and it is within the bounds of freedom of speech, which we all need and should all defend. I signed the petition as a responsible citizen.
Bu nedenlerle beraatimi talep ediyorum.
For all these reasons, I ask for my acquittal.
10 Ocak 2019
January 10, 2019
Your honors,
I stand before you, because I asked for peace and said no more people should die. In December of 2015 and January of 2016, the bad news was unceasing. Civilian deaths, one after the other; the blocking of access to water, food, and medical treatment; the storing of dead bodies in home freezers; a sniper’s killing of Baby Miray in the lap of her aunt; the lying in the open for seven days of Mother Taybet’s corpse; the shooting of relations who tried to reach the body: these are some of the stories that I read. Hearing such news daily, I felt deep sadness; I was in a wretched state, not knowing what I could do. Seeing the text of a petition on the internet, I signed it right way, in hopes that it might do something. I had one idea in mind: to keep more people from dying.
In both your courtroom and others, colleagues charged under the same indictment have explained the details of the events of 2015; I am not going to repeat them. However, from reports published by such independent bodies as the United Nations and the Council of Europe, we have learned that the news we heard in those days was unfortunately correct.
Even though three years have passed, no attempt has been made to contradict the reports. Nonetheless, as a citizen of the Republic of Turkey, I expect the truth of those reports to be confirmed or refuted; if confirmed, I expect the persons who caused these violations of rights to be found and punished.
Perhaps because I am a mathematician, or because I work in a part of mathematics that is close to logic, as I was reading the indictment, what most drew my attention was the logical pattern that it followed.
After graduating from high school, I entered a mathematics department, and from that day since, for thirty years, I have never been separated from mathematics. My undergraduate, master’s, and doctoral degrees are all in mathematics. For me the most beautiful aspect of mathematics is that unsupported claims have no place. What you say in mathematics, you have to prove.
Until I started reading the claims against us, I thought that, just as in mathematics, any legal accusations had to be supported by sound evidence and proof; for if evidence were not a condition for making an accusation, then anybody could accuse anyone of anything.
In the indictment against us, in place of evidence, there are some unexplained personal interpretations. I want to give a few examples from the indictment. “It is understood that the petition’s real aim was …”; “on careful examination it is clearly seen that …”; “in appearance it is legal, but in reality it is illegal”; “in appearance it is A, but in essence it is B”—with such judgments concludes the indictment whereby I am accused.
The second point that I notice is that the indictment is built on an assumption. According to the prosecutor, criticizing the state is equivalent to making terror propaganda. Accepting this as correct, the prosecutor makes his accusations. However, citizens can criticize the state, and when they see something wrong, they can draw attention to the wrong, in order to correct it. Doing this is even necessary, if we want to be a better country. By setting out with the assumption that criticism equals terror, the prosecutor arrives at the conclusion that we made propaganda for terror. However, if we start with a false assumption and follow the rules of logic, we can prove any statement, whether true or false: this is one of the basic facts known to mathematicians and logicians. I thought jurists would know this rule too. In mathematics, while choosing our assumptions, which we call axioms, we are very careful, since otherwise our axiomatic system will collapse; it will be useless, if we can prove all statements, including contradictions.
Now I should like to talk about the strongest basis for the terror propaganda accusation. Supposedly I received an order from someone called Bese Hozat, then behaved accordingly and signed the petition. First of all, I should like to analyze this sentence from the perspective of a logician. This is an existential statement; that is, it asserts that something exists (in this example, the receiving of an order); however, no proof is given. Of course it cannot be proven, because I did not take any orders; but let us leave this aside for the moment. The prosecutor who prepared the indictment does not try to show any evidence here; the only basis for their claim is the observation that this person made a declaration, and our petition followed. In other words, A happened after B, and hence those who did A received their orders from the person who did B.
Your honors, there is a famous example that I am sure you are aware of, but let me recall it here: “After the ice cream sales increased, the number of drownings in the sea also increased.” The logical conclusion that one can derive from this sentence is of course not that eating ice cream leads to drowning. Obviously, the summer came and people use different methods to cool off. The relation between the two events here is not necessitation. According to the logic used by the prosecutor, the relation should have been “eating ice cream entails drowning”; however, in this example, the correct conclusion is, “summer entails eating ice cream and drowning”.
In fact, by studying various data, events that have nothing in common may seem to entail one another. Statisticians have a nice expression for this, they say “Correlation is not causation.” We can restate it as “not every relation is a relation of necessitation.”
As a result, the basis of the accusation of taking orders is not the kind that a logician or a statistician can accept. It has no proof, and it is impossible to prove anyway. I should like to point out once more that I am surprised that an accusation with no basis has a value in law.
Here is another point that I should like to make about this issue. The best defence against an indictment that makes an accusation against me is of course to show that this accusation is false, which is equivalent to proving the negation of the accusation. In other words, to be acquitted, I have to prove that I have not received an order from the person called Bese Hozat. I wish this were a mathematical statement, because in mathematics there are ways to prove non-existence; however, in general, it is impossible to prove a statement of non-existence, if it is about life, earth, or space. This is a well-known fact in philosophy, and one of the best examples was given by the famous logician Bertrand Russell, who happens to be the founder of the International War Crimes Tribunal as well. The striking example given by Russell of a false statement that is impossible to disprove is, “Between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit.” This sentence and the claim that I took an order from Bese Hozat fall into the same logical category. You see how the indictment puts me in an impasse that has been known to philosophers, logicians, and mathematicians for over a century.
The last point that I should like to examine in the indictment is the claim that no country will allow terror propaganda. If we use the prosecutor’s definition of terror propaganda as criticising the state, there are many counterexamples. For instance, in 1960 the “Manifesto of the 121” was announced in France by a group of intellectuals, including Jean Paul Sartre and André Breton, condemning the tortures and human-rights violations committed by the French army in Algeria. They called the war criminal and absurd, and they invited the French people to disobey the army. There was a big reaction to the manifesto in France; its printing in newspapers was banned, and the lectures of a few university professors who signed the petition were suspended. Even though indictments were prepared for 29 signers out of 121, in the end they were not issued; hence, none of the signers were prosecuted.
My second example is from the United States. During the Vietnam War, American universities became the centers of anti-war movements, and many protests took place. Protestors who were violent in the events were arrested; however, not a single academic was prosecuted for making anti-war statements. One of the striking examples from those days is the declaration made by the Harvard University Senate saying, “the most reasonable plan for peace is the prompt, rapid, and complete withdrawal of all US forces. We support a united and sustained national effort to bring our troops home.”
My last example is from Israel. After the acceptance of the Israeli Nation-State Law about six months ago, many intellectuals, writers, artists, and academics published various declarations protesting the law. In one of these declarations, signers said that their state is doing ethnic cleansing against Palestinians, and Israeli courts are legitimizing the destruction of entire villages; hence they called for external pressure before it is too late. No signer was prosecuted in Israel either.
During important events, it frequently happens that politicians and intellectuals have disagreements, and intellectuals feel responsible and warn the politicians. Here, I give only three examples.
Your honors, I have been teaching mathematics to university students for twenty years. In this period, I suppose about three thousand students have taken courses from me. In lectures and in office hours, of course I explain the technical side of mathematics, such as the theorems of our topic, their proofs, and the methods of solving problems; but at the same time, I try to engender the culture of mathematics in particular and science in general. I want students both to observe in me, and to experience for themselves, how a scientist approaches concepts, analyzes problems, and must not break off the pursuit of a problem, though its difficulty may leave everybody not knowing which way to go.
But there is one more thing that I try to teach students, something to which I give more importance than all that I have said. I want them to make a habit of what I can call, briefly, the critical approach. This is something needed not only by scientists, but by everybody. What I mean by the critical approach is, first of all, trusting one’s own mind, instead of giving it over to the command of an authority; not giving up on thinking, and, by investigating the information and ideas that one sees and hears, accepting them or putting them to work only after passing them through the filter of one’s own mind. In this approach, one must believe nobody, no authority, no teacher, no textbook, without investigation. People can be mistaken, books can contain misinformation, something correct under certain conditions can be wrong otherwise. To understand the differences and find the truth, there is no measure we can use, but our own mind.
However, we cannot habituate our students, just by saying words like “Approach everything critically,” or “Use your mind.” Fortunately, in explaining mathematics, quite a few opportunities arise to apply and emphasize the points that I have made.
