7: Arithmoi in Plato

[Laura Gibbs]

Space for thoughts and responses to Chapter 7: Arithmoi in Plato

[Laura Gibbs]

Section C Comments

The very last sentence of section B of this chapter did little to encourage me to carry on - "Only dialectic can open up the realm of true being, can give the ground for the powers of the dianoia and can reveal Being and the One and the Good as they are - beyond all time and all opposition - in themselves and in truth." None of this was on the agenda when I started reading a book which proclaimed to be about "The Origin of Algebra." Now, perhaps Klein will save the day and actually say something about algebra eventually, and maybe "Being and the One and the Good" will have something to do with algebra... but I really would settle for algebra; if I were going to entrust my question for the "Good" to someone, it wouldn't be Plato, or Klein - and as for the "One" and the "Good" being synonymous, I've mentioned along the way several times how this is simply not a rallying cry that appeals to me.

But I did read through the rest of Chapter 7, led on by the endless panoply of terminology (which DOES interest me), so that I could learn something about the "arithmos eidetikos" as promised in the title of this section. The word eidetikos has not come up previously, and when I checked in LSJ, I discovered that it is strictly a technical term which does not seem to have been used outside of strictly technical treatises. But here's what is really strange: Plato is not cited a source for the term. That's very weird - but perhaps that is just an oddity of the LSJ dictionary.

Either way, I can see it is an adjectival form of eidos - so I guess it will mean an arithmos that is also (at the same time?) an eidos... unto itself? That sounds very much like the Pythagorean idea from section A about the root (puthmen) of a kind (eidos) of number. So, I'm hoping and expecting that the puthmen from section A will return again here, and it will turn out that the Platonic idea of arithmos eidetikos is related to that Pythagorean puthmen.

The section begins with a linguistic discussion that is interesting I guess, but which is not distinctive for mathematics, and which is not fleshed out here in sufficient enough detail to really assess. There is a whole range of words which have either contextual or relational properties - nothing unique about math here. And in the world of math terms itself, there are some really interesting linguistic questions which could be raised (and I think should) be raised - the obvious question in Greek is about the distinction I've written about before: numbers one through four are adjectives, but numbers five and above are indelincable (as is the case in Indo-European generally) and with a very uncertain and intriguing linguistic status (indeclinable adjectives? indeclinable nouns?) - I wonder if that is something that Plato pondered in a linguistic moment.

But no matter - the linguistic material here at the beginning (which would be interesting if developed in more detail) is simply a prelude to launching into more philosophical abstractions without having done anything of real significance with the linguistics. The term we end up with - the koinon - is not a useful term for linguistics, although apparently it is of considerable importance for Plato's philosophy. On p. 82, Klein declares that there is a "curious kind of koinonia which shows itself in numbers" - although it doesn't seem really curious at all. Semantically, there are many classifications which have contextual and relational features (the pronouns, as shifters, manifest such traits most profoundly); why numbers in particular should be singled out in this way is not clear to me.

In the same way that the linguistic discussion stopped before it became really interesting, the same is true for the discussion of identity - I actually translated a book about the anthropology of the image in ancient Greece and Rome (Portrait of the Lover, by Maurizio Bettini), so I was surprised to all of a sudden find Klein talking about something here on which I am pretty well-informed (the Greek vocabulary of images - eikon, eidolon, phantasma, etc. is a fascinating subject), but I learned nothing new. Plato's perspective on the images is an interesting one, but I know that it is just one among many in ancient Greece. The way Plato pursues the topic of being-nonbeing in philosophical terms has a very nice parallel in the anthropology of images, where you can find many stories about lifeless images that come to life, images that replace the dead person, and all kinds of variations of Freud's "uncanny" combination of life and death, which is the everyday experience of the philosophical profundity of being and nonbeing, a theme which does indeed pervade the world of images. For what it's worth, the etymology of the word identity is connected with a peculiar problem of counting: as the Orthodox concept of the holy Trinity evolved, they needed a vocabulary to manage to the "three in one" and "one in three" of the Trinity, which is how we got "idemtitas" (same-as-itself-ness, as the Greek theological term was rendered into Latin) which eventually gave rise to our "identity." Without that theological crisis (and a true crisis it was!), we probably would have carried on without any word "identity" at all - even Plato did not manage to come up with such a word, for all that he is interested in very similar questions.

