Download Pete,

Pete,
I’m working on something here that may prove helpful down the line. It concerns what AC Lloyd, in his
The Anatomy of Neoplatonism, calls a P-series; and it may be the connection between the matrix and formal
logic that I’ve been looking for.
Lloyd’s “P-series” seems to be a subset of what mathematicians call a “hyperharmonic series.”
Hyperharmonic series: Σ 1/n p = 1 + 1/2 p + . . . + 1/n p + . . ., n= 1 and p is a positive number; a
polynomial sequence, for example, is such. If p = 1, then we have a harmonic series: 1/1+1/2+1/3+1/4+1/n.
In Scholastic terminology, it’s called an ab uno disce omnes (“out of one instance infer the rest” or “from
one example learn to know all”). A more familiar ab uno case would be the steps by which the “located
monad” builds up to a volume – point to line, line to plane, and plane to body, for all these are virtually
contained within the monad and released for the intellect’s consideration through the geometer’s art. The
monad “proceeds.” If we shift into reverse (i.e., from the complex to the simple) we get a “participatory [or
‘reversion’] series” – body “participates” plane, which “participates” line, which “participates” point. The
“p” in “p-series” could stand for “Pythagorean” or “participation” or maybe, Lloyd just lifted it from
current mathematical jargon.
What has this to do with the price of tea in China? Well, here it is: Neoplatonism reconfigured
Aristotle’s “standard” genus (which is a “whole within the parts,” that is, the genus is potentially [in
the logical sense] and “equally” [univocally] all its particular species) to what is called a “quasigenus,” to wit: a standard genus rediscribed as a “whole before the parts.” In this second
instance, the various species “participate” the genus (hence, it is somehow real because
“participation” is a real relation whereas its counterpart, predication, is a logical one) and
each species does so “unequally” or equivocally. The quasi-genus therefore, forms up a “chain of
being,” i.e., a p-series anchored on the first member. As you probably know, the dissonance
between these two competing accounts of the genus triggered “the problem of universals,” a
quandary that didn’t get sorted out until the late Middle Ages.
Digression: A certain anxiety about this Neoplatonic quasi-genus seems to lurk in the background behind Cajetan’s
“analogy of inequality.” In De Nominum Analogia, Chapter I, paragraphs 5 and 6, we find this:
“Its Names. The logician refers to analogous terms of this type as univocal. The [natural] philosopher, on the
other hand, regards them as equivocal, the difference coming from the fact that the former considers the intentions
expressed by the names and the latter, their natures. That is why Aristotle in X Metaphysics states that the
corruptible [body] and the incorruptible [body] have nothing in common which is univocal, because he scorns unity
which is merely unity of reason or concept. And in VII Physics we are told that in analogy according to genus
equivocations lie hidden because analogy of this type with its unity of concept does not simply imply one nature
but admits within itself many natures that are ordered to one another [by priority and posteriority], as is clear with
respect to the species of any genus, and especially the most special and subaltern species. For every genus can be
called analogous in this way, as is clear from quantity and quality in the predicaments, and body, etc … [My boldface]
“St. Thomas, in I Sentent. Dist. 19 refers to this type of analogy as analogy ‘according to to-be only,’ because the
analogates are considered equal in the formality signified by the common name but are not held equal with respect
to the ‘to-be’ in one thing than in another, as we see so often in metaphysics…”
His anxiety can also be detected in what he says about analogy of inequality, namely, that “it is entirely
foreign” to analogy; and yet he includes this mode as a “necessary moment” in many of his other
discussions in his opuscule. And, of course, how else could anyone construe “analogy according to to-be”
save in terms of “participation”?
A dilemma arises at this point: the very notion of a quasi-genus entails a contradiction. Is the “monad” the
genus itself (“the whole before the parts”) or is it just that genus’ first element or “species”? It seems to be
both. Again, to put the aporia more traditionally: take the genus, “animal;” is the first animal in the p-series
predicated universally of the whole genus or is it just a particular species within it? It can’t be both
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universal and particular at once – nobody has ever observed “animal itself” roaming the range where the
deer and the buffalo play. The best candidate for this type of “first animal” appears to be in Timaeus, where
Plato characterizes the cosmos as a whole to be like a living animal.
