Entries tagged with mathematics

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Excelsior

Archimedes, the Sicilian, asked for a fulcrum situated outside of the earth to move the earth, saying: “Whilst I inhabit it I cannot act upon it.”

(Synesios on Dreams IV, as translated by Isaac Myer.)

Arithmetical truth cannot be defined in arithmetic.

(Informal statement of Tarski's Undefinability Theorem.)

From any given system, one hasn't the perspective to make sense of that system. For that, one needs a perspective outside the system.

This has two implications. First, it makes sense of why the infinite becomes finite in an attempt to know itself: there is nothing outside of God, and so an outside perspective must be constructed, so that part of God may come to know God in part. Second, it perhaps explains why we strive ever higher: if we have questions about the system, it is only by ascending to the next higher system that we can answer those questions, causing us to rise until we return to God.

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How many men did Odusseus leave Troia with?

Okay, this is strictly silly and has nothing to do with spirituality.

Book II of the Iliad tells us that Odusseus came to Troia with twelve ships. It doesn't say how many men those ships had, but it does say the Boiotian ships had six score men apiece; if we assume the same figure for the Kephallenians, then Odusseus initially commanded 1440 men.

Odusseus still had all twelve ships by the time he reached the island of the Circle-Eyes. For sake of argument, I'll assume that he divided his remaining men—however many they were—evenly among his remaining ships when they left Troia, but didn't redistribute them afterwards as they took casualties (all of which came from Odusseus's ship).

When Odusseus reached Aiaia, he divided his men into two halves, one half commanded by himself, and the other by Eurulokhos, his brother-in-law. Eurulokhos's group had 23 men; I presume, therefore, that Odusseus's either also had 23 or else had 24 (with Odusseus taking the odd man out due to seniority). This means that at Aiaia, Odusseus had either 46 or 47 men.

Prior to Aiaia, seven of the men from Odusseus's ship got eaten (one by Antiphates, the king of the Laistrugons; and six by Very-Famous, the Circle-Eye), meaning he had 53 or 54 after the counter-attack by Ismarians; since they killed six men from each ship, the ships presumably had 59 or 60 each when leaving Troia.

If we assume 60 each, then things are too nice and neat to be a coincidence: this would mean that the Troians halved Odusseus's men, while the Ismarians decimated them, which is very convenient from an authorship perspective. If we accept that, then the voyage looks like this:

Odusseus left Ithake with twelve ships of 120 men each (1440).
Odusseus left Troia with twelve ships of 60 men each (720).
Odusseus left Thrake with twelve ships of 54 men each (648).
Odusseus left the island of the Circle-Eyes with eleven ships of 54 men each and one ship (his own) of 48 men (642).
Odusseus left the island of the Laistrugons with one ship of 47 men.
Odusseus left Aiaia with one ship of 46 men.
Odusseus left Skulla and Kharubdis with one ship of 40 men.
Odusseus left Triangle Island alone.

How harrowing! On the other hand, Odusseus was the only eyewitness and a notorious liar, so who really knows?

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The Seed of Life and the Mysteries

There's a very famous geometric pattern inscribed on the Temple of Osiris at Abydos, called the "Seed of Life," which looks like this:

Nobody really knows the formal significance of the shape, but I noticed something interesting about it while I was pondering the mysteries this evening.

Pythagoras was famously the first Greek to formally be initiated into the mysteries of the Osiris cult (though, of course, there must have been prior transmission since the Demeter and Dionysus mysteries are related). A generation later, Empedocles was initiated into the Pythagorean brotherhood, but later expelled for revealing the mysteries in writing. I conjecture that Empedocles' poem was derived from the Osiris cult, since it concerns the same phenomenon (the descent and reascent of the soul) and features the four gods:

τέσσαρα γὰρ πάντων ῥιζώματα πρῶτον ἄκουε·
Ζεὺς ἀργὴς Ἥρη τε φερέσβιος ἠδ' Ἀιδωνεύς,
Νῆστις θ' ἣ δακρύοις τέγγει κρούνωμα βρότειον.
First, hear of the four roots of all things:
shining Zeus and life-giving Hera and Aidoneus
and Nestis, who wets the springs of mortals with her tears.