Yet I still cannot accept that what Plato is describing as his own philosophical vision has anything to do with math as I understand it and as it has evolved in different cultures over time. Counting was discovered long before there was philosophical speculation about the relation of stasis to kinesis - and I don't really see how any of this was the "gift of the gods to human beings." Instead, it looks to me like counting developed quite naturally from the use of demonstrative adjectives, which through prolonged usage (probably over centuries of time) yielded the counting numbers that we now use. The question for me is NOT "how the many are conjoined to form the unity" ... although I am indeed VERY interested in the historical etymology of the word "both" (Greek ampho) and the word "two" (Greek duo), which exist side by side in the language, but which clearly have different origins. About this, of course, neither Plato nor Klein has any interest.

That is because we are looking at objects "outside of change and time" (p. 90) and "everlasting and unchanged" (p. 91). If that is where the eidetic numbers reside, I don't see what bridge I will cross to get there.

Although I do have to admit that I was staggered by Klein's revelation at the bottom of p. 91 that we have no direct evidence for this Platonic concept at all, and instead are relying purely on Aristotle's refutation in order to reconstruct it. Wow, that is quite fascinating indeed - and it certainly explains the gap in the LSJ Dictionary which so puzzled me - Plato did not use the word, so far as we know, having only Aristotle to go on for that, eh? Amazing. If Klein is going to make an argument for the intellectual influence of the arithmoi eidetikoi, he is going to be very hard-pressed to do that. But perhaps he will manage to pull it off. His willingness to rely on Aristotle with such confidence is quite surprising to me, though, since from Kingsley's book on Empedocles, I learned that relying on Aristotle for evidence about pre-Socratic philosophy can be a very risky business indeed.

When, on p. 92, Klein explained that these poorly documented Platonic arithmoi eidetikoi might be comparable to the Pythagorean puthmenes, I felt quite proud of myself for having suspected that to be the case, despite Klein's best efforts to keep us confused about the exposition of concepts in this book.

As for "two" representing the genus of both stasis and kinesis (p. 93), I think I prefer instead the allegories of the Pythagoreans - if you are going to allegorize numbers (which it seems is what Plato is doing, in the great cosmological tradition), I personally find it much more thought-provoking to allegorize them as Cylops and Briareus and Argus, as the Pythagoreans did, rather than allegorizing them with abstractions.

As for the resounding conclusion of this chapter, where we find out that what really matters is not the duplicity, the multiplicity, the oppositions and kinds, but instead only the "One Itself" and the "original, perfect, all-comprehensive Whole"... well, it leaves me cold. I never thought I would say these words but Aristotle will be a welcome relief in Chapter 8, I think... so we have to give Klein credit for making me look forward to Aristotle, ha ha. Since Plato really has not impressed me here, perhaps Aristotle will be able to put things in a way that make sense to my own earth-bound "dianoia" - transcendence was never my strong suit. :-)

[Barry Robbins]

Chapter 7B

In this chapter Klein attempts to clarify the relationship (as understood by Plato) between “logistike” and “dianoia.”

The single word that jumps out at me in this chapter is the word “incidentally” as it is used on page 78: “thus INCIDENTALLY [my caps] laying a foundation for making calculations.” I’ll try to explain what I think the significance of this word is. But first, let me try to say what I think Klein is trying to make clear about the relationship of “logistike” and “dianoia.” But before doing so, please note that it is “logistike” and NOT “arithmetike” that is related to “dianoia.” It is not that activity which attempts to clarify what accounts for each number being a “definite” number that is related to “dianoia,” but rather that which deals the relations of the “definite” things that each number is a number of.

