5: Theoretical logistic and problem of fractions

[Laura Gibbs]

Space for thoughts and responses to Chapter 5: Theoretical logistic and problem of fractions

[Barry Robbins]

This chapter strikes me as pretty straight forward. So I thought I’d try to summarize it before moving on to the next chapter, which, I think, is far more interesting.

So here goes.

Klein’s main purpose in this chapter is to answer the question that he states at the very outset of the chapter:

“What prevents later writers from interpreting the arithmetical theory of relations, i.e., proportions as theoretical logistic? Or, in other words: How did it happen that the Platonic double distinction of theoretical and practical arithmetic, on the one hand, and theoretical and practical logistic, on the other, was reduced to the single distinction between theoretical and practical logistic.”

The first point that Klein makes is that the Platonic distinction between theoretical arithmetic and theoretical logistic was probably Plato’s and “need certainly not to have corresponded to classifications current among the mathematicians of his own or immediately subsequent times” (p 37).

In support of this point, he points out some similarly purely Platonic postulates for a science of astronomy and music “completely freed of sense perception” (p 38). (These are found in Book VII of the Republic and are presented in the course of describing the education of a philosopher.)

So, if there was no tradition for such a distinction, it is not surprising that Plato’s distinction would not necessarily be followed by the later tradition.

The second point he makes is that “even if this demand is taken seriously within the sciences and if one sets about the construction of a theory of relations of numbers which is intended to stand beside the theory of numbers as such, i.e., of their different kinds, it soon becomes clear that such a division in the presentation of those subject matters is hard to maintain.”

In support, Klein cites two passages in Plato’s own works that refer the “knowledge of numbers themselves,” first to logistic (in the Statesman) and then to arithmetic (in the Republic).

So there is already a certain lack of clarity when it comes to the distinction between theoretical arithmetic and theoretical logistic in Plato himself.

The third point, and this, of course, is the most significant, has to do with fractions.

As Klein puts it, “But the crucial obstacle to theoretical logistic — keeping in mind its connection with calculation — arises from fractions, or more exactly, from the fractionalization of the unit of calculation” (p 39).

The remainder of the chapter deals chiefly with a clarification of the problem. Put simply, theoretical logistic, insofar as it deals with unchanging numbers, deals with numbers of “pure” units, i.e., with units that are simply “one” and nothing else. Such “unchanging” units clearly can not be divided into smaller parts. On the other hand, those objects that are treated as units in the course of any calculation (e.g., apples, doughnuts, etc.), because of their “bodily” nature can be divided into fractional parts.

There are two major consequences:

First, “there emerges a remarkable maladjustment between the material in which such calculations take place and that other ‘material’ of ‘pure’ numbers whose noetic character is expressed precisely in the indivisibility of the units” (p 43).

Second, “whether that which distinguishes exact calculation, namely operation with fractional parts of the unit of calculation, can really be sufficiently grounded in the science of the possible relations of numbers, i.e., in the ‘pure’ theory of relations, alone” (p 43).

Klein then indicates that this is NOT accomplished in the “arithmetical books” of Euclid (p 43 & 44), where there is presented a theoretical foundation ONLY for calculations that involve operations in which “it becomes necessary to decompose single numbers into their components (factors), to find the greatest common measure (divisor) of several numbers, to express their ratios in the least terms, etc.” (p 44). No foundation is provided for operations that involve the partitioning of the units themselves, a partitioning that occurs in the course of most calculations.

The final result is that “ ‘Calculation,’ with all its presuppositions, must therefore be referred entirely to the realm of the practical arts and sciences, while the theory of relations loses its fixed place and comes to be assigned now to arithmetic as the theory of the kinds of numbers, now to harmonics as the theory of musical intervals based on ratios of numbers” (p 44).

Accordingly, logistic comes to comprise “approximately the subject matter of present-day elementary arithmetic” (p 45).

Klein concludes the chapter by reminding us that the position that logistic comes to be in is ultimately the result of “that special conception of the 'pure' numbers and their material that governed the Platonic tradition throughout.”

But surely Plato himself must have been aware of these difficulties, and yet he evidently understood the “pure” numbers in such a way as to allow for the double distinction of theoretical and practical arithmetic, on the one hand, and theoretical and practical logistic, on the other.

So clearly it now becomes necessary to try to understand Plato's conception of "pure" numbers better, a conception that, presumably, allows for a theoretical logistic that is also the foundation for practical logistic.

On to chapter 6!!!