It seems pretty reasonable to equate Osiris with Zeus, Hera with Isis, Set with Hades, and Nephthys with Nestis. Now Empedocles talks about how the roots begin united in Love, but peel off one at a time as Strife begins to intervene: first fire, then air, then water, then earth; this is the same as the first part of the Isis myth, where Osiris (fire) is killed, sealed in Set's box (air), dumped in the Nile (water), and encapsulated in a heather stalk (earth). We have a geometric symbol for the same thing: Pythagoras's tetractys, showing the progression of unity (1) into completion (10). It fits very nicely onto the Seed of Life:

Now, the second part of the myth has Osiris chopped into fourteen pieces, but his penis gets eaten by a fish and is never found, so Isis has to make do with the thirteen remaining pieces. Guess how many intersection points the Seed of Life has?

Finally, the last part of the myth has Horus (in place of Osiris) defeating Set and becoming king. This is a myth about the re-ascension of the soul back to its source: the three battles between Horus and Set are the rise from earth to water, water to air, and air to fire. (Diogenes Laertius tells us that Empedocles's Hera is earth, which makes sense to me in a roundabout way since Hera is Isis is Demeter is earth. Notice how, after the first battle, Horus deposes Isis by taking her crown, indicating the soul rising above earth.) Empedocles talks about that, too, since as Strife gives way to Love, the elements re-collapse into themselves in reverse of the way they separated. We might suppose that Pythagoras would have symbolized the regression of the cosmos from completion (10) back into unity (1) with a reverse tetractys, which, too, fits nicely onto the Seed of Life:

So if Pythagoras and Empedocles are (as I conjecture) faithful interpreters of the Isis, Osiris, and Horus mysteries (or if they aren't but my crazed speculation is at least somewhat valid anyway), then the Seed of Life is a nice little mnemonic for the exploration and contemplation of them. Hopefully that's helpful, since I continue to have a lot of contemplation ahead of me...

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Pondering Tutelary Deities

What gods are there? Are there, for example, gods (or perhaps daimons) ruling over, say, dogs, pear trees, races, places?

I should think that the traditional answer is, "yes, of course," but then, where does one draw the line? Is a dog different from a wolf? Well, sure, dogs are domesticated, right? But is a dog or a wolf different than a coywolf? Not really, but if a dog isn't different from a coywolf, and a coywolf isn't different from a wolf, then can you really say a dog and a wolf are different? Difference becomes more a question of degree than a yes/no question, but then how different necessitates a different presiding deity? Is a human different enough from a neanderthal? a chimpanzee? a macaque? a rodent? a lizard? Who could say? I certainly couldn't.

But I jump back to distinctions being a material thing, while unity is a divine thing. Presumably, then, a "species" is a human concept. So I hesitate to think that gods work that way.

I wonder, rather, if the difference between a dog and a man is more like the difference between the numbers 30 and 42: both participate in 2 and 3, but only the former participates in 5 while only the latter participates in 7. That is to say, they share some gods in common, but not others. Maybe dogs are "man's best friend" because we share a lot of gods in common, and that presumably gives us many ways in which we can interact; perhaps a mushroom is more like 38, with which we have only 2 in common and thus few means of interaction. Perhaps there are beings which, like 55, a dog can interact with (having 5 in common) but we humans can't (being relatively prime). Perhaps there are beings like the number 41 which neither can interact with at all.

These numbers are very small and simple to reason about, but presumably the numbers properly analogous to a dog or a human would be mind-bogglingly large, involving many prime factors and many gods. In such a case, very fine distinctions are possible, admitting us to say that, while maybe there isn't a single god of "dogness," there are a collection of gods which, all together, constitute more-or-less the "fingerprint" which we generally recognize as a dog... but the pattern recognition and the label "dog" are both human: they come from within us, and are not part of any real ontological structure at all.

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Olympian Shootout

I just noticed something funny about how I characterize the Intellect as 1, gods as prime numbers, and all other souls as composite numbers.

