4: Theory of proportions

[Laura Gibbs]

Space for thoughts and responses to Chapter 4: Theory of proportions

[Barry Robbins]

Hi, All,

I want to point out a mistake in the text near the bottom of page 32. The text reads "i.e., the tenths, hundreds...." It should read "i.e., the tens, hundreds...." Here's a link to the passage in Domninus to which Klein is referring:

http://books.google.com/books?dq=anecdota+graeca+boissonade&lr=...

The entire Domninus work is quite short (a little over 16 pages) and the Greek is fairly easy and the handwriting is also neat. If I can find the time, I'll provide a translation for everyone.

Also, here's a link to a cool Greek number converter, which also lists the Greek system of numeration used by Domninus, Theon, and Nicomachus:

http://www.russellcottrell.com/greek/utilities/GreekNumberConverter...

This chapter contains a lot of technical language. Here's a link to a site that explains "superabundant" and "deficient" numbers along with some other types of numbers that Klein will make mention of in this chapter:

http://books.google.com/books?id=3mHBstNdFbIC&pg=PA69&lpg=P...

[Ian Carmichael]

Thanks, Barry,

Very helpful glosses.

I hope the real book is in my work pigeon-hole today - I'm finding reading off the screen very distancing - with a tangible book I can "read, mark and inwardly digest" far more easily.
What I'm also finding is that for all Klein's care and precision, I'm left with a book full of allusion; the sense that Klein is always about to explicate something, but then that 'something' is left outside the content of the book - left in the original sources (?). Anyway, "on with the opera!"

[Barry Robbins]

You're welcome, Ian,

Your observation about the way Klein writes is very perceptive. He writes in a tradition that I would describe as philosophically Erotic (I have in mind the discussion by Socrates of the genealogy of Eros in Plato's "Symposium"). And you're right, of course, that the "something" is left outside of the book. Plato tells us in his Seventh Letter that there's a very limited value in written words. The best they can do is provide "tracks" for those who know how to follow. And, of course, those tracks are intended to help us approach something that is not containable by the written word.

I'm glad that you're a fan of opera!

[Ian Carmichael]

Perhaps more Peter, Paul and Mary

/I dig rock 'n' roll music
/I can really get it on that scene
/I think, I could say somethin'
/if you know what I mean
/but if I really say it
/the radio won't play it
/unless I lay it between the lines

[Ed Wall]

I would tend to call the tradition he writes in hermeneutic phenomenological; however, your description of philosophical Erotic seems somewhat apt.

[Laura Gibbs]

Well, here are my two cents' on Chapter 4 here... I know that Barry may find this style of exposition to be erotic, but it doesn't work really well for me, as someone who is a newcomer to this subject matter. If I understand Chapter 4 correctly, Klein is doing a repeat performance of Chapter 2, giving us a later and incorrect Neoplatonic view of the subject before he tells us in a later chapter what the Platonic (correct) view is. So, presumably in Chapter 5 he will enlighten us about the Platonic view which all the Neoplatonists, poor souls that they are, apparently failed to understand.

So, here's what I've learned so far: In Chapter 2, Klein presented the wrong understanding of logistic and arithmetic in the Neoplatonists, and then in Chapter 3 we learned that the Neoplatonists had failed to understand the practical v. theoretical dimensions. Because of this failure they did not realize that there was a practical AND a theoretical arithmetic, and a practical AND a theoretical logistic - they wrongly associated arithmetic with theory and logistic with practice. Now we are on a quest in search of the theoretical logistic.

So, in Chapter 4, I have read through an exposition apparently of why Nicomachus, Theon and Domninus (all neo-Platonists?) are wrong in their understanding of the role of proportion in the theoretical logistic... am I correct that this is the main conclusion of Chapter 4? And that a right understanding of the role of proportion will emerge in Chapter 5...? I hope so!

Klein has started hinting more strongly at the distinction between "noetic" and "aisthetic" realms in this chapter, so I am guessing that might provide the foundation for his critique of the scholars whose (failed) treatises he examines in this chapter. I'm not sure I understand this distinction fully, so I expect to learn more about it. It also appears that the distinction between "according to itself" v. "towards others" is going to be very important, and I am pretty confident that I do understand this distinction, although I was puzzled by the idea of numbers being "prime" to one another; see my question about that below (I would have thought that being "prime" was purely an "in itself" kind of number, not something that is in relation to other numbers).

As a result of Klein's strange style of exposition, starting with the failed Neoplatonists and working backwards from there, I don't think I learned anything useful after all about the theoretical logistic in this chapter, but I did pick up along the way some interesting terms and categories. Klein did not explain them exactly but I used the links Barry gave us (thank you!!!) to find more information about these side topics. I don't know if they will come up again later or not, but I do find them interesting in and of themselves - in fact, I'll confess I find this kind of stuff far more intriguing to me personally than the speculative philosophical investigation which Klein is focused on.

Primes and factors. The ancient interest in odds and evens, primes and factoring continues to be of importance in this chapter. We hear again about how numbers are of two kinds, the odd and the even, and the subspecies of the evens and of the odds: the evens contain the "even times even" and the "odd times even" - while the odds contain, of course, the primes as one of their subspecies. One thing I did not understand, though, was the distinction whereby numbers could be "composite in themselves but prime to one another" (p. 33). I'm guessing that might be important since this distinction between "in itself" and "towards one another" is something that I suspect is going to be of importance later on. So, what does it mean for numbers to be "prime to one another"...? Klein lost me there, since no example was provided and I could not figure one out for myself! I thought that a number was prime in and of itself, and I don't see how being prime is a relational feature of one number to another.

