Polytropy

The main point of this post is to share a passage from an essay by the late William Thurston:

1 What is it that mathematicians accomplish?

… We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.

Therefore, we need to ask ourselves:

2 How do people understand mathematics?

This is a very hard question. Understanding is an individual and internal matter that is hard to be fully aware of, hard to understand and often hard to communicate …

I conclude that there are ethical considerations peculiar to mathematics.

This is brought out by the paper that Thurston is responding to: Jaffe and Quinn, “‘Theoretical mathematics’: toward a cultural synthesis of mathematics and theoretical physics” (Bull. AMS, July 1993). The “synthesis” of the title is not automatic, since the cultures of mathematics and physics differ, particularly regarding what one may assert as if it were true.

The response of Thurston is “On proof and progress in mathematics,” October 26, 1993. I encountered the paper on a blog page featuring quotes on mathematics and the teaching and learning of it.

The blogger, Arkadaş Özakın, is one of the friends (along with Özlem Beyarslan) who help Özge Samancı finish her mathematics degree in her graphic memoir, Dare to Disappoint: Growing Up in Turkey.

Samancı dared to disappoint her father, because she pursued what she wanted in life, which was art. By her report, mathematics had at least taught her how to learn. Perhaps then her teachers achieved what Thurston wrote about – again, this was,

The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.

Etymologically speaking, mathematics is something learned. This is explained in the passage that heads Selections Illustrating the History of Greek Mathematics (two volumes, Loeb Classical Library), edited by Ivor Thomas:

Why is mathematics (μαθηματική) so named?

The Peripatetics say that rhetoric and poetry and the whole of popular music can be understood without any course of instruction (μὴ μαθόντα), but no one can acquire knowledge of the subjects called by the special name mathematics (μαθήματα) unless he has first gone through a course of instruction (μαθήσις) in them; and for this reason the study of these subjects was called mathematics (μαθηματική).

The words are by Anatolius (“bishop of Laodicea about a.d. 280”), as quoted in the Definitions, attributed to Heron.

Can we still ask what those subjects are that require a course of instruction? The sequel gives some indication of an answer:

The Pythagoreans are said to have given the special name mathematics (μαθηματική) only to geometry (γεωμετρία) and arithmetic (ἀριθμητική); previously each had been called by its separate name, and there was no name common to both.

Plato has Socrates talk with Glaucon about those two subjects in the Republic, Book VII (here with Shorey’s translation in the old Loeb edition):

• Arithmetic at 522e:

  ἄλλο τι οὖν, ἦν δ᾽ ἐγώ, μάθημα ἀναγκαῖον πολεμικῷ ἀνδρὶ θήσομεν λογίζεσθαί τε καὶ ἀριθμεῖν δύνασθαι;

  “Shall we not, then,” I said, “set down as a study requisite for a soldier the ability to reckon and number?”

  πάντων γ᾽, ἔφη, μάλιστα, εἰ καὶ ὁτιοῦν μέλλει τάξεων ἐπαΐειν, μᾶλλον δ᾽ εἰ καὶ ἄνθρωπος ἔσεσθαι.

  “Most certainly, if he is to know anything whatever of the ordering of his troops – or rather if he is to be a man at all.”

• Geometry at 527a–b:

  οὐ τοίνυν τοῦτό γε, ἦν δ᾽ ἐγώ, ἀμφισβητήσουσιν ἡμῖν ὅσοι καὶ σμικρὰ γεωμετρίας ἔμπειροι, ὅτι αὕτη ἡ ἐπιστήμη πᾶν τοὐναντίον ἔχει τοῖς ἐν αὐτῇ λόγοις λεγομένοις ὑπὸ τῶν μεταχειριζομένων.

  “This at least,” said I, “will not be disputed by those who have even a slight acquaintance with geometry, that this science is in direct contradiction with the language employed in it by its adepts.”

  πῶς; ἔφη.

  “How so?” he said.

  λέγουσι μέν που μάλα γελοίως τε καὶ ἀναγκαίως: ὡς γὰρ πράττοντές τε καὶ πράξεως ἕνεκα πάντας τοὺς λόγους ποιούμενοι λέγουσιν τετραγωνίζειν τε καὶ παρατείνειν καὶ προστιθέναι καὶ πάντα οὕτω φθεγγόμενοι, τὸ δ᾽ ἔστι που πᾶν τὸ μάθημα γνώσεως ἕνεκα ἐπιτηδευόμενον.

