7: Arithmoi in Plato

[Laura Gibbs]

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

[Laura Gibbs]

I started Chapter 7 with a great deal of excitement - the portion for the weekend was going to be Part A, entitled "The science of the Pythagoreans" ... but I finished reading this with the greatest sense of letdown that I have experienced with Klein so far. I'll keep on going in high hopes that the Plato part of this chapter will be illuminating, but it's not off to a good start. Here are the two problems that are really insurmountable for me:

1) Klein's failure of exposition. Perhaps for people re-reading this book, Klein's failure to offer any kind of expository strategy is not a problem, but for me it is getting worse and worse as I proceed through the book. At the beginning of this chapter, for example, there is some kind of enormous footnote in tiny type which runs to two pages in length, in which Klein makes a tantalizing promise about explaining the difference between our concept of number and the Greek arithmoi - he criticizes other scholars for having MIS-understood the Greeks because of having been "guided after all by our number concept, which has a totally different structure..." - a notion that has cropped up before in Klein's remarks. But I have to ask, since Klein does not seem to have bothered to tell us: what IS "our" number concept, according to Klein? Did I miss his exposition of this crucial concept? Does he think it is somehow so obvious that it does not need expounding? Well, it is not obvious to me, alas. I am willing to wait for him to expound his reconstruction of the Greek concept (perhaps it is indeed coming later in this chapter) - but if the real virtue of this study of Greek arithmoi is to examine the strong contrast with our own concept of our number, why has Klein not explained more clearly just what "our" concept of number is so that we can appreciate, or at least be prepared to appreciate, that contrast? And why was all of this in a two-page long footnote anyway? For scholarship, I'm sure Klein deserves the highest possible marks, but for exposition, I'm afraid he gets very low marks from me.

2) Klein's complete lack of interest in history, culture, and any kind of social context. The portion about the Pythagoreans starts out promisingly with this statement: "The development of the theory of arithmoi is undoubtedly in large part due to the men traditionally called 'Pythagoreans.'" - since Klein himself put the word Pythagoreans in scare quotes, I expected he would go on to say something about how problematic this term is, what his strategy is for dealing with the ancient evidence, and overall what he thinks is the method that will yield the most valuable insights here - because with Pythagoreans, much as with the term "gnostics" that you find in Bible scholarship, the methodological issues are enormous, and demand extremely careful attention to history, culture and social context. But then... nothing. Klein simply plows ahead making statements about these Pythagoreans as if we can take Aristotle's assessment of them at face value (!), extracting a single and coherent perspective on arithmoi in Pythagorean thought. Really??? If Klein does believe such a project is possible, he surely needs to provide some discussion of the historical context which allows him to reach that (unusual) conclusion.

What I did get out of this portion of the chapter... The main notion that Klein seems to want to extract from his reconstruction of Pythagorean thought is that the arithmoi provide a kind of "microcosmos" - a perfect example of "ordering" as such. That is a fine idea, and I can see how that could turn out to be relevant for Platonic thought. I'm less persuaded about the equation of "measurableness" and "countableness" (bottom of p. 67) - but it is an intriguing idea; I have been surprised that Klein has not discussed measurableness before now, and I hope we will learn more about that later on. The idea of the "bodily monads" being a kind of material from which the universe is made sounds suspiciously Aristotelian rather than something that could be labeled Pythagorean, but given Klein's methods here, such confusion is inevitable (Aristotle is the source he cites by far most frequently in this portion of the chapter - and from Kingsley's devastating critique of Aristotle's representation of pre-Socratic philosophy in his book on Empedocles, I myself am reluctant to let Aristotle be the main source for a study of Pythagorean thought...). The newest idea which Klein seems to have introduced here courtesy of the Pythagoreans is this business of what he calls the correlative roots, puthmenes - I'm hoping that is something he will carry on with to develop further. I can't say that I really understand it as he has presented it so far - apparently the word πυθμήν does acquire some specific connotations in Greek mathematical language (here's the LSJ definition), and Klein appears to be adducing it in this technical sense on p. 65: "there is always a first and 'smallest' number (or ratio) of a particular kind such that a particular property belongs to it primarily; this 'first' number or ratio therefore represents the 'root' of its kind." I'm not exactly sure why a "kind" has to have a "root" (or "base"), so I will await further enlightement about that. I've got that word on my Greek radar now! :-)

[Ian Carmichael]

All I can easily say is "Hear, hear". I'm still reading, but Klein's made me work too hard for slim pickings, and I'm currently drifting through his work waiting for him to enlighten me. A writier does have some responsibility to be clear,even though as one philosopher said of G E Moore 'Clarity is not enough.'