In the “divided line” section of Book 6 of the Republic, Socrates distinguishes two types of “noeta” (i.e., things that we have access to with our cognitive capacity as opposed to those things that we have access to through perception, i.e., “aestheta”).

In chapter 7B, Klein focuses on clarifying the first type of “noeta,” and what characterizes the way in which we have access to them.

We are completely familiar with these “noeta,” although we rarely think about this familiarity or are even aware of these “noeta” as “noeta.” These “noeta” are, first of all, what our words really intend whenever we speak, and, further, they enable us to identify everything that we perceive in the visible world. For example, I am now sitting near a swimming pool. I see in front of me eight “lounge chairs.” Though you can’t see them, you “know” what I’m talking about, and if you were here you would easily identify the eight lounge chairs.

The question that Klein raises and attempts to answer is this: “What role does our dianoia play in this process?”

In general, it works like this. Right now I am looking at one of these lounge chairs. I see “one” lounge chair; however, as I look at this same lounge chair I also see “many” other things. I see “green,” I see “flat,” I see “straps,” and many other things. In short, as I look at the very same thing, I see the same thing as “one” and “many.” Now, in general, this situation does not cause us any perplexity whatsoever (this, by the way, is why the examples of large and small given on pages 74-76 are so important, since if you look at Socrates as he stands in the middle of Simmias, who is shorter, and Phaedo, who is taller, the confusion for perception is apparent. Try it and I think that you will agree that what your perception reports to you is that the very same thing is BOTH small and large.). Why not? Because immediately and without, so to speak, any effort, our dianoia separate the various “ones” that I just mentioned. It recognizes that the “one” lounge chair I see is one thing, while the “one” color (green) that I see is another “one” thing. And, as Klein will point out, this is always the way in which our dianoia functions; it always identifies “one” among “many.” Practically speaking, it is impossible see one lounge chair all by itself. What is so interesting, I think, is that you “know” what I mean by “lounge chair;” you know “lounge chair,” but what you know can only be seen when it is accompanied by other individual things that you also “know” in this same way. The lounge chair that you know can only be seen when you look at a lounge chair that also is of a certain color, size, shape, etc.

Klein describes this situation as follows: “the ‘dianoia’ is never directed at a single being as such; rather its view always so encompasses a series of beings that the members of the series are carefully distinguished from, and thus simultaneously related to, each other.” (page 76). What this means, of course, is that dianoia is not able to grasp “one,” except as one among many; Accordingly, it also can not grasp the “oneness” of any number, is never able to grasp what makes it “one” “definite” number, but only that it is a number of “definite” things.

Klein’s insight is that the “dianoia” is always engaged in an activity that “is based on ‘account giving and counting’ (‘logizeshai te kai arithmein’), namely on the ability to recognize ‘many’ as ‘many’…. Klein concludes that the activity of the dianoia is “nothing but — counting” (page 77). Accordingly, since our dianoia is always dealing with “noetic structures that are of oppositional character” (page78), it is able to discover as the true “foundation” of the domain that it deals with the “‘pure’ relations of numbers. i.e., the ratios (‘logoi’) and proportions (‘analogiai’) of the ‘pure’ numbers, because every possible comparison is ultimately founded on these.”

And so theoretical logistike provides the foundation for dianoia, and, only INCIDENTALLY, lays a foundation for the possibility of making calculations. The importance of this is that there is no “maladjustment” (cf., bottom of page 42) between the “material” of theoretical logistic and the “material” of what our dianoia is constantly dealing with – namely the first kind of “noeta,” which are always grasped through the “mirror of our senses” as UNITS, whereas, as we have seen, there is a “remarkable maladjustment” between the material of practical calculations (which is capable of fractionalization) and that of “pure” numbers “whose noetic character is expressed precisely in the indivisibility of the units.” (page 43)

What then provides the foundation for each number being a “definite” number? And how is this related to the “higher” kind of thinking that is directed at the “higher” noeta?

On to chapter 7C!