7: Arithmoi in Plato

[Laura Gibbs]

Space for thoughts and responses to Chapter 7: Arithmoi in Plato

[Ed Wall]

Laura

When I read Klein and others I admit to reading with one foot, so to speak, in the classroom. This doesn't mean that Klein speaks directly to what I observe, but things he says do resonate - or so it seems to me - with my observations. So when - within the first 100 pages - he talks about counting things I think about early stages in counting. When he talks about kinds, I think of cardinality the next big step beyond counting. When he talks about being superior or inferior, I think about ordinality (interestingly ordinals don't seem divisible). And such thinking begins to shake the branches of that comfortable tree in which I am perched as an adult user of mathematics. So, Klein does exactly what I think he wants to. He illuminates for me - that is what phenomenology is all about - number.

What I find interesting about the 4/4 and 5/5 business is that the children - those who haven't 'learned' otherwise yet - view divisibility in very much the way Klein discusses on how in partitioning the one becomes many [40]. They argue that I have with 4/4 four pieces and with 5/5 five pieces. I had always thought that this was because of problems with the idea of 'more', but Klein makes me wonder otherwise. Perhaps the idea of one is more firmly entrenched than I realize and perhaps that has repercussion throughout the grades (I think, to a degree, that Klein alludes to this in his introduction)

[Ron Allen]

Hi Laura:

My comments here may have already been eclipsed by the rest of the thread. Nonetheless, I hear you (or read you), and I myself once read this book for the first time; it was almost worse than "Ulysses", if that means anything.

Klein wrote some of this text as part of an habilitation--a sort of super-Ph.D. dissertation, common in Europe, but not in the US or elsewhere--so that he might get an appointment to the Univ. of Berlin. That was in the early 30s, and political turmoil in Germany, to say the least, precluded Klein's getting a job at Berlin. But the point is that he was writing to other continental thinkers and his style of exposition is probably just exactly what their intellectual diet demanded.

But there are still some parts that cause indigestion to me. And I've read Klein more times than Joyce.

Here's what I think is the Greek vs. "modern" distinction. Since the development of algebra, where we solve equations, like your familiar quadratic or cubic or quartic, we have a way of working with numbers that does not distinguish whether they are integers, rationals, or algebraics. I'm using some modern terminology here. The Greeks and the medieval mathematicians did not grasp the difference between the algebraics and transcendentals, but they at least knew that there were some magnitudes that didn't belong to the first two kinds of quantity.

What was lacking in Greek math (here, I think Klein is somewhat weak in his exposition) was an axiomatic development of the theory of arithmetic that could rival the analytical power of geometry. Uh, you know, that's probably why today (it's out of my space of recent practice) we introduce axiomatics as Euclidean geometry to students in the 2nd year of high school, and an axiomatic treatment of number theory does not come about until maybe the junior year for a *mathematics* major at a university. For some reason, arithmetic--upon which geometry depends--is harder to capture axiomatically than geometry. Well, OK, there's a metaphysical thesis that I'm not qualified to defend, except as an amateur historian.

So at that time, C4 BCE, there was a development of natural number arithmetic on the one hand and of rational and irrational (roots) number theory on the other, with no clear axiomatic system unifying one and the other. It would seem that the theory of whole numbers was as different from the theory of fractions as the theory of plane geometry was from the theory of planetary motions. Yes, to a Platonist, like Plato, there should indeed be a theoretical and a practical logistic, because there was a (rudimentary, separate from whole numbers) theory of fractions and roots and there was also a real-world, material, practice of fractional counting. The discovery of incommensurability only highlighted this divide between realms of intellectual investigation.

But with algebra, and roots of equations, and symbols for quantities divorced from their composite structure, we have a primitive axiomatics for all of quantitative numerical reasoning. Thus, the unity between whole number arithmetic, rational arithmetic, and irrational arithmetic is effected by the study of the solutions of equations--polynomial equations, actually. It's not a clean axiom system like that of Euclid (ha! clean, like Hilbert's!), but at least it's a calculus with definite rules for transitioning from one description of quantities (an equation) to another (an equivalent equation) and a unifying framework for comparing the quantities: positive, negative, zero?, rational, and irrational.