The Neoplatonists got around this problem by using Aristotle’s standard genus in conjunction with
the quasi-genus. That is they in effect used a double-genus or what I call a “cross-eyed” genus or
more charitably, the “global epicenter of dialectical tension.” And here we seem to have a “logical
dyad” in play.
Now here’s the interesting part – What better example of “the whole before the parts” could there be than
the unpacked octave? And if we extend this to the Neoplatonic usage of two paradigmatic genera or, if
these two be joined into one notion, a “bifocal genus,” we have here a logical construct resembling a
double-octave. The “whole within the parts,” Aristotle’s genus, seems to mimic the fully packed octave, all
the more so since nobody has ever managed to pack an octave perfectly. 1 I base this surmise on the fact that
Aristotelian definition (logical species = genus + differentia) resembles the musical arithmetic of “female”
numbers being inseminated by “male” numbers to generate tones. This comparison of musical arithmetic
to biological reproduction is Ernest’s, not mine. In this case the female numbers – to the extent that all
female numbers can be understood as 2 (the other evens are just doublings/halvings) can be considered an
integral genus – they are put for the genus (“genus” is, by the way, cognate with “cunt,” and likely also, the
German Kunst, “art”). The male numbers can be put for the differentiae and the tones, for the species
(“pitch classes”). And Aristotle himself observes that “species are like numbers …”
Now at this point it’s quite interesting that Aquinas considered art-in-general as coming under a doublegenus, too. I’ll quote just one paragraph so you can get the general idea:
Yet the work of art is restricted to a form of double
genus as follows: sometimes [it is restricted] to an
accidental, extrinsic form, as in the art of painting,
sculpting, house building, and the like, and this form is
properly called a “form of art”; and sometimes the work
of art is restricted to a substantial, intrinsic form, as in
agriculture and medicine, and this form is called a “form
of nature.”
I can give you the entire quotation if you like. At this point, I would suspect that Thomas would classify
music under both categories (notice the stark omission of music here) if pressed to do so.
The reason why I think this may sound strange to you is: Elsewhere, Thomas describes logic as “the art of
arts.” If there is a link between the matrix and formal logic (music being logic’s foundation in the soul,
perhaps in the interval between sense and reason), then I infer that music had likely been this “art of arts”
before its shepherding function (scripturalizing mythologies) had been displaced by logic’s directing
function (systemizing philosophy – cf. the old dispute over whether logic is a part of philosophy or just an
instrument for it). And this would grant music and logic a kind of operative equivalence. “Music” for the
Greeks, as I’m sure you know, could also connote all the “fine arts,” namely, those that could not be
prosecuted save under the inspiration of a “Muse.”
Tell me if any of this makes sense from a musician’s perspective. I’m sure that you can improve on it.
John
What I mean here is: Most people don’t know this, but man is the only natural being that really meets the ideal of
Aristotelian definition. This holds because “rational” (or in Greek, “having reason”) is a true differentia – it not only
contracts the genus, “animal,” to the species, “man,” but also explains why man is what he is. With the other animals
we can’t do this quite yet; so we have to make-do with descriptive differentia-substitutes: e.g., “cloven-hoofed,”
“ruminant,” etc., which is akin to tempering in music. Since animals have perceptual awareness, the specific differentia
dividing one animal species from another should be based on the various manners of “perceiving” or “experiencing”
(“animal” literally means “besouled.”), which would then explain why one species by its essential nature divides from
another. So Aristotelian classification has had to “temper” the specifications of its genera, at least when dealing with
natural entities other than man.
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Bryan: Pete had a follow-up question: He had asked: Why the reconfiguration [viz. of the standard to the
quasi-genus]?
