2: Logistic v. Arithmetic in Neoplatonists

[Laura Gibbs]

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

[Barry Robbins]

Hi, Ian,

You're not alone in missing the identification of theoretical logistic. That's Klein's main point so far, I think. In fact, the question that Klein will raise early on in the next chapter is how this could happen. The passage in the Gorgias certainly leaves open the possibility of a theoretical logistic, but all of the Neoplatonists cited by Klein seem to have had difficulty locating it. In the next chapter Klein will explain that this difficulty is not at all surprising, since there's a huge problem with its very existence, if, as Klein will explain, all "theoretical" activity provides the basis for the corresponding "practical" activity. The huge problem has to do with fractions. But I probably should wait until we get to it in the next chapter.

Keep in mind that Klein's big observation is that there has been a fundamental change in the way in which we form concepts. As I think will become clear in chapter 7 (a highly philosophically Erotic chapter), we are constantly in touch with unchanging things without realizing it. When I play tennis and my tennis partner says "nice shot, he has in mind what a "nice shot" is and it's this unchanging principal that he uses to measure what he sees. We do this sort of thing all the time in every imaginable situation. But we are not inclined to reflect on this. Curious, huh?

[Ed Wall]

Yes, the practical problem is, in a sense, fractions

[Barry Robbins]

Hi, Bill,

Welcome aboard! Yes, I think that Klein is intent on clarifying the relationship between what you are calling the "pure" version and the practical application. He will have a lot more to say in the upcoming chapters.

We are certainly in touch with unchanging things all of the time (though we don't even think about it) and, as Plato suggests in Book VII of the Republic (in passages that Klein will deal with), "true" education involves an effort to turn the eye of the soul toward such things. So, who knows, this journey may have a "practical" benefit for your students.

[Laura Gibbs]

Hi Bill, not to worry: I am finding the book very challenging, too. I'm definitely not Klein's "ideal reader," ha ha. But years in graduate school made me into a very good scavenger of information: or maybe even parasite is a better word, since when I read something that I do not understand very well (like when my husband asks me to read some technical meteorological article written by a colleague of his, yikes!), I kind of just feed on what is there, based on what I can understand. Often that comes down to just really pushing at the basic basic basic vocabulary words, even if that is not the main question being addressed in the work at hand.

For example, the one thing that has struck me for the first time is my automatic response to "what is a number?" - my brain immediately sees a WRITTEN number, an Arabic numeral of some kind. In that game kids play, "think of a number between one and twenty" - I think of a numeral - say, 14. Just like that: 14.

Well, that hyperliterary reaction - seeing the "sign" (the written numeral 14) as if the sign were itself the number - is something that exactly parallels the hyperliterary reaction that many people have when you tell them to "think of a word" - depending on how immersed they are in the world of reading and writing (and as a teacher, I am totally immersed in that written world - even more so as an online teacher), people will often think of a word as the thing made up of letters, seeing it in their mind in that written form.

I've spent years pondering the question of how written forms of language kind of take over our brains... so thoroughly that we do not even realize it! Walter Ong's Literacy and Orality book is something I read over 20 years ago, and it still fascinates me.

Well, I had never - and I mean NEVER - realized that the same thing has happened with written numbers. But it has: the numbers in my head are written numerals. So, I am personally hoping that by the time I finish this book it will give me a kind of "brain quake" the way that Walter Ong did, shaking me free to some extent of the signs for numbers, and getting me to realize something that comes before the signs, the numbers themselves. I may be way off there... but that is what I am hoping may happen in the course of this book.