Before this took place with Diophantus, the Greeks had a geometric- and primitive algebra-based theory of rationals and irrationals on the one hand, and a picture-based theory of whole numbers, as a series of units, on the other hand. Two separate practices. Two separate theories. Plato steps in, sees the situation in mathematical practice, then sees the distinction in the realm of Forms, and the fourfold distinction bursts forth. Eh, yeah, you would think that Plato might look first into the realm of Forms and see the distinction and then confirm the disparate practices here on Earth, but, well, that never seems to happen. To this very day.

So, back to Klein's exposition, I accept the principle of the indivisibility of the unit among the Greeks, as Klein lays it out. But, there is still the matter of lag of an axiomatic arithmetic behind that of an axiomatic geometry that bothers me about his account. We used to think of the electric and magnetic properties of matter as distinct and that there were practical and theoretical aspects particular to both phenomena. Now, after Maxwell, we think of them as complementary and unified. Practical and theoretical electric and magnetic--the fourfold distinction. Collapsed by Maxwell in practical and theoretical EM. The "solution theory" of equations--however crude an axiomatic foundation--did the same to Plato's theoretical arithmetic and logistic vs. their practical counterparts.

Now, with equations and their solutions, a solution of x = 2 is just as valid as a solution of y = 3/7 or z = sqrt(3). They are all just quantities. True, some incommensurable with the others, but that's the way these *quantities* happen to behave in the realm of geometry. The myth that 3 = 1,1,1 and that 5 = 1,1,1,1,1 no longer means much, except as a propaedeutic. Three or five could be the solution to an equation just as easily as any proper fraction or any oddball quantity such as sqrt(2). So, this is our modern conception of an arithmos, freed from the structural description as being a pattern, and tied instead to what can be the solution to a (polynomial) equation.

Oof. Too much for one post. Executive summary: (1) quantities as solutions to equations replace the geometric (roots) and the structural (series of units) conception of numbers held by the early Greeks; (2) this undermines Plato's fourfold distinction among later thinkers; (3) we modern folk forget the structural, collection of self-similar units vision of numbers accepted by the early Greeks, where an arithmos was a counting-off, and cling to an abstract quantitative notion instead, manipulating symbols at all times, and forgetting about the "composition" of the whole numbers just as often....

Now, here's the interesting thing: We devolve to the Greek conception of numbers as collection of units with the axiomatization of arithmetic through set theory in the late 19th and early 20th centuries. Klein picks up on this and lays down the links between our very modern, post-Diophantine notion of numbers as series of units, and the primitive conception of natural numbers found among the students at the Academy, the Lyceum, and at Alexendria.

Hope this helps, Laura. Maybe that's an incomprehensible remark at the end, but I can elaborate a little bit. Mayberry's book on set theory and the foundation of math is where I found the hint that Klein was inspired by the Zermelo and von Neumann construction of the natural numbers from set theory axiomatics.

Thanks!
--Ron

[Laura Gibbs]

Your comments here are really fascinating, Ron, and again it just makes me wish there were a book available ABOUT Klein, written by one of the acolytes who have read the book many times. I got more out of your comments here than what I've read in Klein so far after the first couple of chapters.

Klein has not so far made explicit the kind of contrast between Greek math and modern math which you have done here so clearly. I will keep it in mind as I continue reading; perhaps it will help me sort out some of his implications.

Most of all, I am very interested in your notion of two systems here - the geometric (and quasi algebraic) system, and the counting system of arithmoi. Klein has done an incredibly poor job of expounding the world of Greek geometry, at least in the chapters we have read so far, although it certainly comes up here and there (like in the "square" numbers of Theaetetus, etc.). That has been driving me crazy, especially since I have been so fascinated and impressed by what I have been reading about Greek geometry on the side, to compensate for Klein's lack of attention to it. What I have learned from my reading outside Klein is that the Greeks did wonderful computational feats by means of geometry, esp. with regard to irrational numbers, and the effort to approximate them with rational fractions. I've found reading about all of that infinitely more useful to my sense of the Greek mathematical world than reading Klein.