I answered: it seems that the most efficient way for a syncretic metaphysics like Neoplatonism or
Hegelianism to approach its goals is through logical systemization; and for the Neoplatonists, this entailed
“reconciling” Plato and Aristotle through a nimble synthesis of their respective methodologies. Logic is to
ontology [so I had speculated] what tuning theory is to music or, to repeat myself, the genus is to logic
what the octave is to music. Again, the genus is the “logical octave” just as the octave is the “musical
genus.” In many ways dialectic, “the enjoyment of irony in the act of classification” (as I like to put it),
resembles what musicians go through when they turn their thoughts to tunings. And surely, the connection
between Socratic irony and what Ernest calls “scribal jests” cannot be a mere coincidence. By the way, I
am aware that “genus” in music means “diatonic,” “chromatic,” or “enharmonic;” but I don’t know enough
about music to understand why, from a logical standpoint, these three should be called “genera” unless, of
course, they are in some way subaltern rather than summa genera, i.e. than “categories” in Aristotle’s
sense.
If the genus is the “logical octave” then this suggests that the juxtaposition of the standard and the quasigenus to form the double-genus may resemble the (untempered?) double-octave in music. Finally, the fact
that the Pythagorean Greater Perfect System’s axis seems to be the double-octave suggests, to me at least,
that the double-genus plays an analogous role in Neoplatonist logic, not to mention metaphysics. But
again, I don’t know enough about music to decide one way or another.
Anyway, to continue this little story: Since Aristotle favored abstraction from sense-experience and also,
because his logic was widely known to be particularly useful for the specialized natural and mathematical
sciences (e.g., biology and geometry), the Neoplatonists put Aristotle’s logic at the bottom of their system,
where thinking is closest to sense. And in order to “reconcile” Aristotle with Plato, the Neoplatonists
needed to “deconstruct” the Categories in order to make this particular treatise fit the theory of the Ideas
(this they did by unifying Plato’s “five greatest genera” in one highest genus, which they named “Being”) 2.
For the categories comprise the roots of Aristotelian logic’s interface of being and thinking; and
the Neoplatonists had a sharp eye for anything that could conflate the logical and the real. That
Aristotle had insisted that these two realms are different made overcoming this distinction all the
more urgent. They accomplished this through the double-genus, in which “the real is the rational
and the rational, the real.” Indeed, there had to be a double-genus in Neoplatonism just as there
had to be a double-octave in the Greater Perfect System. Like art, systematic philosophy
tolerates no accidents; it cannot be a fluke that Neoplatonism originated in Neopythagoreanism.
Philosophical systems do not evolve by historical chance.
How is this double-genus central to Neoplatonism? According to Lloyd, quasi-genera,
although “prior” [i.e. more inherently intelligible], co-existed with standard genera; and it was
natural for the Neoplatonists to see quasi-genera as transcendent and standard genera, as
immanent. Indeed, Lloyd strongly suggests that logical deduction in Neoplatonism is conserved
by starting in one genus and concluding in the other. Moreover, the double-genus made it
possible to shift gears and, with no great trouble, cross over from epistemology to ontology and
vice-versa. In late Scholasticism this was termed “the convertibility of being and truth.” It may well
be the case that “transcendental names” owe most of their origins to the double-genus; and this
would be mainly due to the manners by which the Neoplatonists were able to equate predication
(a logical distinction) with participation (which is a real distinction).
These two generic constructs not only drove the entire apparatus’ functions – the quasi-genus, as
“reversion,” the standard genus, as “procession,” and both as co-existent phases, as “remaining”
– functions that the Neoplatonists set into one “super genus (i.e. ‘Being’).” But also, the double
The “five greatest genera” were: “Motion,” “Rest,” “Sameness,” “Otherness,” and “Being.” The first four,
because they break down into two contrary pairs, were set under “Being.”
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genus’ two phases, “standard genus”/“quasi-genus,” comprised a dialectical “unit.” This logical
dyad also proved very serviceable in the conflict over universals.
What I mean by the last assertion is this: Aristotle had said that “As bats’ eyes are to the noonday
sun so, too, is the reason in the soul to those things which, by nature [i.e. “in themselves”], are
the most intelligible of all.” In other words, the order of human knowledge is the inverse of an
object’s inherent intelligibility. For example, we find sense-experience easily understandable; but
experienced objects, due to their material component, are less intelligible in themselves than, the
Ideas, which have no matter. The Neoplatonists’ double-genus empowered them to exploit
Aristotle’s analogy and “stand him on his head.”