  “Their language is most ludicrous, though they cannot help it, for they speak as if they were doing something and as if all their words were directed towards action. For all their talk is of squaring and applying and adding and the like, whereas in fact the real object of the entire study is pure knowledge.”

  παντάπασι μὲν οὖν, ἔφη.

  “That is absolutely true,” he said.

  οὐκοῦν τοῦτο ἔτι διομολογητέον;

  “And must we not agree on a further point?”

  τὸ ποῖον;

  “What?”

  ὡς τοῦ ἀεὶ ὄντος γνώσεως, ἀλλὰ οὐ τοῦ ποτέ τι γιγνομένου καὶ ἀπολλυμένου.

  “That it is the knowledge of that which always is, and not of a something which at some time comes into being and passes away.”

  εὐομολόγητον, ἔφη· τοῦ γὰρ ἀεὶ ὄντος ἡ γεωμετρικὴ γνῶσίς ἐστιν.

  “That is readily admitted,” he said, “for geometry is the knowledge of the eternally existent.”

That last point came up recently on this blog, in “Just World,” where I noted how Parmenides too talked of something timeless. Physics then is about something else.

We started out, talking about mathematics as something people do, in time. Plato still provides some corroboration of what Thurston asks and proposes as follows.

Could the difficulty in giving a good direct definition of mathematics be an essential one, indicating that mathematics has an essential recursive quality? Along these lines we might say that mathematics is the smallest subject satisfying the following:

• Mathematics includes the natural numbers and plane and solid geometry.
• Mathematics is that which mathematicians study.
• Mathematicians are those humans who advance human understanding of mathematics.

This account may be not only recursive, but circular. Still, I think the general point is one that I tried to make in a draft essay, which I ended by saying,

mathematics is a way to create a kind of wealth that only increases when shared.

Mathematical wealth does exist, even before being shared. I had written at the beginning,

Like any other science, mathematics can be put to good or bad use. We may be able to inhibit the bad use, if we are clear about we are really trying to do in mathematics. I propose that this is to create something that cannot be hoarded or fought over.

I think the point is good. That doesn’t mean the essay was good. In fact it was rejected by the guest editors of a journal’s special issue on the ethics of mathematics. They had sent the paper to two referees, who appeared to be specialists of analytic philosophy. One of the referees said,

the description of what ethics deals with suggests strongly to me that [the author] has never attended an ethics course in a Philosophy department.

Guilty as charged. The other referee reported,

The article’s style has a kind of almost post-modern stream of conciousness quality. At times it reminded me of some of Montaigne’s more rambling essays.

The present post might elicit a similar response. We do still read Montaigne after more than four centuries. That does not mean we read him because he rambles. The referee continued:

I would question the diction (use of words and phraseology) in various places … The term antithesis, in my mind, conjures up the spectre of a Hegalian [sic] triad (thesis-antithesis-synthesis).

This illustrates what Brendan Larvor says in “Feeling the force of argument” (2009), a paper that I quoted in my draft:

Western philosophy traditionally sees itself opposed to mystery-mongering and intellectual chicanery. Its principal weapon against these two foes is clarity in thought and speech. This self-image is especially prevalent among English-speaking heirs to the analytic tradition, who sometimes signal this self-understanding by practicing a wilful pedantry. (‘You don’t really mean that you’re in two minds. It doesn’t even make sense to say that you’re in one mind. Minds are just not the sort of things one can be in.’ Etc.)

Larvor knows his colleagues. Wilful pedantry was on offer by my first referee, who said of my proposal about what mathematics was for,

I find this a bizarre thing to say. Even if it is true of mathematics (a claim that would need some discussion and clarification) it is not unique to mathematics.

The referee’s own assertion about uniqueness would need discussion and clarification. To the person who discovers it, knowledge of the physical world is less valuable when shared, if the discovery has been made for the sake of selling a product or waging a war. In any case, the referee continued:

Looked at from a different direction, the claim is false because mathematics is not the kind of thing that can have a purpose. Mathematicians can, but mathematics can’t – at least on ordinary ways of thinking about it. I seriously doubt that many mathematicians would define the purpose of mathematics in this way.

Ignoring the question of whether humans are allowed to pass along their own purposes to their creations, I wonder whether having a new idea about one of those creations is really something to be discouraged.

The referee continued in a new paragraph:

In that this paper has claims to be philosophical, in many places it falls short of the practice, commonly found in philosophical writing, of working hard to make clear what the claims it is proposing mean and to distinguish them from other ways they might be understood.

