3: Logistic and arithmetic in Plato

[Laura Gibbs]

Space for thoughts and responses to Chapter 3: Logistic and arithmetic in Plato

[Laura Gibbs]

Given my interest in Olympiodorus's even-odd emphasis in his definition of arithmetic v. logistic in Chapter 2, I was delighted to find out Socrates also used even-odd as the characteristic of numbers as a focus for his definition in the Gorgias, where he says that arithmetic "belongs to that knowledge which deals with the even and the odd, with reference to how much either happens to be." He then says that logistic is something different - because "it studies the even and the odd with respect to the multitude (Greek plethos) which they make both with themselves and each other." Klein notes that while "even" and "odd" show up in this definition, the key term "number" (Greek arithmos) is not to be found. This use of "multitude" (Greek plethos), however, is interesting to me; the Greek word "plethos" CAN be translated as "number" in English, so it is clearly a word worth keeping in mind. I'm guessing Klein will have more to say about it in due course.

But back to arithmetic and logistic: According to Klein, for Plato the arithmetic is about "the art of correct counting," while logistic "is not merely the art of calculation in the sense of operating with numbers..." but instead, it "presupposes knowledge of the relations which connect the single numbers." That is a subtle difference which I'll confess I don't really understand yet, and luckily Klein acknowleges that the arithmetical and the logistical are hard to distinguish on this level (p. 20). Klein notes that the words "to count" (Greek arithmein) and "to calculate" (Greek logizesthai) frequently occur together; I have added these words to the pages for arithmos and logos respectively.

Further complicating things is that apparently there is a practical dimension and a theoretical dimension to BOTH arithmetic and logistic. Thus, it is not just logistic, but the THEORETICAL logistic which is going to lead beyond the realm of objects in the world of sense (where I definitely spend most of my time, ha ha) onwards to the material of the homogenous monads - those hylic monads from Chapter 2 which are the object of my quest in this chapter!

So, here is what Klein concludes about the theoretical logistic in Plato: "theoretical logistic would have to include primarily knowledge concerning all those relations, i.e., ratios (logoi) among "pure" units, on which the success of any calculation depends." There, at last, is the word logos: logos does not mean number, but in its sense of numerical ratio it is being invoked here at the object of knowledge in theoretical logistic.

Theoretical arithmetic, on the other hand, is "knowledge of these pure numbers themselves" - that is, arithmoi themselves, but not the ratio between them (logos).

As the chapter ends, then, I am feeling more confident about understanding the distinction between practical arithmetic and practical logistic (counting versus calculation), and I am starting to understand the distinction between theoretical arithmetic and theoretical logistic (knowledge of the arithmoi versus knowledge of the logoi)... but where are those hylic monads??? I have finished Chapter 3 and I still have not found the object of my quest, alas! But theoretical logistic apparently will point the way to those hylic monads! So, onward I go!

:-)

[Ian Carmichael]

If hylic and homogeneous monads are to be identified - as context would suggest, I'm more anticipating a denouement with primes - but the mystery does persist!

[Barry Robbins]

Hi, Laura,

As I mentioned in my recent post re chapter 2, the hylic monads are not a Platonic concept. This is why they are not mentioned. As Klein indicates in the passage you cite, theoretical logistic would direct its attention to the ratios of numbers of "pure" units (i.e. monads) and would not (and here's where the Neoplatonists cited make their mistake) perform “operations” such as multiplication or division on these units (thereby making them "hylic" monads).

So, I hate to burst your bubble, but I'm afraid that Plato's view of theoretical logistic will point the way away from those hylic monads. Sorry.

[Laura Gibbs]

Thank goodness for your help here, Barry - I can assure you that I am reading very carefully and trying to follow the clues that Klein is leaving for us... but I think you will not be surprised to find out that he is not the easiest writer to follow! I suspect that once we get to the destination and look back, it will all make sense, but he has an awful lot of inter-dependent concepts he wants us to acquire ... and it's not easy to acquire them one at a time since they depend on one another. Before I go on to the next chapter, I'll look back to the passage that made me think Klein was making the hylic monads the way to the theoretical logistic. I now understand hyle to be an Aristotelian thing rather than a Platonic one (as you explained in your other post), but I had not realized that at all before. I am so glad to have that more clearly understood before I move on! Having bubbles burst sooner rather than later is much better indeed! :-)

[Barry Robbins]

Great summary of chapter 3, Laura! Thanks. I just realized that there was a place for posts on chapters other than the introduction. I'm relieved to be moving on.

The subtle difference that you refer to is, of course, of great importance. Klein's example on page 20 is, I think, useful. We all have learned procedures to use when we wish to solve problems involving number of objects. So we would have no difficulty in calculating how many of 211 apples to place in each of 17 boxes if we wished (practically) to put the same number of apples in each box. Think of all of those division worksheets that we had to complete in elementary school.

Klein, however, states that the main thing that we learn through such operations "are the multifarious relations which exist between different numbers;... that 211 apples can be divided into either thirteen or seventeen equal parts." (I suspect that the math teacher you liked so much probably made you more aware of such relationships than simply the procedures to use in order to solve a computational problem.)

It is one thing to know how to calculate with numbers; it's another to know the relationships that exist between numbers, which, interestingly enough, evidently provide the basis for all successful practical calculations. As Klein puts it on page 19, "All meaningful operations on number presuppose knowledge of the relations which connect the single numbers."

logistike, according to Klein, is the "knowledge of the relations that connect the single numbers." This holds regardless of the types of numbers that we are dealing with (i.e., numbers of apples or numbers of "pure units." (More on this later; right now, I am off to the swimming pool.)