[Laura Gibbs]

Hi Ian, just to show that I am not a completely grumpy reader, ha ha, I posted some notes in my blog about an awesome book - Catching the Light: The Entwined History of Light and Mind by Arthur Zajonc. Poor Klein: I suspect I am being harder on him than I normally would be just because Zajonc's book, in contrast to Klein's, suits me so much better as a reader.

Anyway, have you come across this book? I think you would REALLY like it. Of course I always knew that "Fiat lux" had to be something of real significance... and here is a book that gives those words their due in the full context of history and culture. After the flare up we had here at Fireside last year about science and religion as if they were somehow utterly opposed to one another, it is so refreshing to read a book like this by someone who is undeniably bilingual in both! :-)

[Ian Carmichael]

I'll go looking for it. (I think I'm so off-track here, a change is in order - since I posted a contribution to the chapter 6 discussion last night, pithy, germane, succinct, incisive but I must have failed to push the contribute button, and alas, my gems have died on the vine. (I suspect that's indicative of my engagement level!)

I've just read some a nice science/religion mythbusters book: called 'Galileo goes to jail' ed Michael Ruse, and it explores a wide range of the set of misconceptions and myths that have flowered from the 19th century and been tended and handed on...
It still amazes me that there is a common assumption of the necessity and inevitability of opposition between the 'two' - (a point at which 'two' become indeterminate as the questions then arise which religions/theologies, which science/sciences/natural philosophies... which set of religious/scientific hermeneutics... Plethora comes to mind! ...even myriad.)

[Ed Wall]

I must admit to be of the opposite view in that I find the book quite thoughtfully written and although difficult technically speaking quite clear. That is not to say there couldn't be improvements, but that is always the case. However, I, I guess, may not be reading it in the same way some people might. Klein is, in my reading, doing phenomenology. He is trying to give us some insight into Greek mathematical thought. He is not trying to prove one version (or even Greek mathematical thought) is 'right' or 'better,' one version is 'wrong' or whatever. Perhaps, if I say this is all about doing number that might help. There is another aspect I find quite fascinating - as I teach among other things the teaching of mathematics to elementary school teachers - is that some of the same progression with number (developmentally) that is unfolded in Chapter 7 occurs across the primary grades. What is especially fascinating is that teachers (and other adults including myself) are in a very different place than children and the dialogue can be painful for children. Rereading this chapter - which seems incredibly rich (and that means there is a lot of unpacking to be done) - causes me to wonder a bit more about that teacher place.

[Laura Gibbs]

If you could give me the "primary grade school" version of Chapter 7, I would enjoy that very much I am sure! With language, I know that there is much to be gained in terms of linguistic insight from the kinds of "mistakes" that children make in using language, and the generalizations and analogies that they apply in their own speech. So, if there is some kind of connection between children's use of numbers and Klein's exposition of the ontology of numbers, that would be really fascinating! Klein certainly makes no nods in that direction. So, if you can unpack the elementary school version of this chapter to share here, that would be super.

As for Klein not arguing that one version is 'right' or 'better' than another, I have to disagree. He really takes the neo-Platonists to task whenever they fail to understand Plato (or, perhaps more accurately, when they fail to understand Plato as Klein does). In addition, Klein's invocations of the "Truth" and the "Good" et al. did not strike me as being meant ironically. Consider a passage like this (and this is just one of many such): "Only dialectic can open up the real 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" (p. 79). By making this claim about what ONLY dialectic can do, it sure sounds to me like Klein is implying that other, non-Platonic mathematical accounts fail to reach the truth.

[Barry Robbins]

Klein's comments about dialectic are taken directly from Book VII of the Republic (cf., 531d6 and following), where Socrates, in response to Glaucon's comment that the education described so far requires an “enormous amount of work," says that it's only the prelude to what is needed to complete it, i.e., dialectic, which Socrates describes as supplying the coping stone for the education so far described.