[Ed Wall]

Actually I wondered a little bit about that quote on 45 as it is not quite true. The curriculum of what is termed upper elementary is roughly the logistic and the curriculum of lower elementary or what is termed primary is roughly arithmos. This causes some substantial problems for children as they move from one emphasis to another and indeed they often see the partitions as units. So, for example, many will say that 4/4 is more than 5/5.

[Laura Gibbs]

Thanks so much for these notes, Barry - I feel reassured now that I was not simply missing out on something important in the discussion of Platonic terminology in this chapter, and whether the distinctions Plato wanted to make are somehow essential to our understanding. I was left with one main question about terminology as I read through this chapter - and that is about this idea of "noetic" sciences. Can I assume that this use of "noetic" can be taken to be synonymous with "theoretical" as we have been using it so far...? Or is there something distinctive about "noetic" as opposed to "theoretical" (i.e. as in theoretical arithmetic and theoretical logistic as discussed so far).

Meanwhile, the part on fractions really intrigued me - and your comments here helped me integrate my own aimless curiosity on my part into the larger question of the theoretical logistic which Klein is pursuing.

I was especially intrigued by the problem of smaller and smaller units - esp. the passage discussed on p. 39, with philosophers multiplying the "small change" (Greek kermatia) in order to keep it in the form of whole numbers rather than fractions. If you push at that hard enough, of course, making numbers smaller and smaller and smaller, you reach the mysterious zero (just as you reach the mysterious infinity as you make numbers larger and larger). The magnificent problems posed by zero and infinity are such a great topic in the history of math; a few years ago, I read a marvelous book by Charles Seife called Zero: The Biography of a Dangerous Idea which I really enjoyed!

So, for example, I was intrigued by Klein's paraphrase of Plato's notion here on p. 40: "Each single thing can be infinitely partitioned because of its bodily nature as an object of sense. The unit which can only be grasped in thought is, on the other hand, indivisible..." - it was that phrase "infinitely partitioned" which struck me. I wonder just how deeply Plato was inclined to ponder on that "infinite" partitioning process.

Also, just as a side note, on the topic of fractions themselves - I learned something interesting when I was exploring the earliest systems for writing numbers (I've been browsing through a really fine textbook called The History of Mathematics by David Burton - it has lots of practical exercises to do so that you can practice applying some old mathematical systems yourself!) - anyway, in regard to the Babylonian base 60 system, which strikes us as so very odd, apparently some scholars thing that one reason the Babylonians may have adopted such a system is the ease it offers in expressing fractions (or, rather, avoiding the need to express fractions), because 60 has as its factors 1, 2, 3, 4, 5, 6, 10, and 12.

Similarly with 360 degrees in a circle - which is actually a very practical example of changing the "unit" in order not to be having to make small change in Plato's terms! If instead of just have one circle and then having to talk about one-fourth of a circle, or one-sixth of a circle and so on (making small change of the poor circle), we can do much better, avoiding those nasty fractions, with 360 degrees, so we have 90 degrees (instead of "one-fourth-circle"), 30 degrees (instead of "one-twelfth-circle") and so on. I asked my husband about other units that are applied to circles (since 360 degrees probably was very handy before mechanical calculators - but now its advantage in calculations is something of far less value to us), and he told me about radians, a unit of measure that involves π (pi) itself, with 2π radians in a circle, instead of 360 degrees. I checked Wikipedia, and in the "history of the radian" section, we can see that degrees were triumphant for centuries, and that the idea of the radian is something quite modern.

Apparently the poor grad - with 400 instead of 360 as the number for the circle, so that a right angle is 100 grads - is on its way out, at least according to Wikipedia. Although for quick calculation, unaided by a calculator, it is clearly very handy, as Wikipedia explains: "One advantage of this unit is that right angles are easy to add and subtract in mental arithmetic. If one is traveling on a course of 117 grad (clockwise from due North), say, then the direction from one's left is instantly convertible into 17 grads; while the direction from one's right is 217 grads; and the direction from behind one is 317 grads."

I know these questions about practical calculation are not the sort of thing that will be important as Klein's discussion proceeds, but I do enjoy learning about them along the way! :-)

[Barry Robbins]

You are very welcome, Laura. Klein will talk more about "noeta" in the upcoming chapters, especially 7, but, quite simply, things that can be perceived through the senses (I guess that's a bit redundant) are referred to as "aestheta," while those that can be grasped by thought are referred to as "noeta." So I think that you are quite right to take "noetic" here as synonymous with "theoretical."

And isn't it quite amazing, though at the same time completely obvious, that if, for example, you cut an apple down its center, you no longer have "one" but now have "two." instead of having less, you now have more? Of course the unit has changed, since now you see two "pieces," whereas before you saw one "apple." But since this sort of division is evidently not possible with "noetic" units, theoretical logistic can not form a basis for practical calculation.

