Category Archives: Aristotle

Automatia

One day during the Trojan War, Apollo and Athena decide to give the combatants a break. The general conflict is to be replaced with a one-on-one. The Olympians induce Helenus to tell his brother Hector to take on whichever of the Greeks is up for it.

Only Menelaus will accept the challenge at first. His brother Agamemnon makes him withdraw. When none of the other Greeks comes forward, Nestor chides them. After a story of his former prowess, he utters the words that Chapman renders as two couplets:

O that my youth were now as fresh, and all my powers as sound;
Soone should bold Hector be impugn’d: yet you that most are crownd
With fortitude, of all our hoast; euen you, me thinkes are slow,
Not free, and set on fire with lust, t’encounter such a foe.

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Mathematics and Logic

Here is another in the recent spate of mathematics posts. I take up now, as I did in my last post, some material that I had originally drafted for the first post in this series.

Whenever it has been designated for its own post, material can grow, as has the material of this post in the drafting. Large parts of this post are taken up with

  1. the notion (due to Collingwood) of criteriological sciences, logic being one of them;

  2. Gödel’s logical theorems of completeness and incompleteness.

I have defined mathematics as the science whose findings are proved by deduction. This definition does not say what mathematics is about. We can say however what logic is about: it is about mathematics quâ deduction. This makes logic a criteriological science, since it seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are that

  • everything true in all possible mathematical worlds can be deduced;

  • some things true in the world of numbers can never be deduced;

  • the latter theorem is one of those things.

The present post has the following sections.

Entailment

My definition of mathematics is like that of Bertrand Russell in The Principles of Mathematics, whereby the subject consists of deductions in the sense of propositions “p implies q.” However, Russell’s definition leaves us out, as would a definition of physics as consisting of the laws of nature.

Kinds of Science

We can classify sciences as empirical, normative, or criteriological, according as they examine how things are, how we want them to be, or how we want ourselves to be. If this classification is exhaustive, then mathematics is empirical; but it is also in a class by itself.

As scientists, we judge whether we achieve the goals that we set for ourselves in our work. Thus we have freedom of will, and criteriological sciences study this freedom.

Physicist Sabine Hossenfelder denies our freedom, as in a video, “You don’t have free will, but don’t worry,” released during the composition of the present post. The video comes with this summary:

In this video I explain why free will is incompatible with the currently known laws of nature and why the idea makes no sense anyway. However, you don’t need free will to act responsibly and to live a happy life, and I will tell you why.

Raymond Smullyan has a good account of why there is free will, in his dialogue “Is God a Taoist?” reprinted

  • in Douglas R. Hofstadter and Daniel C. Dennett (editors), The Mind’s I: Fantasies and Reflections on Self & Soul (Basic Books, 1981);

  • by me on this blog.

I shall be looking below at Hofstadter’s ideas as expressed in Gödel, Escher, Bach (Basic Books, 1979).

Perhaps Hossenfelder would say Smullyan’s is an argument for responsibility, not freedom. To me it makes no sense to speak of responsibility without freedom, as I said in “Antitheses,” when critiquing Hossenfelder’s 2016 blog post, “Free will is dead, let’s bury it.” By my account now, Hossenfelder is effectively denying that there can be such sciences as history and logic. She might tell me that those sciences do not do what I say they do, or that my understanding of freedom is not hers. My understanding of physics is that it studies things insofar as they do not have free will; thus physics cannot discover an absence of freedom.

Mathematics and Logic

Mathematics is empirical, in its own peculiar way; logic is criteriological, for helping us get straight what we are trying to do with it.

Gödel’s Theorems

Gödel’s one completeness theorem and two incompleteness theorems tell us what we can and cannot hope to prove in mathematics.

Logic of Mathematics

Logic has allowed mathematics to come into its own as the deductive science.


Entailment

Mathematics proves that certain conclusions are necessary conditions of certain assumptions. The other way around, the assumptions are sufficient conditions for the conclusions. In other words, certain postulates entail certain theorems; symbolically, for some instances of p and q,

pq,

p implies q.”

Such an account of mathematics resembles Bertrand Russell’s, in The Principles of Mathematics:

Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form.

Timothy Gowers makes that quotation of Russell, along with one more sentence, to be taken up later, in the Preface of the big book called The Princeton Companion to Mathematics (Princeton University Press, 2008; xx + 1034 pages). I referred to this book in

According to Gowers himself,

The Princeton Companion to Mathematics could be said to be about everything that Russell’s definition leaves out.

Russell’s book was published in 1903, and many mathematicians at that time were preoccupied with the logical foundations of the subject. Now, just over a century later, it is no longer a new idea that mathematics can be regarded as a formal system of the kind that Russell describes, and today’s mathematician is more likely to have other concerns. In particular, in an era where so much mathematics is being published that no individual can understand more than a tiny fraction of it, it is useful to know not just which arrangements of symbols form grammatically correct mathematical statements, but also which of these statements deserve our attention.

In short, the book gives an idea of what mathematics is about.

Kinds of Science

There is a distinction between descriptive and prescriptive science; alternatively, between empirical science and normative science. I have written about a third kind of science, which Collingwood calls criteriological.

  1. Empirical science is about how things are.

  2. Normative science is about how we try to make things.

  3. Criteriological science is about how we try to make ourselves.

The empirical sciences are the natural sciences: the sciences of things that are natural. Natural things cannot be any other way than they already are. At least, they are not trying to be any other way; or if they are, as when they are animals as studied by Tinbergen and colleagues, then they don’t know it.

Engineering and medicine are normative sciences, because they study how to make things be some way, according to the aims of those persons who want them to be that way.

A criteriological science like aesthetics or jurisprudence studies our aims as such. Having itself an aim, which involves being true or correct, a criteriological science comes under its own purview.

One may want to emphasize that a normative science not only studies how to do things, but also does them. One may then call the science an art or, if that usage is old-fashioned, a skill.

Any scientific pursuit may have empirical, normative, and criteriological aspects. It will have the last when establishing a code of conduct for itself.

A reason for distinguishing the kinds of science is the hope of doing better what we do, when we know what it is.

Richard Feynman tells a story about confirmation bias in replications of the Millikan Oil Drop Experiment. When researchers followed Millikan’s example in inferring from their experimental data a unit of charge (namely the charge on a so-called electron), they threw out results that were too far from Millikan’s, until they understood that his were off.

The story is currently quoted on Wikipedia. Feynman continues it, in “Cargo Cult Science,” as reprinted in “Surely You’re Joking, Mr. Feynman!” (Norton, 1985):

But this long history of learning how to not fool ourselves—of having utter scientific integrity—is, I’m sorry to say, something that we haven’t specifically included in any particular course that I know of. We just hope you’ve caught on by osmosis.

If the ethics of physics is indeed not taught in physics courses, presumably this is because it is not strictly part of physics. Physics is a natural science; ethics is not. However, every scientist needs to be ethicist enough to understand Feynman’s advice:

The first principle is that you must not fool yourself—and you are the easiest person to fool. So you have to be very careful about that. After you’ve not fooled yourself, it’s easy not to fool other scientists. You just have to be honest in a conventional way after that.

