Author Archives: David Pierce

Mathematician & logician; amateur of philosophy; relation of journalists; alumnus of St John’s College (USA); living in Ankara & Istanbul since 2000


This post is based on recent readings, often on or through Twitter, especially of

  • Lilith Saintcrow on “Domestic abusers, white supremacists, and religious bigots”;

  • C. S. Lewis on gulling the educated, and objectivity as a dubious value;

  • Marilynne Robinson on consensus as concealing the objectively true;

  • Neil deGrasse Tyson on objectivity as a good value;

  • Plato on seeming wise, without being so;

  • Mark Vernon on imagination in William Blake;

  • whoever wrote an “Open Letter Concerning Transphobia in Philosophy,” signed by many professional philosophers;

  • Kathleen Stock, the subject of the “Open Letter”;

  • Agnes Callard on how philosophers shouldn’t be signing petitions;

  • Rebecca Reilly-Cooper, on the incoherence of the notion of gender identity;

  • Aaden Friday, on what’s wrong with Reilly-Cooper and other such women;

  • Brian Earp, on declaring pronouns;

  • John Steinbeck, on being a man;

  • Christa Peterson, on what gender identity might be.

I have edited and augmented this essay since originally posting it on January 9, 2021; the current version is from January 19.

A lot of old PSA’s about drugs are on YouTube and the Web Archive, and sometimes they are linked to by articles that ridicule them. There is one that I have not been able to find, perhaps from around 1970, in which parents confront their teenager with the drug paraphernalia that they have found in his room. The boy storms out of the house, saying, “You don’t understand!”

There’s a lot that I don’t understand. I must not, since it seems childish, but is coming from adults. Some of these adults stormed the US Capitol the other day; others encourage them; still others are professors of philosophy.

“Human egg and sperm cells.”
Asimov’s New Guide to Science (1984), page 600

I encountered a relevant thread of tweets by Lilith Saintcrow; here’s a selection.

[Violently abusive white supremacists] absolutely do not believe their own bullshit, but it’s useful for them to pretend they do.

[One of them] fixed me with his baleful, watery stare, and said, “Obama was born in Kenya, you know.”

Gene absolutely, positively did not believe that Obama was born in Kenya. But he would continue to say he believed it, no matter who asked, to the end of his life.

Because he thought saying he believed it absolved him of responsibility.

Now. There are pollsters asking violent white supremacists and their fellow travelers who was responsible for yesterday’s attack on state and federal Capitol buildings. The people saying “Biden was responsible” absolutely know it’s not true.

Domestic abusers, white supremacists, and religious bigots all operate off the same thin but very useful playbook that exploits other people’s politeness and (I’ve got to say it) “civility.”

The playbook exploits our politeness, our civility, and, perhaps, our education.

“Why you fool, it’s the educated reader who can be gulled.”

I have heard Noam Chomsky say something like that: that the people who are more subject to propaganda are the people who read more. Perhaps one should add, “or watch television more,” and now, “or read Facebook and Twitter more.” In any case, the actual words that I quoted, “Why you fool …,” are from Chapter 5 (out of seventeen), “Elasticity,” of C. S. Lewis’s novel, That Hideous Strength: A Modern Fairy-Tale for Grown-ups. The speaker in the novel goes on to ask,

When did you meet a workman who believes the papers? He takes it for granted that they’re all propaganda and skips the leading articles. He buys the paper for the football results and the little paragraphs about girls falling out of windows and corpses found in Mayfair flats.

That was published in 1945. I don’t know what Lewis wants Miss Hardcastle to mean by “paragraphs about girls falling out of windows.” Such paragraphs could themselves serve as propaganda: propaganda of the normalizing kind, gulling you into thinking that a certain kind of horrible event is nothing to be alarmed about.

Şule Çet “fell out of a window” of the 20th floor of a building in Ankara in May of 2018; she was twenty-three. It was suicide, said the two men she had been with. A male judge might accept such a story, and we don’t have jury trials in Turkey. There is still the possibility of public pressure, which mounted in this case, and the two men were ultimately convicted of kidnapping, rape and murder.

There is a Wikipedia article about the case, documented by articles in English and German. Only articles in Turkish – such as one at Bread and Roses, and another at Vikipedi – name the location of the crime. This is said to be Yelken Plaza, although I had remembered it to be Armada Shopping and Business Center, which my wife and I used to pass daily on the commute by bus between home and university; that was in the aughts, before we moved to Istanbul. Since an armada is a fleet of ships, and yelken means sail, and the two complexes are near each other, Yelken and Armada may have a single owner, who or which however is currently unnamed in the Vikipedi article on the latter.

The more you think, the more you may be trapped by thought: thought on questions that can be answered by Wikipedia articles, or thought on what truth is anyway. Here is Marilynne Robinson, in an essay called “The Tyranny of Petty Coercion”:

It is consensus that conceals from us what is objectively true. And it is consensus that creates and supports “truths” that are in fact culturally relative. And, interestingly, it is consensus that is preserved when the objective truth is disallowed on the grounds that “truth” is merely the shared understanding of a specific group or culture.