As for the Neoplatonists, I think that Klein's intention was not so much to take them to task as to understand why they interpreted the passages from the Gorgias and Charmides as they did.

Klein begins the chapter by stressing that “the fundamental phenomenon that we should never lose sight of in determining the meaning of ‘arithmos’ is counting, or, more exactly, the ‘counting off’ of some number of things.” He next makes the point that whenever we count any number of objects we treat them as “uniform.” So, for example, if you count the apples that you purchased at the store today, you will treat all of them as “of the same kind” (i.e., apple), regardless of any differences in size, color, etc. that they may display. Klein then refers to passages in Plato and Aristotle to make this more clear and to reach the conclusion on page 48 that “a number is always indissolubly related to that of which it is the number.” In other words, there can be no number that is not a number of “something.” And, again, this follows from the ordinary phenomenon of counting (“arithmein”). Very simple stuff, really, and stuff the any child would find quite obvious, I think.

Now, if we can see that this is, indeed, a characteristic of “every” number, then, of course, it follows that those numbers (i.e., the “pure” numbers) that we have at our disposal before we begin to perform operations with any collection of objects must also consist of “definite” things. Or do you disagree? If so, please explain, since it seems pretty obvious to me.

And in response to any possible suggestion that such “pure” numbers are different in this regard and should be spoken of as “abstract” or “unspecified,” (page 48) Klein stresses that, “even a ‘pure’ number, i.e., a number of ‘pure’ units, is no ‘less concrete’ or ‘specified’ than a number of apples. What distinguishes such a number in both cases [i.e., a numbers of ordinary objects and “pure” numbers] is its twofold determinateness: it is, first of all, a number of objects determined in such and such a way [e.g., apples, tables, pure units], and it is, secondly, an indication that there are just so and so many [e.g., four, five, ten] of these objects.” (page 48)

Klein next addresses the second determinateness of every number. i.e., that every number (including “pure” numbers) is also a “definite” number of definite things. It is this “definiteness” which makes any number “one” number. It is this “definiteness” which allows any multitude of objects (apples or pure units) to be understood as “one” number. What is so interesting about this “definiteness” (as Klein will discuss later in the chapter) is that it is independent of the character of the particular units of the number.

And it is this "definiteness" that is responsible for the curious definition of arithmetic and logistic given by Socrates in the Platonic passages commented on by the Neoplatonists. Curious because Socrates does not use the word "arithmos," but instead the phrase the "even" and "odd." "Even" and "odd" are, of course, those "eide" that primarily account for the "definiteness" of any number. Klein points out that by defining arithmetic and logistic in this way, Socrates allows for the broadest understanding of them. In other words, these definitions apply BOTH to "practical" and "theoretical" arithmetic and logistic, since even and odd apply BOTH to numbers of perceptible objects (e.g., apples) as well as those numbers of "pure" units that we have at our disposal whenever we need to count or calculate and which allow us to do so with accuracy. On the other hand, use of the word "arithmos" would produce an ambiguous definition, since the character of what the “arithmos” is a number "of" would determine whether the arithmetic and logistic were practical or theoretical. In other words, it would leave open the various interpretations found in the Neoplatonic commentators.

Therefore, from the definitions in the passages under consideration, one may conclude that Plato allows for both “practical” and “theoretical” arithmetic and logistic. And the beauty of these definitions is that they are applicable to ALL of arithmetic and logistic, with the theoretical activity providing the foundation for the practical. So far so good, right?

But here’s the seemingly insuperable problem: the practical activity of calculation very often requires working with fractions, and, therefore, given the indivisible character of pure units, how can theoretical logistic provide the foundation for practical logistic?

So it's not at all surprising that the Neoplatonic commentators had difficulty preserving the original intent of the Platonic definition, even though it is precise and unambiguous. We too would be very hard pressed to do so. And yet Plato certainly was aware of this problem. So there must be something that we are not seeing. Klein, I think, has done a great job of preparing us for the Platonic solution. (By preparing, I mean helping us to see the problem and become aware of our own lack of understanding its solution.)

So how did Plato deal with the problem in such a way as to allow theoretical logistic to provide the foundation for practical logistic? Chapter 8 holds the answer, but, once again, it is an enormously difficult chapter, that requires, I think, a special willingness to take very seriously (as children might) the simple phenomena that Klein draws upon from Book VII of the Republic.