I would like to add something that’s not essential to the science, but something I kind of believe, which is that you should not fool the layman when you’re talking as a scientist.

I quoted that last sentence and more in “Be Sex Binary, We Are Not,” because I thought Feynman’s advice was being ignored in a Scientific American blog post.

All of this puts me in mind of Plato’s Gorgias dialogue, in which the title character agrees with the account of his profession that Socrates offers (455A; Lamb’s translation in the Loeb Classical Library; emphasis mine):

And so the rhetorician’s business is not to instruct a law court or a public meeting in matters of right and wrong, but only to make them believe; since, I take it, he could not in a short while instruct such a mass of people in matters so important.

Gorgias boasts that if a town is hiring a physician, then a rhetorician can pad his résumé better than any actual physician, and get the job; however, teachers of rhetoric such as himself should not be blamed when their students misuse what they have learned (456A–D). Socrates presses him (459B–C):

Then the case is the same in all the other arts for the orator and his rhetoric: there is no need to know the truth of the actual matters, but one merely needs to have discovered some device of persuasion which will make one appear to those who do not know to know better than those who know.

Gorgias again boasts in reply (459C):

Well, and is it not a great convenience, Socrates, to make oneself a match for the professionals by learning just this single art and omitting all the others?

Socrates asks if the rhetorician is as ignorant of justice and goodness as he is of medicine and other arts. Gorgias is reluctant to admit it, but says that the student of rhetoric will have to learn the difference between right and wrong (460A). But now Gorgias has contradicted himself, since nobody can do wrong who knows what it is, and yet Gorgias has admitted that his students may do wrong.

Everything that we do, we do initially for its own sake, because we cannot know what will come of it. As infants we have to utter sounds at random before we can learn to control them and use them to get what we want. We do research out of pure curiosity, before learning how the research can serve other interests.

Then one can find ways to abuse research, or one’s standing as a scientist. “For example,” says Feynman (loc. cit.),

I was a little surprised when I was talking to a friend who was going to go on the radio. He does work on cosmology and astronomy, and he wondered how he would explain what the applications of this work were. “Well,” I said, “there aren’t any.” He said, “Yes, but then we won’t get support for more research of this kind.” I think that’s kind of dishonest.

“Kind of”?

Logic

Logic aims to provide, for science in general, and especially mathematics, a code of conduct that is agreeable to the practitioners: a code not for dealing with the public, but for establishing theories and, in mathematics, proving theorems. That is what I would say, though perhaps the current Wikipedia article called “Conceptions of logic” does not really cover this conception.

In the broad sense, “logic is the analysis and appraisal of arguments,” as the main Wikipedia article on the subject says. I would emphasize that the analysis and appraisal are of arguments as such, arguments quâ arguments. These are intended to have a certain effect, and they may succeed or fail in achieving this end, according to the people making the arguments.

For example, in Propositions 5, 6, 18, and 19 of Book I of the Elements, Euclid proves that, in a triangle,

  1. equal sides subtend equal angles,

  2. equal angles are subtended by equal sides,

  3. the greater side subtends the greater angle,

  4. the greater angle is subtended by the greater side.

The last of these follows from the first and third by pure logic, as Euclid’s proof reflects; but so does the second, and it is not needed for the proof of the third. Logically then, Euclid could have dispensed with his proof of the second. If logic were normative, it might convict Euclid of the style error of proving what didn’t need proof. As a criteriological science, logic can only ask whether Euclid would accept style advice on this point.

Logic is called that and not logics, although there are other sciences called ethics and physics, because Aristotle wrote collections of books called, respectively, Ethics and Physics, but not a collection of “logics”; he wrote Analytics instead. Collingwood points this out in a footnote on the first page of An Essay on Metaphysics (Oxford, 1940). He accounts for logic and ethics as criteriological sciences in Chapter X (pp. 108–9; bold emphasis mine):

the Greeks … constructed a science of theoretical thought called logic and a science of practical thought called ethics. In each case they paid great attention to the task of defining the criteria by reference to which theoretical and practical thought respectively judge of their own success. In view of this … these sciences have been traditionally called normative sciences. But the word ‘normative’ may prove misleading … as if it were for the logician to decide whether a non-logician’s thoughts are true or false and his arguments valid or invalid, and for the student of ethics to pass judgement on the actions of other people as having succeeded or failed in their purpose. This suggestion is incorrect. The characteristic of thought in virtue of which a science of thought is called normative consists … in the necessity that in every act of thought the thinker himself should judge the success of his own act … I propose to substitute for the traditional epithet ‘normative’ the more accurate term ‘criteriological’.

The chapter is called “Psychology as the Science of Feeling,” because that is what psychology was created to be, when people recognized that, not being self-critical, feeling was not thought and therefore must be studied by a science different from logic and ethics. Psychology is non-criteriological, and you can understand this, even if you think that everything in nature behaves “teleologically.” Things can have purposes or ends, but if the things don’t know this, then you won’t be studying them criteriologically.

Collingwood traces the origin of psychology to the sixteenth century, but gives no details. The Oxford English Dictionary traces it more specifically to sixteenth-century Germany, and in particular to Melanchthon, though the earliest recorded use of the term in English is from a 1693 translation of Steven Blanchard’s Lexicon Medicum. Wikipedia spells the author’s surname as Blankaart, but traces the word “psychology” to a Croatian humanist, Marko Marulić, author of Psichiologia de ratione animae humanae, 1510–17.

Collingwood’s next chapter is called “Psychology as the Pseudo-science of Thought,” because a program to study thought empirically can only be a mistake or a fraud. Such a program was nonetheless urged in the eighteenth century, and possibly for the good reason that logic and ethics had ceased being strictly criteriological, but had become normative in the sense of imposing standards from outside (pp. 114–5):

It might very well be true that a revolt against the old logic and ethics had been desirable and had proved beneficial; for it might very well be true that people who professed those sciences had misunderstood their normative character, and had claimed a right of censorship over the thoughts and actions of other people; and for the sake of scientific progress such tyranny might very well have to be overthrown. When it is a case of overthrowing tyranny one should not be squeamish about the choice of weapons. But the tyrannicide’s dagger is not the best instrument for governing the people it has liberated.

I have elsewhere criticized Collingwood’s violent language, along with my own grandfather’s writing that “we were too squeamish” to fight the Vietnam War properly.

I find in a recent Guardian Weekly an example where a criteriological science, here economics, is treated as normative. Given courtroom evidence, people were asked to make a case, either for the plaintiff or the defense. Then they were asked to predict what the judge had done. According to Tim Harford in “Feeling is believing” (Guardian Weekly vol. 203 no. 14, 18 September 2020; online as “Facts v feelings: how to stop our emotions misleading us”),

Their predictions should have been unrelated to their role-playing, but their judgment was strongly influenced by what they hoped would be true.