This makes sense. If all truth is known by consensus, then consensus is what you are bound to accept. However, there is still another kind of truth, which consensus may distract us from: the objective truth.

I am generally suspicious of the word “objective,” which can itself can be used to gull or coerce you into accepting some other people’s consensus. “If you would only consider the matter objectively, you will find that you agree with us”: I can hear this, annoyingly, in my mind’s ear. Why should I be objective?

Neil deGrasse Tyson gives a reason, in “What Science Is, and How and Why It Works,” dated January 23, 2016. The bold emphasis is mine; italic, his:

Science especially enhances our health, wealth and security …

The scientific method, which underpins these achievements, can be summarized in one sentence, which is all about objectivity:

Do whatever it takes to avoid fooling yourself into thinking something is true that is not, or that something is not true that is.

This approach to knowing did not take root until early in the 17th century, shortly after the inventions of both the microscope and the telescope. The astronomer Galileo and philosopher Sir Francis Bacon agreed: conduct experiments to test your hypothesis and allocate your confidence in proportion to the strength of your evidence …

Objective truths exist outside of your perception of reality, such as the value of pi; 𝘌 = 𝘮𝘤²; Earth’s rate of rotation; and that carbon dioxide and methane are greenhouse gases. These statements can be verified by anybody, at any time, and at any place. And they are true, whether or not you believe in them.

Bacon was born in 1561; Galileo, in 1564. However, if doing science in the most basic sense means working not to fool ourselves, then Socrates was engaged in this, two millenia earlier.

According to the words that Plato attributes to Socrates in the Apology. the oracle at Delphi had told Chaerophon that nobody was wiser than Socrates. Socrates didn’t believe it, and so he

approached one of those individuals people suppose to be wise … and as I conversed with him, I formed the conclusion that, while this person seemed wise to lots of other people, and especially to himself, in reality he wasn’t; upon which I made a concerted attempt to demonstrate to him that he only thought he was wise, but really wasn’t. Well, this made him hate me …

Plato has Socrates say that in the Apology (Ἀπολογία), translated by Christopher Rowe as Socrates’ Defence, Penguin Little Black Classics No 52.

A century after the Athenian democracy put Plato’s teacher to death, Euclid worked out a method, which we still use, for not fooling ourselves in mathematics. The deductive method is the reason why I call mathematics pacifist in principle. Like Tyson’s “objective truths,” mathematical truths “exist outside of your perception of reality” and “are true, whether or not you believe in them.” However, the way to resolve a mathematical disagreement is not with microscope, telescope, or other instruments, which we may not all have access to; in mathematics, we can work things out with chalk and slate, or pencil and paper, or even just larynx and air.

Tyson mentions “health, wealth and security,” which he says are “greater today for more people on Earth than at any other time in human history.” Health, wealth, and security are goods, and we have ways of measuring them objectively; nonetheless, I would say, goodness as such has no objective measure. Mark Vernon has some words for this situation in his Aeon essay dated 4 September 2020 called “The Four-fold Imagination”:

In his poetry, [William] Blake describes four states of mind that exhibit distinctive attitudes towards the imagination …

Ulro is the state of single vision. ‘In Ulro, that which can’t be expressed quantitatively does not exist,’ writes the Blake scholar Susanne Sklar. Wisdom is sought in logic and numbers, and debate is met with the cry to gather more evidence. It’s the state of mind experienced by the subterranean Newton, whose devotion to measurement curtails his awareness …

It’s a mentality that has proven highly desirable. During a pandemic, it motivates what it takes to flatten the curve or increase the amount of testing and, more generally, it organises life around metrics such as economic growth and GDP. Only, it’s never quite clear what these seeming goods are for …

I only propose that objectively measurable things may be a proxy for real goods; but sometimes they are counterfeit. Objectivity itself becomes counterfeit in That Hideous Strength, where C. S. Lewis has Miss Hardcastle’s colleague Frost say, in Chapter 12, “Wet and Windy Night”:

The great majority of the human race can be educated only in the sense of being given knowledge: they cannot be trained into the total objectivity of mind which is now necessary. They will always remain animals, looking at the world through the haze of their subjective reactions.

Since the listener, Studdock, believes himself to be educated, he will now want to display his objectivity by accepting what Frost says; at least, that is Frost’s hope.

So I am skeptical around the word “objective,” though Marilynne Robinson may have used it well to distinguish what is true from what is accepted by consensus. The way I think of “objective,” Robinson herself thinks of “bashing”:

To say that the disparity between rich and poor in this country exceeds any previously known in American history (putting aside the marked economic disparity between plantation owners and slaves) is to say something falsifiable – that is, for practical purposes, verifiable, and in any case arguable. But such statements are now routinely called “Bush bashing.”

Harper’s printed that in August of the year, 2004, when George W. Bush would win a second term in the office of US president, despite the crime of invading Iraq.