Perfect, superabundant, and deficient numbers. Very fun stuff here:
Perfect numbers are the sum of their factors. 6 has three factors: 1, 2, and 3 - and six is also the sum of those factors. 28 is also a perfect number. They are extremely rare. The Greeks knew only the first four of them, which are: 1, 28, 496 and 8128. (Euclid had a formula for this which you can read about at Wikipedia.)
Superabundant numbers are numbers where the sum of the factors exceeds the number - for example, 12 has as its factors 1, 2, 3, 4, and 6 - for a total of 16, which is greater than 12.
Deficient numbers are numbers where the sum of the factors is less than the number. Primes are the most extremely deficient of numbers, since they have no factors to sum except the number 1. Poor prime numbers! I always thought of them as being very cool, not as being deficient, ha ha.

Superparticular numbers and superpartient relations; sesquitertian, superquintipartient, double sesquiquartan. There is a good article at Wikipedia explaining the system of superparticular numbers and their relations and how this is connected to musical harmonics.

Figurate numbers. The link Barry provided has some great visual representations of these numbers: triangular, quadratic (square) and oblong (rectangular).

Also, after Barry's comments about the Aristotelian tendency of the term, I'm not sure if "hyle" is going to be important later on - but I think I understand those "hylic monads" after all, which makes me feel better, ha ha. The multitude of units (πλῆθος μονάδων) is what allows us to judge the quantity expressed by a number - and those are the hylic monads! In other words, quantity is constituted by the "hyle" (ὕλη) of a number, that is, by its multitude of those hylic monads. At least, that is what I got out of the passage from Domninus at the bottom of p. 32. It may have turned out to be a red herring in the long run, but I feel like I understand the term "hylic monad" at last. :-)

[Barry Robbins]

Just a quick post now to clarify "composite in themselves but prime to one another." Take the numbers nine and eight, both of which are composite (i.e., not prime) numbers. However, in relationship to each other they are considered prime, since they have no common factors other than the unit.

Here's a translation of the relevant passage in Domninus (416-417)

“Of numbers, some are measured by the unit alone as a common measure, as five and seven. For there is no number, one and the same, that measures them; however, other numbers are also measured by a number, either by one or by more [numbers]; by one, as 6 and 9 (for they are measured by only one number as a common measure); by more, 6 and 12; for these are measured by 2 and by 3. Those that are measured by the unit alone as a common measure are said to be prime (protoi) to one another, but numbers that are measured by a number as a common measure, either by one or by more, are called composite (synthetoi) in relation to each other.”

Also, thanks for hanging in there, Laura. This is by far, far, far, the most tedious chapter in the book, and, I admit, not especially erotic. However, it's all up, up, up from here.

(I'll write more later in response to the other points you made in your post, Laura, and, hopefully, will find something erotic to say about this chapter. I do sense your frustration.)

[Laura Gibbs]

Aha, that makes perfect sense. I like the idea of a number being "prime to itself" and also "prime to other numbers" - I can see how that would be a really useful concept in thinking about factoring relationships between numbers.

And for all that I get confused by Klein's larger goals (which I think I will only understand when I get through the book!), I love this way of learning incidentally about Greek math. The book that you linked to was really fun for me to read: I'd always known that Pythagorean math was strange stuff, but I'd never been exposed to any of the details - so that was really intriguing, especially the part about superabundant numbers being like the hundred-handed monsters, and the deficient numbers being like the Cyclops without the requisite number of body parts, etc.

[Ian Carmichael]

I think I'll choose other books for my Erotic input - still waiting on the hard-copy version of Klein.

[Ed Wall]

Perhaps another take on Chapter #4. There seem to be some substantial things happening here and among them is the difficulty to maintain distinctions - a mathematical problem by the way - among the theoretical and practical arthimos and logistic. The point being difficulties, as Klein puts it, "in the whole conception of theoretical logistic" (36). However, all this seems not some sort of rebeginning, but an effort to illuminate the Platonic definitions more clearly.

[Barry Robbins]

Here's a link to a passage (from what looks like a very cool book) explaining the difference between an arithmetic proportion and a harmonic proportion.

http://books.google.com/books?id=AFTXCMQrYBQC&pg=PA17&lpg=P...

[Ian Carmichael]

Nice-looking book Barry, but the link doesn't get me in, just to the book description. I have, however found what looks like a helpful quote from D E Smith's History of Mathematics
"The early writers often used _proportio_ to designate a series,
and this usage is found as late as the 18th century. The most
common use of the word, however, limited it to four terms. Thus
the early writers spoke of an arithmetic proportion, meaning
b-a=d-c, as in 2,3,4,5; and of a geometric proportion, meaning
a:b=c:d, as in 2,4,5,10. To these proportions the Greeks added
the harmonic progression 1/b - 1/a = 1/d - 1/c, as where a=1/2,
b=1/3, c=1/4, and d=1/5. These three names are now applied to
series. To them the Greeks added seven others, all of which go
back at least to Eudoxus (c. 370 B.C.). The Renaissance writers
began to exclude several of these, and at the present time we
have only the geometric proportion left, and so the adjective
has been dropped and we speak of proportion alone."

The full context in which the quote is embedded is here