Polytropy

In Book I of Plato’s Republic, Socrates distinguishes between two ways to respond to a disagreement. The two parties can:

1. Have a debate, to be judged by a third party.
2. Work with one another to resolve the disagreement.

The latter would seem to be dialectic, although Socrates does not call it that. I said this last time, when I also noted that Socrates does refer to dialectic as such in Book V; but I deferred investigation of the passage till now.

In the passage now in question, at 454a, Socrates reiterates the distinction that he made in Book I – or perhaps he makes a new one, but dialectic is still one of the two responses that are being distinguished:

“Oh, Glaucon,” I said, “the power of the contradicting art (ἡ δύναμις τῆς ἀντιλογικῆς τέχνης) is grand.”

“Why so?”

“Because,” I said, “in my opinion, many fall into it even unwillingly and suppose they are not quarreling but discussing (οὐκ ἐρίζειν ἀλλὰ διαλέγεσθαι), because they are unable to consider what’s said by separating it out into its forms. They pursue contradiction (τὴν ἐναντίωσιν) in the mere name of what’s spoken about, using eristic, not dialectic (ἔριδι, οὐ διαλέκτῳ), with one another.”

The translation is by Bloom, who notes that the passage gives us the first appearance of the word “dialectic” in the Republic. Cornford’s translation does not use this word:

It is extraordinary, Glaucon, what an effect the practice of debating has upon people.

Why do you say that?

Because they often seem to fall unconsciously into mere disputes which they mistake for reasonable argument, through being unable to draw the distinctions proper to their subject; and so, instead of a philosophical exchange of ideas, they go off in chase of contradictions which are purely verbal.

I have tried to bold the same parts that I did in Bloom’s translation; however, Cornford seems to have nothing for Bloom’s “eristic.” This word in the Greek is a case of ἔρις rather than ἐριστικός, just as Bloom’s “dialectic” is from διάλεκτος, rather than διαλεκτικός. Nonetheless, Bloom’s note on the passage is as follows.

“Eristic” means “contentiousness,” argument carried on for the sake of winning; formally it looks like “dialectic,” which is friendly conversation; but eristic is not carried on for the sake of truth. This is the first use of the word “dialectic” in the Republic and provides a first common-sense view of its meaning.

More precisely, the passage gives us the first use of the word “dialectic” in Bloom’s translation. As I noted last time, Cornford’s first use is in Book VI, 511b, where the word names that which takes in the ultimate part of the Divided Line:

Then by the second section of the intelligible world you may understand me to mean all that unaided reasoning (αὐτὸς ὁ λόγος) apprehends by the power of dialectic (τῇ τοῦ διαλέγεσθαι δυνάμει) when it treats its assumptions (τὰς ὑποθέσεις), not as first principles, but as hypotheses in the literal sense (τῷ ὄντι ὑποθέσεις), things ‘laid down’ like a flight of steps up which it may mount all the way to something that is not hypothetical (μέχρι τοῦ ἀνυποθέτου), the first principle of all; and having grasped this, may turn back and, holding on to the consequences which depend upon it, descend at last to a conclusion, never making use of any sensible object, but only of Forms, moving through Forms from one to another, and ending with Forms.

What Cornford translates as “dialectic” is the infinitive διαλέγεσθαι. Bloom translates it the same way:

Well, then, go on to understand that by the other segment of the intelligible I mean that which argument itself grasps with the power of dialectic, making the hypotheses not beginnings but really hypotheses – that is, steppingstones and springboards – in order to reach what is free from hypothesis at the beginning of the whole. When it has grasped this, argument now depends on that which depends on this beginning and in such fashion goes back down again to an end; making no use of anything sensed in any way, but using forms themselves, going through forms to forms, it ends in forms too.

Near the beginning of Book I (328a), when urging Socrates to come to his house, Polemarchus says “we’ll talk”: διαλεξόμεθα, a finite form of the verb διαλέγεσθαι: this is the first reference in Bloom’s Subject Index under “Discussion, dialegein, sometimes also translated dialectic.”