There is talk that is reasonably dismissed as “bashing.” It’s just hard to stay alert to all of the possibilities, all the time. Even trying to stay alert may be a trap. Robinson has a possible explanation. We are “acculturated to distrust strong emotion”:

Why critics are so flummoxed I can only speculate. Perhaps it is because most of the people in this country who take on public issues are educated and middle class. As is true of their kind anywhere, they are acculturated to distrust strong emotion, so they are effectively rebuked when they are accused of harboring it. Oddly, they seem often to be shamed out of defending the poor and vulnerable on the grounds that they themselves are neither poor nor vulnerable, as if there were properly no abstract issues of justice, only the strategies of interest groups or, more precisely, of self-interest groups. That their education and experience prepare them to think in terms larger than their own immediate advantage makes them an “elite,” and ipso facto they are regarded as a self-interested subgroup of a particularly irksome kind.

As far as I can tell, some trans rights activists are trying to flummox some feminists in a complementary way, by accusing them of not caring for people whom they see as different from themselves.

According to an “Open Letter Concerning Transphobia in Philosophy,” dated this month (January 2021),

We write to affirm our commitment to developing a more inclusive environment, disavowing the use of professional and cultural authority to further gendered oppression.

Declaring themselves to be “professional academic philosophers,” the signers of the letter are thus using their own authority to condemn Kathleen Stock, professor of philosophy at the University of Sussex. She was recently made officer of the Order of the British Empire, which I see is an honor that C. S. Lewis declined. According to the “Open Letter,”

Trans people are already deeply marginalized in society, facing well-documented discrimination, ranging from government policy to physical violence. Discourse like that Stock is producing and amplifying contributes to these harms, serving to restrict trans people’s access to life-saving medical treatments, encourage the harassment of gender-non-conforming people, and otherwise reinforce the patriarchal status quo. We are dismayed that the British government has chosen to honour her for this harmful rhetoric.

I understand blaming Donald Trump for the storming of the Capitol. I do not understand blaming “discourse like that [which] Stock is producing and amplifying” for such harms as are described. The signers hasten to add,

We do not say Stock should not be permitted to say the things she does. We believe in the principles of academic freedom …

However, they go on to say,

Academic freedom comes with responsibility; we should not use that freedom to harm people, particularly the more vulnerable members of our community.

Freedom is indeed connected with responsibility; ultimately, I would say, freedom is responsibility. However, Agnes Callard seems right that, in philosophy at least, the best response to objectionable statements is not signing a petition, but making an argument:

We’d never approach questions such as “Are possible worlds real?” or “Is knowledge justified true belief?” by petition, so why are we tempted to do so in the case of questions around sex, gender and hurtful speech? The answer is that the latter question involves real feelings and real people, and it is about something that is happening now – for all these reasons, it strikes us as being of grave importance. The petition writers are thinking to themselves, this time it really matters. I think it is a mistake for a philosopher to take the importance of a question as a reason to adopt an unphilosophical attitude toward it.

That’s from “Why Philosophers Shouldn’t Sign Petitions” (New York Times, August 13, 2019).

If you need to get people to see something, “Then argue for it!” as Callard says: “If you strip the list of signatures off your petition, you’ll find that you have an argument on your hands.” In that spirit, though not being a professional philosopher, I wrote “Academic Freedom,” March 25, 2016. It so happens that this was in response to the Turkish government’s persecution of academics who had signed a petition.

In many social interactions, we don’t tell truths that hurt; but sometimes we do, as Socrates did. Apparently a lot of professionals may disapprove. I note that one signer of the “Open Letter Concerning Transphobia in Philosophy” is Jason Stanley, whom I respect for his work on and against fascism.

Another philosopher has an essay called “Gender is not a spectrum,” dated 28 June, 2016. It is among the articles that I save in pdf format, and I was reminded of it by a recent tweet of Nigel Warburton, who wrote that the essay had “had the most page views of all the pieces I’ve commissioned/edited for [Aeon].” The subtitle of the essay reads,

The idea that ‘gender is a spectrum’ is supposed to set us free. But it is both illogical and politically troubling.

There are properties measured along a spectrum, while still being described in binary terms. Height is an example, where you can be short or tall. But you are short or tall, only in comparison with others.

If gender, like height, is to be understood as comparative or relative, this would fly in the face of the insistence that individuals are the sole arbiters of their gender … Further … when observing the entire population, only a small minority of people would be accurately described as Tall or Short … if we extend the analogy to gender, we see that being non-binary gendered is actually the norm, not the exception.

In short, if gender is a spectrum, then we are all non-binary.

The philosopher here is Rebecca Reilly-Cooper. When I searched DuckDuckGo for her name, I found a condemnatory essay, “Anti-Trans ‘Feminists’ Are More Dangerous Than Religious Zealots,” dated November 15, 2017.

In place of the word “bashing,” which Marilynne Robinson questioned, the essay about “anti-trans ‘feminists’” uses “fear-mongering.” So does the “Open Letter” about Kathleen Stock:

our concern is that some – apparently including the British government – have a tendency to mistake transphobic fearmongering for valuable scholarship, and attacks on already marginalized people for courageous exercises of free speech.

Be it noted that the “Open Letter” needed a correction:

Erratum: the original version of this letter incorrectly stated that Stock opposes the UK’s Gender Recognition Act. This was an error; it should have said that Stock is well-known for opposing amendments to the Gender Recognition Act that would have made it easier for people to self-identify their gender.