Plato, after all, is a crucial figure who stands exactly at the boundary of orality and literacy: Socrates, Plato's teacher, was an oral teacher... and Plato put Socrates's dialogues (oral forms of discourse) into written form... but at the same time Plato wrote (!) about how writing itself (!) is a dreadful thing, with speech being far more true, and writing only a dumb (in all the sense of that word) imitation of speech. Seeing how that tension between the written representation and the "thing itself" plays out in the world of mathematics - especially Plato's mathematics - is something I am hoping to do with this book... either directly, or indirectly, as I learn more about just what Klein wants to teach us. :-)

[Barry Robbins]

Hi, Laura,

No, you are not way off at all; as usual, you are right on. After all, it's "the numbers themselves" and not their written representations that are the object of "arithmetice." Also, our tendency (which you described very clearly) to be conscious of the "numerals" rather than the numbers themselves is, I think, another consequence of the modern mode of conceptualization and its origin in algebraic symbolism.

When we visualize the numeral "7," for example, we are completely out of touch with any actual "arithmos" seven, which always consists of seven "definite objects." Accordingly, we are also completely removed from an awareness of anything upon which further reflection (i.e., theoria) could be "grounded." If all I have in mind are "numerals," what on earth would I look towards in trying to answer the question, "What is a number?" Any "concept" that I might come up with would NOT be based on those numbers that we are perfectly familiar with and, indeed, which we were especially aware of when we first learned to count.

In contrast, the ancient mode of conceptualization always begins with a phenomenon that is perfectly familiar to us. So, in the case of numbers, the phenomenon that Klein will stress is "counting." If we reflect on this phenomenon, it becomes quite clear that every number is always a number of "definite things." This being the case, the next phenomenon to reflect upon is that we have certain numbers "at our disposal 'before' we begin counting or calculating and which must be independent of the particular things that happen to undergo counting" (page 49). Further reflection on this phenomenon leads us to ask the question, "of 'what' are these the numbers" (page 49). This, I think, is where it gets very interesting, especially since, in response to the same question, and based upon the same phenomenon, Plato and Aristotle will come up with very different answers and understandings, and these different understandings will have a huge consequence on the later tradition. (But more on this later.)

By the way, I know that you will appreciate the fact that the setting for most of Plato's Phaedrus (the dialogue that I know you will absolutely love and which we must, must, must study after Klein's book) is a plane-tree outside of the city of Athens. Socrates and Phaedrus engage in a conversation about Eros (also about the relationship of spoken speech to written speech) while seated beneath this tree. And do you know the Greek word for this tree? You'll love this - it's "Platanos." Get it? (The written dialogue of Plato provides a sheltered place for Socratic conversation.)

[Laura Gibbs]

I have another VERY profound connection to Plato's Phaedrus which makes it inevitable that I will have to read it some day or another, and that is the fact his "Phaedrus" provides the name of the protagonist in Robert Pirsig's Zen and the Art of Motorcycle Maintenance which is certainly the very most important book in the last three decades of my life (I first read it when I was 16). :-)

As for numbers and counting, I am starting to wonder if the questions we are going to explore here in Klein's books are going to help me get a sense for the real importance of Greek mathematical science. I will confess that I have not much admiration for ancient Greek linguistics - ancient Sanskrit linguistics beats ancient Greek linguistics hands down in terms of its richness and depth and subtlety - but nevertheless, for all that it seems impoverished to us today, the mere EXISTENCE of Greek linguistic thought is quite remarkable. For all the thousands and thousands of languages that have been spoken on this planet, the eruption of linguistic self-awareness is extremely rare. I'm starting to guess that the same thing might be true of mathematics. Counting I would guess is a universal human experience, just as language is - but mathematics as a theorizing about counting, a theorizing about the numbers themselves, might be just as rare as linguistics, and something that very much interests me. :-)

[Ed Wall]

In some ways the mathematics coming out of India and China is as rich as or richer than Greek mathematics. However, methodologically speaking, in the modern world we do a mathematics which we tend to attribute to Euclid.

[Ed Wall]

There are, interesting enough, written number like 2 and spoken ones like two.

[Ed Wall]

I tend to think that Klein is just trying, in a certain sense, to get us to see (or be more aware) logos a bit more clearly.