Thus, where Bloom says of 454a in Book V, “This is the first use of the word ‘dialectic’ in the Republic and provides a first common-sense view of its meaning,” it would be more accurate to replace “and” with “that.” A common-sense view of dialectic was given earlier at 348a–b in Book I, but the word “dialectic” was not used.

As for eristic, Bloom’s Subject Index has an entry for this, citing only

• 454a–b in Book V, where, again, Socrates distinguishes quarreling (eristic) from discussing (dialectic);
• 499a in Book VI, where the topic is what we call the philosopher-king and the difficulty of having one; here, Bloom translates the neuter plural τὰ δὲ κομψά τε καὶ ἐριστικὰ as “the subtleties and contentious quibbles.” (Note 1)

Socrates on Sex and Gender

Socrates distinguishes quarreling from discussing, in order to be able to discuss sex and gender, as in 454d–e, here in Bloom’s translation (with my bullets):

• if either the class of men or that of women shows its superiority in some art or other practice,
  • then we’ll say that that art must be assigned to it. But
• if they look as though they differ in this alone, that the female bears and the male mounts,
  • we’ll assert that it has not thereby yet been proved that a woman differs from a man with respect to what we’re talking about; rather,
  • we’ll still suppose that our guardians and their women must practice the same things.

That Socrates says this doesn’t mean he believes it. That Plato has Socrates saying it doesn’t mean he believes it. I will go so far as to assert that what Plato has Socrates say is worthy of attention.

For once, Bloom turns out not to be as literal as could be. Here is the Greek:

τὸ τῶν ἀνδρῶν καὶ τὸ τῶν γυναικῶν γένος,

• ἐὰν μὲν πρὸς τέχνην τινὰ ἢ ἄλλο ἐπιτήδευμα διαφέρον φαίνηται,
  • τοῦτο δὴ φήσομεν ἑκατέρῳ δεῖν ἀποδιδόναι:
• ἐὰν δ᾽ αὐτῷ τούτῳ φαίνηται διαφέρειν, τῷ τὸ μὲν θῆλυ τίκτειν, τὸ δὲ ἄρρεν ὀχεύειν,
  • οὐδέν τί πω φήσομεν μᾶλλον ἀποδεδεῖχθαι ὡς πρὸς ὃ ἡμεῖς λέγομεν διαφέρει γυνὴ ἀνδρός, ἀλλ᾽
  • ἔτι οἰησόμεθα δεῖν τὰ αὐτὰ ἐπιτηδεύειν τούς τε φύλακας ἡμῖν καὶ τὰς γυναῖκας αὐτῶν.

Bloom’s English is less than literal, because it

• does not reflect the parallelism of the μέν–δέ construction as well as it might;
• makes a disjunction (“men or … women”) where Socrates has a conjunction (“men and … women”);
• introduces superiority where Socrates mentions only difference.

The last point is the most important. Here is Shorey’s translation:

• if it appears that the male and the female sex have distinct qualifications for any arts or pursuits,
  • we shall affirm that they ought to be assigned respectively to each. But
• if it appears that they differ only in just this respect that the female bears and the male begets,
  • we shall say that no proof has yet been produced that the woman differs from the man for our purposes, but
  • we shall continue to think that our guardians and their wives ought to follow the same pursuits.

One could assert that men should rule on either of two grounds:

• They are better at it.
• It is their job.

Socrates is disputing the latter. He goes on to say that men are usually better than women in doing things; the point is that no job of government belongs exclusively to a man or woman as such (455d, Bloom):

Therefore, my friend, there is no practice of a city’s governors which belongs to woman because she’s woman, or to man because he’s man (γυναικὸς διότι γυνή, οὐδ᾽ ἀνδρὸς διότι ἀνήρ); but the natures are scattered alike among both animals; and

• woman participates according to nature in all practices, and
• man in all, but in all of them
• woman is weaker than man.