A number of philosophers signed the letter before the correction was made. According to one of the ultimate signers, Christopher Bertram,

some people (particularly philosophers) are more fussy about whether particular details are right than others.

Those words were tweeted in response to Nathan Oseroff-Spicer, who had found it worthwhile to tweet,

After a quick search it’s clear [certain philosophers] are not currently listed as signatories on the open letter concerning transphobia in philosophy.

Bertram began his response to this by saying, “I’ve signed,” without specifying when. I do not see the name of Oseroff-Spicer on the letter; perhaps he did not qualify as one of the “professional academic philosophers” in whose name the letter was written. In any case, I saw his tweet first as an image, included for ridicule in another tweet.

Here are the words of Aaden Friday in the aforementioned essay on “anti-trans ‘feminists’” in general and Rebecca Reilly-Cooper in particular:

Her fearmongering about the elimination of cisgender men and cisgender women is identical to the fearmongering religious zealots use against gay marriage – if marriage isn’t confined to one man and one woman, then the word becomes irrelevant.

It’s not identical. It may be similar. The question then is how deep the similarity lies. Evidently Aaden Friday does not want to take up such a question.

Some people declare their pronouns in their Twitter profiles; other people say they will unfollow people who do this. Declaring pronouns may be harmful to people who are not ready to be out about their own gender, or even to people who are, but don’t think it’s relevant; Brian Earp discusses the possibilities in a recently drafted essay, “On Sharing Pronouns.” Aaden Friday implicitly declares their pronoun to be just that, “they/their.” They give reasons in another essay, “Trans-Exclusionary Feminists Cannot Exclude My Humanity”:

The fluidity of gender is complicated; it is messy and it is beautiful. If I’ve learned anything, it’s that I cannot say with any real sense of authenticity or certainty that I know who or what I am – not fully. I lived as a cis heterosexual man for the first 22 years of my life. I then lived as a cis homosexual man for another decade. Today I am something much closer to myself.

I identify as non-binary because at this point in my life – as I deconstruct obstructions that have confined my existence thus far – I understand that there is a deeper truth found within that I have yet to unearth.

As a child, I recognized female empowerment as personal …

Friday’s essays are in The Establishment, which “champions the voices and stories of those who have been marginalized by mainstream media.” The publication was paying its writers, but ran out of money.

Friday seems unable to distinguish argument from attack. This is from “Anti-Trans ‘Feminists’ Are More Dangerous Than Religious Zealots”:

Progressive, socially liberal women who increasingly and persistently attack, dehumanize, and slander trans and gender non-conforming people, are – in my eyes – more dangerous than religious zealots who do the same. The zealots’ views are harmful and put people in danger, there is no question there, but they can be spotted a mile away … These women, on the other hand, claim progressive and feminist values. They fight for the autonomy and agency of individuals, and they vehemently oppose any and all restrictions placed upon women because of their gender. These so-called progressive feminists, who know precisely how reprehensible it is to vilify, attack, and discriminate against a person based on their gender, are doing exactly what has been done to them, all the while claiming to be the victim.

Again I say that the women in question are not doing “exactly what has been done to them.” They may be doing something similar, the way immigrants to America, once established, may fight to keep out new immigrants. Friday continues:

These women have a range of titles and backgrounds: journalist, author, activist, lecturer, scientist, feminist. They are all cisgender, and they all want to debate the identity of transgender people …

Some persons may like debate for its own sake. Others, myself included, want to learn things, such as what “identity” and “transgender” even mean, especially if they are going to be used in actual laws.

I say we want to learn; and yet Aaden Friday is suffering,

because what [Reilly-Cooper] did was erase my existence, and the existence of many others who are neither man nor woman. Her fear is our lived existence, yet our realities aren’t real enough for her to grasp.

On behalf of trans people, one of the signers of the “Open Letter” is suffering; for as Liam Kofi Bright tweeted on January 17,

talking about trans issues (especially in philosophy, closest to home) is incredibly stressful. Trans people are having to put up with a level of vitriol just for being that I honestly don’t think I could take, it’s just so absurd and unfair to them …

… there’s something about how trans people are themselves singled out and targeted for just utterly unfair and oppressive social treatment even in normal conversational contexts, that is especially intense.

I am not sure what Bright means here, perhaps because I am not interacting with people the way Bright is. It is dismaying to learn what Masha Gessen tells about how,

Every night, when I walk my dog, several strangers, similarly tethered, will ask me the same two questions: “Boy or girl?” and “How old?” The pragmatic meaning of these questions escapes me.

That’s from “We Need to Change the Terms of the Debate on Trans Kids,” The New Yorker, January 13, 2021. Gessen observes,

I began my own transition at fifty, long after experiencing the misery of pregnancy and the incomparable joy of breastfeeding. I have no regrets. Had I had the option of transitioning as a teen-ager, I would have chosen to do so – and I am almost certain that I would have had no regrets then, either, because I would have had a different life.

In youth, I was asked whether I was a guy or a girl, presumably because of my long hair, and perhaps also my slight build. I was addressed as “Ma’am” from behind once or twice, but I think the speaker was more embarrassed than I when I turned my head.

