Polytropy

This post involves:

• “the” philosopher –
  • Aristotle;
• two mathematicians –
  • Euclid,
  • David Hilbert;
• three persons associated with Black Mountain College –
  • Josef Albers,
  • Dorothea Rockburne,
  • Max Dehn;
• one person (in addition to myself and Dehn) associated with St John’s College –
  • David Bolotin.

I am going to elaborate on the following.

• I have been reading Aristotle’s work on the anima, psychê, or soul – De Anima.
• By viewing color as deceptive, Josef Albers differs from Aristotle, particularly in the interpretation of my teacher David Bolotin.
• The art of Dorothea Rockburne was influenced by mathematician Max Dehn at Black Mountain College, where Albers also taught.
• Dehn was a student of David Hilbert, who tried to bring Euclid up to modern standards of rigor.
• The demonstration of the second part of Euclid’s Elements Proposition VI.1 (that parallelograms under the same height are to one another as their bases) cannot be made to work in three dimensions, and Dehn proves that it cannot; this is why Euclid uses what we call infinitesimal calculus for his demonstration of Proposition XII.5 (that triangular pyramids under the same height are to one another as their bases).
• One can construct a work of either conceptual art or chindōgu (i.e. useless invention), based on a correspondence of sums and products, as are both the slide rule and continued proportions. (See also Discrete Logarithms.)

I read De Anima with a Catherine Project group in fourteen sessions. There was one session a week, with some breaks, from December 12, 2024, till the following April 3. Earlier, starting November 7, we had four meetings on selections from the Physics.

For the reading of De Anima, I acquired the 2018 translation of my teacher David Bolotin (born 1944).

For an abortive reading in 2004, I acquired the Oxford Classical Texts edition of De Anima by W. D. Ross. I referred to that book sometimes for the recent reading. Ross’s text seemed likely to be the source of any Greek version that I found online, in case I wanted to make blog posts as for the Nicomachean Ethics in 2023–24. However, in Bolotin’s judgment (page viii),

Ross’ distrust of the manuscript tradition, especially regarding Book III of De Anima, led him to tamper with the text in dozens of passages, with the result that Aristotle’s thought, especially in those passages, becomes inaccessible to those who rely on him.

Presumably Ross would explain what he had done in his apparatus criticus, but I did not want to get into that. This is a reason why I did not make any blog posts about De Anima before this one.

The later parts of this post are based on an email that I sent on February 4 (2025) to my Euclid reading group (meeting since winter, 2023). That group includes the leader of the De Anima group, who mentioned that she was going to be studying Josef Albers.

David Bolotin studied with Leo Strauss, as did Allan Bloom (1930–92). To his translation of Plato’s Republic (read here in 2021), Bloom added an “interpretative essay” more than half as long as the Republic itself. I am sorry Bolotin did not do the same with De Anima, especially since his interpretation would have been based on a career spent reading and talking about Aristotle – him and everybody else who is read and discussed at St John’s College.

Perhaps it was to avoid prejudicing College discussions that, to his translation of De Anima, Bolotin did not append an interpretation, except in copious and sometimes lengthy footnotes. These may clear up local difficulties, without supplying “global” remarks that a student might quote without understanding.

For the recent reading, I sometimes consulted the Penguin edition of De Anima, which has a lot of commentary; however, this is by a 31-year-old.

Bolotin does say in his Introduction (page vii) that

especially when taken together with its sequel, the Parva Naturalia, [De Anima] comes closer than any other work I know to presenting the truth about soul, and its relationship to the world as a whole. Most importantly, it seems to me, it gives the best account, and in fact the only adequate account I am aware of, of how the world as in ordinary experience it is present to us – i.e., to the soul, and in the first place to its faculty of sense perception – can be the true world, which it is, and not, in particular, a dubious and largely mistaken interpretation of data that is found in consciousness. But to understand how Aristotle conceives of the world, in its relation to our grasp of it, requires much more effort than is usually thought …

If you put that much effort into the reading, I think what you come up with is your own conception of the world. That doesn’t mean it’s not Aristotle’s too.