The fallacy being objected to would seem to be the following.

• A and B are different;
• A does X;
• therefore B cannot do X.

Socrates puts more elaborately at 454a–b, in the continuation of the passage we started with; Glaucon speaks first, as follows, in Bloom’s translation:

“This is surely what happens to many,” he said. “But this doesn’t apply to us too at present, does it?”

“It most certainly does,” I said. “At least we run the risk of unwillingly dealing in contradiction.”

“How?”

“Following the name alone, we courageously, and eristically (ἀνδρείως τε καὶ ἐριστικῶς), insist that a nature that is not the same must not have the same practices (τὸ μὴ τὴν αὐτὴν φύσιν ὅτι οὐ τῶν αὐτῶν δεῖ ἐπιτηδευμάτων τυγχάνειν). But we didn’t make any sort of consideration of what form of different and same nature, and applying to what, we were distinguishing when we assigned different practices to a different nature and the same ones to the same.”

“No,” he said, “we didn’t consider it.”

Given the subject being discussed, we might use the etymological translation of ἀνδρείως as “manfully,” in place of Bloom’s “courageously.”

Sex and Gender Today

If A and B are different, and A does X, it is a fallacy to conclude that B cannot do X. People do commit the fallacy, as for example in case

Socrates himself uses the example of hair, in the continuation of the last passage:

“Accordingly,” I said, “it’s permissible, as it seems, for us to ask ourselves whether the nature of the bald and the longhaired (φαλακρῶν καὶ κομητῶν) is the same or opposite. And, when we agree that it is opposite, if bald men are shoemakers, we won’t let the longhaired ones be shoemakers, or if the longhaired ones are, then the others can’t be.”

“That,” he said, “would certainly be ridiculous.”

It is a curiosity that κομήτης can refer to a long-haired star, namely a comet, which is why we call it that.

In the nineteenth century, ten years into the reign of Queen Victoria, Charlotte Brontë called out the fallacy above in case

Here is Jane Eyre (1847; vol. I, ch. XII):

Women are supposed to be very calm generally: but women feel just as men feel; they need exercise for their faculties, and a field for their efforts, as much as their brothers do; they suffer from too rigid a restraint, too absolute a stagnation, precisely as men would suffer; and it is narrow-minded in their more privileged fellow-creatures to say that they ought to confine themselves to making puddings and knitting stockings, to playing on the piano and embroidering bags. It is thoughtless to condemn them, or laugh at them, if they seek to do more or learn more than custom has pronounced necessary for their sex.

It is my impression that there are men today who

• want “to confine themselves to making puddings and knitting stockings, to playing on the piano and embroidering bags,” or perhaps to wearing women’s clothes, but
• want therefore to be taken for women.

This seems to be an application of the fallacy above, recast in the following form.

• A does X;
• B does X;
• therefore A and B are the same.

Passionate disagreement arises here. (Note 2) We have seen how Socrates lays out two or three possible responses:

• debate,
• quarreling (eristic),
• discussion (dialectic).

Another possible response is the authoritarian peremptory lecture, as for example in the San Francisco Chronicle (June 1, 2023):

Ask many Americans these days and they will insist upon the scientific validity of a binary definition of sex. This line of thinking holds that there are only two sexes available for humans to inhabit: male or female.

As a doctorate-carrying scientist, however, I attest that this is false.

Thus Ash Zemenick in “Sex and gender are binaries? Sorry, that’s a scientific falsehood.”

There are emergencies when ignorant people need to shut up. We may be able to make them shut up by asserting our own authority the way Zemenick does. (Note 3)

Nonetheless, I hope my own students accept nothing from me, merely because I am the teacher. They should learn to use their own authority to confirm my assertions – or to correct them, when I am in error. But then, my students and I are doing mathematics, where everybody who takes an interest in an assertion ought to have, or to be given, the resources to check the assertion.

More precisely, (Note 4) students should accept nothing from me as being automatically true. They should however accept certain hypotheses.