In mathematics, nobody can make anybody accept a false statement or proof. There is nothing like “command acceptance of a theorem.” We have to prove our claims. Actually, proving them is not enough, but other persons have to be convinced. If they are not convinced, they ask about the points they do not understand, and you have to explain them. This time new questions may arise, and you have the burden of answering them. For this reason, you cannot fight in mathematics: if you see an error, you say so; if the other person can correct it, they do so; if they cannot, they withdraw their claim; or perhaps what you called a mistake was not, and then you accept that you were in error. Of course, such a discussion can happen only in an environment where there is freedom of expression. In the ideal university, if even a first-year student notices that a professor is in error, they can comfortably say so; anybody can dispute with anybody. The university is where everybody is looking for the truth.
Reason, scientific conduct, honesty, freedom of thought and expression: they are the sine qua non, not only of mathematics, but of the whole of academia. New ideas are born only in free minds, in a free environment. If we want to advance as a country, we must defend freedom of thought and expression everywhere, without compromise.
To sum up my words, I shall say that I consider this accusation of “signing on orders” as the biggest denigration and insult that can be made to a person who puts reason and rational thought at the center of her life. I reject this insult.
Not only to a scientist or an academic, but to any adult, claiming that they are taking orders is the biggest insult. Of course, people can ask others’ opinions and thoughts, but they should make their decisions after filtering through their own minds and consciences—especially people who are responsible to society through their professions, such as judges, prosecutors, and lawyers like you, who administer justice; doctors, in whose hands we put our lives; teachers, who raise the next generations; and many others.
At this point, with love and respect, I should like to mention our teacher Cahit Arf, who contributed a lot to the academic world of Turkey both directly and indirectly. Even though I was not his student, I was a student of his students and colleagues. In the late 1970s, when Cahit Arf was a dean at METU, he did not agree to follow the orders of the Chief of the General Staff, and he protected the university and his students against pressures.
Before I finish my statement, I should like to say a few more words about our petition. In the petition we make a call to resume the peace negotiations. Does a text which makes terrorism propaganda also make a call for peace? I leave it to you to decide. Also we mentioned in the petition that if the peace negotiations start, then we will volunteer to be observers in the process. Hence it is another irony that while we are openly declaring that we should like to give as much support as we can for achieving peace again, we are being accused of making propaganda for terrorism. If the petition is making propaganda of something, then it is for peace; never ever for terrorism. It contains no insults, and it is within the bounds of freedom of speech, which we all need and should all defend. I signed the petition as a responsible citizen.
For all these reasons, I ask for my acquittal.
January 10, 2019
In my 1997 doctoral dissertation, the main idea came as I was lying in bed one Sunday morning. I write about this now, because an email friend expressed curiosity. I am also testing the assertion of a mathematician friend:
You may revisit some corner [of mathematics] after being away for 30 years, and discover that everything there is the same as it was when you left it.
What was a correct mathematical argument, when I completed my dissertation, is correct now. This has to do with the universality of mathematics, which I described, though not by that term, in “Some Say Poetry.”
Our right to decide what is mathematically correct is inseparable from our responsibility to find agreement with interested others. Here then is an antithesis. What I thought right in the 1990s, I am likely to find right still, because my work was checked, and in some cases corrected, by me, an advisor, an examining committee, and a referee, in the manner also discussed in “Antitheses.”
Nonetheless, as Aristotle says in the second sentence of the Physics,
πέφυκε δὲ ἐκ τῶν γνωριμωτέρων ἡμῖν ἡ ὁδὸς καὶ σαφεστέρων
ἐπὶ τὰ σαφέστερα τῇ φύσει καὶ γνωριμώτερα
The natural way is from what is clearest and most knowable to us
to what is clearest and most knowable by nature.
This is consonant with Alexandre Borovik’s observation in another recent blog post:
This is the Catch-22 of learning mathematics: only at the next stage of learning it becomes possible to tell whether the learner mastered the previous stage.
What we have known may still become clearer when we look back at it.
Art and physics are academic fields; but unlike mathematics, they produce work that anybody can enjoy. With art this is obvious; with physics, there are products like the photographs of nebulae that Christopher Hitchens commends to our attention, in the quotation from God Is Not Great that I made at the end of my last post, on Gödel’s Incompleteness Theorem.
Mathematics can give you beautiful images, as of fractals like the Mandelbrot set. However, these are not what most of mathematics is about. I appreciate how Timothy Gowers declines to share such images in A Very Short Introduction to Mathematics (2002):
Very little prior knowledge is needed to read this book—a British GCSE course or its equivalent should be enough—but I do presuppose some interest on the part of the reader rather than trying to drum it up myself. For this reason I have done without anecdotes, cartoons, exclamation marks, jokey chapter titles, or pictures of the Mandelbrot set.
Mathematics originates in geometry. Etymologically speaking, this is surveying. Each of the propositions in Euclid’s Elements has a diagram, which anybody can look at. You can treat the diagram as a map of something on the ground. However, the fundamental concept of Greek geometry is the ratio, and this cannot be drawn in a diagram. A ratio is not really an object, but a relation between two objects, namely magnitudes. We can draw two magnitudes, and then draw two more magnitudes that have the same ratio; but we cannot point to what is the same.
Euclid points to things with letters. Letters can label points, or line segments, or angles, or regions. As far as I have seen, letters do not label ratios, except in the last proposition of Book VII of the Elements. Therefore I suspect that this proposition was tacked on later, when a more modern, symbolic conception of mathematics started to develop.
Euclid gives two different accounts of when two ratios are the same. Before these, historically, there was a third account. I talked somewhat about these matters in “The Geometry of Numbers in Euclid.” Thus my doctoral work took up an ancient theme: when two things are the same, how can you tell? For me, those things were not ratios, but elliptic curves.
A curve is the solution-set of a polynomial equation in two variables. If we allow any number of variables, and more than one equation, the solution-set is a variety. A variety has a function-field, namely the field of polynomial functions that can be defined on the variety. The function-field is indeed a field, because any two members have a sum, a difference, and a product, and members other than zero have reciprocals.
In papers of 1986 and 1992 in the Journal of Symbolic Logic, Jean-Louis Duret took up the question of whether you can distinguish two varieties by the first-order theories of their function-fields. This means looking only at what can be said about sums and products of individual elements of the function-fields. Duret showed that you can make the desired distinction, unless one of the varieties is an elliptic curve with complex multiplication. I ended up showing that Duret’s theorem still holds, even for some elliptic curves with complex multiplication. By a theorem of algebraic number theory, there are only thirteen of these curves.
We are working here with varieties whose defining polynomials have coefficients from an algebraically closed field, such as the field ℂ of complex numbers. This means every polynomial equation in one variable has a solution. The field ℂ has characteristic zero, by default, since it has no prime number as a characteristic. Duret considered only this case for his elliptic curves. However, for every prime number p, such as 2, 3, 5, 7, 11, and so on, there are fields, and therefore algebraically closed fields, having characteristic p. This means the pth multiple of unity is zero in the field. For example, in the two-element field, effectively studied by George Boole, 1 + 1 = 0, so the characteristic is 2. Only as a post-doc was I able to generalize, to arbitrary characteristic, Duret’s result concerning elliptic curves with no complex multiplication.
An elliptic curve is not an ellipse, but somehow arises from trying to compute lengths along an ellipse. I shall not go into this.
We can obtain an elliptic curve as the quotient of the additive group of complex numbers by a certain kind of discrete subgroup. In a simpler illustration of the process, we can obtain a circle as the quotient of the additive group of real numbers by a discrete subgroup. I shall spell this out below. Meanwhile, the desired discrete subgroups of ℂ are lattices. An example is the lattice ⟨1, i⟩ of Gaussian integers, namely the complex numbers k + mi, where k and m are integers. If L is a lattice, then, multiplying each point of the lattice by the same integer n, we obtain the lattice nL, which is a sub-lattice of L. In some cases, we can obtain a sub-lattice by multiplying by a complex number. For example, in the case of the Gaussian integers, the number i works. The corresponding elliptic curve is then said then to have complex multiplication.
If L and M are lattices, and α (alpha) is a nonzero complex number for which αL is a sub-lattice of M, then multiplication by α is an embedding of L in M. The index of the embedding is the size of the quotient M/αL. For example, multiplication by an integer n embeds a lattice in itself with index n^{2}. The thought I had in bed one Sunday morning was that, if there are two different embeddings of L in M, and the indices of these embeddings are prime to one another, then the function fields of the corresponding elliptic curves are, to a certain extent, logically indistinguishable. If L is required to have complex multiplication, there are only the thirteen cases that I mentioned, where M would have to be isomorphic to L itself, in order for the embeddings of co-prime index to exist. Thus there are lots of examples of non-isomorphic elliptic curves that we do not know how to distinguish with first-order logic. This is actually an example of what makes logic interesting: that it cannot say everything.