Do you remember how, in the first book of the Iliad, Hephaistos urges Hera to reconcile with Zeus by saying that He is as strong as all the other gods put together? Well, let us suppose a being's "strength" consists of how many souls participate in it: all souls participate in the Intellect, half of souls participate in the greatest god, a third of souls participate in the next greatest god (some of which also participate in the greatest), a fifth of souls in the next greatest (some of which also participate in the two greatest), etc.

As the greatest god, Zeus would correspond to the number 2, and so half of all souls participate in Him. The other half, therefore, participate in all the other gods put together. Thus, Hephaistos' statement is literally true under my model!

If we limit ourselves to the first twelve gods (e.g. first twelve primes), then 50% of all souls participate in Zeus; ~35% participate in at least one of the other Olympians (but not in Zeus); and ~15% participate only in gods not among the Olympians. So unless Hera managed to get Thetis on her side, along with all the other Oceanids, Naiads, Nymphs, and so on, She and the other Olympians wouldn't stand a chance!

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Isis on the Regions of the Cosmos

There are in the universe, four regions, governed by a fixed and immutable law: heaven, the ether, the air, and the most holy earth. Above, in heaven, dwell the Gods, ruled as are all the rest, by the Maker of the universe; in the ether are the stars, governed by the great fire, the sun; in the air are the souls of the genii, governed by the moon; upon earth are men and other animals governed by the soul who, for the time, is their king. [...]

This expanse, my son, is divided into four provinces, and into sixty regions. The first province from the earth upwards comprehends four regions, and extends as far as certain summits or promontories, which it is unable to transcend. The second province comprises eight regions in which the motions of the winds arise. Be thou attentive, my son, for thou hearest the ineffable mysteries of the earth, the heavens, and of the sacred fluid which lies between. In the province of the winds fly the birds; above this there is no moving air nor any creature. But the air with all the beings it contains distributes itself into all boundaries within its reach, and into the four quarters of the earth, while the earth cannot lift itself into the mansions of the air. The third province comprehends sixteen regions filled with a pure and subtle element. The fourth contains thirty-two regions, in which the air, wholly subtle and diaphanous, allows itself to be penetrated by the element of fire. Such is the order which, without confusion, reigns from depth to height;--to wit, four general divisions, twelve intervals, sixty regions, and in these dwell the souls, each according to the nature thereof. They are indeed all of one substance, but they constitute a hierarchy; and the further any region is removed from the earth, the loftier is the dignity of the souls which dwell therein.

(Kore Kosmou ["The Daugher of the Cosmos," that is, "On the Soul"] II–III.)

The material world has, of course, its four elements; Isis is saying as you go up the hierarchy, the scale of each world doubles. This smells suspiciously similar to the notion of Flatland: four elements is, of course, the number of quadrants in a two dimensional world; doubling this is equivalent to adding a dimension, comprising the octants in a three dimensional world; doubling this is equivalent to adding another dimension, comprising the sixteenth parts in a four dimensional world; etc. It touches nicely, I think, on how we are bound by time; how angels exist in time but may freely traverse it (that fourth dimension, a prison to us, can be walked back and forth by them); etc.

Obviously, as the Absolute comprises all that exists, the dimensions must go infinite as one gets high enough, so this repeated doubling must be a mere model rather the reality of the matter; nonetheless, it points at the notion that the divine worlds are far, far more vast and interesting than the dusty, gray wastes which we are equipped to inhabit.

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Always Cooperate

A soldier named Nobushige came to Hakuin, and asked: "Is there really a paradise and a hell?"

"Who are you?" inquired Hakuin.

"I am a samurai," the warrior replied.

"You, a soldier!" exclaimed Hakuin. "What kind of ruler would have you as his guard? Your face looks like that of a beggar."

Nobushige became so angry that he began to draw his sword, but Hakuin continued: "So you have a sword! Your weapon is probably much too dull to cut off my head."

As Nobushige drew his sword Hakuin remarked: "Here open the gates of hell!"

At these words the samurai, perceiving the master's discipline, sheathed his sword and bowed.

"Here open the gates of paradise," said Hakuin.

(Nyogen Senzaki, 101 Zen Stories LVII "The Gates of Paradise.")