The precision described here is expected in mathematics, it seems to me, but I was not writing mathematics. I was writing about mathematics, for mathematicians, along with others who are interested in what we do.

Yet again, Thurston says of what we do,

The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.

Unless Thurston means to say that there is no difference between

1. doing mathematics and
2. getting people to understand it,

then one should be able to do well at the former, without the latter. Collingwood addresses this point:

Suppose a man devotes his life to the study of pure mathematics. Is he to be condemned for living on a selfish principle? Not, as my friends readily admitted, on the ground that pure mathematics cannot feed the hungry. Pure mathematics, apart from any consequences which may ultimately come of it, is pursued because it is thought worth pursuing for its own sake. In order to judge its social utility, then, you must judge it not by these consequences but as an end in itself.

What is more, you cannot judge the social utility of a pure mathematician by asking whether he publishes his results. Unless there is value in being a pure mathematician, there is no value in publishing works on pure mathematics; for the only positive result these works could have is to make more people into pure mathematicians; and a society which does not think it a good thing to have one pure mathematician among its members will hardly think it a good thing to have many.

That’s from “Monks and Morals” in The First Mate’s Log (1940), reprinted in Essays in Political Philosophy (1995). Collingwood’s friends here are Oxford students, some from North America, who have asked him to join them on a yacht cruise in the Mediterranean in 1939. When they stop in Italy, they have to deal with Fascists. On the island of Santorini, the students are disturbed that the Monks of the Prophet Elijah sing beautifully, but to no audience.

At least the monks sing to one another. So do mathematicians.

Maybe Thurston’s remark applies, mutatis mutandis, to other areas of research, as one of the referees said of my similar remark. I’m not sure. The measure of success in physics, chemistry, or biology would seem to be agreement with experience. This experience is not with other people, but with what is seen in telescopes or particle accelerators or test tubes or petri dishes or the Galapagos Islands.

Physicists who don’t worry about agreement with experience are ridiculed in books such as Not Even Wrong and apparently now The Ant Mill – on which see Robert A. Wilson’s blog post.

In “The tool/weapon duality of mathematics, revisited,” Alexandre Borovik would seem to say mathematics is special, perhaps in another way:

To bring some clarity in the debate, perhaps it is time to cut the knot: Mathematics is morally and ethically neutral.

Or would one say this of any pursuit? Maybe I don’t understand what is meant. I think Borovik says in the essay that while mathematics can be a tool or a weapon, there’s no point asking which one, if you haven’t actually done the mathematics in the first place. Here, you have to suit yourself, as Özge Samancı urges in Dare to Disappoint. This much makes sense.

After quoting Vladimir Putin as saying,

Today, I have already heard our participants in combat operations saying that the today’s warfare is a “war of mathematicians,”

Borovik says,

The basis for an individual decision of a mathematician facing a war of mathematicians – to fight, or become a conscious objector, or a saboteur – could be ethics, nationalism, religion, ideology, politics. In this toxic bouquet, ethics is the sweetest scent, and also is the least relevant.

This seems to me like saying, “We can prove this theorem by logic, directly, by contradiction, by analysis into cases.” Every proof is logical, though the logic may be flawed. Every decision is ethical – perhaps ethically bad.

Maybe ethics here means what is elsewhere in the paper called “normative ethics.” I gather this is something imposed from on high, like the Ten Commandments, or a course in ethics required of all faculty by a university administration or a state legislature.

On that point, in “Is there ethics in mathematics?” (shared also by Borovik on Substack), Roman Kossak seems right to say,

• One cannot learn how to act ethically by taking a course in ethics. No amount of learned wisdom can prevent one from making ethically questionably decisions (case study: Heidegger).
• However, one learns how to behave ethically by observing how other people behave (role models).

Still, we grow up seeing both good and bad examples of behavior. Don’t we have to decide which is which? The examples themselves do not teach us whether they are good or bad. Thus I would seem to be agreeing with Kant in Groundwork for the Metaphysics of Morals (§ 2, 408):

one could not do morality a worse service than by trying to derive it from examples. For every purported example of morality would first have to be judged according to principles of morality, to see whether it is actually worthy to serve as a foundation …

I happened to work through the Groundwork recently in a group of amateurs. The quote is from the Oxford World’s Classics edition.