To be precise, in the situation described, we cannot distinguish the curves using sentences of the form
∀x_{1} … ∀x_{n} ∃y φ(x_{1}, …, x_{n}, y),
where φ is quantifier-free. We don’t know (or at least I don’t know) whether more complicated sentences would work.
I said I would elaborate on obtaining circles and elliptic curves. The range of the function f defined by
f(t) = (sin t, cos t)
from the real line ℝ to the Cartesian plane ℝ × ℝ is the circle defined by
x^{2} + y^{2} = 1.
The function f is periodic, with period 2π (two pi). Thus f is a one-to-one correspondence between
The real numbers t in the interval correspond to the sets of real numbers of the form
t + 2kπt,
where k ranges over the set ℤ of integers. The sets
{t + 2kπt : k ∈ ℤ}
then make up the quotient denoted by
ℝ / 2πℤ.
In this quotient, addition and subtraction remain well-defined; thus the quotient is an additive group.
By means of the analytic functions of sine and cosine, we have obtained a geometric object, a circle. We may observe further that the cosine function is the derivative of the sine function.
Likewise, for every lattice L of ℂ, there is a function, the Weierstrass ℘ function, whose period is the given lattice. The function is the P function, but the P is given its own special form, even in Unicode, as here. The ordered pair of ℘ and its derivative ℘′ satisfy an equation
y^{2} = 4x^{3} − ax − b.
This is the equation that defines the elliptic curve corresponding to L. The equation being cubic, every straight line cuts the curve in three points. The sum of those three points is zero, in the addition in the curve that corresponds to addition in ℂ / L. This makes elliptic curves especially interesting.
For the record, when I was first writing the present account, here in Istanbul, I felt an earthquake, and so I went to report my experience.
]]>This is an appreciation of Gödel’s Incompleteness Theorem of 1931. I am provoked by a depreciation of the theorem.
In the “Gödel for Dummies” version of the Theorem, there are mathematical sentences that are both true and unprovable. This requires two points of clarification.
The example will be the sentence written symbolically as
∀x Q(x, p),
read out as
“For all numbers x, Q is true of x and p,”
for a certain binary predicate Q and a certain number p.
The sentence ∀x Q(x, p) will be true, because its very meaning is that the sentence ∀x Q(x, p) has no proof in the given system. In the “Gödel for Dummies” version, the sentence means, “I have no proof.” If this were false, then the sentence would have a proof and would therefore be true; thus it is true.
The sentence ∀x Q(x, p) does not refer to itself as such. The reader may skip the details, which we shall take up again only later; but
Thus the meaning of ∀x Q(x, p) is that no x is ever the Gödel number of a proof of the sentence φ(p), when φ(y) is the formula of which p is the Gödel number. In short, φ(p) is unprovable. We now observe that φ(p) is precisely the sentence ∀x Q(x, p).
That was a simplified version of Gödel’s Theorem. Such a version can be found, for example, in the two-page article “Gödel’s Theorem” by Peter J. Cameron in the Princeton Companion to Mathematics (2008). There is additional interest in the full version of the Theorem, which is reviewed by C. Smorynski in a 46-page article, “The incompleteness theorems,” in the Handbook of Mathematical Logic (1977). For the present article, my main source is Gödel’s own article, translated as “On Formally Undecidable Propositions of Principia Mathematica and Related Systems I” in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (third edition, 1976).
Even in the simplified version, is Gödel’s result surprising? At least one person thinks not. In an interview by John Horgan called “Philosophy Has Made Plenty of Progress” (Scientific American, November 1, 2018), Tim Maudlin says,
Absolutely no one should have ever been surprised that mathematical truth cannot be equated with theoremhood in some finite axiomatic system. An infinitude of mathematical truths are uninteresting trivia, with no obvious route to being proved. Example: [to be considered later].
… All Gödel did was find a clever way to construct a provably unprovable mathematical fact, given any consistent and finite set of axioms to work with. The work is clever but in no way profound. It should have come as no surprise at all.
Gödel’s “clever way to construct a provably unprovable mathematical fact” laid the foundations of computer science, proof theory, recursion theory, and even, in a different sense, my own field, model theory. Perhaps one thinks these pursuits were going to develop anyway.
I see two important technical errors in what Maudlin says.
Several more points should be clarified.
Before the Incompleteness Theorem could be proved, somebody needed to bring the two notions of consistency together with a Completeness Theorem. Gödel did this too.
The fifth proposition of the first book of Euclid’s Elements is that the base angles of every isosceles triangle are equal to one another. By what we can now recognize as a rule of inference called instantiation, we can conclude from the proposition that if, in a particular triangle ABC, the sides AB and AC are equal, then so are the angles ABC and ACB. If we can prove that ABC is indeed thus isosceles, then, by another rule of inference, called modus ponens, the base angles of ABC must be equal.
Those rules of inference are nothing surprising, and Euclid does not spell them out. However, Euclid’s “common notions”—rules such as “Equals to the same are equal to one another”—can be understood as the axioms of his proof-system.
We can apply a proof-system to additional axioms, or perhaps postulates, so as to derive theorems from these. Thus we can understand the Elements as an application of a certain proof-system to Euclid’s explicit postulates, such as the Parallel Postulate (“If a straight line cross two others and make the interior angles on the same side less than two right angles, then the two straight lines, extended, will meet on that side”).
For the Incompleteness Theorem, Gödel derives a proof-system from the Principia Mathematica (1910–3) of Whitehead and Russell, and he applies it to the Peano Axioms for the counting numbers.
Actually Gödel works with the natural numbers, starting with zero. We start the counting numbers with unity: this is what Peano himself did, and I shall do the same, though there are technical reasons why having zero can be nice.
Unity is a counting number, and every counting number has a successor. Concerning these two conditions, we have the following postulates.
Those are the Peano Axioms, which Peano published in 1889, using notation that Russell and Whitehead would adopt for the Principia Mathematica. Dedekind published equivalent postulates in 1888, but without notation that caught on.
Dedekind understood the postulates better though, as I have discussed in an article called “Induction and Recursion” (The De Morgan Journal 2, no. 1 [2012], 99–125). The third of the Peano Axioms above is the Induction Axiom. The three Peano Axioms together (but no two of them alone) entail the Recursion Theorem.
For example:
Perhaps that is the simplest non-trivial proof by induction.
n + 1 = Sn
and, for all counting numbers k,
n + Sk = S(n + k).
There is a similar recursive definition of multiplication.
The Induction Axiom is second-order, because when written out formally as
∀X (1 ∈ X & ∀y (y ∈ X ⇒ y + 1 ∈ X) ⇒ ∀y y ∈ X),
the postulate uses a variable, namely X, that stands for sets of numbers. The variable y stands for individual numbers; a formula using only such variables is first-order. Peano introduced the symbol ∈, which we can read as “is in”; the symbol is based on the Greek epsilon, the initial letter of ἐστί, meaning “is.” For the arrow ⇒ of implication, Peano used a different symbol (a reversed letter C, thus Ɔ, later stylized as ⊃).
The Recursion Theorem is also second-order, since it uses variables for functions of numbers. Using Greek letters η and ζ (eta and zeta) for such variables, we can write the Recursion Theorem as
∀x ∀η ∃ζ (ζ(1) = x & ∀y ζ(Sy) = η(ζ(y))).
Gödel proves the Incompleteness Theorem using second-order variables. Then he shows that, by introducing second-order constants, + and ×, for the recursively defined functions of addition and multiplication, along with appropriate axioms that govern their behavior, we can do everything in first-order logic.
The first-order sentences that are true in a particular structure constitute the complete theory of that structure. The structure is then a model of that theory. If we let N be the set of counting numbers, then the complete theory of the structure (N, +, ×) is not recursively axiomatizable. In other words, the indicated complete theory is undecidable: there is no algorithm for deciding whether a proposed first-order theorem about (N, +, ×) is true. This is Gödel’s 1931 result. Equivalently, any recursively axiomatized theory for which the structure (N, +, ×) is a model must be incomplete.