Alone in immunity from magic is he who, though drawn by the alien parts of his total being, withholds his assent to their standards of worth, recognizing the good only where his authentic self sees and knows it, neither drawn nor pursuing, but tranquilly possessing and so never charmed away. [...] Thus this universe of ours is a wonder of power and wisdom, everything by a noiseless road coming to pass according to a law which none may elude—which the base man never conceives though it is leading him, all unknowingly, to that place in the All where his lot must be cast—[but] which the just man knows, and, knowing, sets out to the place he must, understanding, even as he begins the journey, where he is to be housed at the end, and having the good hope that he will be with gods.

(Plotinus, Enneads IV vi "Problems of the Soul (2)" §44–5.)

If you're losing the game, try instead playing the different game that is one level up.

(Mark Dominus.)

Have you ever played Nicky Case's little explanatory toy The Evolution of Trust? If you haven't, you should.

I think Case makes an error, by equating the Golden Rule with the "tit-for-tat" strategy; the Golden Rule isn't about treating others the way you are treated, it is rather about treating others the way you wish to be treated. Therefore, the Golden Rule is more of a mirror: if one is kind, then the Golden Rule is the "Always Cooperate" strategy, while if one is cynical, then the Golden Rule is the "Always Cheat" strategy.

As Case demonstrates, "Always Cooperate" never seems to survive. (And is destroyed all the more rapidly in times like these where "Always Cheat" is on the ascendant!) And yet, we try to be good and kind anyway. Why is that?

Is it because "Always Cooperate" is the strategy of angels? If one wishes to be among them, it makes sense to practice in preparation...

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Tetractys After Sengai

(With my respects to Sengai Gibon and D. T. Suzuki.)

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Levels of Being

hypostasis	element	cardinality¹	computability²		vehicle³	temporality⁴	sufficiency⁵	veracity⁶
Good	fire	transcendent	transcendent			transcendent	transcendent	transcendent
Intellect	air	uncountably infinite	definition			immutable	providential	definition
Soul	water	countably infinite	hypercomputable		soul	eternal	self-sufficient	truth
Nature	earth	finite	Turing-computable	{	imagination	immortal	content	right opinion
Nature	earth	finite	Turing-computable	{	body	mortal	insatiable	variable opinion

I am following the intuition that the continuum hypothesis is true and relates to the universe of discourse in a Neopythagorean system. In brief, the Good cannot be discriminated at all; the nous is too big to be measured even in theory; the world of soul is infinite in size even if it can be measured in theory (e.g. it is measurable, but only by the nous); and the world of matter is puny and boring.
I have already discussed the computability of the various levels of reality.
The term "imagination" is due to St. Synesius. Plutarch calls it "the mind" (which I really like but don't use because the Neoplatonists give that name to the nous), while Plotinus calls it "the lower soul." Prophyry and Proclus are more persnickety and call the soul-body, the "luminous vehicle;" the imagination-body, "the pneumatic vehicle;" and the physical-body, "the shell-like vehicle."
The Intellect exists outside of time and can comprehend all things simultaneously. The soul exists outside of time but can only comprehend things one-at-a-time (and this sequential comprehension is what gives rise to time). The imagination exists within time but, if it is idealized, can live forever. The body is, of course, mortal. The body and imagination come into being more-or-less together, but the imagination-body dies slower than the physical body does (owing to its lesser neediness).
The natural bodies always require external sustenance; the imagination-body ideally always has enough, while the physical-body never does.
The nous is the definition of all truth. Souls have a perfect understanding of truth from a given perspective. The imagination-body ideally has an intuition of truth (that is, it knows the right answers but does not know why), while the physical-body has only guesswork.

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Why the Geomantic Judge is Always Even

You cast sixteen random numbers (each 1 or 2) to generate the mothers of a geomantic chart. Let's call them A, B, C, D, etc. To generate the daughters, you simply rearrange these. To generate the nieces and the court, you add these together (modulo 2) in a pairwise fashion:

D H L P	C G K O	B F J N	A E I M	M N O P	I J K L	E F G H	A B C D
C+D G+H K+L O+P		A+B E+F I+J M+N		I+M J+N K+O L+P		A+E B+F C+G D+H
A+B+C+D E+F+G+H I+J+K+L M+N+O+P				A+E+I+M B+F+J+N C+G+K+O D+H+L+P
A+A+B+C+D+E+I+M B+E+F+F+G+H+J+N C+G+I+J+K+K+L+O D+H+L+M+N+O+P+P

What happens if we add each of the lines of the judge together? We would end up with A+A+B+B+C+C+D+D+...: that is, each number appears exactly twice. (This is because one copy comes from the mothers and the other comes from the daughters.)