Will Klein ever discuss the two separate practices as you have described them? He seems focused on the ontological minutiae of the "picture-based theory of whole numbers, as a series of units" about which, honestly, there is not that much to say, if you strip off all the Platonic window-dressing... and he has said precious little about the other system, what you call here "a geometric- and primitive algebra-based theory of rationals and irrationals" - in my reading aside from Klein, I am clearly MUCH MORE interested in the geometric and primtive algebra-based theory. And honestly, unless Klein finally says something about that... or at least acknowledges its existence... I'm not sure what the point is for me to continue with this book.

Well, given that I am certainly NOT Klein's intended audience (I figured that out pretty quickly!), I am little better than an intruder here... but your comments have given me some things to think about when I go on to the next chapter. THANK YOU VERY MUCH!!! Klein is very lucky to have such devoted commentators. He needs them badly. :-)

[Ron Allen]

Hi Laura:

I think everyone that reads this book for the first time has some trouble with the style, the subject matter, and the line of argument. I did.

You might read the last paragraph of Chapter 9. There Klein clearly (ha!) states what his plan shall be. The ancient arithmos, which used to be seen as a structure, becomes a plain "number" and then a "general magnitude". He goes on to say that the structure of the mathematical objects is transformed into a "symbol-generating abstraction". So all of algebra becomes a symbol manipulation of abstract quantities, and this is the modern conception of numbers...not as assemblages of units. (That conception does not come back until the development of arithmetic from axiomatic set theory in the late 19th and early 20th centuries (Zermelo and von Neumann).

And with the indefinite divisibility of the unit that that implies, it's essentially Plato's practical logistic, only much more mature and powerful.

What I seem to get out of Klein's exposition is that the arithmoi continue in mathematical philosophy as assemblages of units in later NeoPlatonism. But Plato also has this notion of arithmoi eidetikoi, as opposed to the arithmoi mathematikoi (which are also noetic), and opposed to the arithmoi aisthetoi (based on sensory objects), which he needs to assume for the internal coherence of his Theory of Ideas. These (the eidetikoi) are analogous to the Pythagorean decad, of course, and there is this mystical derivation of the origins of numbers, but they are also *not* collections of units. I could be wrong about that, but I think that's what Plato wants and Klein says. So you would have to see The Five not as 5 = 1,1,1,1,1 but as some kind of integrated penta-whole, if that makes sense. Your dianoia can help you out with this! So they become the conceptual model, an intellectual stepping-stone as it were, to the new modern conception of numbers as magnitudes. It's ironic. At that point, Plato's theoretical arithmetic has actually helped make the transition away from his own conception of theoretical logistic. All that's left is the ideal arithmoi mathematikoi and eidetikoi from the Theory of Ideas, and the algebraic methods that treat all numbers as divisible magnitudes.

I hope I'm getting this right.

Klein talks more about geometry in the second Part of the book. I'm generally of the opinion, though, that his presentation in the first Part is missing something by not elucidating the contrast in levels of axiomatic development between geometry on the one hand and arithmetic on the other. It's probably a research problem. I believe one of the scholarly types (Knorr?) once made a thorough search through Greek math for signs of the notion of equivalence relation--which is crucial to axiomatizing arithmetic--and didn't find it.

Thanks!
--Ron

[Laura Gibbs]

Hi Ron, I am very much in agreement with your assessment of the problem of exposition in this book - and I just don't have it in me to carry on to Part Two when Klein finally presents something like an overall picture. As a result of not being a Plato expert to begin with, I've gotten so little out of the first part, that I doubt I have accumulated enough to Klein's haphazard account so far to be able to appreciate his big picture if/when he finally gets to it.

Somebody really needs to write a good book about the philosophy AND science AND anthropology AND language of math in ancient Greece. It would be such a valuable book and well-received, I am sure - Heath is a maddening book in its own way, and very very very dated, but it comes closer to meeting my needs than Klein does, given my non-philosophical background.

[Ron Allen]

Hello again Laura:

One fairly recent book that I like is Serafina Cuomo's "Ancient Mathematics", London: Routledge, 2001. It does not span quite as many centuries as does Klein, stopping with the C6 CE, i.e. to about the end of the western Roman Empire. One thing that is attractive, I think, is that it presents things in a chapter-pair format with a historical & factual chapter followed by an interpretive chapter that analyzes the previous one. There are three maps at the very beginning with all the places mentioned. Amazing.