By contrast, Presburger gave, in 1929, a complete axiomatization of the complete theory of (N, +); Skolem gave, in 1930, the same for (N, ×). Since then, many decidable complete theories have been found. They are the subject of Abraham Robinson’s 1956 book Complete Theories, and they are a subject of model theory, which again is that branch of mathematical logic that I happen to specialize in.
Tim Maudlin specializes in the philosophy of physics. His belittling of accomplishments in other fields is unbecoming. Mathematics is intimately connected with physics, and the two fields have been indistinguishable for much of their history; but they are different.
Maudlin may be satisfied with general inductive proof. Others of us are not.
One might attempt to characterize an inductive proof as having the form, “Such-and-such has happened in n cases; therefore it will always happen.” For example, when we draw k points on the circumference of a circle, then connect every pair with a straight line, these lines divide the circle into 2^{k − 1} regions in at least 5 cases, when k is 1, 2, 3, 4, or 5. Therefore, “by induction,” the same will happen, no matter what number k is.
It doesn’t happen. With 6 points we get 31 regions, not 32. But even if the suggested result always occurred, the proposed proof would be inadequate, even as an inductive proof in the informal sense.
Inductive or not, every proof needs to be backed up with some kind of reason. Different fields of inquiry will allow different kinds of reasons. I can say no more than this, except that, in mathematics, in an inductive proof, the reason will again be a two-part deduction:
Elsewhere in the interview by John Horgan, Maudlin says he believes—though he has no proof—that
the fundamental physical law—when presented in the right mathematical language—will be so compellingly simple that we would think that any other structure would be unnecessarily complicated.
There seems to be no question of whether there is a fundamental physical law. However, it seems to me that, if there were a single “fundamental physical law,” it would effectively be an axiom from which every scientific truth could be proved. In that case, with Maudlin’s own interpretation of Gödel in mind, might we not suggest that physicists are wasting their time and our money, trying to find a fundamental physical law, since “absolutely no one” should expect it to exist?
Physics is mentioned in the 1958 book of Ernest Nagel and James R. Newman called Gödel’s Proof:
… although certain parts of physics were given an axiomatic formulation in antiquity (e.g., by Archimedes), until modern times geometry was the only branch of mathematics that had what most students considered a sound axiomatic basis.
But within the last two centuries the axiomatic method has come to be exploited with increasing power and vigor. New as well as old branches of mathematics, including the familiar arithmetic of cardinal (or “whole”) numbers, were supplied with what appeared to be adequate sets of axioms. A climate of opinion was thus generated in which it was tacitly assumed that each sector of mathematical thought can be supplied with a set of axioms sufficient for developing systematically the endless totality of true propositions about the given area of inquiry.
Gödel’s paper showed that this assumption is untenable …
Should mathematicians really have known all along that “this assumption is untenable”? They would first have had to recognize the assumption as such.
From the mid-nineteenth-century discovery of non-Euclidean geometry, one may induce that Euclidean geometry without the Parallel Postulate is incomplete. One can deduce the same thing, by using Euclidean geometry to construct a model of Lobachevskian geometry, where the Postulate is denied. In one such model, you let an infinite straight line divide the Euclidean plane into two parts. You throw out one part, along with the points on the dividing line itself. In the remaining “half-plane,” semicircles with centers on the boundary line are to be considered as “straight” in Lobachevski’s sense. So are straight lines at right angles to the boundary line; but no other lines, whether straight or curved, are to be considered as “straight” in Lobachevski’s sense. Euclidean full circles in the half-plane are Lobachevskian circles, though the centers will be different; right angles stay the same. In this way, as Euclidean geometry is consistent, so is Lobachevskian geometry, though it denies the Parallel Postulate; thus this postulate is not implied by Euclid’s other postulates.
We say Euclidean geometry is consistent, because we think that it has a model. In the seventeenth century, Descartes used this model to establish the consistency of algebra. The discovery of non-Euclidean geometry casts doubt on whether Euclidean geometry is consistent. We can now turn Descartes around. David Hilbert does this in The Foundations of Geometry (1899). If (Ω, +, ×, <) is an Archimedean ordered field in which Ω contains the square root of each sum a^{2} + b^{2} of squares of elements of Ω, then the set Ω^{2} of ordered pairs of elements of Ω serves as a plane in which Hilbert’s axioms for Euclidean geometry are satisfied.
Hilbert observes the possibility of an Axiom of Completeness, which would ensure that Ω must be the set R of real numbers. However, the completeness here is not of a theory; it is completeness of the ordering of the field. An ordered set is complete if every nonempty subset that has an upper bound has a least upper bound. This is a second-order condition.
As Tarski showed in the 1940s, the first-order theory of the real ordered field (R, +, ×, <) is decidable; equivalently, the field has a recursive axiomatization. The axioms are those of a real closed field. But there are other real closed fields besides R.
By contrast, the ordered field of real numbers is completely determined by the axioms of a complete ordered field; in a word, these axioms are categorical. So are the Peano Axioms. However, in each of these two cases, one of the axioms is second-order. The Archimedean Axiom, which Ω above satisfies, is also second-order; however, the unique complete ordered field R is automatically Archimedean.
There is a third kind of completeness, enjoyed by a consistent proof-system when, for every sentence σ that is true in every model of a set Γ of axioms, there is a proof of σ from Γ. By the Completeness Theorem, proved by Gödel in his doctoral dissertation of 1930, a complete first-order proof-system can be extracted from the Principia Mathematica. Consequently, if this proof-system cannot derive a contradiction from a given set of axioms, then this set must be consistent in the sense of having a model.
At the beginning of the 1930 paper based on his dissertation, Gödel explains the Completeness Theorem as follows, for the case when Γ above is empty:
Whitehead and Russell, as is well known, constructed logic and mathematics by initially taking certain evident propositions as axioms and deriving the theorems of logic and mathematics from these by means of some precisely formulated principles of inference in a purely formal way (that is, without making further use of the meaning of the symbols). Of course, when such a procedure is followed the question at once arises whether the initially postulated system of axioms and principles of inference is complete, that is, whether it actually suffices for the derivation of every logico-mathematical proposition, or whether, perhaps, it is conceivable that there are true propositions (which may even be provable by means of other principles) that cannot be derived in the system under consideration.
It seems that in fact the question of completeness did not arise at once, at least not for Russell and Whitehead, though it did for Hilbert and Ackermann in 1928. Nonetheless, Gödel continues:
For the formulas of the propositional calculus the question has been settled affirmatively; that is, it has been shown that every correct formula of the propositional calculus does indeed follow from the axioms given in Principia Mathematica. The same will be done here for a wider realm of formulas, namely those of the “restricted functional calculus” …
The restricted functional calculus is a calculus of what we now call first-order formulas. In the same paper, Gödel proves also the countable case of what we now call the Compactness Theorem: that if every finite subset of a set Γ of sentences has a model, then Γ itself has a model. Malcev will later establish the general case, where Γ may be uncountable. The general form of the Completeness Theorem follows from the special form, where Γ is empty, and Compactness. Conversely, since proofs are finite, Completeness implies Compactness.
There can be no compactness theorem for second-order logic, and hence no completeness theorem. For example, to the axioms for a complete ordered field, we can add infinitely many axioms, whereby the ordered field has an element a that is greater than 1, 2, 3, and so on, and is thus infinite. A model of these axioms would be non-Archimedean, and then its ordering would not be complete (since there could be no least upper bound on the set of finite elements). So the enlarged set of axioms has no model, although every finite subset has one, namely R itself with a great enough.
In Hilbert’s Foundations of Geometry, the distinction between first- and second-order logic has yet to be recognized. In the Principia Mathematica, Russell and Whitehead work with variables of arbitrarily high order. For the Incompleteness Theorem, as we said, Gödel works first with the whole system of the Principia Mathematica, before showing that first-order logic is enough.
Pace Mr Maudlin, a lot of distinctions have to be discovered, before the question can even be raised of whether there are undecidable first-order theories.
The physicist Richard Feynman had a way of teasing mathematicians about their proofs. Possibly Maudlin imitates Feynman, who told his mathematician friends,
I bet there isn’t a single theorem that you can tell me—what the assumptions are and what the theorem is in terms I can understand—where I can’t tell you right away whether it’s true or false.
Could Feynman cut up an orange and rearrange the pieces into another solid sphere, as large as the sun? He thought not. The mathematicians said he could. However, this result, the Banach–Tarski Paradox, requires infinite divisibility of matter (also the Axiom of Choice). Feynman was correct, if his friends were talking about real oranges, made up of atoms. “So I always won,” he said:
If I guessed it right, great. If I guessed it wrong, there was always something I could find in their simplification that they left out.