Now, consider what it means to add X+X in geomancy. An active line has one dot, so 1+1=2. A latent line has two dots, so 2+2=4=2. So if you ever add a number to itself, you always get 2.

So A+A+B+B+C+C+D+D+...=2+2+2+2+...=2, regardless of what A, B, C, D, etc. are. Therefore, your judge will always have an even number of dots. This means it will always be one of Populus, Fortuna Minor, Amissio, Conjunctio, Carcer, Acquisitio, Fortuna Major, or Via. ∎

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The Matter with Matter

Note to future me: this blog post by Mark Dominus describes a silly math problem which dovetails nicely with my idiosyncratic pseudo-neo-Pythagorean spiritual model:

One represents the nous.
All other positive integers represent souls.
Prime numbers represent gods.
Composite numbers represent all other souls.
And... zero represents matter.

Zero doesn't technically exist: it's the limit of existence. Now, I'm one of those horrible people who think Cantor was crazy and insist that you can't get to the end of an infinite sequence (grumble grumble unless you're the nous), but the simple end-run around this is that zero is a multiple of all numbers and no number is a multiple of zero, so it must be "below" every other number, regardless of how far you choose to enumerate them (whether "infinitely far" or not).

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The Computability of Plotinus' Metaphysics

Quite some time ago, I was speculating on the computability thesis and how it relates to Plotinus' metaphysics. I think Enneads VI 6 really solidified everything in my mind, but while I was discussing this all with a friend yesterday, it occurred to me that I never wrote up where my thoughts ended up. I figure I'll briefly remedy that.

There are many models of computation—that is, "how can one define an algorithm?"—but it turns out that we've proven all of them that we can actually make equivalent to a particularly famous one called a Turing machine. (You can think of it as a simple kind of computer that can do one thing at a time.) The fact that they're all equivalent led several famous mathematicians to propose that something is computable if and only if it can be computed by a Turing machine (or one of these other, equivalent, systems).

I note that doing one-thing-at-a-time is characteristic of souls, and Plotinus says that this is why time exists in the sensible world (e.g. because it is generated by one-thing-at-a-time souls). Consequently, let us assume that the computability thesis is true for the sensible world.

The next level above souls, Mind, differs from them in that it doesn't operate sequentially: rather, it operates comprehensively, in an everything-at-once manner. The obvious variation on such a Turing machine is that, instead of doing operations one-at-a-time, it does all-operations-at-once. (The reason we can't build such a machine is that the number of operations may become infinite, and one cannot get to the end of an infinite sequence of steps in a finite amount of time. However, we're saying here that the Mind can because time doesn't exist.) Such hypercomputers have been discussed by mathematicians, and unless I misunderstand the literature, this one is effectively equivalent to what's called an Oracle machine (which is just a machine you can instantly get the "right" answer to whatever problem you want). Therefore, I think the psychic world is hypercomputable in the sense of Oracle machines. (I therefore half-joke that heaven, for a computer programmer, is to finally get to solve all those pesky NP-complete problems that have been bugging us for generations.)

The next world up is the mental world, where the Mind itself exists. Plotinus already answered this one for us, since the Mind is the definition of truth. That is, this is simply the world in which computation itself is defined. That is, it is the world in which all questions, all answers to those questions, and all means of finding those answers, are all defined. Therefore it is above computation of all kinds.

Finally, of course, in the highest world, the One itself, computation isn't its own thing because what would you distinguish it from? Everything in the One is just the One.

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Concerning Twelve

Why are there twelve gods? I suppose the traditional explanation would be that there are twelve signs to the zodiac, though I haven't seen any explanation of how or why the Babylonians came to this number. (There are twelve months in a year? Except a third of the time, when there aren't?) The philosophers, for their part, say it's because there are four functions with three phases each, making twelve total.