One may raise such concerns about Kant as Brendan Larvor does, again in “Feeling the force of argument”:

This idea, that humans are essentially rational, did not originate in the eighteenth century. However, it is the central thought of the movement we now recall as ‘the Enlightenment’. It is the motif of Kant’s major works, and finds political expression in his essay What is Enlightenment? The first paragraph reads:

Enlightenment is man’s release from his self-incurred tutelage. Tutelage is man’s inability to make use of his understanding without direction from another. Self-incurred is this tutelage when its cause lies not in lack of reason but in lack of resolution and courage to use it without direction from another. Sapere aude! ‘Have courage to use your own reason!’ – that is the motto of enlightenment.

This idea is attractive because it is egalitarian – all humans are rational, not merely a lucky few. All that enlightenment in Kant’s sense requires is ‘resolution and courage’. In other words, becoming rational largely consists in removing self-imposed impediments to the use of one’s reason. This, though, is a poor model for thinking about teaching philosophy, because it assumes that formal rationality is already present in students and lecturers alike, and needs only to be drawn out and exercised. Experimental psychology has shown this to be false.

“This idea is attractive because it is egalitarian” – what if we are not egalitarian? A number of heads of state are maintaining power by dividing us from them: anti-egalitarianism.

I think Kant argues that to be rational is to be egalitarian; however, we are not wholly rational.

Larvor cites a book called The Psychology of Judgment and Decision Making. As I understand from Collingwood, there can be no such thing. As instances of thought, judgment and decision-making are the province of the criteriological sciences of logic and ethics. As an empirical science, psychology can study only feeling.

On the other hand, thoughts are accompanied by feelings, which I guess is Larvor’s point. It’s even in his title, “Feeling the force of argument.”

I had a great feeling one Sunday morning. I was lying in bed, and the insight came to me that would pretty much constitute my dissertation.

I had another such feeling, on a later Sunday afternoon, when I was a postdoc at MSRI. I was in a Berkeley cinema, and a solution came to me of the problem I was working on.

A psychologist might study such experiences of insight. I imagine they are like Poincaré’s experience, described by Robert Pirsig in Zen and the Art of Motorcycle Maintenance (ch. 22):

He left Caen, where he was living, to go on a geologic excursion. The changes of travel made him forget mathematics. He was about to enter a bus, and at the moment when he put his foot on the step, the idea came to him, without anything in his former thoughts having paved the way for it, that the transformations he had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. He didn’t verify the idea, he said, he just went on with a conversation on the bus; but he felt a perfect certainty. Later he verified the result at his leisure.

The feeling of insight doesn’t make the insight correct. The discovery I made in the Berkeley cinema turned out to be wrong. I didn’t learn this till after getting it published in a peer-reviewed journal. Somebody who then read the paper sent me a counterexample to my theorem. Thus the error was in the statement of the theorem, not only the proof.

That person had a student who had successfully defended his dissertation, even though it contained an error similar to mine.

I did ultimately discover and publish what would seem to be the correct theorem. People have cited it, though I don’t know how carefully they have checked it for themselves.

It seems to me that words of Kant (Groundwork § 1, 399–400) apply mutatis mutandis to mathematics:

an action from moral duty has its moral worth not in the purpose that is to be attained by it, but rather in the maxim according to which it is decided upon. The worth therefore does not depend on the realization of the object of the action, but merely on the principle of willing according to which the action is done, without regard for any object of the faculty of desire.

Kant may be famous for inscrutability, but then so are mathematicians. I would translate the general point about ethics to mathematics thus.

A theorem has its worth in its truth. This truth is determined only by the correctness of its proof, not by peer review or publication or social acceptance or even by the tremendous feeling of enlightenment that the theorem or its proof provides. Computer verification may provide evidence of truth, but not infallible evidence, since computers are physical objects like us.

One may disagree with Kant or with my mathematical version. A debate on these things plays out in Ulf Persson, “A conversation with Reuben Hersh”:

Ulf Persson

… What is true in mathematics is not up to our discretion, certainly not as individuals.

Reuben Hersh

But in practice truth is agreed on by a process of social confirmation. I can give you a specific concrete example. As I told you before, I worked on linear partial differential equations with constant coefficients. My work was later extended by Heinz-Otto Kreiss to the case of variable coefficients. His theorem was quickly accepted as a known result that anyone else can freely quote and use. The proof is long and complicated. I could never really understand it all …

UP

… everything you say on the practice of mathematics we agree on. But I think that there is something beyond the practice of mathematics, beyond the human fallible way of doing mathematics.

RH

… Mathematical Platonism is a … fallacy. It arises from the unfounded idea that there must be something to mathematics beyond the practice of mathematics.

Call me a mathematical Platonist then. There is no physical test for mathematical truth. There is only a social test – practically speaking. Still, does a mathematician want to have a published theorem, or a correct theorem? There’s a difference.