Myself, I am not sure what it would mean for there to be a true world, if there must then also be a false world. In the sonnet pictured at the top of this post (and quoted also at the bottom), John Donne refers to viewing heaven. He refers to it as not happening yesterday, and perhaps then heaven is not a part of “ordinary experience.” It is still part of the world, somehow.

It is a relief now to pass from De Anima to something easier: the Poetics, which may be the subject of future posts.

In the former work, I thought, Aristotle rarely if ever said anything definitive. For example, he would say motion was not caused by desire (433^a6–7) –

ἀλλὰ μὴν οὐδ’ ἠ ὄρεξις ταύτης κυρία τῆς κινήσεως;

later, that it was (433^a21, ^b10) –

ἕν δή τι τὸ κινοῦν, τὸ ὀρεκτικόν;

εἴδει μὲν ἓν ἂν εἴη τὸ κινοῦν, τὸ ὀρεκτικόν, ᾗ ὀρεκτικόν.

I do not say that those passages are in contradiction with one another, although superficially similar passages in a work of mathematics might be. How can we say anything definitive about the anima, the psychê, the soul?

I can say that while the soul does mathematics, it is not studied by mathematics or an experimental science such as physics.

In the Republic, Socrates analyzes the soul into three parts, which I would call appetite, passion, and intellect; however, he is a fictional character in a novel – in a work of poetry, in the broad Greek understanding.

It makes sense that De Anima is difficult. I wonder then whether it imitates its subject. In Poetics chapter I, § 2, Aristotle introduces the concept of imitation, representation, mimêsis, by saying of several of what we call the performing arts, “they all happen to be imitations” (πᾶσαι τυγχάνουσιν οὖσαι μιμήσεις). However, this is matter for another post.

I am looking back now at my own paperback copy of Interaction of Color (Yale, 1975; “Unabridged text and selected plates”), which for some reason I bought when I was a student at St John’s College. Josef Albers opens with the assertion,

In visual perception a color is almost never seen as it really is – as it physically is … In order to use color effectively it is necessary to recognize that color deceives continually.

One can ask how this is to be reconciled with Aristotle, De Anima II.6.2, where apparently color as such does not deceive, though other things do:

λέγω δ’ ἴδιον μὲν

ὃ μὴ ἐνδέχεται ἑτέρᾳ αἰσθήσει αἰσθάνεσθαι,

καὶ περὶ ὃ μὴ ἐνδέχεται ἀπατηθῆναι,

οἷον ὄψις χρώματος

καὶ ἀκοὴ ψόφου

καὶ γεῦσις χυμοῦ,

ἡ δ’ ἁφὴ πλείους [μὲν] ἔχει διαφοράς,

ἀλλ’ ἑκάστη γε κρίνει περὶ τούτων,

καὶ οὐκ ἀπατᾶται

ὅτι χρῶμα

οὐδ’ ὅτι ψόφος,

ἀλλὰ τί τὸ κεχρωσμένον ἢ ποῦ,

ἢ τί τὸ ψοφοῦν ἢ ποῦ.

By a special object of a particular sense I mean

that which cannot be perceived by any other sense

and in respect to which deception is impossible;

for example, sight is of colour,

hearing of sound

and taste of flavour,

while touch no doubt has for its object several varieties.

But at any rate each single sense judges of its proper objects

and is not deceived as to the fact

that there is a colour

or a sound;

though as to what or where the coloured object is

or what or where the object is which produces the sound,

mistake is possible.

The Greek there is from Mikrosapoplous; the English, the 1907 Loeb translation by Hicks, because I was able to cut and paste it. Another translator would seem to distinguish Aristotle’s thought from Albers’s:

Over the years, Bolotin has come to think of Aristotle as superior to other philosophers. While Descartes has argued that the world as it appears to us is illusory, and Kant that we know only the world of appearance (rather than things in themselves), Aristotle was right, in Bolotin’s view, to think that the world as it is given to us is the true world, or reality in the truest sense. And if this is so, Bolotin claims, Aristotle’s account of soul, and its relation to the world, must be right – at least in its fundamentals.