This is from “Surely You’re Joking, Mr. Feynman!” (1985; page 85).
By way of showing that the Incompleteness Theorem should have been obvious, Maudlin proposes the question of whether, in the decimal expansions of two real numbers, there is a difference in the first digit, and the next two digits, and the next three digits, and so on. Approximately eight times out of nine, all of these differences will occur. In this case, in Maudlin’s terminology, the original numbers fail to “match.” Can we then prove the failure? Maudlin says,
No amount of just grinding out the digits and checking will ever prove it: there are always more digits to check. And I see zero prospect of any other way to prove that they don’t match. So if they don’t match, that is an unprovable mathematical fact.
Maudlin sees no way to come up with a proof; therefore, for him, there is no proof. This is an induction. Mathematics asks for deductions.
I recall a fellow mathematician to have said that a conjecture ought to be more certain than a theorem. You can prove a theorem without having any intuition for the result; but your proof can always contain mistakes that you did not notice. If you make a conjecture, your confidence in it ought to be greater than your confidence in all of the steps of a proof that convinces you of some other assertion. That is one opinion. I believe the person who expressed it had confidence in the Conjecture of Birch and Swinnerton-Dyer. This did not mean he was going to derive consequences from the conjecture and assert them as new theorems.
As for Gödel’s Completeness Theorem, one might try to argue that there are uncountably many pairs of real numbers that do not “match,” but only countably many proofs; therefore there must be examples of pairs of real numbers that do not “match,” although we cannot prove the failure. As an alternative to Gödel’s proof, such an argument itself fails, because only countably many real numbers can be defined. For example, if you are going to express a real number by its decimal expansion, you need a rule for generating the digits; and there can be only countably many such rules.
There can be only countably many recursively axiomatized complete theories. However, there are uncountably many countable complete theories, since the structure (N, 1, S, P) has a different theory for each choice of P as a subset of N. We can conclude that one of these theories, even uncountably many of them, must be undecidable; but we do not yet know which. Gödel, again, gives us an example, namely (N, +, ×), of a structure with an undecidable theory.
Let us look at Gödel’s argument in more detail.
If you are reading these words on a computer screen, you know that, behind the scenes, each letter has a certain code, which is a number. We express mathematics using formulas; we can understand these as strings of symbols or characters from a certain list; we can give each of those characters a number. One example of a formula is a sentence expressing Fermat’s Last Theorem, proved by Wiles:
∀x ∀y ∀z ∀w (x^{w} + y^{w} = z^{w} & w ≠ 1 ⇒ w = 2).
Let us call this an arithmetical formula, because the variables range over the counting numbers: arithmoi for the Greeks. Unity was not an arithmos, because it was just one; but we count unity as a number. Gödel counts zero as well, but I prefer to leave this out.
We should understand the typographical raising of an exponent as having its own symbol, such as a circumflex (ˆ). Then the symbols that we used above are on the list
∀, x, y, z, w, (, ˆ, +, =, &, ≠, 1, ⇒, 2, ).
We can understand our formula itself as the list
∀, x, ∀, y, ∀, z, ∀, w, (, x, ˆ, w, +, y, ˆ, w, =, z, ˆ, w, &, w, ≠, 1, ⇒, w, =, 2, ).
We number the list of all symbols that we may want to use. For writing down numbers themselves in arithmetical formulas, the ten usual digits suffice; moreover, since any number can be written as a sum
1 + 1 + … + 1,
the symbols + and 1 suffice. Still our list of symbols may be infinite, since there is no bound on the number of variables that we may want to use. We could however let every variable be x followed by some number of “primes,” as in x′, x′′, x′′′, and so on. In any case, every symbol on our list can be given its own number, even if the list is infinite. Using these numbers, we can convert any formula into a list of numbers. This will always be a finite list.
We can convert such a list into a single number, in a reversible way. Gödel’s method relies on the Fundamental Theorem of Arithmetic, that each number has a unique prime factorization. If we list the prime numbers as p_{1}, p_{2}, p_{3}, and so on, then any finite list
n_{1}, n_{2}, …, n_{k}
of numbers corresponds reversibly to the product
p_{1}^{n1} × p_{2}^{n2} × … × p_{k}^{nk}.
In this way, given an arithmetical formula φ, we convert it into a list of numbers, and then into a single number. Let us denote this last number by
‹φ›.
We now call this the Gödel number of φ.
Given some true arithmetical sentences, we can derive other true sentences in a formal way, by such rules of inference as were mentioned earlier. For example,
If we choose a specific list Σ of true arithmetical sentences, called axioms, then a proof from Σ is a finite list of sentences, each one being either an axiom or a sentence derived from previous sentences on the list by a rule of inference. The last sentence on the list is what the proof proves. If a proof Σ is the list
σ_{1}, …, σ_{k},
so that Σ proves σ_{k}, then we define
‹Σ› = p_{1}^{‹σ1›} × p_{2}^{‹σ2›} × … × p_{k}^{‹σk›}.
Our list of axioms may be infinite. Still we shall require the list to be recursive; again, this means there will be some clear rule for determining whether a given sentence is on the list. Equivalently, the set of Gödel numbers ‹σ› of sentences σ on the list should be recursive.
The notions of recursive set, formula, and proof are so bound up together that, for any counting number m, a recursive set of m-tuples of counting numbers is precisely a subset R of N^{m} such that, for some formula φ(x_{1}, …, x_{m}), for all m-tuples (a_{1}, …, a_{m}) in N^{m}, the following two conditions hold.
The first condition is that φ defines R. We need the second condition, because the set that a formula defines is not always recursive: this will be a consequence of Gödel’s Theorem.
In forty-five steps, Gödel shows that there is a formula B(x, y) defining a recursive set, and for all counting numbers a and c, the sentence B(a, c) is true if and only if, for some proof Γ of some sentence σ, we have
a = ‹Γ›,
c = ‹σ›.
There is then another formula, Q(x, y), also defining a recursive set, and for all counting numbers a and b, the sentence Q(a, b) is true if and only if
Now let p be the number ‹∀x Q(x, y)›. Suppose, if possible, that there is a proof Γ of ∀x Q(x, p). Then
Since Q, and therefore ¬Q, defines a recursive set, the true sentence ¬Q(‹Γ›, p) has a proof. We can extend this to a proof of ¬∀x Q(x, p). Again, this is based on the assumption that there is a proof of ∀x Q(x, p) as well.
Thus, if ∀x Q(x, p) is provable, then so is its negation. Assuming our proof-system is consistent, we can conclude that ∀x Q(x, p) has no proof. In particular, for each counting number n, ¬B(n, ‹∀x Q(x, p)›) is true. This sentence is equivalent to Q(n, p), which therefore has a proof.
According to Gödel,
We can readily see that the proof just given is constructive; that is, the following has been proved in an intuitionistically unobjectionable manner.
The “following”—what we have proved—is that, if either of ∀x Q(x, p) and ¬∀x Q(x, p) has a proof, then from it we can obtain proofs of
Not all of these sentences can be true in N. If the theorems of our proof-system must be true in N, then neither ∀x Q(x, p) nor its negation is provable in the system.
If we are not worried about being “intuitionistically unobjectionable,” we may just observe that, if the sentence ∀x Q(x, p) is false, this means that the same sentence actually has a proof; thus, if our proof-system is consistent, the given sentence must be true, while it has no proof.
We saw earlier the Recursion Theorem, whereby, if a is a counting number, and f is a function of counting numbers, then there is another such function, g, given uniquely by the requirements
g(1) = a,
∀x g(Sx) = f(g(x)),
where S is the function converting x to x + 1. Functions built up in this way, from constant functions and S, by repeated application of recursion in the sense above are by definition recursive (now they are called primitive recursive). If f is recursive, the solution set of the equation f(x) = 1 is by definition recursive. There are clear procedures for
We have seen that addition is recursive. We can understand the formula
x + y = z
to be an abbreviation of the formula
∃η (η(1) = Sx & ∀w η(Sw) = S(η(w)) & η(y) = z).
This is how formulas defining recursive sets are obtained. However, the example here is second-order. We can simplify it to the formula
∃η (η(1) = Sx & ∀w (w < y → η(Sw) = S(η(w))) & η(y) = z).
That x + y = z means there are numbers u_{1}, …, u_{y} such that
u_{1} = Sx,
∀w (w < y → u_{w + 1} = S(u_{w})),
u_{y} = z.