This is all part of the ethics of mathematics. I mentioned Poincaré. Jaffe and Quinn do too, in the paper that Thurston was responding to. In the following, they use the word “theoretical” to mean speculative, non-rigorous:

Algebraic and differential topology have had several episodes of excessively theoretical work. In his history [of algebraic and differential topology 1900–1960], Dieudonné dates the beginning of the field to Poincaré’s Analysis Situs in 1895. This “fascinating and exasperating paper” was extremely intuitive. In spite of its obvious importance it took fifteen or twenty years for real development to begin. Dieudonné expresses surprise at this slow start, but it seems an almost inevitable corollary of how it began: Poincaré claimed too much, proved too little, and his “reckless” methods could not be imitated. The result was a dead area which had to be sorted out before it could take off.

Maybe the writers are too hard on Poincaré. At least Thurston would seem to think so. All I know is that when Thurston’s student Bill Goldman taught us topology in graduate school, he said its original name had been analysis situs.

Larvor’s paper, “Feeling the force of argument,” has a section called “Collingwood to the rescue”:

we would like to think about teaching philosophy using resources drawn from philosophy itself, rather than having to borrow from psychology or educational theory. At the same time, we would like our philosophical pedagogy to avoid the shortcomings we found in mainstream English-speaking philosophy, hampered as it is with its Enlightenment residue. Happily, such resources do exist … the view of language that R. G. Collingwood set out in his Principles of Art.

… His chief aim in his philosophical psychology is to develop a theory of imagination, because for Collingwood, imagination is the capacity that allows creatures who feel to become creatures who think, write books and create art

– and create mathematics, I suppose! Larvor continues:

… It is a great mistake, he argues, to suppose that ‘intellectualised’ language or ‘symbolism’ (that is, language used to articulate thoughts) makes sense in isolation from its expressive function …

… Collingwood’s chapter on language ends as this paper begins, with the emotional aspect of mathematical work. Here, he insists that ‘the emotions which mathematicians find expressed in their symbols are not emotions in general, they are the peculiar emotions belonging to mathematical thinking’ …

It all makes sense. I’m thinking also that Plato was there all along, and I wonder if Kant forgot him, because he says (Groundwork § 2, 428),

the inclinations themselves, as sources of need, are so far from having an absolute worth that would make them desirable in their own right, that it must rather be the universal wish of every rational being to be entirely free from them.

Maybe the wish here is like that of the inmate freed from the Cave, in the allegory recounted in Republic VII, earlier than the passages on arithmetic and geometry quoted above. Having seen the light of day, one does not want to go back into the darkness, even if it is the darkness of a cinema, with images playing against one wall. Still, one has to go in, and perhaps this is not only because one’s compatriots inside require it, for their own sake. The intellect cannot live in isolation from the appetites and the passions.

Anyway, as Roman Kossak says,

Ethics is a one of the hardest parts of philosophy and one cannot seriously discuss any particular issues concerning ethics without at least acknowledging some of the general difficulties.

I guess I’m trying to do that. Kossak goes on to quote Chiodo and Bursill-Hall as saying,

the conclusion we feel is the most important for further study of ethics in mathematics is that we need the skills and knowledge of social scientists – sociologists, anthropologists, and psychologists …

No we don’t, as I think Kossak says near the end:

Ethical behavior and its various patterns are not shaped by a particular profession or field of study. The ideals of higher education cannot be taught by sociologists, cannot be manufactured on request. There must be a better way. Certainly, it’s worth pursuing.

Indeed. The ethics of academia have to be worked out by academics; the ethics of mathematics, by mathematicians. On that last point though, Kossak seems to disagree:

I do care about ethical conduct in one’s personal life and in professional career and I am very open to discussing ethical issues with anyone. I even sat at some tables of small power, but I cannot move to Level 1, because I do not believe that there are issues inherent in mathematics that would not be covered by much more general ethics considerations.

Borovik talks that way too:

Perhaps I have to make a clear statement that I am not talking about the pond life of academia where moral issues are safely covered by the obvious rules of normative ethics not specific to mathematics: do not plagiarise, do not steal results from your PhD students and do not sexually harass them, etc. – and so often are blatantly ignored. I leave it to readers to continue the list if they wish so …

I don’t think moral issues can have been “safely covered” if they “so often are blatantly ignored.”

After a majority of students fail my exam, is it OK for me to think, “Well, I presented the material in class; if they don’t want to learn it, that’s their problem”?