Psychologists call this “motivated reasoning”. Motivated reasoning is thinking through a topic with the aim, conscious or unconscious, of reaching a particular kind of conclusion.

I have said it before: all reasoning is motivated reasoning. We engage in it for a reason! In the present “real court case about a motorbike accident,” it was for the judge to decide what “should have” happened; it was for the experimental subjects of Linda Babcock and George Loewenstein to argue, hypothetically, about what “should have” happened; it was not for the two economists to say what those subjects “should have” decided.

Logic has been wrongly made normative if used to “correct” Mick Jagger for singing,

I can’t get no satisfaction.

The pedant may assert that what Jagger is “really” saying here, “logically,” is,

I always get satisfaction.

That is not what he is saying. He is saying what he means, or what his persona as rock-n-roll singer means; and what that persona means, in “standard” English, is,

I can never get any satisfaction.

And this is only an approximate translation. The full translation, into English, would just be the original lyric, which is already in English.

Mathematics and Logic

In the address that I discussed in “What Mathematics Is,” Euphemia Lofton Haynes identifies logic and mathematics; but there’s a difference. If mathematics is one of the sciences fitting the three-part classification above, it is a descriptive or empirical science. However, again, logic is the criteriogical science of how sciences justify their findings.

My specialty within mathematics is model theory, which is said to be a part of mathematical logic. It is just mathematics though, albeit mathematics made possible by the development of symbolic logic.

A senior colleague in model theory once suggested to me that logic is about something, namely reason, as physics is about the physical world; but mathematics as such is about nothing. This sounded reasonable; however, when I asked this person about it in a later year, he did not particularly remember what he had said (and this is why I am not naming him).

Logic is associated with mathematics, because mathematics is the science that logic has best illuminated. I have written about Wigner’s notion of an “unreasonable effectiveness of mathematics in physics.” One might speak also of the unreasonable effectiveness of logic in mathematics.

I ended “What Mathematics Is” by saying that logically possible worlds are worlds that can be deduced from postulates. This was glib. The properties of a mathematical world are deduced from the postulates of that world. The properties cannot be observed in the conventional sense, by sense, such as hearing, seeing, and touching. To say that a mathematical world exists in the first place is to say that its postulates are consistent; and this we can prove, only by assuming the consistency of some other postulates.

Today those other postulates are normally the Zermelo–Frankel axioms of set theory, with the Axiom of Choice, composing the collection called ZFC. However, by Gödel’s Second Incompleteness Theorem, ZFC does not entail its own consistency, if it is consistent.

Gödel’s Theorems

I have already reviewed both of Gödel’s incompleteness theorems. I do it here now, somewhat differently: usually more tersely, but sometimes in more detail, or in another way.

Gödel’s First Incompleteness Theorem is that there is not, and cannot be, an algorithm for identifying all true statements about numbers. By numbers I mean counting numbers: 1, 2, 3, and so on. By a statement about them, I mean, technically, a formula that has no free variables, in a sense to be made more precise later. Meanwhile, in our context, a formula is an expression built up from polynomial equations by finitely many applications of the logical operations of

  • conjunction (forming pq, “p and q,” from p and q);

  • negation (forming ¬p, “not p,” from p);

  • disjunction (forming pq, “p or q”; but this is already ¬(¬p ∧ ¬q));

  • implication (forming pq; but this is ¬pq);

  • universal quantification (forming ∀x p, “For all x, p,” from p and a variable x).

  • existential quantification (forming ∃x p, “For some x, p”; but this is ¬∀x ¬p).

In our polynomial equations, the only parameter is 1. However, from 1 we can build up the other counting numbers as the sums 1 + 1, 1 + 1 + 1, and so on, these themselves being polynomials, namely constant polynomials.

The definition of truth is fairly straightforward. An equation of constant polynomials is true or false in the obvious sense; an equation of arbitrary polynomials is true if it is true for all values of its variables; the definition of truth for more complicated formulas follows accordingly.

There is a stronger notion. A formula is logically true if it is true in the sense above, regardless of how values are assigned to variables, sums, and products. The formula

x = yx + z = y + z

is logically true. The statement

xy xy

is true, but not logically true, since it is false if every variable, and in particular x and y, can take only the same value. However,

x x = x

is logically true.

There are algorithms for generating some true statements about numbers:

  1. Start with some statements known to be true of numbers: these statements are now postulates.

  2. Derive new formulas by means of some rules of inference that are known to preserve truth. Such rules will normally include

    • modus ponens: from p and pq, derive q;

    • generalization: from p, derive ∀x p;

    • logical axioms: from nothing at all, derive

      • tautologies, such as p ∨ ¬p (these are the logically true formulas that involve no quantifiers);

      • other formulas that are logically true, such as

        • laws of equality such as x = x,

        • x pp(c), where p(c) is the result of replacing each free occurrence of x in p with c, this being a constant polynomial.

The derived formulas that are statements are theorems, because they have proofs, which show how to derive them from the postulates by means of the rules of inference. The theorems constitute a theory of the counting numbers.

We are now using the word “theorem” in two ways.

  1. Gödel’s theorems are logical theorems.

  2. The statements about numbers that are theorems in the sense just defined are mathematical theorems.

One may distinguish between

  • the logical axioms, which derive mathematical theorems from nothing;

  • the other rules of inference, which have to start with something.

The logical axioms and the rules of inference then constitute a proof system. Before the incompleteness theorems came Gödel’s Completeness Theorem, whereby there are proof systems that yield every logically true statement as a theorem.

We shall henceforth assume that our proof system is complete in this sense.

Gödel’s First Incompleteness Theorem is now that from no postulates can all true statements about numbers be derived as theorems.

We are not allowed to take all of those true statements as postulates in the first place. I meant to suggest this by saying that our postulates must be known to be true of the counting numbers. In particular, though the postulates may be infinitely numerous, there has to be a mechanical rule for writing them down. In technical terminology, to be made a bit more precise later, the postulates have to be recursively enumerable.

The proof of Gödel’s First Incompleteness Theorem relies on the possibility of turning every formula into a number itself. This number is the Gödel number of the formula. Strictly, it is the set of Gödel numbers of our postulates that has to be recursively enumerable.

Instead of assigning to every formula a number, we could assign a set; then, by Gödel’s method, we could prove that no theory of sets, such as ZFC, is complete. The method applies to any mathematical structure whose elements we can manipulate in a way that mimics how we form postulates and apply rules of inference to them.

Assigning Gödel numbers to formulas is then like assigning letters to sounds, the way the Phoenicians did in creating the alphabet.

A better metaphor is turning grammatical sentences into nouns, as with use of quotation marks or a determiner such as “that” or “whether”:

  • Are you coming?

  • I said, “Are you coming?”

  • I want to know whether you are coming.

  • I know that you are coming.

We assign Gödel numbers to formulas quâ strings of symbols, and we can recover the formulas from the numbers. In the same way, we assign Gödel numbers to lists of formulas. Such a list could be a proof. Whether it is a proof is something that can be recognized from the Gödel number. Saying that a certain formula about numbers has a proof now means saying that a number with a certain property exists; that the formula has no proof means the number doesn’t exist.