And this is how they did it: Aristotle had said that the universal, in being abstracted from
experience, is “posterior” in the sense that it is not “innate.” But to be posterior “for us”
epistemologically (according to Aristotle’s “bat metaphor”) is to be prior metaphysically, prior as
regards “the things themselves” – for the order of human knowledge is the inverse of the order of
how things really are. So the standard genus is the “copy” of which the quasi-genus is the
“original.” And I’ve got to say that, whether or not you agree with them, you have to admire the
Neoplatonists for their technical artistry.
The following is part of a reply to Ernest in another email:
2. A quick summary of the predicables. Aristotle distinguishes two kinds of predicables: those
which are said of a subject as a whole and those which are said of a subject’s constituent parts.
An example of the first is “John Brown is a man;” an example of the second, “John Brown’s body
lies a-moldin’ in the grave.” The predicables’ overall task is to generate Porphyrian Trees, ten in
all according to the ten categories and in principle, able to classify all things under heaven.
There are two major lists of predicables: Aristotle’s is “genus,” “definition,” “proprium (essential
property),” and “accident (inessential property);” Porphyry’s is “genus,” “differentia,” “species,”
“proprium,” and “accident.” It’s important that we remember that these terms, especially genus
and species, are not the same as biological species and genera.
a) In either list, the most important are genus and differentia, which comprise the “extremes,” i.e.
the extremes of definition. The first list, Aristotle’s, conceals the differentia within “definition,”
since definition occurs when we reduce a genus, via a series of differentiae, to its component
species. Missing in this first list, and present in the second, is “species.”
i) Why? Probably because a completed definition is a genus’ lowest species; and definition, or
what the craft calls the “logical species,” had been most important to Aristotle in this context.
Porphyry, I speculate, included species but omitted definition because he wanted to sneak the
quasi-genus into the mix; for “species” can be ambiguous, as we will see below. For example, he
defines “species” as “that which” is predicated of individuals (or as he has it, diversified in number
but not in kind, and also in regard of what they are) – whereas Aristotelian “species” can be at
any level in a genus (e.g., “animal” is relatively a species in respect to the higher genus,
“organism,” but “dog” is also a species of “organism” through the lower genus, “animal,” and of
course, a species of “animal” as well). With Porphyry, “species” always imports the suggestion
that there are always “individuals” immediately underfoot – so it is possible that, if “animal”
is a species of “organism,” then there always remains an implication that “animal” is real, namely,
“animal itself,” to use the Platonist lingo. For Aristotle, “animal” is simply a logical note to help in
the task of classification. This has immense metaphysical consequences (the so-called “problem
of universals”). Especially intriguing is Plotinus’ earlier, clever description of “differentia” as “the
complement of substance” – intriguing, because the differentia is a predicable and therefore,
logical; and substance is a category, i.e. mode, of being and therefore, real. So how could a
logical distinction be a complement of a real distinction? This implies that if we remove a
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differentia we destroy its “complement,” a substance; that is: to remove a “being of reason” is to
destroy a real being – and this is not only odd, but starkly surreal. But if possible, it would justify
the alleged co-existence of the standard genus and the quasi-genus in Neoplatonism.
ii) Aristotle’s list also omits “differentia” for the reason given above – redundancy (and also, he is
strangely rather quiet about it, too – he has hardly any commentary dedicated to differentiae).
“Proprium” is an attribute that belongs to one species only, e.g., “endowed with natural rights.”
“Accident,” which is not the same as a category of accident (e.g., quantity, quality, etc.) is “what
can belong or not belong to a thing,” e.g., whether a man is black or white, and therefore has no
impact on the thing’s definition.
iii) Aristotle’s predicables shape only concepts, Porphyry’s can also shape things as well, which
obviously would benefit greatly the Neoplatonist metaphysics’ famous inclination to merge the
real with the rational. Generally, most logicians understood the predicables as midway between
“nominalism” and “realism,” and the Neoplatonists had the constant option to shade things either
way.