I am looking at these things, because I chanced upon an email I had sent in 2017 to somebody called David Peifer, author of “Dorothea Rockburne and Max Dehn at Black Mountain College” (Notices of the American Mathematical Society, December, 2017).

At BMC, Rockburne apparently found Dehn’s formal mathematics lessons beyond her. Nonetheless, she accompanied Dehn on his walks in the mountains. He talked about mathematics, and she later read about it on her own. Peifer tries to relate some of Rockburne’s artworks to mathematics. The connection may be clarified by these words of Rockburne’s from her website (ultimately from a 1972 interview; I am correcting what seem to be typos):

My interest in Set theory is not that Set Theory has to do with my art, because it doesn’t. I am an artist and it is one of my tools, the way graphite is. The usage of it comes from personal experience. In college I had the good fortune to meet a theoretical mathematician. Mathematics didn’t interest me but somehow Max Dehn’s enthusiasm was contagious. He erased the panic and showed me how to put one foot in front of the other. He introduced me to math as a consistent history of thought, the thing I responded to in art.

Then too, I was angered by the fiction I read because to read novels, on no matter what level requires some empathy with the people who are being portrayed. Women are usually depicted as plodders, fools or victims. I couldn’t in any sense identify with them and started to read books on mathematics. Math by contrast was straight, simple thinking and it never enclosed its own situation. If it did it was only a situation to be set aside for later consideration in relation to something else which would again open the total concept. I was excited by this and bored by art school instruction. I knew, though that I was an artist and not a mathematician.

I wonder whether Rockburne had read Middlemarch (which I am reading in another Catherine Project group) or Jane Eyre.

Like Albers, Dehn was a refugee from Nazi Germany. According to Wikipedia, when he got to the US, he worked at various places, including St John’s College in Annapolis, before moving to Black Mountain College in 1945.

Dehn had been a student of David Hilbert, whose Foundations of Geometry is intended to update Euclid:

Geometry, like arithmetic, requires only a few and simple principles for its logical development. These principles are called the axioms of geometry. The establishment of the axioms of geometry and the investigation of their relationships is a problem which has been treated in many excellent works of the mathematical literature since the time of Euclid. This problem is equivalent to the logical analysis of our perception of space.

This present investigation is a new attempt to establish for geometry a complete, and as simple as possible, set of axioms and to deduce from them the most important geometric theorems in such a way that the meaning of the various groups of axioms, as well as the significance of the conclusions that can be drawn from the individual axioms, come to light.

To fellow mathematicians, starting in 1900, Hilbert proposed 23 problems, and apparently Dehn was the first to solve any one of them.

We know from Elements I.37 that triangles on the same base are equal if that base is parallel to the straight line through the apices of the triangles.

This is because each triangle is half a parallelogram, and under the hypothesis, the parallelograms are equal by I.35: the proof is that if we

• add the same triangle to each parallelogram, and
• remove the same triangle from each combination,

then what we get is two congruent triangles.

All of this gives us Euclid’s first geometrical result about proportion, namely Proposition VI.1: triangles and parallelograms under the same height are in the ratio of their bases.

In a similar way, now adding and removing the same solids, we are going to get Propositions XI.29 and 30, whereby parallelepipedal solids (στερεὰ παραλληλεπίπεδα) under the same height and on the same base are equal.

The parallelepipedal solids are still equal if they are on equal bases: that’s Proposition XI.31. Finally, as we got part of VI.1, so we get Proposition XI.32: parallelepipedal solids under the same height are in the ratio of their bases.

A parallelogram can be divided into two triangles that are congruent and therefore equal.

Likewise, a parallelepiped can be divided into six tetrahedra, or what Euclid calls pyramids having triangular bases (πυραμίδες τριγώνους ἔχουσαι βάσεις). Each of the six is congruent to another, in the sense of being a mirror image, but there may be no other congruences among the six. Therefore we do not have an immediate proof of Proposition XII.5, that tetrahedra under the same height are in the ratio of their bases.

Euclid’s proof is by what we today call infinitesimal calculus.