In describing logical axioms, I said p(c) was the result of replacing each free occurrence of x in p with c. The rules are:

  1. Every occurrence of a variable in a polynomial equation is free.

  2. Being a free occurrence in a formula is preserved when the formula is negated or conjoined with another, but not when the variable is quantified.

Thus in the formula x = x ⇒ ∃x xx = x, the first two occurrences of x are free; the remaining four are not free. To say that a variable is free in a formula is to say that it has a free occurrence in the formula; it may also have a non-free occurrence, though we can usually arrange for this not to happen, as for example by writing the formula above as x = x ⇒ ∃y yy = y.

A formula with a free variable is effectively a predicate with unspecified subject. Given a formula p with a free variable, we can take any counting number a and, using Gödel numbers, form the statement that p(a) has no proof. We shall be interested in the case where a is the Gödel number of p.

Another level of abstraction is possible. We write the Gödel number of any formula p as ⌜p⌝. There is a formula φ with one free variable such that, for all statements s,

φ(⌜s⌝) is true if and only if s is a theorem.

There is now a formula ψ with a free variable such that, for all formulas p with a free variable,

¬φ(⌜p(⌜p⌝)⌝) is ψ(⌜p⌝).

If we think of forming the Gödel number of a formula as quoting the formula, then ψ is the predicate, “yields an unprovable statement when predicated of the quotation of itself.”

We can predicate ψ of the quotation of itself in this sense, forming the statement ψ(⌜ψ⌝). If we write this statement as σ, then it is also ¬φ(⌜σ⌝). Briefly, σ says, “I have no proof.” It doesn’t really have a first-person pronoun though; what σ says is,

“Yields an unprovable statement when predicated of the quotation of itself” yields an unprovable statement when predicated of the quotation of itself.

Upon reflection, we see that the particular predication that this statement is about—which is a predication that has no proof—is just the statement σ itself. If σ were false, then φ(⌜σ⌝) would be true, and thus σ would be a theorem. Since theorems are true, σ must be true; but then it cannot be a theorem. This proves Gödel’s First Incompleteness Theorem.

My use of the quotation of predicates is based on the dialogue that begins on page 431 of Douglas R. Hofstadter’s book of xxii + 778 pages called Gödel, Escher, Bach: An Eternal Golden Braid (Basic Books, 1979). The dialogue is called “Air on G’s String,” and in it, the Tortoise recounts to Achilles the receiving of an “obscene” phone call. Written out and punctuated correctly, what the caller says is,

“Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.

It’s a way of saying “I am lying,” without actually saying “I.” It seems I have remembered it from reading Hofstadter’s book when the paperback edition came out, in 1980, when I turned fifteen.

I enjoyed reading the book then. Its main benefit may have been to lead to me to learn more about Zen Buddhism, first through Reps and Senzaki, Zen Flesh, Zen Bones (1957).

I cannot have understood Gödel’s Theorem from Hofstadter. I am still trying to understand it now. In graduate school, a professor told me he hoped Hofstadter himself had not understood Gödel’s Theorem, because teaching it with a book like Hofstadter’s would be unconscionable.

I am not naming the professor, because I may not have thoroughly understood and properly recalled his meaning. For the same reason, I am not naming the tutor who told me, in my freshman year at college, that Gödel’s Theorem need not rely on self-reference, contrary to Hofstadter’s evident belief.

That tutor may have been alluding to how Gödel’s Theorem derives from the Halting Problem, as follows.

In principle we can make a list of all computer programs that can be set to work on single numbers as input. We can denote the nth program on the list by {n}. Working on a number k, this program may halt and produce some output, or it may never stop running. For example, the program may seek the least twin prime that is greater than k; if the Twin Prime Conjecture is false, and k is large enough, then the program will never halt.

The set of all numbers at which a given program halts is, by definition, a recursively enumerable set. For example, the set of twin primes is recursively enumerable, whether it is finite or infinite. We can list the elements of a recursively enumerable set in stages. At stage k, on all numbers that do not exceed k, we have run the program for k steps, and we have written down the numbers for which the program has halted so far.

There is a recursively enumerable set of numbers whose complement is not recursively enumerable. This means, by definition, that the original set is not recursive. I looked at this set at the end of “Hypomnesis,” which is about a logic meeting in Delphi in 2017; but let me define the set again here. There is a program that does at k what {k} does. This program must be {m} for some m. But then the set of numbers where {m} does not halt cannot be recursively enumerable; for if it were, then it would be, for some k, the set of numbers where {k} halted. In this case {k} would halt at k if and only if it did not.

For each number k, there is a statement pk that the program {k} does not halt at k. Then the set of k for which pk is true is not recursively enumerable. Hence also the set of Gödel numbers ⌜pk⌝ of the statements pk that are true is not recursively enumerable. However, the set of Gödel numbers of all of the statements pk is recursively enumerable. Therefore the set of Gödel numbers of all true statements about the counting numbers cannot be recursively enumerable.

Now we have proved Gödel’s First Incompleteness Theorem a second time. The second proof does not involve a proof system. In particular, it does not give us a particular statement that cannot be proved; it just shows that such a statement will always exist, regardless of our proof system. Much less does the proof give us a statement that says, “I cannot be proved.” The proof still uses self-reference, in the sense of applying a program to its own number. It doesn’t really need Gödel numbers in the precise sense; however, the whole notion of a computer program is an analogue of Gödel numbering.

After the dialogue of the Tortoise and Achilles that I have referred to, Hofstadter has a chapter called “On Formally Undecidable Propositions of TNT and Related Systems.” The title comes from that of Gödel’s 1931 paper by substitution of TNT (“typographical number theory”) for Principia Mathematica. Hofstadter says his chapter will be “more intuitive” than Gödel’s article, and

I will stress the two key ideas which are at the core of [Gödel’s] proof. The first key idea is the deep discovery that there are strings of TNT which can be interpreted as speaking about other strings of TNT; in short, that TNT, as a language, is capable of “introspection”, or self-scrutiny. This is what comes from Gödel-numbering. The second key idea is that the property of self-scrutiny can be entirely concentrated into a single string; thus that string’s sole focus of attention is itself. This “focusing trick” is traceable, in essence, to the Cantor diagonal method.

We used the diagonal method to produce a recursively enumerable set that is not recursive. Indeed, we can think of a table in which the entry in row k and column m is

  • 1, if {k} halts at m;

  • 0, otherwise.

Each row then is a string of 0s and 1s. The string of entries on the diagonal is also row m, if again {m} is the program that does at each k what {k} does. The string that we get from that row by interchanging 0 and 1 can therefore occur as no row; thus there is no program that halts precisely where {m} does not.