[Bill Pearsall]

Good Evening - I am enjoying all the insights you folks are providing - I have finished chapter 5 and found the ideas about the one being "impartable and indivisible" versus a single thing that can be "infinitely partitioned" are not as simple as they first appeared to me - this caused me to finally begin reading about neoplatonism which led to the writings of Plotinus - I only thought I knew something about the history of mathematics but this is certainly interesting - Klein brings up criticisms of the Pythagoreans and I am wondering if I need more knowledge of these people to understand this book or will the information in this book be sufficient to the task? - I am very curious to see where this will lead me - and thank you for sharing your insights

[Laura Gibbs]

Hi Bill, I am very intrigued by the Pythagoreans, too! There were some very interesting tidbits in the book that Barry had linked to for the superabundant and deficient numbers here:
http://tinyurl.com/lr5ggt
(not all of the book is available in preview mode, but a lot of it is - and it's good proof of the power of book preview for advertising; I've added this book to my Amazon wishlist!)

I am still hoping that perhaps Peter Kingsley will someday write a book all about Pythagoras. Kingsley is someone with a real interest in the mystical and religious dimension of ancient thought - although he is trained as a hardcore academic philosopher, he gave that up in pursuit of more mystical pursuits (and his later books abandon footnotes and all the scholarly apparatus which he was still using in his earlier works). His book on Empedocles - Ancient Philosophy, Mystery, and Magic: Empedocles and Pythagorean ... - is hands down one of the most interesting books I have ever read, and it has some discussion of the legacy of Pythagoras in there, but not on Pythagoras per se.

Does anybody else have some reading recommendations on Pythagoras and the Pythagorean religion? The Stanford Encyclopedia of Philosophy has a detailed and VERY informative article online. The article makes it clear that even identifying Pythagoras as a figure in the history of mathematics is a very precarious business - for all that schoolchildren learn the "Pythagorean theorem" and think of him as a mathematician nowadays! :-)

[Ron Allen]

Hi Laura:

Precarious indeed. Since Burkert's book was published in '62 and translated into English a decade later, scholarship has not been very kind to Pythagoras--to say the least. You might have a look at Burnyeat's reviews of a couple of recent books about Pythagoras in the London Review of Books for 22-Feb-2007: http://www.lrb.co.uk/v29/n04/burn02_.html

The books reviewed are C. Riedweg, Pythagoras: His Life, Teaching and Influence (which is kinder to the traditional interpretations of Pythagoras & his school) and C. Kahn, Pythagoras and the Pythagoreans.

Klein's book came out in the 30s, so he writes as if all the things that Burkert & Burnyeat argue are pure myths were really factually based. Can't be blamed, of course; the "debunking" of Pythagoras hadn't happened yet. But, it's interesting to read Klein's Chap. 7 with the modern critique in mind.

Thank you, by the way, for kicking off this discussion. The Klein list on yahoo has been dormant for some time.

--Ron

[Laura Gibbs]

Aha, Burkert, that's right - I totally forgot about that venerable old book! And what a great article from LRB; thanks so much for the link! After reading through this, it makes me wish even more that Kingsley would tackle Pythagoras as a topic - his Empedocles book was able to pull off some amazing research into the tangled connections between the Empedocles legends and the larger philosophical tradition as we understand it, while also making forays into the realms of what we would call mysticism that Aristotle had so rigorously walled off for the future. Kingsley has a sincere and deep interest in mysticism, something that Burnyeat clearly doesn't have, as we can see from the end of the article - although Kingsley shares with Burnyeat that fascination with the historical detective work required to write about the ancient Greek world. It's not something I have the scholarly patience to accomplish myself, but I love benefiting from the intense research that others are inspired to do in such quests!

[Ian Carmichael]

The SEP is a superb resource, isn't it, Laura. I'm hanging in, browsing the comments, but Klein is still on the waters. If my suppliers don't stump up soon, I'll have to fold my tent, I fear!

[Laura Gibbs]

Further reinforces my commitment to public domain texts... it's not that I don't read anything 20th-century (I do, sometimes)... but the sheer pleasure of being able to give EVERYBODY access to texts online is something that is unrivaled as an educational premise. Luckily, with Aesop's fables, I am not in any kind of dire straits by limiting myself to the treasure trove at Google Books and other online libraries... I finally found this weekend a gigantic 600-page long 1544 edition of Latin fables by Camerarius online, another treasure for browswing this summer!!! :-)