By the Chinese Remainder Theorem, there are numbers b and d such that each u_{k} is the remainder of b after division by 1 + kd. This allows us to express x + y = z in a first-order way. In return though, we need both addition and multiplication, in order to be able to do this. However, if we allow ourselves symbols for these operations, then we can define all other recursive functions and relations using first-order formulas.
They will not be arbitrary first-order formulas either, but their quantified variables will be bounded, as in
∃x (x < z & R(x, y)).
Conversely, the sets defined by such formulas are recursive.
Recursive functions and sets are now called primitive recursive, because as Gödel observed in lectures at the Institute for Advanced Study in Princeton in 1934, there are general recursive functions that are not recursive in the original sense (as Ackermann had shown in 1928).
There is a Second Incompleteness Theorem. The set of all Gödel numbers of sentences being recursive, there is a formula F defining it. Let τ be the sentence
∃y (F(y) & ∀x ¬B(x, y)),
with B as earlier. Then τ expresses the consistency of our proof-system. Thus we have shown
τ ⇒ ∀x Q(x, p).
Our proof of this can be formalized within our proof-system (though as Gödel admits, this point needs fleshing out). The proof of ∀x Q(x, p) cannot be formalized. Therefore a proof of τ cannot be formalized, if it is true. If consistent, our system cannot prove this.
The theory of (N, +, ×) has no recursive axiomatization; but again, the theories of some interesting structures do have. This is what makes interesting the mathematics that I do.
John Horgan’s interview of Tim Maudlin is called, again, “Philosophy Has Made Plenty of Progress.” According to Maudlin,
Overwhelmingly most philosophers are atheists or agnostics, which I take to be convergence to the truth.
According to Christopher Hitchens in God Is Not Great (2007),
Not all can be agreed on matters of aesthetics, but we secular humanists and atheists and agnostics do not wish to deprive humanity of its wonders or consolations. Not in the least. If you will devote a little time to studying the staggering photographs taken by the Hubble telescope, you will be scrutinizing things that are far more awesome and mysterious and beautiful—and more chaotic and overwhelming and forbidding—than any creation or “end of days” story …
Twenty years ago, in 1998, when I was a post-doc in California, a friend sent me prints of a couple of those Hubble photographs. I have them still. They are awesome and mysterious and so on, as Hitchens says; but I leave off the comparative “more.” More awesome to me is the power of the mind, in its ability to create myths, to take and interpret photographs, and especially to discover, prove, and make use of such results as Gödel’s Incompleteness Theorem.
]]>The Antitheses are the six parallel teachings, delivered by Jesus of Nazareth in the Sermon on the Mount, as recounted in Chapter 5 of the Gospel According to St Matthew, starting at verse 21. I summarize:
For better or worse, these are part of the cultural heritage of many of us; they are at least a commentary on the cultural heritage (the Mosaic Law) of more of us.
I write now specifically, because I think the Antitheses can illustrate or illuminate some contemporary philosophical concerns, as seen in a few articles that have come to my attention. Topics include
Of particular concern to me is that theorists of punishment would ignore forgiveness, and at least one theorist would think there could be responsibility without freedom.
I shall bring in that work of William Blake whose title names a synthesis of antitheses: The Marriage of Heaven and Hell. Meanwhile, by way of introduction, I shall consider
A friend who read an earlier draft of this essay suggested that it could use some “thematic discipline.” To my mind this would mean breaking the work into pieces, each to be expanded into its own essay. I am loath to think that anything here should simply be deleted, although I might have less interest in some of those separate essays than in others.
Another friend independently alerted me to a poem called “Heart’s Horizon,” by conductor Vladimir Fanshil:
In this world
Fences and lines
Are couches of society:
To protect, divide & define
My hearts fence
Is the horizon
& when I reach it
there I’ll draw the line.
Everything is interconnected, and fitting things into fences and lines does violence to them. Nonetheless, in part because another friend said he simply didn’t know what I was talking about, I have divided the essay into sections, which can be reached with the links above, if not by just reading through.
From Wikipedia I have learned for the teachings above the term “Antitheses,” and that the term originated with the Gnostic heresiarch called Marcion of Sinope. In the New Oxford Annotated Bible (Revised Standard Version, 1973), the heading of the notes for Matthew 5.21–48 is simply, “Illustrations of the true understanding of the Law.”
This “true understanding” is not necessarily a new understanding. In the same edition of the Bible, a note at Proverbs 25.21–22 refers forward to the Sixth Antithesis. A note here could just as well refer back to the verses of Proverbs, which read, in the King James Version,
If thine enemy be hungry, give him bread to eat; and if he be thirsty, give him water to drink:
For thou shalt heap coals of fire upon his head, and the Lord shall reward thee.
I thank the blog Adventures in Ankara for reminding me of these verses, which would seem to contain the germ of (at least) the last two of the Antitheses. As is said in the second verse of the same chapter of Proverbs,
It is the glory of God to conceal a thing: but the honour of kings is to search out a matter.
Etymologically speaking, a thesis is something given a position. An antithesis stands in opposition to a thesis. The opposition expressed by the prefix “anti-” may have varying degrees. With respect to a given place on earth, the antipodes are as far away on the planet as can be; but the small island of Antikythêra, which gives its name to a remarkable ancient mechanism found in the sea nearby in 1902, is only 38 kilometers from Kythêra, towards which tended, by the account of Hesiod, the foaming genitals of Uranus, severed by Cronus, before they came ashore at Cyprus and gave birth to Aphrodite.
According to the Oxford English Dictionary, an antithesis is (1) “an opposition or contrast of [two] ideas …” or (2) the second of these two ideas (or of the two clauses expressing them). For the six teachings of Jesus in Matthew 5, I shall use the term Antitheses, not necessarily as implying a Manichaean irreconcilable opposition, though this may have been the meaning of Marcion.
We might understand the Antitheses as instances of a more fundamental antithesis: that of the letter and spirit of the law. There is a related antithesis of
Being about retribution, the Fifth Antithesis especially, though not exclusively, suggests the antithesis of punishment and forgiveness. What Jesus says is,
Ye have heard that it hath been said, An eye for an eye, and a tooth for a tooth:
But I say unto you, That ye resist not evil: but whosoever shall smite thee on thy right cheek, turn to him the other also.
Rules such as “An eye for an eye” mean presumably that retaliation for an injury should be measured. When we try to mete out punishment, we may exceed or fall short of the just amount. At the very least, the Fifth Antithesis is a warning against excessive punishment. A warning against the reverse is probably not needed.
A Latin antithesis may clarify what Jesus is trying to do:
Verba volant, scripta manent.
The spoken word flies, the written word remains.
This can have two meanings:
Written words like “An eye for an eye” come down to us from the Books of Moses. The proper or intended meaning may not have come along with them.
That is pretty much what Socrates tells Phaedrus, in the Platonic dialogue named for him (at 275D, here in the 1914 Loeb translation by Harold North Fowler):
Writing has this strange quality, and is very like painting; for the creatures of painting stand like living beings, but if one asks them a question, they preserve a solemn silence. And so it is with written words; you might think they spoke as if they had intelligence, but if you question them, wishing to know about their sayings, they always say only one and the same thing.
Writing about such subjects as justice may be an amusing pastime; “but,” says Socrates (at 276E),
in my opinion, serious discourse about them is far nobler, when one employs the dialectic method (ὅταν τις τῇ διαλεκτικῇ τέχνῃ χρώμενος) and plants and sows in a fitting soul intelligent words which are able to help themselves and him who planted them …
I propose to consider the Antitheses dialectically. The Fifth Antithesis finds justice in a dialogue between punishment and forgiveness. The dialogue is not really between abstractions; it is between ourselves and within each of us.
To engage in dialogue needs a certain degree of self-control. One has to be patient in listening to one’s misguided opponent. The law imposes external control. The thesis of the Sixth Antithesis refers to this control:
Ye have heard that it hath been said, Thou shalt love thy neighbour, and hate thine enemy.
Jesus’s apparent source in Leviticus 19:18 is more explicit:
Thou shalt not avenge, nor bear any grudge against the children of they people, but thou shalt love thy neighbor as thyself: I am the Lord.
This verse is cited in The Greek New Testament (fourth revised edition, edited by Aland et al., 1993), which sets quotations from Hebrew scripture in bold. What Jesus says about loving neighbors is in bold; hating enemies, not. The antithesis begins:
But I say unto you, Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you …
Nothing is bold here, but Aland et al. cite Exodus 23:4–5:
If thou meet thine enemy’s ox or his ass going astray, thou shalt surely bring it back to him again.