[Barry Robbins]

Obviously I've got my chapters mixed up. (I don't have Klein's text handy). Where I wrote "chapter 7" read "chapter 6" and where I wrote "chapter 8" read "chapter 7." i should have posted my last comments in the chapter 6 blog, but I started out by responding to your comments here, Laura, and didn't realize that this is the chapter 7 blog. Sorry.

[Laura Gibbs]

Hi Barry, no problem! All the comments pop up in the RSS feed no matter which forum they go into, and one nice thing about Klein's technical vocabulary is that it makes it easy to use the "search" thingy to find something from earlier on, no matter where it was posted. :-)

[Laura Gibbs]

I think you need to do an annotated edition of Klein's book, Barry, because you are quite clear in every way he is not. But I would still maintain that this sure sounds like abstraction to me: 'the paradigm for such a koinoia, which is no longer accessible to the logos or, therefore, to counting is the eidetic "two" which consists of stasis and of kinesis; kinesis is that which, in confrontation with stasis is the "other" without which even stasis itself cannot "be," since precisely only "both together" amount to "being."'

At least when I count: one flower, two flowers, three flowers, four flowers, I'm clear on arithmoi, no problem... but I can assure you that I have no perception - none, zero, nada, nothing at all - of this "eidetic two" with its wholeness of being as it consists of stasis and kinesis etc. etc. etc. etc.

I already understood before chapters 6 and 7 that arithmos is about the numbers for counting off definite things. I already understood that there were two main kinds of arithmoi, the even and the odd, plus their sub-kinds. I understood that fractions were a problem - and I know this from computational mathematics too. Just this weekend, for example, I read a wonderful book about the history of the compass and marine navigation, and it turned out that the "knots" used to estimate a ship's speed were literally knots in a rope, based on a system of 48 knots rather than the more mathematically accurate 51 knots, because by having access to the the whole factors of 48, it avoided the need for fractions - based on a system of 48, it was easy to give the speed in knots AND fathoms, which would have been a fractional nightmare if there were 51 knots to the mile.

Anyway, I'll proceed to chapter 8 on the assumption that my understanding of arithmos can survive without taking on the burden of the dialectic...

[Ed Wall]

Laura

Briefly Developmentally: (Group), count, cardinality, ordinality (actually this last gets little play as teachers are somewhat oblivious to this). Early on the One has tremendous force and it is unclear how the Many and the One are assimilated by children. However, even in third grade (and I am sure much later conceptually), there are cogent arguments by children that, in fact, 4/4 is less than 5/5. I do need to say again that this is a parallel progression and, while I suspect that there are 'similar' things going on for children as for the Greeks, they are not the 'same.' I find that Klein is helpful in thinking about this parallelism.

In my reading, I don't think that Klein makes the claim that Plato reaches the 'truth.' As far as the quote goes - and much of this seems to be from the Theaetetus - it is hard to tell whether Klein believes this or not. If you have read any Merleau-Ponty, there are similar difficulties. Unfortunately, I admit to doing other reading in the period Klein was writing and I may bring biases which argue strongly against a 'right' or 'better' and strongly for an illumination of some fundamental difficulties - that are very much still prevalent - with number. I also bring experiences teaching in elementary school and college that point to some cracks which have been, so to speak, plastered over. So, as I said before, this is only my reading.

[Laura Gibbs]

It sounds like Klein can be a great catalyst for some folks (not me, alas) in terms of making connections between the things Klein discusses (in his oblique and obscure way) and other questions of interest. Unfortunately for me, Klein-qua-Klein, in and of himself, does not have a lot to offer, and the ways in which he resonates with my own interests are counterproductive, rather than the other way around. That's why I've given up on the book, after making what I think was a valiant effort for several weeks...

I'm personally fascinated by the vocabulary used for counting cardinally and ordinally (just looking at the difference between the English words "one" and "first" can give some wonderful insights into the different mental systems at work there - and other languages supply yet more material to ponder - looking at that vocabulary is interesting in any language), and I'm very intrigued by the idea of 4/4 being somehow less than 5/5. I fail to see where Klein even hints at such delightful topics or has any interest in them at all - perhaps that becomes more clear later in the book; it certainly had not emerged by p. 100 when I abandoned ship. :-)