2: Logistic v. Arithmetic in Neoplatonists

[Laura Gibbs]

Space for thoughts and responses to Chapter 2: Logistic v. Arithmetic in Neoplatonists

[Laura Gibbs]

I will admit to being surprised both by the title and contents of Chapter 2 - "the opposition of logistic and arithmetic in the Neoplatonists" - since I expected from the Introductory chapter that Klein would want to take us BACK to Plato (and perhaps even Pythagoras) and then move forward, but as I got to the end of this chapter, I think I understood his strategy: he wants us to see that even though there appears to a clear opposition between logistic and arithmetic in the Neoplatonists, this is not so easy to discern and understand, and to figure out just why it might have turned out a bit muddled, we will have to go back to Plato himself in the next chapter.

Before sharing comments about the chapter and the definitions it surveys, I wanted to say something first about Greek arithmos and the word number in English - and the terrible dilemma faced by the English translator of Klein's book - did people notice the Translator's Note at the very beginning? She explains:

The Greek word arithmos is rendered in the German text as Anzahl: "a number of [things]" to distinguish it from our modern Zahl: "number." Since English approximations to Anzahl are either obsolescent (e.g. "tale") or awkward (e.g. "counting numter," "numbered assemblage"), Anzahl, like Zahl, has been rendered simply as "number," although it is a chief object of this study to show that Greek "arithmos" and modern "number" do not mean the same thing, that they differ in their intetionality, for the former intends things, i.e., a number of them, while the latter intends a concept, i.e., that of quantity.

Wow: as someone who has done a fair amount of scholarly translation work (Italian to English), I can really sympathize with a fundamental dilemma like this, which strikes to the heart of a work like this.

As for the word she dismisses - English "tale" - that is actually a really fascinating thing to reflect on. Think about it: the meaning of the Greek word "logos" is ALSO tale. One of the most basic meaning of the word "logos" is a story or myth or fable (this is the use of the word I am most familiar with myself, since it comes up continually in Aesop's fables, which are called indifferently either "mythos" OR "logos" in Greek, and both words mean "story, tale").

Well, I had never reflected on the fact that like Greek "logos," the English word "tale" is a word that comes both from storytelling and narrative (the primary meaning it has today) but ALSO from the world of counting!!! Here is the information from the very useful etymonline.com website:

O.E. talu "story, tale, the action of telling," from P.Gmc. *talo (cf. Du. taal "speech, language"), from PIE base *del- "to recount, count." The secondary Eng. sense of "number, numerical reckoning" (c.1200) probably was the primary one in Gmc., cf. teller (see tell) and O.Fris. tale, M.Du. tal "number," O.S. tala "number," O.H.G. zala, Ger. Zahl "number." The ground sense of the Mod.Eng. word in its main meaning, then, might have been "an account of things in their due order." Related to talk and tell.

So, Greek "logos" is a word that can mean both "counting" (reckoning, etc.) and ALSO "storytelling" - and the English "tale" works the same way (and is etymologically related to German Zahl). In the Romance languages, the same: just think about Spanish "cuento" - a story, but Spanish "cuenta" which is the account or bill that you would ask for in a restaurant.

I always knew about the Romance language parallel, but I had not picked up on the Germanic one until reading the translator's note here at the beginning of this book.

So, I'll share some notes later here about the actual definitions of aristhmetic and logistic in a separate post, but I wanted to contribute here my own personal connection to this, in that I am very much interested in how counting and telling are two closely related human activities, as evidenced by these basic word groups in Greek and also in the German and Romance languages.

:-)

[Ian Carmichael]

And indeed, in the days when there were bank staff, the UK and Australian use at least referred to them as "bank tellers", I suspect that was lost in the US use of 'bank clerk' (although still concealed in the acronym ATM.)

[Laura Gibbs]

Ha ha, Ian, I had to laugh when I read your reply - my husband brought up EXACTLY that example when I was telling him about English "tale" at dinner - he had never heard the use of "tale" to mean a reckoning, but he immediately said, "Oh, I guess you mean something like banktellers."

Can you believe it: I study storytelling, and I must use the word "storyteller" thousands of times in a year... and I never connected bank-tellers with story-tellers.

DOH. I cannot believe I never even noticed that, staring me right in the face!!! :-)

[Ed Wall]

I tend to read Klein the same way as you are doing here. Perhaps it would be useful to say that I tend to read him as trying to get a grasp on logos by bouncing it off, so to speak, various traditions including the modern.

[Laura Gibbs]

So, now that I've said my piece about logos and logistic and telling/storytelling (see above), I should try to say something about this chapter! I'm guessing that unlike Barry and Lee and people who have a sustained interest in the history of philosophy, the term "Neoplatonist" is one that doesn't have a lot of resonance for me. I'm definitely out of my depth in this chapter, in that I don't have any substantial context in which to assess how this particular study of Neoplatonic definitions of arithmetic and logistic fit into overall tendencies and trends in Neoplatonic philosophy.