You may recall that I've been toying with the pseudo-neo-Pythagorean conception of unity being representative of the Intellect while numbers are representative of souls, with prime numbers being representative of indivisible natures (e.g. gods). Well, I woke up with an odd math problem in my head: what is the point at which there are more composites than primes? After all, the first number (two) is prime, so initially there are more primes than composites; but since every even number and some odd numbers are composite, there are eventually more composites than primes; so what is the point at which it crosses over?

It turns out that it's very easy to find by hand, and as you've probably guessed from my first paragraph, it's the twelfth number: at this point, there are six primes (2, 3, 5, 7, 11, 13) and six composites (4, 6, 8, 9, 10, 12). After that, the composites dominate and the primes never catch up.

What's curious about this, to me, is that if you consider the Olympians, six of them are first-generation (Hestia, Demeter, Hera, Poseidon, and Zeus are Cronus's children, while Aphrodite was created from Uranus by Cronus' actions) and six of them are second-generation (Artemis, Apollo, Athena, Ares, Hephaestus, and Hermes are all Zeus's children). This is just like how of the first twelve numbers, six of them are prime, while the other six are generated from the first six.

This is not to say that looking at the prime numbers is where the Greeks (or, for that matter, the Babylonians) got their cosmogony; it seems quite likely to me that the number theory antedates the theology, and anyway the particular genealogy of the composites doesn't match Hesiod. (All of the second-generation Olympians are sired by Zeus, but there is no prime shared among all six composites.) So I don't have my answer for "why twelve?" But it is nonetheless interesting that my odd little number speculation has a parallel to the theology.

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On Thrice-Greatest Hermes

You cannot make a polygon with one or two sides: the first polygon is the triangle, which has three. One and two thus, in a sense, only exist in potential: they cannot take shape. But one plus two is three, so three gives one and two shape and makes the potential, actual. Indeed, the ancients didn't even consider one to be a number at all: Euclid, for example, says that number is the "multitude made of units," [Elements VII, def. 2] thus making one the measure of number and so beyond number. In that sense, one is doubly potential: it is number in potential and shape in potential. Two occupies a middle ground, being number in actuality but shape in potential. Three is finally what is both number in actuality and shape in actuality.

This is the kind of thing Porphyry talks about when he says that because "incorporeal forms and first principles could not be expressed in words, [the Pythagoreans] had recourse to demonstration by numbers." [Life of Pythagoras XLIX] So, metaphorically, one is heaven, spirit: that which is beyond and ever unreachable, even as it acts as a template. Two is earth, matter: both imminent in a sense and beyond in a sense, being infinitely divisible and never properly graspable. Three is the combination of the two, the things that exist from them, finally actual and sensible.

And the Neoplatonists loved to read these things into myth. Zeus is heaven, as the seed of all things. Maia is earth, that which receives and so gives form (but not form itself). Hermes is the result of their union, bringing the potential into actual, and so mediating between heaven and earth, and heralding the intelligible to the sensible.

Thus Zeus is spirit is one, Maia is matter is two, and Hermes is things is three. It is no mistake that Hermes is the patron of storytellers, for stories must have a beginning, middle, and end—three parts—in order to be complete; similarly, he is called thrice-greatest, because he brings perfection or completion or form to that which comes before.

Now, consider that you, yourself are a product of heaven and earth, possessing a spiritual soul and a material body. That means you are that which gives actuality to the potential. That means you are the mediator between potentials. That means you, yourself, are Hermes. When the Hermeticists say that Thrice-Greatest Hermes is their teacher, what they are really saying is that they are self-taught: truth does not come down from on high, it comes from within.

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Idle Speculation on Number Theology

(As a disclaimer, before I begin, I should note that I am not a magician and everything I say here is entirely speculative. Don't rely on it for anything other than ideas!)

I have an idiosyncratic theory (per Plotinus and the other Neopythagoreans that I have read) that various types of numbers fall into three different categories based on their essential potency:

The weakest numbers are the even numbers. This is because they may be divided into two equal groups and set against each other, tying their energies up and preventing action. (It is for this reason, I think, that the Pythagoreans considered even numbers passive.)
The middle class of numbers are the odd composite numbers. These may not be set against themselves—there's always a "tie breaker," a majority, and so there is always direction and the possibility of action—but they may still be divided is some way or other, and thereby weakened.
The strongest numbers are the odd prime numbers. These may not be divided at all, and thus always possess their essential unity. (And, as Proclus tells us, unity is power.)