Hypothetical Science

We saw hypotheses in the passage on the last part of the Divided Line. The word itself comes from the Greek ὑπόθεσις. From corresponding Latin parts we have “supposition,” meaning the same thing: something placed underneath, so that we can build an argument on it. Possible mathematical hypotheses include:

1. A certain triangle has two equal sides.
2. A straight line can be drawn from any one point to any other.
3. From each pair in an infinite collection of pairs, one of the two members can be chosen, all at once.

Hypotheses of the first two types are what Collingwood alludes to in An Essay on Philosophical Method (2005/1933), where he says of the theorem “that every square has its diagonals equal” (pages 117–8),

What is necessary is not to believe that a square anywhere or in any sense exists, but to suppose it. Given a rough chalk diagram, we must be able to suppose these broad and crooked marks to be straight lines, suppose these lines equal, and suppose each to be connected with its neighbours by a right angle.

The ability to use a chalkboard (or something like it) may be “a conditio sine qua non of geometrical knowledge,” but is “not geometrical knowledge itself.”

In mathematics we frame a supposition and then see what follows from it; this complex thought is called in logic a hypothetical proposition; and it is of such propositions that the body of mathematical knowledge is composed.

Somebody who knows mathematics and delights in “the power of the contradicting art” may dispute Collingwood’s description of mathematical practice. We don’t just “frame a supposition and then see what follows.” However, our students may get this impression, as when we demand of them, “Let f be a symmetric bilinear form on a finite-dimensional vector-space V,” so that we can show that f has a diagonal matrix with respect to some basis of V.

Collingwood’s purpose is to distinguish hypothetical sciences, such as mathematics, from philosophical sciences, such as ethics and logic. The latter are categorical, albeit in a way that Collingwood defines only negatively: they are categorical,

• not in the sense of delivering authoritarian peremptory lectures, but
• in the sense of not being hypothetical.

The argument for why ethics and logic are categorical is that they themselves are instances of what they study. Indeed, according to the terminology that Collingwood will use in later work, these sciences are criteriological, because they give an account, a logos, of criteria that we already use in assessing what we do and say. Every science consists of things that we do and say. Thus it is no mere hypothesis that a philosophical science is about something: it is about itself, if nothing else.

I tried to talk about all of that in “A New Kind of Science.” The distinction between hypothetical and philosophical sciences corresponds to the distinction between the third and fourth sections of the Divided Line. As Collingwood has it (EPM, page 123), referring to the passage that we have already quoted,

That the aim of philosophy is in the last resort to formulate its thought categorically, is a principle repeated by philosophers of all times in the most varied manners. Thus Plato, discussing the difference between dialectic and mathematics in a passage already quoted (Rep. 511 b), explains that whereas the starting-points of mathematical reasoning are mere hypotheses, dialectic demands for itself a ‘non-hypothetical starting-point’ (ἀρχὴ ἀνυποθέτος).

I don’t want to go much further with the Divided Line. I reviewed the layout in an earlier post with an obvious title, “The Divided Line.” I asked whether the social contract that Glaucon describes in Book II of the Republic should be counted as an hypothesis. Thus I started to question whether the third section of the Line comprises only mathematical objects. Cornford suggests that it does, in a table that I reproduce approximately as follows. (Note 5)

I am counting the sections from the bottom up. Although Cornford puts the Good on his table, I don’t think it constitutes a fifth section of the line; it illuminates the line, or the upper part, as the Sun illuminates the lower part.

The third section of the line comprises the objects of hypothetical sciences. Empirical science is one of these, as Collingwood also shows. Biology studies things that do happen, but interprets them by means of hypothetical “fixed points”:

… when a pathologist or bacteriologist talks of tuberculosis, he is talking of facts that are actual, of a disease that exists and from which people really die. But the scientist is not concerned simply with bare facts in all their multiform variety. He is also and especially concerned with a certain framework into which he fits them, grouping them round fixed points and treating these fixed points as foci of his thought.