So, with that caveat, here's what I found really profound (personally speaking) in this chapter: the discussion in Olympiodorus about oddness and evenness as being fundamental characteristics of numbers: "arithmetic [arithmetike] concerns itself with the kinds [eide] of numbers [arithmoi]; logistic [logistike], on the other hand, with their material [hyle]. There are two kinds of number: the even [artion] and the odd [peritton]..." Olympiodorus later goes on to say that "logistic concerns itself with their material, investigating not only the even and the odd as they are by themselves but also their relation to one another in respect to their multitude. For multiplication is either among the same kind or is of different kinds with one another, the former, when I multiply even with even or odd with odd, the latter, when I multiply odd with even or even with odd."

That was something I really had not thought about since elementary school, and that "WOW, HOW COOL!" sense of figuring out that if you multiplied an even by an even, you always got an even, and if you multiplied an odd by an odd, you always got an odd - BUT if you multiplied an even by an odd, you also got an even, every time. I remember an amazing teacher in second grade (I think he must have been a student teacher, looking back on it) who did incredible things to our minds. We used number lines in class (I am very much a product of the "new math" circa 1970), and he showed us how you could draw the number line the other way and make zeroes. That totally rocked my world. He taught us how to subtract 5 from 2... and told us not to tell ANYBODY that we knew how to do it, because it was a big secret. And I also remember that he told us about odds and evens, and their multiplication properties, and he asked us, if we thought it was ALWAYS true that odds and evens multiplied this way, and I can remember spending hours and hours writing out examples to test this... but never making that breakthrough that would allow me to express the general principle, aside from just working out example after example after example of multiplication, to see what happened when I multiplied two numbers, odd or even.

Then, we moved, and in my new school I annoyed my math teacher in third grade so much that she made me go do math with the fourth graders, and I asked the fourth-grade teacher about the even and odd thing and that is when I learned a tiny bit of algebra, since he showed me how I could understand the even and odd thing as 2x*(2y+1). I came home and made my dad teach me how that worked, and glory hallelujah my dad did teach me since he is a serious math geek. Then, alas, I was doomed to be deeply bored in math class until high school because I wanted to study the X's and the Y's and not any stupid "number numbers" as I called them... and to this day, I still remember the total amazing thrill that it was to figure out how the numbers could be understood in a way (the X and Y way) that I had not understood before or been able to figure out on my own.

But, in terms of the philosophical labels I guess I am just as confused about how that works as Olympiodorus himself was!!! At the end of the chapter Klein says that Olympiodorus is mixing up "the immutable, i.e. noetic" constituent of the numbers and the "changeable" constituent. Klein then says:

In the number "six," for instance, the multitude "six," with its quantity, its hyle, must be distnguished from its eidos, namely the "even-times-odd" (artioperitton - since six is composed of the even factor two and the odd factor three). Arithmetic treats of the eidos, logistic of the multitude of "hylic" monads.

I need to understand what hylic monads are... and I definitely don't understand that yet... but that's a good question to carry me over into the next chapter!

Meanwhile, artioperitton - what a helluva word!!! It would have to be translated into English as "even-odd" but of course we have no such word in English. In the Greek-English dictionary, the definition provided is: "even-odd, of even numbers, the halves of which are odd" - VERY COOL. The Greeks really did have a staggering vocabulary...

I should say something about the very cool etymology of "odd" in Greek - perissos, or perittos (like the way a "peri-meter" goes around the outside of a space, or something "peri-pheral" is carried beyond the topic). It literally means "beyond, something over and beyond" which is such a cool way to see what odd numbers are: it's like they are even, but then they overflow the container of evenness, and become odd. The verb perisseuein means "to be over and above a certain number, be more than enough, abound". :-)

[Ian Carmichael]

Hmm, the "hylic" monads - are these the unconfused, that decompose to odd-odd, even-even, or are they more primitive, namely the primes, into which each multitude seems to be analysed?
(This is more a note to self than an insight masqueraded as a question!) We shall see on the next round of the journey.
I too was a little surprised to see us meet the Neo-Platonists before Plato - it seemed an abrogation of Klein's intention to lead us through the conceptual developments trying not to read back later concepts. Yet a neo-Platonic understanding of Plato is an anachronistic reading of Plato, its just not modern.

[Barry Robbins]

I think that I can say something useful about “hylic monads,” at least I will try. But first, I want to thank you, Laura, for your “philosophia” toward them. You helped me to realize that I had never given the phrase enough thought during my other journeys through Klein’s text, and, accordingly, never quite understood the point of Chapter 2.

First, a few preliminary remarks about “hyle.” Though it also meant something like ”forest,” it came to mean something more like “timber” or “fire wood,” and this, I think, is the meaning that is at the root of the use of the term in Aristotle and also the Neoplatonists. So, what is timber? Well, imagine you are walking down the street and you see a vacant lot with piles of planks of wood stacked neatly. You probably would conclude that some sort of building was about to be constructed there. Aristotle will go to great lengths to make the case that those planks of wood ARE the building that will be constructed there, “potentially” (dunamei). In fact, Aristotle, as I recall, defines “hyle” as “potentiality.” It is a crucial concept for his understanding of everything, and it’s apparent that the Neoplatonists drew from it.