It's not as specialized as Klein's book, though. It diverges into technology, sociology of mathematics, and even covers some of the non-mathematical texts, such as Polybius and glimpses of mathematical practice from Aristophanes. So it has some of your "AND and AND" requirements, and that might be your ticket.

Thanks!
--Ron

[Ian Carmichael]

Beaut! I'll see if I can extract it from my local library.

[Laura Gibbs]

Agreed! Thanks so much, Ron - it sounds like that is the book those among us who are less Platonic should have started with! I'll see if I can scrounge a copy to read sometime soon. Thanks!!! :-)

[Barry Robbins]

Hi, All,

I'm not sure that my last post got through, so I am re-posting it here. Forgive the redundancy if it did.

Chapter 7B

In this chapter Klein attempts to clarify the relationship (as understood by Plato) between “logistike” and “dianoia.”

The single word that jumps out at me in this chapter is the word “incidentally” as it is used on page 78: “thus INCIDENTALLY [my caps] laying a foundation for making calculations.” I’ll try to explain what I think the significance of this word is. But first, let me try to say what I think Klein is trying to make clear about the relationship of “logistike” and “dianoia.” But before doing so, please note that it is “logistike” and NOT “arithmetike” that is related to “dianoia.” It is not that activity which attempts to clarify what accounts for each number being a “definite” number that is related to “dianoia,” but rather that which deals the relations of the “definite” things that each number is a number of.

In the “divided line” section of Book 6 of the Republic, Socrates distinguishes two types of “noeta” (i.e., things that we have access to with our cognitive capacity as opposed to those things that we have access to through perception, i.e., “aestheta”).

In chapter 7B, Klein focuses on clarifying the first type of “noeta,” and what characterizes the way in which we have access to them.

We are completely familiar with these “noeta,” although we rarely think about this familiarity or are even aware of these “noeta” as “noeta.” These “noeta” are, first of all, what our words really intend whenever we speak, and, further, they enable us to identify everything that we perceive in the visible world. For example, I am now sitting near a swimming pool. I see in front of me eight “lounge chairs.” Though you can’t see them, you “know” what I’m talking about, and if you were here you would easily identify the eight lounge chairs.

The question that Klein raises and attempts to answer is this: “What role does our dianoia play in this process?”

In general, it works like this. Right now I am looking at one of these lounge chairs. I see “one” lounge chair; however, as I look at this same lounge chair I also see “many” other things. I see “green,” I see “flat,” I see “straps,” and many other things. In short, as I look at the very same thing, I see the same thing as “one” and “many.” Now, in general, this situation does not cause us any perplexity whatsoever (this, by the way, is why the examples of large and small given on pages 74-76 are so important, since if you look at Socrates as he stands in the middle of Simmias, who is shorter, and Phaedo, who is taller, the confusion for perception is apparent. Try it and I think that you will agree that what your perception reports to you is that the very same thing is BOTH small and large.). Why not? Because immediately and without, so to speak, any effort, our dianoia separate the various “ones” that I just mentioned. It recognizes that the “one” lounge chair I see is one thing, while the “one” color (green) that I see is another “one” thing. And, as Klein will point out, this is always the way in which our dianoia functions; it always identifies “one” among “many.” Practically speaking, it is impossible see one lounge chair all by itself. What is so interesting, I think, is that you “know” what I mean by “lounge chair;” you know “lounge chair,” but what you know can only be seen when it is accompanied by other individual things that you also “know” in this same way. The lounge chair that you know can only be seen when you look at a lounge chair that also is of a certain color, size, shape, etc.

Klein describes this situation as follows: “the ‘dianoia’ is never directed at a single being as such; rather its view always so encompasses a series of beings that the members of the series are carefully distinguished from, and thus simultaneously related to, each other.” (page 76). What this means, of course, is that dianoia is not able to grasp “one,” except as one among many; Accordingly, it also can not grasp the “oneness” of any number, is never able to grasp what makes it “one” “definite” number, but only that it is a number of “definite” things.