(We may as well classify 1 with the prime numbers, even though it is not technically prime, since obviously it is as indivisible as it gets; further, within a class, smaller numbers have greater potency than larger numbers.)

One thing that is common in folk magical practice is the combining of harmonious elements into a whole (e.g. in an amulet, etc.). I wondered to myself whether it were possible to combine elements that were each individually strong (e.g. odd, prime) and strong in combination (e.g. odd, prime). For any two elements it is not, and the proof is trivial: two odd numbers, when summed, form an even number. Therefore, one must add an even number to an odd number to get another odd number, thereby introducing a weakness. However, it is possible with three elements: a couple trivial examples are 1+3+7=11 and 3+5+11=19.

I might suppose a prime number of ingredients, combined in prime terms into a prime total, might be more potent than other combinations. But an esoteric number theory of this sort does not seem to be well-developed: all systems I've seen only assign meanings to 1-10 (and higher numbers are considered in light of those). I'd be curious to see a theory treating the primes as conveyors of essential meaning, but there is an infinity of those...

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More Squares

I've posted about Napoleon's problem before: can you inscribe a square in a given circle using no more than six circles total (e.g. including the given one)? It's tricky, but in fact you can, and there's a bunch of ways to do it. If your given circle has a radius of 1, the resulting square has a side length of √2.

Later on, I found a way of circumscribing a square around the given circle, still using six circles. This square has a side length of 2.

Later still, I found a way of making this strange one, again using six circles. It's got a side length of √6.

Now, I've played around quite a bit and figured that those were the only three sizes to find—I hadn't tried to prove it or anything, just hadn't seen anything else come up for a while. But I was playing around some more today, trying to see if there's a way to make an octagon out eight circles (my previous best was nine), and I was very surprised that there's another six-circle square out there!

Bonkers! That square has a side length of 2√2. Finding it inspired me to write a quick computer program to look for every possible size of square that can be made in only six circles, but unfortunately there are no others. I've caught 'em all.

Since I had the program put together, I went ahead and searched to see if I could make any eight-circle octagons from any of those squares, but nope, it's not possible. (There may yet be eight-circle octagons that don't start from a square, but this strikes me as pretty unlikely.) Regardless, it's amazing to me that there was more to find over ground I've gone over so many times already! It seems to be good practice to periodically double-check things one thinks one understands.

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Eye

An eye in seven circles. (Appropriate, that.)

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Metamorphosis

A refinement of my butterfly using 8 circles.

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Enneads VI 6: On Numbers

This was a very valuable tractate to me, I think, clarifying a lot of the moving pieces in Plotinus' cosmology and helping to see how they all fit together. It's another one I recommend in full, though I imagine it's not very accessible without first having read a lot of the others: Porphyry says he placed the more difficult essays later in the text, and that seems the case to me as well.

VI 6: On Numbers
We have elsewhere expressed a definite ordering of principles: the One, the Intellect, the Soul, etc., and there we stated that the Intellect was dual. This must imply that Number transcends the Intellect, for how can something be "dual" before there is "two?" Consequently we must assume that Number is latent within the One, and the Intellect is the expression or unfolding of Number. That is to say, the Intellect is the reflection of Number, and each Idea is the reflection of a particular number.
This is why number is such a slippery thing: if you have ten objects, those are merely objects; they each exist due to their participation in Being, and they each have their own unity due to their participation in the One. But when we count them, it is we who are making the statement: the objects each have their individual existences, but the ten exists within our minds, which is itself a reflection of the true Ten in the Intellect. You have given Life to that ten by counting them, but if you had merely been content with your own Being, resting within your own unity, you wouldn't have been counting anything! So while Number has true existence (in the One), knowledge of Number is a lesser thing (in the Intellect), and the process of working with Number is a lesser thing still (in the Soul).
What of "infinity?" We take it that the Intellect is limitless in scope, and that all numbers must exist there; but it is not possible to count them up, since individual souls must be limited by whatever number is involved in their composition. Thus infinity exists, but you cannot, say, count to it.