Thus tuberculosis is “a certain set of standard symptoms with a standard history, to which all these varieties [of clinical phenomena] more or less conform”; they may not exactly conform, any more than a chalkboard square is exactly what Euclid constructs in Proposition 46 of Book I of the Elements.

Biology

Likewise, no organism of a species that reproduces sexually may conform exactly to a textbook definition of male or female. However, as far as I know, the following happens.

• An organism itself can tell the difference between males and females of its species, in the sense that
  • the organism competes with members of its own sex,
  • in order to mate with a member of the opposite sex.
• The features that allow successful mating are inherited by the offspring.

These assertions may not always be true, but they are essential hypotheses for explaining the origin of species. At least, that is what I hypothesize, as a biological layman, about the theory of sexual selection as understood by experts. There is also natural selection, namely selection of who gets to mate whom by external forces, not by the potential mates themselves.

That’s biology. Nonetheless, according the San Francisco Chronicle article by Dr Ash Zemenick, who earned a Ph.D. in ecology in 2017, after a B.S. in ecology and evolutionary biology in 2011,

Biological sex can be defined in many ways. And when it is accurately defined, it’s never binary. How so, you might ask? Let me explain.

First, let’s start with what we can see – external genitals – the penis, testes, vulva and vagina. They must be binary, right? That’s how we assign sex at birth! I’m so sorry, but, nah. Think about the frequency of people with naturally red hair. Roughly the same frequency of people are born with intersex characteristics …

The condescension here, the “I’m so sorry, but, nah” – this may betray the author’s struggle with doublethink. (Note 6) Dr Zemenick is avoiding the whole truth, and he knows it, because (I presume) he learned about sexual selection during his studies, and he is able to corroborate the account that I gave. Perhaps the account can be corrected, the way Newtonian physics is corrected by Einsteinian physics:

Einstein has been accepted over Newton, not as a replacement so much as a refinement. The Newtonian view of the universe can still be used as a simplified approximation that works well enough in ordinary life …

Thus Isaac Asimov in Asimov’s New Guide to Science (1984; page 390). (Note 7)

Dr Zemenick must know that the theory of sexual selection requires no particular definition of the sexes; this comes with further research. Meanwhile, sexual selection requires that organisms recognize a distinction between sexes, in the way I described. Likewise, we humans recognize a distinction between grammatical and ungrammatical sentences, even if we cannot explain what it is. (Note 8)

Zemenick also knows that sex is not “assigned at birth” by some external authority; rather, it is intrinsically determined at conception. Sex can be inferred, if not in in utero, then at birth, by a look at the genitalia of the child in question. Sometimes the inference is mistaken, or cannot be made in the first place; however, as far as I know, this happens far more rarely than a red-haired child is born. (Note 9)

I cannot tell the difference between male and female pigeons, unless I see one of them strutting around with chest puffed up, and another standing by. I then assume that the former pigeon is male, the latter female. My hypothesis is confirmed if the former is able to jump on top of the latter. As far as I know, the bird on top, in one instance of mating, will never be on the bottom in another instance – or if it is, still, the bird will have offspring from only one form of mating. Thus I have an empirical criterion for distinguishing pigeons as male and female. If I were an ornithologist, I might look for an alternative criterion in the body of the pigeon. I believe chicken farmers are able to use such a criterion to distinguish female chicks from males; this is important if the farmers are looking to raise laying hens.

With a microscope, one can look further. Zemenick takes up this option for humans.

Next, let’s consider the chromosome. XX and you’re a girl, XY and you’re a boy, right? Not necessarily. To put it briefly, chromosomes can have really beautiful and varied arrangements. People can be born not just with XX and XY, but also XYY, XXX, XXY or XO. Chromosomal sex is not binary.

Dr Zemenick leaves out the possibility of being a girl with XY chromosomes, or a boy with XX. As for the “really beautiful and varied arrangements,” one might see them in diagrams such as the one in Scientific American with the article “Beyond XX and XY: The Extraordinary Complexity of Sex Determination.” However, as far as I know, people with unusual chromosomes have health problems, concerning reproduction if not otherwise.