The problem of adolescence is being neither child nor adult: thus, in particular, being neither man nor woman. Words in a Steinbeck story, “Flight,” might have unnerved me, had I read them back then:

A boy gets to be a man when a man is needed. Remember this thing. I have known boys forty years old because there was no need for a man …

Yes, thou art a man, my poor little Pepé. Thou art a man. I have seen it coming on thee. I have watched you throwing the knife into the post, and I have been afraid …

Pepé goes on a journey. Pepé is a man now. He has a man’s thing to do.

The “man’s thing to do” is to flee the avengers of the man he has thrown his knife at in anger.

Reilly-Cooper feels herself to be dealing with children:

… if you want to call yourself a genderqueer femme presenting demigirl, you go for it. Express that identity however you like. Have fun with it. A problem emerges only when you start making political claims on the basis of that label – when you start demanding that others call themselves cisgender, because you require there to be a bunch of conventional binary cis people for you to define yourself against; and when you insist that these cis people have structural advantage and political privilege over you, because they are socially read as the conformist binary people, while nobody really understands just how complex and luminous and multifaceted and unique your gender identity is.

Why does she not just let the young people be? Is it really a problem to call herself “cis” out of politeness?

It is a problem, when the notion of “gender identity” gets written into law without being defined. Reilly-Cooper points this out, in the lecture on YouTube that Aaden Friday is actually responding to, called “Critically Examining the Doctrine of Gender Identity.” According to Friday,

Like many other avowed feminists, Reilly-Cooper is bent on “proving” the absurdity of trans identity. More than that, she seeks to reveal how cis, white women like herself are actively harmed by policies and laws which aim to protect transgender individuals from discrimination and ensure their equal access to services.

I may be wrong, but I think Reilly-Cooper is bent on finding the truth. You can ask why this truth and not that. There are people who are keen on sharing the stories of Karen White and Barbie Kardashian for their shock value. Reilly-Cooper is using her training as an analytic philosopher to ask what “gender identity” is. If you can have a penis while being a girl, so that your penis must therefore be a girl’s penis, what is the criterion whereby you are a girl in the first place? If you have a female brain in a male body, what makes the brain female?

Maybe some criterion can be found, but it does not seem to have been found so far. Why would you look for one anyway? It could be like searching for a biological distinction between races.

As far as I understand, while there may be superficial differences between persons originating in different parts of the world, there are no differences such as those between horses and donkeys, which prevent them from having fertile offspring. Genetic variation is as great within any one human population as it is in the species as a whole.

“Yes, but …” some people say. Maybe we can correlate lots of little variations, in order to establish a biological distinction between races. As far as I understand, when people try to do this, they get different numbers of races.

For a long time, there has been no question of how many persons it takes to conceive a child. In the distant past, one person may have been thought sufficient: the mother. Then it was surmised that some unique one of the men she had slept with was the father. I don’t know how scientific this surmise could have been. The notion that the man had “sown the seed” of the child was not too accurate, but it has stuck in the terminology. Ultimately, with the help of the microscope, it was understood that the man had supplied a so-called spermatozoön (from σπέρματ- “seed” and ζῷον “living thing”); and the woman had supplied an ovum. The ovum and spermatozoön together created the seed of the child, and this seed was then planted in the woman’s womb, so to speak.

There’s no reason why the microscopic distinction between egg and sperm should determine your role in society. I understand second-wave feminism to argue this, and I agree with it. To argue against it is traditionalist or conservative or right-wing or reactionary.

That doesn’t make the argument wrong. As Socrates shows in various dialogues, it can be hard to know what we really want. Maybe men and women are mistaken, when we want to take roles traditionally reserved for the other sex.

Maybe it was good to have a “handsome pilot” and “pretty stewardess” in Richard Scarry’s Best Word Book Ever. I enjoyed that book as a child, and I have thought since that Scarry was clever to depict humans as rabbits, foxes, and other beasts, who didn’t need humanoid skin-tones. This was in a day when the big box of crayons included a pink one labelled “flesh.”

Scarry’s book came out in 1963, and Crayola had supposedly “phased out” the “flesh” crayon in 1962; however, I could have sworn using that crayon in my childhood, which did not begin till 1965. My memory could be off; however, CNN commentator Elliot Williams remembers the crayon too, “probably” from a time after Star Wars had come into existence in 1977.

Maybe reinforcement of stereotype would be a good thing, and there is a way that our genes or our gametes can show this. I think some people really want there to be a way.

Other people seem to think that, if you want to be a flight attendant, this may be a sign that you are a woman, regardless of your anatomy. They don’t want to assert, as men such as Boy George have done, that everybody is allowed to wear dresses and make-up; they apparently think that these things are indeed for women. Who then is the conservative?

I may just have given a parody of trans activism. According to Christa Peterson, in a blog post on Kathleen Stock from January 11 that has been tweeted by Jason Stanley and others,

The misportrayal of gender identity as simply how well you align with gender stereotypes turns being trans into a mere anti-feminist confusion that should be easy to “cure” – as though the source of trans people’s identities is a false belief that you have to be a woman to wear makeup and dresses and trans people simply need to be informed that men can in fact be feminine and women masculine.