If thou see the ass of him that hateth thee lying under his burden, and wouldest forbear to help him, though shalt surely help with him.
Where then has Jesus found the teaching about hating one’s enemy? Perhaps he has been reading Book I of Plato’s Republic, where Polemarchus cites Simonides as saying, “It is just to render to each his due” (331D, in the 1930 Loeb translation by Paul Shorey). Polemarchus interprets this to mean, “friends owe it to friends to do them some good and no evil” (332A), while “there is due and owing from an enemy to an enemy what is also proper for him, some evil” (332B).
Doing evil to enemies is giving in to one’s passions. One should rather think things through. Socrates sets an example by observing that we can be mistaken about our enemies (334C). Plato sets an example by writing dialogues, rather than lectures. Jesus gives lectures, but in speech, not writing. What we have of these lectures lies in four different canonical Gospels, as if to say that no one piece of writing holds the truth, but is only the starting point for dialectic.
My attention was drawn to the Antitheses by a reference to one of them, late in Collingwood’s New Leviathan of 1942. Several years ago, in January of 2014, I started working through this book, writing a blog post about each chapter as I read it. The parts of the book are Man, Society, Civilization, and Barbarism. I was first interested mostly in the first part, which is an account of how we mature individually into free rational beings. One part of the process is converting appetites into desires.
As I have read the chapters of the New Leviathan, fitfully over the years, the outside world has seemed to become more barbarous, even with the kind of barbarism that was ravaging Europe as Collingwood wrote in the early 1940s. Collingwood ends with four chapters, reviewing four examples of barbarism. The last of these is Nazism. One of the earlier examples is the so-called Albigensian heresy, understood as a form of Manichaeism.
The Albigensians refused to swear oaths. They might thus seem faithful to the Fourth Antithesis:
Again, ye have heard that it hath been said by them of old time, Thou shalt not forswear thyself, but shalt perform unto the Lord thine oaths:
But I say unto you, Swear not at all; neither by heaven; for it is God’s throne:
Nor by the earth; for it is his footstool: neither by Jerusalem; for it is the city of the great King.
Neither shalt thou swear by thy head, because thou canst not make one hair white or black.
But let your communication be, Yea, yea; Nay, nay: for whatsoever is more than these cometh of evil.
However, Jesus prefaces the Antitheses by saying,
Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.
Fulfilling the law is not the same as making it more strict.
The law establishes bounds on our freedom. The point is not to encourage license within those bounds. In one of the Abrahamic religions, the law about four wives means a man must not take a fifth wife, not that he ought to take even a second wife. Vegetarian Muslims can show in the Quran how God has instituted animal sacrifice, not for God’s own benefit, but for the benefit of humans. If you are going to make sacrifice, there is a proper procedure; but killing animals is not obligatory.
“Be ye therefore perfect, even as your Father which is in heaven is perfect”: this is how Jesus sums up the Antitheses. Perfection does not mean adhering to the strict letter of the external law, even when the law itself is understood in the strict sense that Jesus’s words suggest. If you refuse to divorce your spouse, or to say the words “I swear,” this does not make you perfect, or even necessarily more perfect than otherwise.
To be more perfect, understand the commitment involved in marriage, and do not speak so carelessly that somebody might not trust you when you are being serious. If you think your marriage is failing anyway, or if you are asked to swear an oath, it is an historical fact that churches have worked out accommodations for divorce and swearing. These are not necessarily abuses of the Gospel.
The quest for perfection is dialectical. This is the theme of the New Leviathan, where Collingwood observes that the perfect state is neither a democracy nor an aristocracy. These are abstractions, and neither can be understood in isolation from the other. They are antitheses, we might say, though Collingwood does not use this term for them. In democracy, everybody rules; but who is everybody? Should the ruling class include children, or animals? In aristocracy, the best rule; but who are the best? They are not necessarily the progeny of those who are already considered the best.
Everybody should be best. Everybody should be perfect. This is the teaching of Jesus. If they keep this in mind, the democrat and the aristocrat can work together, dialectically. That is Collingwood’s liberal dream, which I can share, even as we live in a world where some persons refuse to engage in dialogue. They may still want to debate you, formally; but this can be a trap. I pause to note words of Laurie Penny, from “No, I Will Not Debate You” (Longreads, September 18, 2018):
The far right does not respect the free and liberal exchange of ideas. It is not open to compromise, and it does not want a debate. It wants power …
Steve Bannon, like the howling monster from the id he ushered into the White House, exploits the values of the liberal establishment by offering an impossible choice: betray their stated principles (free, open debate) or dignify fascism and white supremacy … Either way, what matters to them is not debate, but airtime and attention. They have no interest in winning on the issues. Their image of a better world is one with their face on every television screen.
Formal rules granting freedom of speech are good; but once they are written down, somebody can figure out how to abuse them.
For real dialectic, members of a deliberative body need not belong to a particular sect, or read a particular book. There are conflicting interpretations anyway of what are supposedly holy scriptures. There is even conflict over whether the scriptures need be interpreted.
Formal adherence to external rules is not enough for the good life. This is my interpretation of the Gospel; but I think it is true independently. I see an instance of the idea in an article by Brian Earp called “The Unbearable Asymmetry of Bullshit” (Quillette, February 15, 2016).
Natural science has a system whereby we conduct experiments, write them up, and submit the results for peer review. As a mathematician, I can say that we conduct experiments, if these are understood to include attempts to prove theorems.
The system of science is intended to bring us a more perfect understanding of the world. The system can fail. Thus I once wrote up a proof that I thought was correct. An anonymous referee did not disagree, and the proof was published. Then somebody discovered a counterexample. I found my error, and with more work, I found the correct theorem and proof. However, I already had a record of fallibility. Still, a referee accepted my new result. This was published, and other researchers have used the theorem. We might all still be wrong. The only proof that we are all correct is go back over the published proof, to make sure that there was indeed no mistake.
The system of science assumes good faith on the part of scientists. We may not have it. I think this is Earp’s main point. In medicine, for example, though even in physics, somebody with an agenda can abuse the system to promote an ill-founded idea that can somehow profit that person. For a recent example, see in the New York Times, “He Promised to Restore Damaged Hearts. Harvard Says His Lab Fabricated Research” (October 29, 2018).
I connect all of this to the Antitheses of the Gospel. Righteousness is not just following the letter of the law, be this law the Torah or the scientific method.
I learned of Earp’s article through his retweet of a tweet by an entity called neuro.social.self. Thinking I might learn of other interesting papers through this entity, I started following it; I also replied to its original tweet, suggesting the connection between Earp’s paper and the Gospel. Then the entity blocked me (as I pointed out in another tweet). I can only imagine that the entity is directed by a doctrinaire Atheist, offended by my allusion to religion. However, nothing I have said about the Gospel, or shall say, asserts or assumes the existence of supernatural powers, unless perhaps those powers be understood as ourselves. As Blake says, at the end of a passage quoted below, “All deities reside in the human breast.”
The Antitheses can clarify Stoicism. Brian Earp has written about this doctrine too, in “Against Mourning” (Aeon, August 21, 2018). I would summarize his explication in Biblical language: Ye have heard that it hath been said by them of old time, Thou shalt not mourn your dead: But I say unto you, That ye train your emotions, to know that what ye love can ever be taken from you.
Stoicism does not ask you to affect indifference. If you have not prepared yourself for the worst, then there is something wrong with not mourning, when the worst happens.
If a street sign says, “No Parking 9 AM to 5 PM,” this means you will not get a ticket at other times. It does not mean you ought to park then, or even have a car at all, much less feel self-righteous about it. Likewise should you not feel self-righteous about never breaking your word, or not swearing at all. Just do not give anybody reason to doubt your word, and do not be offended if they ask you to swear anyway.
I interpreted the Fifth Antithesis as punishment versus forgiveness. Collingwood analyzes these two concepts in the penultimate chapter, called “God’s Redemption of Man,” in his first book, Religion and Philosophy of 1916. Punishment is suffering, undergone by the wrong-doer. Ideally it is moral suffering, brought on merely by verbal condemnation of the wrong-doing, with no incidental bodily pains or restrictions. Forgiveness is restoring the wrong-doer to society, without inflicting those incidental pains. For the young Collingwood,
[Punishment and forgiveness] refer to the same attitude of mind, but they serve to distinguish it from different ways of erring. When we describe an attitude as one of forgiveness, we mean to distinguish it, as right, from that brutality or unintelligent severity (punishment falsely so called) which inflicts pain either in mere wantonness or without considering the possibility of a milder expression. When we call it punishment, we distinguish it as right from that weakness or sentimentality (forgiveness falsely so called) which by shrinking from the infliction of pain amounts to condonation of the original offence.