Hofstadter continues:

In my opinion, if one is interested in understanding Gödel’s proof in a deep way, then one must recognize that the proof, in its essence, consists of a fusion of these two main ideas. Each of them alone is a master stroke; to put them together took an act of genius. If I were to choose, however, which of the two key ideas is deeper, I would unhesitatingly pick the first one—the idea of Gödel-numbering, for that idea is related to the whole notion of what meaning and reference are, in symbol-manipulating systems. This is an idea which goes far beyond the confines of mathematical logic, whereas the Cantor trick, rich though it is in mathematical consequences, has little if any relation to issues in real life.

One could question Hofstadter’s (or anybody’s) presumption to

  • understand Gödel’s proof “in a deep way,”

  • be able to recognize a master and a genius,

  • know all about “real life.”

For one thing, mathematics is a part of real life, as opposed to illusory life, unless indeed one harbors illusions about what one knows, or tries to create illusions in others’ minds about what one knows.

In his article “Gödel’s Theorem” in the Princeton Companion, Peter J. Cameron also says that the theorem is based on two ideas. Cameron’s first idea is Hofstadter’s, which is Gödel numbering; but Cameron’s second is just self-reference, as in the forming of the statement ψ(⌜ψ⌝).

I don’t know that either person’s “two ideas” are really two. Gödel numbering is an embedding of the logic of arithmetic in arithmetic itself. Embeddings as such are everywhere in mathematics: there’s the embedding of the plane in space for example, or of the counting numbers in the integers or the positive rational numbers. Gödel numbering is special for letting arithmetic be about not just numbers, but itself. Our second proof of Gödel’s theorem seemingly used diagonalization in place of Gödel numbering; but a diagonal argument relies on the possibility that rows and columns of the same table can “speak” about one another in the sense of bearing the same serial number. Moreover, the numbering of programs is like Gödel numbering in the sense that we can mechanically obtain the program {k} from k; this is why there can be a statement, in our formal sense, that {k} does not halt at k.

It remains to establish Gödel’s Second Incompleteness Theorem, that the consistency of our postulates for the numbers is not a theorem. The consistency is a statement though, namely the statement that Λ has no proof, where Λ is a logically false statement, such as ∃x xx.

We want then to show that

¬φ(Λ)

is not a theorem, since this is the statement that Λ has no proof. We already know that

¬φ(σ)

is not a theorem, where σ is ψ(⌜ψ⌝), where ψ(⌜p⌝) is ¬φ(⌜p(⌜p⌝)⌝), so that ¬φ(⌜σ⌝) is just σ. Thus it is enough to show that the implication

φ(⌜σ⌝) ⇒ φ(⌜Λ⌝)

is a theorem. There is a proof of this statement from

φ(⌜σ⌝) ⇒ φ(⌜¬σ⌝),

and of this statement from

φ(⌜σ⌝) ⇒ φ(⌜φ(⌜σ⌝)⌝).

This last statement is a theorem, regardless of what σ is. So Gödel’s Second Incompleteness Theorem holds. (The sketch of the proof is based on that of C. Smorynski in “The Incompleteness Theorems” in Jon Barwise, editor, Handbook of Mathematical Logic, Elsevier, 1977.)

As telling us what we can and cannot prove in mathematics, Gödel’s Completeness Theorem and First Incompleteness Theorem are theorems of logic. We can turn the latter into a mathematical theorem, and this is how the Second Incompleteness Theorem is proved.

Logic of Mathematics

In the first section, “Entailment,” I left off the last sentence of Gowers’s quotation of Russell:

In addition to these [propositions of the form “p implies q”], mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.

That was in 1903, Russell having been born in 1872; Gödel would be born in 1906.

What Russell says about truth in the book is provoked by a paradox concerning the rule of modus ponens that Lewis Carroll presented in the dialogue, “What the Tortoise Said to Achilles.” Along with his own dialogues involving the same characters, Hofstadter will reprint Carroll’s dialogue in Gödel, Escher, Bach. The paradox is that modus ponens is about implication, but is also itself an implication. It is that p and the implication pq together imply q. If we now write the rule as the same kind of implication, namely

p ∧ (pq) ⇒ q,

then, to use the rule, we need another rule, that p and pq and the rule just given together imply q: in short,

p ∧ (pq) ∧ (p ∧ (pq) ⇒ q) ⇒ q.

But now this is not enough. If we define propositions pn recursively so that

p1 is pq,

pn+1 is pp1 ∧ … ∧ pnq,

then all of these together cannot compel us to accept q.

That is so, I would say, because logic is not normative, but criteriological. It cannot tell us what we must do; it observes what we do do. One thing we do is draw conclusions that accord with the rule called modus ponens.

In the first proposition of the first book of the Elements, given a segment ΑΒ and the possibility of drawing circles, Euclid constructs a triangle ΑΒΓ in which ΑΓ = ΑΒ and ΒΓ = ΒΑ. Since ΑΒ and ΒΑ are the same segment, we know that ΑΓ = ΒΓ. Thus the sides are all equal to one another, and therefore, by definition, the triangle is equilateral.

The sides ΑΓ and ΒΓ are equal to one another, because

  • each is equal to the same side, written indifferently as ΑΒ or ΒΑ;

  • equals to the same are equal to one another.

This is only an explanation of something that we already know.

We have to know too that the point Γ exists in the first place as an intersection of the circles, each of which has one of Α and Β as center and passes through the other.

Some modern readers complain that Euclid applies no explicit rule whereby those circles must intersect. Such readers are like the visitors that Ruth Fuller Sasaki describes in “Zen: A Method for Religious Awakening” (1959; reprinted in The World of Zen, edited by Nancy Wilson Ross, 1960; I bought the book in 1981):

How many hours have I not spent in my Kyoto temple listening to people, usually Americans recently come to Japan, tell me just what Zen is. To such visitors I have nothing to say; to those who do not understand, I am always searching for a way to give a clue to what Zen is about.

Mathematicians such as Timothy Gowers are keen on searching for a way to give a clue to what mathematics is about. To me the most important clue lies in a chapter of Gowers’s little book called Mathematics: A Very Short Introduction (Oxford, 2002; xiv + 133 pages). Chapter 3 is called “Proof,” and there Gowers observes,

the steps of a mathematical argument ean be broken down into smaller and therefore more clearly valid substeps. These steps can then be broken down into subsubsteps, and so on. A fact of fundamental importance to mathematics is that this process eventually comes to an end. In principle, if you go on and on splitting steps into smaller ones, you will end up with a very long argument [that] starts with axioms that are universally accepted and proceeds to the desired conclusion by means of only the most elementary logical rules (such as ‘if A is true and A implies B then B is true’).

I would propose a couple of adjustments.

  1. Some axioms, such as the Axiom of Choice, are not universally accepted. If you prove a proposition P using this axiom, then everybody will agree that at least you have proved the proposition that AC implies P.

  2. Possibly not everybody will agree, even then, if for example you have used the logical rule of the excluded middle (“either Q or not-Q”). Intuitionism rejects this rule. But an Intuitionist should still be able to tell whether an argument is correct within a system that does allow use of the rule of the excluded middle. The Intuitionist will then prefer to find an argument, a “constructive” argument, that does not need this rule, and a mathematician of any school can confirm the construction.