Peterson describes herself as “a philosophy PhD student working on questions about moral language and thought.” I agree with her that it’s important to get straight what people mean by being a woman. As was mentioned in the question period after Reilly-Cooper’s lecture, there are signs in some women’s bathrooms, telling users not to question anybody else’s right to be there, regardless of appearances. This conflicts with the advice that girls are also given, at least at home: be wary of strange men.

There can be a problem with making that advice public, when it gives men an excuse to abuse women who are not wary. “If only you’d looked after your daughter (Siz de kızınıza sahip çıksaydınız),” said one of Şule Çet’s killers to her parents in court. If society just said, “Hey girls, you’re on your own,” that might at least be honest.

Aaden Friday cites statistics of hate crimes and murders of trans people, then says,

As these statistics show, trans individuals, especially trans women of color, are being targeted simply for existing and living their truths. Their suffering is real; the idea that cis women are being harmed by their identity is not.

I don’t know how mere numbers would show why trans persons are targeted. In any case, nobody causes her own harrassment or murder. This should not need saying, but sometimes it does need saying, and Aaden Friday may be saying it. But then they say something that doesn’t make a lot of sense: that there is real suffering, but an unreal idea.

I don’t know what it would mean for an idea not to be real. If the idea can be expressed, then it exists and is thus real. It may however be mistaken or just absurd.

Friday’s point may be that being the target for violence is real suffering, but merely knowing, as a cis woman, that there are trans women is not real suffering.

Nonetheless, knowledge of the thoughts of Kathleen Stock and Rebecca Reilly-Cooper does seem to cause suffering.

Friday continues:

Cis women aren’t being murdered because trans and gender non-conforming individuals exist and, in some locations, receive protection under the law.

Such an assertion would seem unobjectionable; but why do you make it? How do you know it’s true? If even professional philosophers can assert that what some of their colleagues say causes harm, how do we know that the existence of trans persons is not driving other persons to murder?

In the earlier quotation from their letter, the professionals say that feminist discourse is “serving to restrict trans people’s access to life-saving medical treatments.” I am not sure what this is about. The treatments referred to may involve sex reassignment, which could be considered life-saving in the sense of preventing suicide. In any case, the professional philosophers seem to be saying that the words of one of their colleagues are causing death.

What then is causing the murders of women such as Şule Çet? Could it be words that make light of women’s concerns?

Aaden Friday continues:

Cis women are not being denied entrance or access to restrooms, locker rooms, and changing rooms because trans and gender non-conforming people exist and need access to those same spaces for the same reason cis women need them.

Why do cis women need those spaces? One reason is to get away from men. This may be why trans women need those spaces too. However, by some definitions, those trans women, or at least some of them, are men too. So here is a conflict.

Philosopher Holly Lawford-Smith has an extensive analysis, with the conclusion, stated near the beginning,

Gender neutral bathrooms alongside the standard male and female bathrooms are a very good idea. They work for almost everyone.

Lawford-Smith had the essay on Medium, along with a number of others, including a good one on the mistaken “adoption analogy” proposed by another philosopher, Sophie Grace Chappell. Trans women are not to women as adoptive parents are to parents. Parents have the project of raising children, regardless of how they got the children; but women as such have no project. As Lawford-Smith points out, “There’s no such thing as a bad and a good woman qua woman.”

Some conservative persons would probably disagree there. Let them disagree, and let them post their disagreement on Medium if they want. However, Lawford-Smith’s articles are no longer there. The author was, as she says,

Permanently suspended from Medium for ‘Hate speech’. No details provided, appeal rejected without explanation. I was on the platform from October 2018 – October 2020 and had posted 35 gender critical essays.

By their own standards, the philosophers who signed the “Open Letter Concerning Transphobia in Philosophy” would seem to be fueling misogyny, which tries to silence colleagues such as Lawford-Smith. You can look up her old Medium essays on her own website; but the place where I first learned about her was Twitter, and she has been banned from there as well.

Before then, with Jason Stanley, she had an exchange, which I entered with a question. Twitter has erased Lawford-Smith’s part of that exchange, but Stanley answered me,

we are doomed because different factions of the left are at war, whereas for the most part different factions of the right, eg libertarians, ethno-nationalists, and religious conservatives, are unified despite quite profound differences.

It would seem to me that, by signing the “Open Letter Concerning Transphobia in Philosophy,” Stanley himself is engaging in the factional dispute he decried.

I continue to hope that rising above factionalism is possible. Sophie Grace Chappell’s adoption analogy may be misguided, but I think it is offered in good faith. I proposed an analogy between race and gender identity: they are both hypotheses, made to explain things that appear to be true or that people want to be true.

In earlier era, an hypothesis made to explain why things burned: they were giving off phlogiston. When it transpired that substances like magnesium got heavier when they burned, it was inferred that phlogiston could have, not gravity, but “levity.” Ultimately the hypothesis was dropped.