Thurston observes,

Organizers of colloquium talks everywhere exhort speakers to explain things in elementary terms. Nonetheless, most of the audience at an average colloquium talk gets little of value from it.

How one gives a talk is an ethical issue. It is of peculiar importance in mathematics, at least if one agrees with Thurston that the whole point of mathematics is to communicate it. In physical sciences, experiments may speak for themselves, but I don’t think proofs do.

On that subject, Borovik says,

Mathematics, as we know it, was born as a weapon of subjugation and tyrannic control.

Some of the subject-matter of mathematics was born that way. Herodotus says the Greeks learned geometry from Egypt, because Pharoah needed it for measuring fields for taxation. But this is geometry in the original sense of surveying. Euclid turned it into mathematics.

I am aware of vehement (even hysterical) disagreement with that last assertion, and I looked at it in “More of What [Mathematics] Is.”

The rule learned in school today as 𝐴 = π ⋅ 𝑟² was established by Euclid in the form,

Circles are to one another as the squares on their diameters.

This may seem obvious, once one understands what it means. Nonetheless, Euclid’s proof is as rigorous as the one we might write today, using the letters ε and δ. I don’t know why Euclid thought he needed to give such a rigorous proof. Today we may observe that we need a proof, since we know that there are non-Euclidean planes, and the theorem is false there. Euclid must have had a sense that there were such planes, since he ruled them out by means of the Parallel Postulate.

We are in a field where we may still cite results from over two millenia ago. Other sciences can’t do that, as far as I know, unless perhaps some of Aristotle’s biological observations are brought up as a curiosity (as for example in Korda et al., “The History of Female Ejaculation,” Journal of Sexual Medicine, 2010, DOI 10.1111/j.1743-6109.2010.01720.x).

The persistence of mathematics tends to confirm what Roman Kossak wrote:

I found the following statement in a popular Polish mathematical magazine for high school students Delta.

… I believe that mathematics is the structure of the world …

… To me, it represents well what made mathematics attractive for me and for many of my fellow students.

Some physicists think their subject is the structure of the world. I think Gödel’s Incompleteness Theorem ought to dissuade them, and I included a bit about this in an essay on the Theorem that appeared this summer in the Journal of Humanistic Mathematics. Before that, I tried to put the essay on the ArXiv, but it was not allowed, because

Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv.

See also “Gödel and AI” on this blog.

Apparently Steven Weinberg is famous for saying,

The more the universe seems comprehensible, the more it also seems pointless.

This is unwarranted, since physics is not designed to see a point in what it studies. Mary Midgley makes that point in Evolution As a Religion (revised edition 2002).

Biology does recognize organisms as having a point, namely to survive and reproduce. Apparently there’s a word “teleonomy” for this. From his own work on the subject, Jacques Monod concluded, in Chance and Necessity (1971),

man must at last wake out of his millenary dream and discover his total solitude, his fundamental isolation. He must realize that, like a gypsy, he lives on the boundary of an alien world …

Midgley remarks,

But ‘discovering his total solitude’ is just adopting one imaginative stance among many possible ones. Other good scientists, very differently, have used the continuity of our species with the rest of the physical world to reprove human arrogance …

I might double the length of this post with such investigations. They are relevant to the ethics of mathematics, if only as a warning not to think expertise in our subject makes us expert in others. I don’t know how liable we are to that failing.

In the horror of the last US presidential election, Terence Tao found encouragement to keep doing mathematics:

there is one precious thing mathematics has, that almost no other field currently enjoys: a consensus on what the ground truth is, and how to reach it … if my students can learn from this and carry these skills – such as distinguishing an overly simple but mathematically flawed “solution” from a more complex, but accurate actual solution – to other spheres that have more contact with the real world, then my math lectures have consequence. Even – or perhaps, especially – in times like these.

I would disagree only with Tao’s use of the term “real world,” which applies better to mathematics than to politics.

I discovered recently that Chance and Necessity had been reviewed in 1975 by Wilfrid Hodges, author of the big red book called Model Theory that came out while I was a graduate student.

Casual readers will get the impression that Monod’s remarks about teleonomy and purpose are deep. One of life’s profoundest mysteries seems to be relentlessly ebbing away, chapter by chapter, leaving us all in the dry and empty desolation of a purely mechanical universe. This impression, which I suspect Monod wishes to encourage, is almost certainly mistaken. Closer inspection shows that Monod’s remarks about teleonomy confuse three quite distinct things …

If I have confused various things in this post, I’m sorry!