Klein’s insight is that the “dianoia” is always engaged in an activity that “is based on ‘account giving and counting’ (‘logizeshai te kai arithmein’), namely on the ability to recognize ‘many’ as ‘many’…. Klein concludes that the activity of the dianoia is “nothing but — counting” (page 77). Accordingly, since our dianoia is always dealing with “noetic structures that are of oppositional character” (page78), it is able to discover as the true “foundation” of the domain that it deals with the “‘pure’ relations of numbers. i.e., the ratios (‘logoi’) and proportions (‘analogiai’) of the ‘pure’ numbers, because every possible comparison is ultimately founded on these.”

And so theoretical logistike provides the foundation for dianoia, and, only INCIDENTALLY, lays a foundation for the possibility of making calculations. The importance of this is that there is no “maladjustment” (cf., bottom of page 42) between the “material” of theoretical logistic and the “material” of what our dianoia is constantly dealing with – namely the first kind of “noeta,” which are always grasped through the “mirror of our senses” as UNITS, whereas, as we have seen, there is a “remarkable maladjustment” between the material of practical calculations (which is capable of fractionalization) and that of “pure” numbers “whose noetic character is expressed precisely in the indivisibility of the units.” (page 43)

What then provides the foundation for each number being a “definite” number? And how is this related to the “higher” kind of thinking that is directed at the “higher” noeta?

On to chapter 7C!

[Laura Gibbs]

Since Klein decided to preface his discussion of Plato with a discussion of Pythagorean material, I'm assuming there will be strong connections here. The main message I got from the Pythagorean material was the cosmological dimension of numbers (the being - ousia - of all things is number, p. 64) and also the importance (not yet explained) of the root (puthmen) of the various kinds of numbers. Then, as Klein starts the section on Plato, he calls this section "logistike and dianoia." So, in addition to the cosmology and "roots" I'm bringing over from the first part of the chapter, I've now got logistike (old term) and dianoia (new term) on the horizon.

I will confess to being increasingly confused, though, about the term "pure" numbers (scare quotes are Klein's) which has been appearing with greater and greater frequency. Are these the asensory, purely noetic numbers? How do the "pure" numbers relate to the eide and the puthmenes? Is the word "pure" Klein's contribution, or does it correspond to something in Greek? I would be very glad if I got an answer to that question as we proceeded through this section, too.

As Klein begins this section, he reassuringly tells us (p. 69) that "Plato's philosophy was decisively influenced by Pythagorean science" (I'm not sure about the contrast here between philosophy and science, but I'm assuming it is not terribly important). But Klein then pulls the rug out from under my feet again by saying (p. 70) "Plato assigns to mathematical objects a totally different place and a totally different mode of being than they can possibly have in Pythagorean science." Totally different...? I'm guessing this is not mere hyperbole for hyperbole's sake, but it makes me optimistic that I will gain a very new and crystal clear understanding of mathematical objects (are those arithmoi? again, the terminology makes me uncertain) by the end of this section.

Apparently this difference is that Plato makes the numbers separable from the objects of sense - now, given that the Pythagoreans were very interested in the kinds (eide) of numbers, I guess the kinds are NOT noetic, and are somehow sensible. But compare what Klein says on p. 72: the pure numbers ARE the kinds, the eide. Uh, that doesn't seem to fit: since the Pythagoreans DID discuss the eide of numbers with considerable interest, what is the Platonic shift then? Are the asensory objects going to be the noetic arithmoi? That is what I need to understand apparently as I read through the chapter.

As the story unfolds on p. 71, I learn the arithmoi and the act of counting is "grounded" in these nonsensible units (which are apparently not exactly eide after all)... but then the shift that bothers me so much emerges: there is a hierarchy of value here - because it's pretty clear that the sequence and precedence which Plato insists on is not one of chronological sequence (since the pure numbers are timeless, and Plato seems to have no sense of historical cultural development generally speaking), but instead a sequence of value. "The being of that which is the foundation takes precedence over that which is so founded..." (p. 71) - and why is that? It is not just prior, but higher: "There is a higher kind of reflection in which this supposing is raised to the rank of a conscious procedure" (p. 73).