Perhaps the hypothesis of gender identity can be saved. According to Christa Peterson, in the blog post already mentioned (with my bold emphasis),

Like literally everything in the mind, how exactly gender identity is cognitively realized is theoretically complicated. Trans people do not need a theory of gender cognition to validate that their experiences are real and trustworthy. But because it has been actively mystified, we should return to earth: the idea that people have a cognitively deep representation of their own gender that is not reflectively revisable fits very naturally with modern cognitive science. We aren’t behaviorists anymore. We know that what we and other animals learn from our infinitely complex environment depends on what we’re cognitively set up to take from it, and we take to gender like fish to water. A representation of our own gender being among the deep reference points that orient us in the world would not mean any particular gender norms are innate. It need not contain any substantive content about gender role at all; it could, for example, instead be a means of picking out people as who we are co-gendered with, orienting us to particularly learn social behaviors from them. Or it could be an innate framework that is then environmentally filled in during early childhood, like other innate conceptual structures. (See, for example, Susan Carey’s The Origin of Concepts.)

If I understand correctly, there are various things a gender identity could be, without being something that imposes specific societal roles. That’s fine, let the research continue. Still, the rhetoric here recalls Marilynne Robinson’s criticism of consensus as concealing what is true.

Is it really true that “We aren’t behavorists anymore”? If I may quote Asimov’s New Guide to Science (1984), my source also for the image of egg and sperm near the beginning of this post,

In the early decades of this [the twentieth] century, the American psychologist John Broadus Watson built a whole theory of human behavior, called behaviorism, on the basis of conditioning. Watson went so far as to suggest that people have no deliberate control of the way they behave; it is all determined by conditioning. Although his theory was popular at the time, it never gained wide support among psychologists. In the first place, even if the theory is basically correct … behavorism is not very enlightening on those aspects of human behavior that are of most interest to us, such as creative intelligence, artistic ability, and the sense of right and wrong.

It seems to me that, for philosophers at least, failure to be enlightening is no bar to proposing a theory like physicalism; but I have gone into that in another post. As for behaviorism itself, it would seem to be a version of the denial of free will that some philosophers and physical scientists continue to engage in; but I have looked at that too, in one post and another.

One may recognize that behaviorism cannot explain everything, while continuing to find it useful in some contexts. For some people, as far as I can tell, gender is such a context, since they explain a rash of girl-to-boy transitions as social contagion. Thus I think Christa Peterson begs the question to say, “We aren’t behavorists anymore.”

Also in her blog post, I just don’t understand the point of the first sentence that I bolded (“Trans people do not need a theory of gender cognition to validate that their experiences are real and trustworthy”). Does she mean to deny any responsibility for actually making an argument? Without having a theory of gender identity, we may all somehow know which sex ought to be on our birth certificate and which bathroom we ought to be using. But Peterson seems to be begging the question of whether that knowledge can be in error, or whether it can be confirmed by anybody else.

I first learned about Peterson’s post from a tweet by Simone Webb, a “recent PhD in Mary Astell and the late Foucault.” That tweet was quote-tweeted by Dominic Berry, “Historian & philosopher of science, technology & engineering,” who announced on the first of the year, in reference to the Aeon editor referred to earlier,

As Warburton has consistently platformed philosophers with trans exclusionary views, I am making him part of the social media firewall I have around myself. I’ll be ceasing to follow anyone still following him by end of Monday. That’s about 140 of you.

I don’t know whether Berry carried out the threat, which seems puerile to me. I started following Warburton when I saw it, though he tweets a lot, and I often end up muting the person who does that.

On January 11, Kathleen Stock tweeted her appreciation of an “Open Letter Concerning Academic Freedom.”

Stock went on to write about the campaign against her in “The sinister attempts to silence gender critical academics,” The Spectator, 13 January 2021:

The spectacle of paid thinkers, whose entire training emphasises the importance of sober argumentation, signing a document which wouldn’t look out of place in the Salem Witch Trial archive, makes one question particularly pertinent: what’s actually going on here?

That is indeed what I wonder too.

Leiter Reports: A Philosophy Blog turns out to have posted, on January 6, “The “Anti-Kathleen Stock” open letter is full of mistakes, fact-free speculation and misleading innuendo,” addressing some points that I did (namely the error in the original “Open Letter Concerning Transphobia in Philosophy,” the question of which “life-saving medical treatments” the letter refers to, the “smear” about “transphobic fearmongering,” and Christopher Bertram’s apparent indifference to getting “particular details” right).

Peterson quotes Brian Leiter of Leiter Reports as calling a critic of Stock “real vile and stupid piece of garbage … If anyone knows who he is, message or e-mail me.” Peterson comments,

This is silencing. Philosophers go to great lengths to avoid becoming a target for Leiter. It is professionally threatening.

That does sound right.

Peterson reports also on how Stock’s use of the expressions “morons,” “dickhead,” and “fuck off” represented the kind of vitriol that she objected to receiving; but when somebody pointed this out, Stock tweeted back, “Fuck off you dickhead moron xxxx.”

Apparently Stock thought better of that tweet, which she deleted later.

More is going on behind the scenes than is clear from popular articles.

Law and History

I learned about Peter Turchin recently through his profile in the Atlantic by Graeme Wood. I had learned about the Atlantic article from historians on Twitter such as James Ryan, who does “Turkish history and other stuff,” according to his own Twitter profile, and who tweeted in response to Wood’s article,

This is really interesting research, but, uh, it is only history in the way that a particle physicist does history.