Definition itself is what Aristotle called “the formula of the essence;” it analyzes what a thing is
(i.e. what its name means) and is itself an isolated concept. Existence can be logically comprised
only in the judgment or proposition, which contains concepts as its elements. The proposition
asserts a fact, the syllogism, which holds three propositions or judgments, demonstrates a cause.
b) I did a little research on the internet and to my pleasant surprise I read that the Greeks had
recognized “octave species,” which seems to be a fancy label for “scales.” Now the use of
“species” (Porphyry’s addition to the list of the predicables) could be somewhat vague here, but
examples would be the Dorian, Mixolydian, etc. modes. By parity (or perhaps parody) of
reasoning, if there are musical “genera” and “species” there may also be some apt analogues for
the remaining predicables lurking about.
For example, what would be a musical “differentia” be like?
i) I am of course purely conjecturing; but it seems to me that “doubling for fit” in the matrix and
wielding the predicables in logic share something in common, namely, that both are more logical,
i.e. “technological,” than natural. Both operations, “doubling for fit” and dividing raw intuitions via
the predicables, are a kind of “making sense” that we perform. Likewise, the overtone series and
the categories are real.
ii) Now let us briefly imagine that the plucked fundamental and the category of substance are coanalogous; and let us add to this Plotinus’ characterization of the differentia as the “complement
of substance”:
“Essential qualities are those that are complements of substances. Complements are properties the loss
of which destroys their subjects. Properties that can be gained and lost without the subject being
destroyed would not be essential. Hence the differentia is included under the definition of substance,
since it is a complement of substance, and the complements of substances are substances.”
This involves a prior distinction made in the meaning of “subject.” Plotinus seems to be construing
“subject” two ways: as 1) “matter deprived of all quality” and 2) “matter determined by a common
or particular quality.” Understood as “differentia” this would seem to suggest the -1/+1 that you
talk about. Add to this Aristotle’s dictum that “species are like numbers; thus, just as adding or
removing a unit constitutes a new number so, too, adding or removing a differential note
constitutes a new species.” The disparity, though, with Plotinus’ formulation here is that “matter
deprived of all quality” seems to indicate not a -1 but a 0; and matter as “determined” commonly
and particularly, as an alternation between whole and part; and this seems to hearken us back to
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the predicables’ initial division mentioned above (page 2). In other words, the differentia seems to
be more complicated than it had appeared to be at first. But for the moment this will have to do.
And just as spiral fifths/fourths tuning selectively patterns the “primal order” implicit in the
overtone series for doubling so, too, the categories fall into a primordial series that needs
“treatment” through the predicables for the sake of our understanding.
Aristotle says that, first, there is being, that is the composition of act and potency, and then this
overall “ratio” (act/potency) “falls into the categories,” i.e. it spontaneously distributes itself into ten
ultimate and peculiar versions of act/potential (for which reason they are called the “categories of
being”). The initial act/potency is, respectively, existence (that a thing is) and essence (what a
thing is) – in other words, “x is” versus “x is y.” 3
The categories represent ten ultimate “flavors” that descend from this original act/potentiality,
which we order for the sake of understanding via the predicables. Both “doubling for fit” and
employing the predicables reduce “raw nature” to patterns that we can deal with.
The octave is a quasi-genus insofar as it is the dynamic limit of the overtone series, for this series
spontaneously arranges itself according to natural priority/posterity; by the same token, the
standard genus is the predicables’ upper logical limit – we can make no univocal predications
above the level of the summa genera (which the categories are called when examined through
logic’s “spectacles”) – and its lower logical limit is the differentia. This is why I compared it with
the “demiurgic unit.” The differentia is simultaneously like 1 and by extension, “male primes;”
the differentia “inseminates” the genus to produce logical species, namely, definitions.
3 Logicians call the first example the “is” of existence; the second, the “is” of predication. Both meanings of
“is” are different yet coextensive, and this partial correspondence attunes logic to real being – somewhat like
“sympathetic vibration.” The Neoplatonists were the first to make this essence/existence distinction explicit.
So we can lay out a “continuous metaphysical analogy” in this way: Essence is to existence as potentiality is
to act as matter is to form.
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