Plotinus uses the term "henad" a lot in this tractate, and by it he just means a reflection of the One at a given level of existence: the One is the original henad (and, of course, the henad at its own level of existence), the Intellect is the henad of the Intellectual level of existence, the Soul is the henad of the Psychic level of existence, etc. But of course, the whole universe is reproduced at each level from this henad—at least, in the best way that henad is able. Indeed, even individual souls are the henad of the microcosmic universe within, which is the basis of Plotinian mysticism: it is why you can look within yourself to find the All. Nowadays we might simply say that the cosmos is fractal at all scales.

This principle is usually stated in Neoplatonism as "All is in all, but each in a manner appropriate to each." This formulation is commonly cited from Proclus (Elements of Theology CIII) but is due to Porphyry (Sentences X, but be advised that many manuscripts badly mangle it: hence, for example, Thomas Taylor's translation is nonsensical). §7 contains the principle in Plotinus' own words.

I had been wondering about Turing Machines and Oracle Machines before. It's nice to see Plotinus directly answer my questions in §18.

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How the One Becomes Many

Tao gives birth to One.
One gives birth to Two.
Two gives birth to Three.
Three gives birth to everything.
But everything carries yin and embraces yang.
And yin and yang, together, are One.
(Tao Te Ching, 42)

I am One transformed into Two;
I am Two transformed into Four;
I am Four transformed into Eight;
I am, after this, One.
(The Coffin of Petamun)

As all things were produced by the mediation of the one,
so all things were produced from the one by adaption.
(The Emerald Tablet of Hermes Trismegistus)

I was trying to explain Neoplatonist metaphysics to a computer programmer friend the other day and stumbled onto a useful analogy which might help to explain the idea of hypostases.
Are you familiar with the fundamental theorem of arithmetic? It states that every integer greater than one can be uniquely represented as a product of prime numbers, that is, that every number is equivalent to 2^a·3^b·5^c·7^d·11^e·13^f⋯. Wikipedia gives the useful example that the number 1200=2⁴·3¹·5².
So, suppose you have a box somewhere which can contain exactly one number. It doesn't matter how large this number is, just that it can only contain one of them. If you want to store a single number, great, you can just stuff it into the box and you're done.
But what if you want to store two numbers? Well, thanks to the fundamental theorem of arithmetic, we can cheat: suppose we call the first a and the second b, then we can find 2^a·3^b. This is fine since the result will be a unique number, we can stuff it into the box, and we'll always be able to pull a and b back out of it again if we need to.
If we want to store three or ten or a hundred numbers, it's the same thing again: we need more prime numbers, but we'll still get one resulting number in the end. The more numbers we want to store, the more ridiculously huge that result will be, but the box doesn't care how big it is, and so it's no problem. In fact, we can do this no matter how many numbers we have.
So in this way, a single box is, in fact, equivalent to a row of pseudo-boxes. But wait, there's no reason we have to stop there: each pseudo-box in the row is equivalent to a row of its own, right? So by repeating the process on each pseudo-box in the row, that single real box is equivalent to a two-dimensional grid represented by rows and columns of pseudo-pseudo-boxes. And we can repeat the process again on each pseudo-pseudo-box, making something of a cube of pseudo-pseudo-pseudo-boxes. And we can repeat the process again, and so on. We started with a zero-dimensional grid of boxes (that is, a point), but through a transformation process, found we could produce a one-dimensional grid of boxes (that is, a line), or a two-dimensional grid of boxes (that is, an area), or a three-dimensional grid of boxes (that is, a volume), or as many dimensions as you like. And these productions are inherent simply from the original box existing—it doesn't involve doing anything, it's just a matter of how we interpret it.
The connection to Neoplatonism is that the only thing that exists is the One, which is our single box, the only one that really exists; but this box is virtually a line of boxes, which is like all of the distinct ideas of the Intellect; but this line of boxes is virtually a grid of boxes, which is like all the distinct beings of the Soul; but this grid of boxes is virtually a volume of boxes, which is like all the various lives of those souls; but this volume of boxes is...