In response to that, a thread began:

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The Asıl of the Iliad

Etymologically speaking, the asıl of a thing is its root. The Arabic root of the Turkish word means bitki kökü, “vegetable root,” according to Sevan Nişanyan’s Turkish etymological dictionary.

In the Iliad, why is Achilles so affronted by Agamemnon as to refuse to help the Greeks, even as their attack on Troy is becoming a defensive war, at the wall that they have erected about their own ships? If the answer is to be found through study, then Book IX of the Iliad is what to study.

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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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Pacifism is properly pacificism, the making of peace: not a belief or an attitude, but a practice. Mathematics then is pacifist, because learning it means learning that you cannot fight your way to the truth. Might does not make right. If others are going to agree with you, they will have to do it freely. Moreover, you cannot rest until they do agree with you, if you’ve got a piece of mathematics that you think is right; for you could be wrong, if others don’t agree.

The book *Dorothy Healey Remembers,* with photo of subject

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Being hosted by, thus using the content management system, this blog has posts, pages, and media. This directory is for the pages and the verbal media (namely pdf files).

The posts, such as this one, have initial publication dates and can be seen at, in reverse chronological order. I have them listed in forward order on my About page. As I explain there, I try to keep track of posts with tags and categories. Moreover, if one post revisits a theme of another post, I try to link to that post, which will then show at the bottom which posts are linked to it.

A dream, never to be realized, would be to have all of my ideas as well-organized as in Wittgenstein’s Tractatus Logico-Philosophicus.

I don’t know how the random visitor can find pages, although search engines find some of them. Much less do I know how one would find media, although the media allowed by WordPress now include pdf files. I have uploaded a number of these, and created a number of pages. and in this post (which I hope to remember to keep up to date), I try to classify them, if only to remind myself what they are.

It would be possible to have all top-level pages included automatically in the menu which now forms a horizontal list at the top of each post and page.



Poetry, in the broad etymological sense of something made; call it conceptual art, or whatever you like, but it’s all referred to in the post “Discrete Logarithms”:



Pages describing (as well as listing) categories of posts


For my courses I normally prepare pages on my department’s server; but since I cannot access this from home, I may also use the blog.

  • Aksiyomatik Kümeler Kuramı Özeti (“summary of axiomatic set theory”: an attempt in fall 2019 to supply just that in html; I had not yet discovered the usefulness of pandoc as described in “LaTeX to HTML”)

  • Analitik Geometri Özeti (“summary of analytic geometry,” for a course in spring 2020; as the Covid-19 lockdown took hold, the page just became the course page)

  • Ordinal Analiz (“ordinal analysis,” that is, set theory with emphasis on the ordinals as a structure analogous to the linearly ordered set of real numbers studied in so-called real analysis; the post “Ordinals” also takes up the analogy; I made the page for a course in Şirince, in case I wanted to change the page while I was there, though in the event I didn’t; notes from the second week, in English, are on a departmental page, along with the syllabus for a summer course in 2020 that was cancelled)

  • Öklid (Resources for the course Öklid geometrisine giriş, “introduction to Euclidean geometry,” fall 2020)

  • Ayşe Berkman’ın yedek sayfasıdır (for her spring 2020 course during the lockdown)

Writing of others

Sometimes annotated by me:

Articles on Collingwood

This article gathers, and in some cases quotes and examines, popular articles about R. G. Collingwood (1889–1943).

  • By articles, I mean not blog posts like mine and others’, but essays by professionals in publications that have editors.

  • By popular, I mean written not for other professionals, but for the laity.

Continue reading

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.


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.


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,


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 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


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


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

Multiplicity of Mathematics

I continue with the recent posts about mathematics, which so far have been as follows.

  1. What Mathematics Is”: As distinct from the natural sciences, mathematics is the science whose findings are proved by deduction. I say this myself, and I find it at least implicit in an address by Euphemia Lofton Haynes.

  2. More of What It Is”: Some mathematicians do not distinguish mathematics from physics.

  3. Knottedness”: Topologically speaking, there is a sphere whose outside is not that of a sphere. The example is Alexander’s Horned Sphere, but it cannot actually be physically constructed.

  4. Why It Works”: Why there can be such a thing as the horned sphere.

When I first drafted the first post above, I said a lot more than I eventually posted. I saved it for later, and later is starting to come now.

Continue reading

Why It Works

The last post, “Knottedness,” constructed Alexander’s Horned Sphere and proved, or sketched the proof, that

  • the horned sphere itself is topologically a sphere, and in particular is simply connected, meaning

    • it’s path-connected: there’s a path from every point to every other point;

    • loops contract to points—are null-homotopic;

  • the space outside of the horned sphere is not simply connected.

This is paradoxical. You would think that if any loop sitting on the horned sphere can be drawn to a point, and any loop outside the horned sphere can be made to sit on the sphere and then drawn to a point, then we ought to be able to get the loop really close to the horned sphere, and let it contract it to a point, just the way it could, if it were actually on the horned sphere.

You would think that, but you would be wrong. Continue reading