Gowers goes on to say:

What I have just said in the last paragraph is far from obvious: in fact it was one of the great discoveries of the early 20th century, largely due to Frege, Russell, and Whitehead (see Further reading). This discovery has had a profound impact on mathematics, because it means that any dispute ahout the validity of a mathematical proof can always be resolved

… the fact that disputes can in principle be resolved does make mathematics unique. There is no mathematical equivalent of astronomers who still believe in the steady-state theory of the universe, or of biologists who hold, with great conviction, very different views about how much is explained by natural selection, or of philosophers who disagree fundamentally about the relationship between consciousness and the physical world, or of economists who follow opposing schools of thought such as monetarism and neo-Keynesianism.

I have become an exponent of this idea, that there is one field of human endeavor where all disputes can be settled amicably.

My wife used the idea in her courtroom defense against the accusation of being a terror-propagandist.

I would summarize the idea as being that mathematics is the deductive science. Gowers doesn’t say it that way, though he uses the word deduction:

No mathematician would ever bother to write out a proof in complete detail—that is, as a deduction from basic axioms using only the most utterly obvious and easily checked steps.

It seems to me that everybody who learns anything about mathematics should learn its deductive nature. Otherwise they haven’t really learned mathematics. The idea of deduction will however need building up over the years of one’s education. It has already needed building up in the millenia since Euclid.

Revised and corrected, January 12, 2021

Map of Art

The bulk of this post is a summary of the chapter on art in Collingwood’s Speculum Mentis: or The Map of Knowledge (1924). The motto of the book is the first clause of I Corinthians 13:12:

Βλέπομεν γὰρ ἄρτι δι’ ἐσόπτρου ἐν αἰνίγματι

For now we see through a glass, darkly

The chapter “Art” has eight sections:

  1. Art as Pure Imagination

  2. The Work of Art

  3. The Monadism of Art

  4. Meaning in Art

  5. Knowledge as Question and Answer

  6. Art as a form of Error

  7. The Dialectic of Art

  8. Play

The summary that I give below is arranged according to these sections.

Art is one instance of an activity that may succeed or fail. In the strictest sense, every activity has this property; for there is a distinction between what we are doing and what we are trying to do.

I begin by observing that, with reference to quoted words of Agnes Callard about Aristotle. Callard happens to be named in the acknowledgments of Lost in Thought by Zena Hitz, whose words about analytic philosophy I shall ultimately quote as well.

Editing this post from a certain email that I drafted on July 26, I have removed (as not having been intended for publication) some words of other persons that I had quoted from a 2012 email discussion of art and Speculum Mentis.


On July 21, Agnes Callard had a good essay in the New York Times, “Should We Cancel Aristotle?” Sure, Aristotle liked slavery and stuff; but reading him – reading him literally – doesn’t mean we agree with him. Neither does it make us agree with him, unless perhaps he is right after all. In Callard’s words:

Yet I would defend Aristotle, and his place on philosophy syllabuses, by pointing to the benefits of engaging with him. He can help us identify the grounds of our own egalitarian commitments; and his ethical system may capture truths – for instance, about the importance of aiming for extraordinary excellence – that we have yet to incorporate into our own.

And I want to go a step further, and make an even stronger claim on behalf of Aristotle. It is not only that the benefits of reading Aristotle counteract the costs, but that there are no costs. In fact we have no reason at all to cancel Aristotle. Aristotle is simply not our enemy.

I, like Aristotle, am a philosopher, and we philosophers must countenance the possibility of radical disagreement on the most fundamental questions. Philosophers hold up as an ideal the aim of never treating our interlocutor as a hostile combatant. But if someone puts forward views that directly contradict your moral sensibilities, how can you avoid hostility? The answer is to take him literally – which is to say, read his words purely as vehicles for the contents of his beliefs.

Aristotle urges “the importance of aiming for extraordinary excellence.” As philosophers, he and Callard “hold up as an ideal the aim of never treating our interlocutor as a hostile combatant.” They have aims; we have aims; and yet we may not achieve them.

In particular, the words of Aristotle are not simply “vehicles for the contents of his beliefs”; not boxcars in which the Philosopher loaded his beliefs in order to deliver them to us. Before the loading, what could the beliefs have been? The words represent an attempt to work out the beliefs. The attempt may have been only partially successful.

The idea is in Collingwood, who says for example in The Principles of Art (1938):

The proper meaning of a word … is never something upon which the word sits perched like a gull on a stone; it is something over which the word hovers like a gull over a ship’s stern. Trying to fix the proper meaning in our minds is like coaxing the gull to settle in the rigging, with the rule that the gull must be alive when it settles: one must not shoot it and tie it there. The way to discover the proper meaning is to ask not, ‘What do we mean?’ but, ‘What are we trying to mean?’

The word under consideration, by Collingwood then and by me now, is art.

A beginning of such considerations is recorded in “Poetry and Mathematics,” a post beginning with talk about writings in a couple of Australian literary publications. One of the essays was about poetry and mathematics, by a poet who had studied mathematics.

I found some reason to observe how Michael Oakeshott had worked on the theme of Collingwood’s Speculum Mentis: or The Map of Knowledge of 1924.

The book can be classified, with Religion and Philosophy of 1916, as belonging to Collingwood’s juvenilia. This could have made it more accessible than a mature work; but it didn’t.

Hegel’s Phenomenology of Spirit was inaccessible. In my day at St John’s College, we read short excerpts of this book, and if memory serves, one of our two seminar leaders admitted from the start that he didn’t know what the book was about.

The other seminar leader had read the tome in graduate school, but perhaps didn’t feel much more confident about it. He did suggest that just reading the whole thing might be the way to go, if reading little bits at a time did not clear it up.

Nonetheless, reading Speculum Mentis section by section seems to clear some things up for me.

Here is my attempt to summarize the chapter called “Art.”

“Art” is Chapter III, the first two chapters being

  • “Prologue,” which begins, “All thought exists for the sake of action”;

  • “Speculum Mentis,” which begins, “Our task, then, is the construction of a map of knowledge” – spoiler alert, the map never gets constructed.

Chapter III has eight sections. Collingwood will not explain till § 4 that the first three sections give a one-sided account of art, based on Vico and Croce: an account that, since the Renaissance, has tended to replace the ancient account.

The mode of thinking in the chapter then would seem to be hypothetical, so to speak, as art itself is.

§ 1. “Art as pure imagination”

“Art is the simplest and most primitive, the least sophisticated, of all possible frames of mind,” not because only children and “savages” make art or are even best at it, but because art imposes no requirement of literal truth.

For example, it is irrelevant to Cymbeline the play, as a work of art, whether there really was a king called Cymbeline who acted as Shakespeare’s character does.

“A philosophical theory must be capable of being conceived as a whole, a historical narrative, of being narrated as a whole – narrated, that is, as true – a work of art, of being imagined as a whole.”