According to his homepage, in his doctoral research Zemenick “unraveled the complex interactions between plants, flower visitors, and floral microbes.” Perhaps not the microbes, but the plants and the “flower visitors,” such as bees, butterflies, and hummingbirds: they all reproduce sexually, as far as I know. At its theoretical base, sexual reproduction involves a union of two gametes, male and female. Zemenick knows this, but makes a feeble attempt to discount it:

What are gametes? Reproductive cells. Eggs and sperm. Sounds binary, right? As a human, you either produce eggs or sperm, yeah? Nah … some people … don’t produce or have gametes at all. Therefore, there are three states: no gametes, eggs or sperm. It’s a triplet, a trifecta. Gametic sex is not binary.

Instead of the words in “tri-,” which don’t fit the setting, why not use the word that does fit? Gametic sex is ternary: this would seem to be the natural conclusion to Zamenick’s line of thought. (Note 10)

The man does not want to go there. He does not want to limit himself to any particular number of sexes, or even to suggest that there is a definite number at all. Although he doesn’t say sex is a spectrum, he mentions a distribution, as in, “the levels of estrogen and testosterone in bodies is a distribution.” That doesn’t mean there aren’t two sexes; it may mean hormone levels are not a great way to tell the difference.

Zemenick is not interested in telling the difference. What he wants to talk about is

the lived experiences of your community members – they may be trans men, trans women, nonbinary folks and two-spirit people, amongst many other genders that are just a Google away.

Thus he passes to gender, which is perhaps a topic for sociology or history, rather than biology. Zemenick refers to it as

sex of the mind, or what I’d like to call cerebral sex. This is one way to think about gender – how we, in our heads, think about ourselves and how we fit into our society and culture.

If you are going to call this cerebral sex, and not simply cerebration, then I might expect it to concern your biological sex, as known by you in the sense required by the theory of sexual selection. However, Zemenick goes on to suggest that there may be no connection:

While a doctor may need to know your genital or hormonal sex, society at large should always respect, define or treat people based on their cerebral sex.

I don’t think Zemenick’s training as a scientist gives him authority to make such an ethical assertion. It would be a contradiction in terms, if he meant to suggest that you should be allowed to hang out naked in a spa reserved for women, as long as you though you belong there. Reserving a space for some class means gatekeeping of some kind, as I tried to argue in “Parenthood and Sex.”

Zemenick’s “cerebral sex” would seem to be what others call gender identity. A way I can make sense of it is to look again at mating as involving two steps:

1. Competing with members of one’s own sex, in order to be able to pass to the next step.
2. Choosing a mate among available members of the opposite sex.

It can happen in step 2 that the mate you seek is from your own sex; then you are homosexual. Likewise, in step 1, you may choose your competitors from the opposite sex; then you are transsexual. (Note 11) All of this would seem to make sense biologically, because it can be observed in any sexually reproducing species, in principle; however, I do not know whether it actually is observed. In any case, what it means ethically is not clear. (Note 12) In “Be Sex Binary, We Are Not,” I noted how studies of primates might be used, albeit not alone, to justify human abortions.

According to Zemenick, “the most reductive definition of sex” is the one in terms of gametes. That he would say this may explain why he avoids talking honestly about biology. Is it reductive to define a human being as a member of the species Homo sapiens? It would be reductive to conclude from this that sociology, or history, is a part of biology.

It is likewise reductive to assign jobs according to sex when they have nothing to do with sex. We have looked at Socrates’s rejection of this. Again, what Socrates says need not represent Plato’s final word on a subject; and even if it did, we need not accept it. Still, the fact that Plato wrote dialogues is a good recommendation of the dialectical method.

Collingwood talked about “grouping [bare facts] round fixed points and treating these fixed points as foci of [our] thought.” It seems to me the dialogues of Plato would make good fixed points.