So, “material” is not a bad translation, if we focus on the “potential” of the material and not on any specific form that it is already in.

Klein begins chapter 2 by reminding us of a “fundamental distinction” that is found in all of Greek “science” and specifically in Neoplatonic mathematics – namely between the contemplation of things that are subject to change and things that are not subject to change. Now, if something in not subject to change, you can’t “do” anything to it. You certainly can not turn it into something else. There is no “potentiality” in such things to become anything other than what they are. (And this is why the same distinction may also be spoken of in terms of theory and practice.)

Klein then goes on to discuss four Neoplatonic commentaries on a passage in Plato’s Gorgias, a passage in which Socrates speaks about the relationship of arithmetic and logistic.

Now, as will become clearer in chapter 3 when Klein cites the actual passage from the Gorgias, Socrates’ statement (according to Klein) allows for a “theoretical” logistic, which would, speaking tentatively, involve a “theory of proportions.” (i.e., would endeavor to understand the “ratios” that exist among numbers that are not subject to change; e.g., the number ten is related to the number five as two is related to one.)

Now, what’s important here is that such an endeavor would NOT involve multiplication or division or any other “operations” on numbers. Such an endeavor would not DO anything to numbers that would cause them to become other numbers than what they are. Accordingly, it would not treat the quantity of numbers as “hyle,” for becoming something else. And since the units of any number make up its quantity, theoretical logistic would NOT deal with “hylic monads.”

This is the point that, according to Klein, was misunderstood by all four of the commentators. Proclus and the Charmides scholium did not allow for a theoretical logistic (i.e., one that would deal with unchanging numbers) at all, while Olypiodorus and the anonymous scholium to the same passage in the Gorgias do speak about a logistic that deals with unchanging numbers, but one which deals with them “practically.” This is why Klein point out that “Olypiodorus uses multiplication to exemplify the state of affairs.”

So, in summary, hylic monads, as understood by Olympiodorus, are the units of unchanging numbers that can be operated on by multiplication, division, etc.

I hope this helps.

[Laura Gibbs]

Wow, Barry, this is super - I am sure it will be a huge help when I move on to the next chapter but, even better, this really helps me already to understand something about Aristotle and "hyle" that had never clicked with me before. Thinking about it in terms of "(building) materials" really makes sense to me!

As for changeless things: that is not a category that I normally have in mind (it is VERY philosophical indeed) - and it is good to be warned about that before I proceed on to the next chunk of reading. I had not made the connection between the theoretical and practical dimensions which Klein raised in chapter 3 with the parallel distinction between unchanging and changing objects. I am really struck by this idea that the study of the ratios (logoi) among the numbers is different from the manipulations involved in other kinds of mathematical operations.

So, THANK YOU for these good thoughts to keep in mind as I proceed further!!! This is a very challenging book for me to read, and it is great to have these signposts to watch for when I start in on the next chapter. :-)

[Barry Robbins]

You are very welcome, Laura. By the way, not to be to flippant, but unchanging things are the ONLY things that you can have in mind. More about this when we get to chapter 7.

[Ed Wall]

A lot of "early' Greek mathematics appears in the elementary school. The sad thing can the move towards symbolization - I'm not saying it is bad - as so many things become covered up. For example, algebra is not need to "prove' the even/odd business and children's relationships with numbers can begin to fade with the introduction of algebra.

[Bill Pearsall]

Hello to all - I apologize for being so late on this post - I was engaged in a brief seminar and just started the book last Sunday - however, while I believe I have some idea of what Klein is writing about, my level of comprehension is not in the same ballpark with your observations - I am afraid I cannot add much (if anything) to the discussions - my knowledge of is neoplatonism is deeply deficient, so I am reading very quickly on this background material while I continue with the main book - is Klein talking about the development of the practical applications of mathematics versus a "pure" version? - this falls into discussions students and I have in my classroom, though they approach the topic as if there is nothing but practical applications and they have little interest in that - I use the idea of something deeper (pure) as a lure - please tell me if I am way off the mark or simply to lightweight to continue - I do enjoy your insights and they have sent me off to new areas of study in order to better appreciate this book

thanks
Bill

[Ian Carmichael]

Hi Bill,

Don't worry mate, you're not the only scrabbler!

It seems to me that Klein is reporting the later (neo_Platonic) understanding of Greek mathematics as distinguishing the science of number from the art of calculation (arithmoi vs logistic), but in fact, he claims there is a fourfold nuance in the original Platonic view - practical and theoretic both for arithmoi and logistic.
In my mind I've assumed that the theoretic arithmoi is concerned with the numbers as they are in themselves (even, odd, triangular, square..), and the practical arithmoi is concerned with numbers as they are in their essential relationships (relative primes, can't think of another..)
Somewhere I've missed the identification of theoretic logistic - even though it's important for the society elite [I mustn't be cut out for the elite!], and practical logistic is of course the art through which business and statecraft is quantified and controlled.

[It seems to me the neo-Platonists have promulgated a fortuitous (mis)understanding of the Platonic vision!]