“Art, then, is pure imagination.”

§ 2. “The Work of Art”

Nonetheless, art is trying to be something: not literally true, but beautiful. Art can fail to be this. The possibility of failure “is the standing refutation of all emotional and sensationalistic theories of art.”

Nonetheless, beauty is not a concept. There are no laws or principles for achieving it. If you try to explain the beautiful in terms of its form, you fail.

This is a paradox, whose resolution is that beauty is not a concept, but “the guise under which concepts in general appear to the aesthetic consciousness.”

The paradox seems real to me. A way I try to understand the resolution is to think of Collingwood’s 1916 essay “The Devil,” in which evil is found to be neither the negation of good, nor the opposite of good, but the counterfeit of good. As with being beautiful, so with being good, there is no law for it. “It is a duty, indeed it is the spring of all moral advance, to criticise current standards of morality,” and yet the essence of evil is also to engage in this criticism.

Meanwhile, being an act of imagination, the work of art is not a physical object. The physical object may help the rest of us achieve the aesthetic activity that the artist has accomplished; but this is not really true, since “one never sees anything in anybody’s work but what one brings to it.”

I would object here that Collingwood (or rather Croce, as Collingwood seems to write in his guise) discounts the possibility that the physical work of art teaches us how to see. We shall come back to this in § 4.

§ 3. “The Monadism of Art”

“Every fresh aesthetic act creates a new work of art, though one such act may last for five years at a time.”

“Works of art always ignore one another and begin each from the beginning: they are windowless monads.”

“Art in its pure form is therefore unaware even that it is imagination; the monad does not know that it is windowless; the artist does not say ‘I am only imagining’, for that would be to distinguish imagination from knowledge, and this he does not do. Hence the aesthetic life of children and uneducated people results in what an unintelligent critic calls lying and hallucination.”

I would suggest, as an example, that the current POTUS does not lie, because he has no conception of the truth. He has been allowed to continue living in the dream-world of a child (and this is a terrible indictment of the United States).

§ 4. “Meaning in Art”

Now Collingwood points out what I said at the beginning, that the foregoing account of art is one-sided.

The “special problem of the philosophy of art to-day” is to reconcile that account with the older one, whereby art could teach moral, religious, or philosophical truths.

Imagination must go to work on something:

“Our dreams have a certain continuity with our waking life, and imagination never cuts itself wholly adrift from fact.”

The idea of art as a kind of dreaming comes back in § 7.

§ 5. “Knowledge as Question and Answer”

To imagine is to suppose, which is to question. To answer is to assert.

“Questioning is the cutting edge of knowledge; assertion is the dead weight behind the edge that gives it driving force.”

I note that Pirsig uses a similar metaphor in Zen and the Art of Motorcycle Maintenance. What he calls Romantic Knowledge is the leading edge of the train of knowledge; Classical Knowledge is the train itself, the engine and cars.

§ 6. “Art as a form of Error”

Art wants only to ask questions. In life you can’t do that. “The artist is an artist only for short times; he turns artist for a while, like a werewolf.”

Thus the life of art is unstable. For example, a school of art, once founded, declines. (Herbert Read opens his Concise History of Modern Painting—which I first read while working on a farm in 1988—by quoting Collingwood on this decline.)

Collingwood does not spell out the reason that I would give for the decline. Founding a school of art (such as Impressionism, or Abstract Expressionism) means figuring out how to do something (e.g. paint shadows with colors, or just splash paint on canvas). This means finding a technique. Once you have the technique, you can go on applying it without any fresh act of imagination.

§ 7. “The Dialectic of Art”

I think Collingwood will suggest such an idea about technique in The Principles of Art. There he finds it important to distinguish art from craft. They are not really separable in fact, and I think Collingwood says this now (in § 7 of Speculum Mentis), though without referring to craft as such:

“The cleavage between means and end, technique and inspiration, talent and genius, materials and result, is inevitable in the life of art, and only those who idealize that life by looking at it from the outside deny the dualism. The artist knows that he can only get his work of art by passing through a non-aesthetic world which is that of facts, training, daily life, and so forth.”

In The Principles of Art, the presence of distinctions between (i) means and end and (ii) raw material and finished product will be two of the differentiae of craft as distinct from art.

Collingwood considers dreams as works of art. A dream has a structure: this is why we can call it a dream, one dream, as opposed to some kind of un-unified mélange. The structure is not “unconscious,” since everything we can know about the dream comes from our being conscious of it. But the structure is implicit until made explicit by psycho-analysis.

Knowing the possibility of such analysis, if you proceed to construct your “dreams” consciously, you have become an artist.

I put “dreams” in quotes because I take the word here to refer to any work of the imagination. The artist “conceives [the work of art] in advance of imagining it, in the sense that at any given moment in the process of creation he has in his mind a criterion which enables him to distinguish between the right and the wrong way of continuing the process of imagination itself.”

The artist’s awareness of such a criterion is why Collingwood will later (in The Principles of Art) coin the term “criteriological” for such sciences as aesthetics (also logic, ethics, and economics; I add the example of grammar, which I used in my best attempt to explicate the notion of criteriological science).

In art there is a distinction between form and content, but if it is “developed,” the work of art ceases to be that: for example, poetry becomes prose.

I mentioned the Australian poet (her name is Anupama Pilbrow) who had studied and written about mathematics. There may be poetry in mathematics, but the way mathematics is communicated is entirely prose.

§ 8. “Play”

As art is the most rudimentary thought, play is the most rudimentary action.

We can find reasons for play, whether childish or adult; but “all such explanations of play are in part mythological and forced.”

“God may be pictured as an artist, or as playing, with far more verisimilitude than as a scientist or a business man.”

Still people will ask why we play. “What is the use?” they ask.

It is no excuse “that the overstrained spirit must be allowed some relief from the burden of its responsibilities; for these responsibilities, properly understood, are nothing but its own highest and freest life, and to face them is to find, not to sacrifice, our happiness.”

Life is always an adventure, and “play, which is identical with art, is the attitude which looks at the world as an infinite and indeterminate field for activity, a perpetual adventure.”


In her recent book Lost in Thought: The Hidden Pleasures of an Intellectual Life (Princeton University, 2020), St John’s College alumna and tutor Zena Hitz says that in grad school, one thing she learned was

the delightful mental gymnastics of analytic philosophy, in which any manner of thesis whatsoever is defended, explored, and almost always refuted.

That is the comic version of a tragedy that Collingwood tells of a senior colleague at Oxford, now counted among the analytic philosophers:

[H. A.] Prichard, developing his extraordinary gift for destructive criticism, by degrees destroyed not only the ‘idealism’ he at first set out to destroy but the ‘realism’ in whose interest he set out to destroy it, and described a path converging visibly, as years went by, with the zero-line of complete scepticism.

That’s from An Autobiography (1939). Speculum Mentis is itself autobiographical, in the sense of reviewing careers that Collingwood might have selected. The book covers in turn art, religion, science, history, and philosophy. In 1939 (the autobiography being with the publisher), Collingwood wrote his wife from Java that he wished he had been a novelist.