Hilbert asked whether, as for I.37, there was an elementary proof, by adding and subtracting solids, that tetrahedra under the same height and on on equal bases are equal.

Max Dehn proved that there wasn’t.

On a piano, if you press one key, then the twelfth key over, you have changed tones by an octave.

The same happens if you pluck one string, then, holding it down in the middle, pluck one of the halves.

In short, distances along the keyboard correspond to ratios of parts of a single string.

This is the principle of the slide rule, as well as of the continued proportions, or geometric sequences, that Euclid studies in Elements Book IX. The same number of steps along a geometric sequence always corresponds to the same ratio. Algebraically, the principle is

(“multiplying powers of the same base is the same as adding the exponents”).

Pierre de Fermat (1601–65) observed that, for example, in the continued proportion or geometric progression

the 3rd term is 1 more than a multiple of the prime number 13, and then so is the 6th term; moreover, 3 is a factor of 13 − 1. This is not an accident, but

Every prime number is always a factor of one of the powers of any progression minus 1, and the exponent of this power is a divisor of the prime number minus 1 … I would send you the proof if I were not afraid to be too long.

(That is from a letter to Bernard Fránicle de Bessy, translated in D.J. Struik, ed., A Source Book in Mathematics 1200–1800.)

If instead of 3 in the progression above, we start with 2, then not until the 12th term, which is 4096, do we get 1 more than a multiple of 13. Since 12 = 13 − 1, this means 2 is a primitive root of 13: every non-multiple of 13

• differs from a power of 2 by a multiple of 13, that is,
• is congruent, modulo 13, to a power of 2.

That 3 is a primitive root of 17 can used for a “circular slide rule” that performs multiplication modulo 17 as follows. If you want to multiply, say, 13 and 16, then, on the ring below, note that from 1 to 13 there are four steps, and four steps beyond 16 in the same direction is 4, which is therefore the remainder of 13 × 16 after division by 17.

Here is the text of the last of John Donne’s Holy Sonnets, pictured at the top of this post.

OH, to vex me, contraryes meet in one :
Inconſtancy unnaturally hath begott
A conſtant habit ; that when I would not
I change in vowes, and in devotione.

As humorous is my contritione
As my prophane Love, and as ſoone forgott :
As ridlingly diſtemper’d, cold and hott,
As praying, as mute ; as infinite, as none.

I durſt not view heaven yeſterday ; and to day
In prayers, and flattering ſpeaches I court God :
To morrow I quake with true feare of his rod.
So my devout fitts come and go away
Like a fantaſtique Ague : ſave that here
Thoſe are my beſt dayes, when I ſhake with feare.

The photograph is of a booklet that I made comprising all of the sonnets. I had a stationer

• print out my pdf file onto two sides of each of six sheets of A4 paper,
• cut the stack of sheets to size A5,
• turn one of the new stacks under the other, then
• staple and fold the new stack into size A6.

I have since corrected the error on the last line. I wanted to have the poems in a handy format, to read perhaps when walking through the forest. In the fall of 2024, I had joined a group to read Rumi, Love Is a Stranger, translated by Kabir Helminski (Brattleboro, Vermont: Threshold Books, 1993). The translator says in his Introduction,

More than fourteen years have passed since I began to translate some of Rumi’s poetry … this city of Sufi verse is now a more familiar place … This world is as real and present to me as the outer world I live in, but even more so, for it is a world of meanings and not just things.

What can this mean? Such assertions make me nervous, since a sudden physical calamity might change one’s whole outlook.

One of the poems in Love Is a Stranger is called “A World with no Boundaries” and includes the lines (on page 63),

A fountain of refreshment
is in the head and the eyes –
not this bodily head
but another pure, spiritual one.

I would question the idea that there is something “impure” about the body we have.

Later, in “Empty the Glass of Your Desire” (page 75):

Abandon life and the world,
and find the life of the world.

Is that what Jesus says (Matthew 16:25)?

For whosoever will save his life shall lose it: and whosoever will lose his life for my sake shall find it.

In any case, from Rumi, I found myself turning to the Psalms and then Donne.