Collingwood would later disavow the strict identification of such antitheses as punishment and forgiveness. They are rather “overlapping classes,” a notion he would develop in An Essay on Philosophical Method of 1933. Meanwhile, he said in the same chapter of Religion and Philosophy:
Granted, then, that in any given situation there can be only one duty, it follows necessarily that if of two actions each is really obligatory the two actions must be the same. We are therefore compelled to hold that punishment and forgiveness, so far from being incompatible duties, are really when properly understood identical.
They are not identical. We have already seen the distinction between them in the earlier quotation from Religion and Philosophy. What is the same, if anything, is what punishment and forgiveness seek, as it were, when they are in dialogue with one other.
I make bold to suggest that another Aeon article might have benefitted from considering forgiveness as an antithesis to punishment. In a dialogue with Gregg D. Caruso called “Just Deserts” (October 4, 2018), Daniel C. Dennett first says roughly what I say (or at least think) when people deny the existence of free will. When we grow from birth through adolescence and beyond, something happens to us. We gradually acquire or develop something, which is commonly called freedom of will. At least most of us acquire it, more or less. The process, again, is described in the first part of the New Leviathan. Some of us, in various ways, never grow up, and thus never become free. Here is how Dennett puts it:
A key word in understanding our differences is ‘control’. [Gregg,] you say ‘the way we are is ultimately the result of factors beyond our control’ and that is true of only those unfortunates who have not been able to become autonomous agents during their childhood upbringing. There really are people, with mental disabilities, who are not able to control themselves, but normal people can manage under all but the most extreme circumstances, and this difference is both morally important and obvious …
Caruso agrees that people have different levels of “rational control.” Still, he says,
As a freewill skeptic, I maintain that the kind of control and reasons-responsiveness you point to, though important, is not enough to ground basic-desert moral responsibility—the kind of responsibility that would make us truly deserving of blame and praise, punishment and reward in a purely backward-looking sense.
Caruso goes on to talk about retributive justice. Dennett agrees that it is “a hopeless muddle.” By Caruso’s account, of “the various justifications one could give for punishing wrongdoers,”
One justification, the one that dominates our legal system, is to say that they deserve it. This retributive justification for punishment maintains that punishment of a wrongdoer is justified for the reason that he/she deserves something bad to happen to them just because they have knowingly done wrong. Such a justification is purely backward-looking.
This is a crude understanding of retribution. It is the understanding expressed in the formula, “An eye for an eye, and a tooth for a tooth.” The response of Jesus is, “resist not evil.” This is the Fifth Antithesis. Forgiveness is the other side of punishment, needed to fill out its meaning. Again, by his own account, Jesus has come not to destroy the law, but to fulfill it. The Greek for fulfill here is πληρῶσαι, which would seem to be cognate with each part of our verb ful-fill.
What the wrong-doer deserves is not “something bad,” but punishment. It remains to be seen whether punishment is something bad. Collingwood spells out the problem in Religion and Philosophy:
Punishment consists in the infliction of deserved suffering on an offender. But it is not yet clear what suffering is inflicted, and how it is fixed, beyond the bare fact that it must be deserved … Punishment is fixed not by a self-evident and inexplicable intuition, but by some motive or process of thought which we must try to analyse.
In the Republic again, Socrates gets Polemarchus to agree that “when [men] are harmed it is in respect of the distinctive excellence or virtue of man that they become worse” (335C) and, this virtue being justice, “It is not then the function of the just man, Polemarchus, to harm either friend or anyone else” (335D).
As was said above, punishment is properly effected, merely by condemnation of the crime. Here is Collingwood again: “The pain of punishment is simply the pain of self-condemnation or moral repentance.”
Forgiveness is the other side of punishment. Responsibility is the other side of freedom. In a crude sense, we can inflict merciless punishment. Also crudely, we can have freedom from responsibility. Children especially have this kind of freedom. In a stricter sense, freedom is precisely a responsibility for one’s actions. Freedom is vague and ambiguous; or it is an abstract ideal, which, to be properly understood, must be paired with an antithesis, called responsibility or obligation.
Nonetheless, Sabine Hossenfelder grants the existence of responsibility, even while saying, “Free will is dead, let’s bury it.” This is the title of a post on her blog, Backreaction. The essay proper begins,
I wish people would stop insisting they have free will. It’s terribly annoying. Insisting that free will exists is bad science, like insisting that horoscopes tell you something about the future—it’s not compatible with our knowledge about nature.
According to our best present understanding of the fundamental laws of nature, everything that happens in our universe is due to only four different forces: gravity, electromagnetism, and the strong and weak nuclear force. These forces have been extremely well studied, and they don’t leave any room for free will.
Those forces may “leave no room” for free will; and yet they somehow leave room for responsibility:
Even if you don’t have free will, you are of course responsible for your actions because “you”—that mass of neurons—are making, possibly bad, decisions. If the outcome of your thinking is socially undesirable because it puts other people at risk, those other people will try to prevent you from more wrongdoing. They will either try to fix you or lock you up. In other words, you will be held responsible. Nothing of this has anything to do with free will. It’s merely a matter of finding a solution to a problem.
On the contrary, if others are trying to do something to you, they are exercising their free will. Hossenfelder’s notion of responsibility makes no more sense than what I would call the freedom to incur responsibility. This freedom does make sense to me, although Hossenfelder rejects it as “bad science.”
Natural science has found a miraculously successful method of understanding the natural world. The method starts by assuming that there is no such thing as responsibility or freedom. Nothing in nature is trying to do anything; nothing is trying to accomplish anything; nothing is to be blamed for what does happen. At least, this is the modern hypothesis; we have not always made it. Things just happen now; and the way they happen can be described by mathematical laws.
I come not to destroy those laws, but to fulfill them. We have chosen, in our freedom, to adorn the world with these laws. Here I have used a word from Plate 11 of William Blake’s great work of antithesis, The Marriage of Heaven and Hell:
The ancient Poets animated all sensible objects with Gods or Geniuses, calling them by the names and adorning them with the properties of woods, rivers, mountains, lakes, cities, nations, and whatever their enlarged & numerous senses could perceive.
And particularly they studied the genius of each city & country, placing it under its mental deity.
Till a system was formed, which some took advantage of & enslav’d the vulgar by attempting to realize or abstract the mental deities from their objects; thus began Priesthood.
Choosing forms of worship from poetic tales.
And at length they pronounced that the gods had orderd such things.
Thus men forgot that All deities reside in the human breast.
For “ancient Poets,” read Galileo and Newton. Deities reside in the human breast; so does physics. It is only enslaving the vulgar to insist that, by divine command as it were, everything shall conform to the hypothesis on which natural science is founded: the hypothesis that there is no freedom or responsibility.
In Blake’s terms, David Chalmers seems to have been taken advantage of, or to be taking advantage, in his 1995 paper called “Facing Up to the Problem of Consciousness.” Here he lays out the “hard problem of consciousness”:
It is widely agreed that experience arises from a physical basis, but we have no good explanation of why and how it so arises. Why should physical processing give rise to a rich inner life at all? It seems objectively unreasonable that it should, and yet it does.
If “arising from a physical basis” means being explicable by physics, as founded on the hypothesis above, then our experience does not arise from a physical basis. Here though I assume our experience includes trying to do things, even things like physics, with varying degrees of success. Trying is an exercise of freedom.
My perusal of Chalmers’s article suggests that exertion of will is not part of his conscious experience. His experiences are visual sensations, along with:
the sound of a clarinet, the smell of mothballs. Then there are bodily sensations, from pains to orgasms; mental images that are conjured up internally; the felt quality of emotion, and the experience of a stream of conscious thought.
Perhaps Chalmers will find a “physical basis” for these things, if they really are merely things undergone or received. But are there not experiences that we actively create?
There is an experience of struggling to prove a theorem, but making no progress; then suspecting that the theorem is false, but finding no counterexample; then wondering what to do next. We can be struck on one cheek, be inclined to strike back, but then offer the other cheek to be struck. Is all of this to be understood, merely as part of a “stream of conscious thought”? If so, then what the stream flows with is like the water that, even in infancy, we learn to make and thus control.
]]>