Zena Hitz starts her book with a Prologue (“How Washing Dishes Restored My Intellectual Life”), which reviews her life story: growing up in San Francisco in a house full of books; heading east to Annapolis for college; grad school in philosophy; a reconsideration after 2001/09/11; staying with philosophy and getting a job in it; joining the Roman Catholic Church; investigating some convents and moving to a sort of Catholic commune (Madonna House) in Ontario; finding her calling to be back at St John’s.

On Being Given to Know

  1. What if we could upload books to our brains?
  2. What if a machine could tell us what was true?

We may speculate, and it is interesting that we do speculate, because I think the questions do not ultimately make sense—not the sense that seems to be intended anyway, whereby something can be got for nothing.

View from Şavşat

Continue reading

On Causation

Causation seems commonly to be understood as a physical concept, like being a fossil. The paleontologist seeks the one right answer to the question of when a particular dinosaur bone became part of the fossil record; likewise readers of international news seem to think there is one right answer to the question of whether Donald Trump or Ali Khamenei caused the shooting down of Ukraine International Airlines Flight 752 on January 8, 2020.

There is not one right answer. If you are Trump, you caused 176 civilian deaths by attacking the Iranians and provoking their response. If you are Mitch McConnell, you caused the deaths by inhibiting the removal of Trump from office. If you are Khamenei, you did it by meeting Trump’s fire with fire.

Being a cause does not mean you deserve condemnation or praise: that is another matter.

Causation is relative. This is an observation by R. G. Collingwood in An Essay on Metaphysics (1940). Continue reading

Logic of Elliptic Curves

In my 1997 doctoral dissertation, the main idea came as I was lying in bed one Sunday morning. Continue reading

NL XXVI: Democracy and Aristocracy

Index to this series

Executive summary (added September 12, 2018, edited January 26, 2019).

  1. Aristocracy and democracy are abstractions (§§26. 1[0]–28). Inseparable from one another, they are properly understood as correlative rules for the ruling class:

    1. Don’t let in anybody who is unqualified.
    2. Don’t keep out anybody who is.
  2. By what Collingwood will call, in Chapter XXXI of the New Leviathan, the Principle of the Limited Objective, (which was recognized by the Early Church Fathers,) the ruling class should be prepared to solve, not every problem (as Plato wanted), but those problems that are expected to come up (§§26. 3[0]–34).

  3. The democratic and aristocratic principles have been at work throughout the history of Europe: in Greece, in Rome, under the feudalism of the Middle Ages. The French Revolution was not only democratic, but also aristocratic, for aiming to give power to the bourgeoisie, not the entire population (§§26. 4[0]–66).

  4. Adapted from the literary concept of peripety, the concept of revolution actually has no place in history, where there are no heroes or villains, just human beings, partly good and partly bad (§§26. 7[0]–82).

  5. The concept of a revolution in history is even dangerous, if it leads you to think a political problem can have been solved once for all (§§26. 9[0]–96).


There can be no pure democracy, not for a length of time, not if we understand a democracy to be a society whose ruling class is the whole of it. Even the most extreme democrat of ancient Greece never contemplated citizenship for women, slaves, or resident foreigners. We may be more liberal today, at least regarding women and slavery. Still we do not open the ruling class to foreigners, such as myself where I am; nor do we open it to children. We could do so, in the sense of extending the franchise to all human residents. Both possibilities were discussed favorably in the early 1990s in The Nation, as I recall, in one case by a legal minor. However, perhaps most children could not be given such adminstrative duties as used to be assigned by lot to Athenian citizens. Even if they could, what of the animals that live among us—shall they be citizens?


Edward Hicks (American, 1780-1849),
The Cornell Farm, 1848, oil on canvas,
Gift of Edgar William and Bernice Chrysler Garbisch,
National Gallery of Art Continue reading

What It Takes

This essay ends up considering arguments that natural science—especially mathematical physics—is based on absolute presup­positions whose mythological expression is found in Christianity—especially the doctrine of Incarnation.

I take note along the way of continuing censorship of Wikipedia by the Turkish state.

The post falls into sections as follows.

  • Where to start. To the thesis that everybody can be a philosopher, an antithesis is that persons with the professional title of philosopher ought to know the history of their subject.

  • Ontology. Disdain for this history may lead to misunderstanding of Anselm’s supposed proof of the existence of God.

  • Presupposition. To prove anything, you need a pou sto, or what Collingwood calls an absolute presupposition.

  • Progression. Newton rejected antiquated presuppositions.

  • Reaction. Coal-burners and racists reject new presuppositions.

  • Universality. From the 47th chapter of the Tao Te Ching (in the translation of Gia-fu Feng and Jane English):

    Without going outside, you may know the whole world.
    Without looking through the window, you may see the ways of heaven.
    The farther you go, the less you know.

    Thus the wise know without traveling;
    See without looking;
    Work without doing.

  • Religion. To say that we can know the laws governing the entire universe is like saying a human can be God.

  • Censorship. Thus everybody who believes in mathematical physics is a Christian, if only in the way that, by the Sun Language Theory, everybody in the world already speaks Turkish.

  • Trinity. That the university has several departments, all studying the same world—this is supposed to correspond to the triune conception of divinity.

This post began as a parenthesis in another post, yet to be completed, about passion and reason. To anchor that post in an established text, I thought back to David Hume, according to whom,

Reason is, and ought only to be[,] the slave of the passions, and can never pretend to any other office than to serve and obey them.

This might express something I said in my previous post: “Reason is the power of testing what we want.” However, I had not really read Hume since college. I thought more about things that had not ended up in the previous post—which was called “Effectiveness” and concerned the article of Eugene Wigner with that word in its title. As I thought and wrote, it seemed I was putting so much into a parenthesis that it could be another post. True, the same might be said of many things in this blog. In any case, the parenthesis in question became the present post.

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War and Talk

This is a foray into the mystery of how things happen, based the 164th of the 361 chapters of War and Peace. This chapter contains, in a one-sentence paragraph, a summary of Tolstoy’s theory of history:

Each man lives for himself, using his freedom to attain his personal aims, and feels with his whole being that he can now do or abstain from doing this or that action; but as soon as he has done it, that action performed at a certain moment in time becomes irrevocable and belongs to history, in which it has not a free but a predestined significance.

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Freedom to Listen

“It’s a free country, so shut up!”

On Thursday, February 16 of this year (2017), at Bosphorus University, a talk on the subject of freedom of speech was given by a Guardian columnist who was a history professor at Oxford. This was Timothy Garton Ash, who observed that freedom of speech and of the press had been severely curtailed in Turkey. For a defender of the regime, the accusation might be belied by the speaker’s freedom to make it. Academics can still come from abroad and give their critical talks. However, as Professor Garton Ash detailed, many Turkish academics have been fired from their positions; many journalists have been imprisoned; other journalists cannot get their articles published. Continue reading