Yet as Klein then explains in broader terms, not just mathematical, how the dianoia operates, it does so with the tools made available to it by a CULTURE - I don't see anything timeless about that at all. As Klein looks at his finger and decides to call it small or thin or hard (p. 75) he is appealing to a set of cultural values, contextualized in a language, in the usage of that language, in its conventions. Admittedly, this is what makes mathematics so intriguing: more than perhaps any other human symbolic system, mathematics can lay claims to a kind of transcultural universalism (although not everyone will agree). But instead of asking that question - i.e., to what degree is mathematics (UNLIKE other symbolic systems) a transcultural phenomenon, Klein goes down a different path, as if all culture can be equally mathematicized. Yet how we regard our fingers, our hands, our bodies a culturally diverse and symbolically complex phenomenon... although that is not the kind of thing that interests Klein at all.

At least insofar as I have a philosophical method, I guess you can call me a structuralist - because the only system I have studied in any kind of rigorous way is language, and structural linguistics is the mode of language analysis that makes the most sense to me. Structural linguistics is based on comparisons and opposition, just as Klein adduces here - but it seems to me that this whole discussion misses the insight I've gained from structural linguistics: structural opposition (big-small) both creates meaning, and at the same time hides it. The structured oppositions allow meaning to rise up from the infinitely diverse range of phenomena so that we can "make sense" of it - but those structural oppositions are themselves arbitrary. In English, the opposition between "b" and "p" is phonemic; because it is part of the structure of English phonology, we HEAR that difference. Yet in other languages, such as Arabic, where there is no structural opposition between "b" and "p", they do NOT hear the letter "p" - or, rather, the sound goes into their ears, but it does not yield the sensory perception of "p" - because the b/p opposition does not exist. There are, however, phonological features of Arabic which are imperceptible to us as English speakers, and that is one of the things which makes it so very hard for us to learn to speak Arabic well.

Klein (following Plato?) seems to act as if these oppositions that yield meaning are timeless and universal. That simply makes no sense to me at all.... and since there is no proof offered for it, except an appeal to personal experience, I don't have anything to help lead me over that bridge: my personal experience is of diversity, not universality. At moments, it seems like we are talking about a real structure in the way that I understand it (p. 78, for example: dianoia "cannot recognize one element of language, i.e. a single sound rendered by a letter, without the remaining sounds") - but what is the good of acknowledging the structure if you do not at the same time acknowledge the arbitrariness of that structure and the arbitrariness of the sign system?

After this long discussion of things OTHER than numbers (the fingers of the hand, the beautiful/ugly, the good/foul), there is a small bit about numbers on p. 79 - but then it appears that the real discussion of numbers is now going to begin in section C, which has the subtitle "arithmos eidetikos" - eidetikos sounds promising, and will perhaps help explain what seems to be my confusion about the eide of numbers, and whether the "pure" numbers are the eide (as Klein seems to say on p. 72), or whether the "pure" numbers are something else. Perhaps they are these arithmoi eidetikoi.

So, per usual, my experience of postponement continues... hopefully I'll be able to read the rest of Chapter 7 later today. :-)

[Barry Robbins]

Just a quick response, Laura, since I have to catch a flight back to the states early tomorrow morning and I haven't packed yet.

My text on page 72 reads "...but these reflections...are aimed not at these particular objects but at the 'pure' numbers, OR [my caps] their eide...." Is this the passage you meant? Sorry if I missed it; I only had time to take a quick look at the page.

As for the "pure" numbers, they are simply those numbers that are numbers of "pure" units, those numbers that you have available right now and which you "know" and which, further, allow you to recognize, for example, five apples as being five. Right this moment you "know" that ten is twice five. Right now you "know" ten and you "know" five. What is so interesting is that, as a rule, we don't give much thought to such numbers; in particular we don't ask the question "What are these numbers that I know numbers OF?"

So, I guess, "yes" these are what you referred to as "asensory, purely noetic numbers." Why their units must be "pure" was discussed by Klein in chapter 6. I think that I also referred to it in an earlier post. These "pure" units are what those numbers that you"know" are numbers of.

Now, in the same way that people attempting to classify all articulate sounds discovered that every articulate sound is either a vowel or consonant (I'm sure there are exceptions, and since this is your field of expertise, Laura, please bear with me on it, but you get the idea), along with many sub-classifications so as to comprehend all articulate sounds, those who are interested in studying all numbers discovered "eide" that allow every number to be classified as "one" definite number. Odd and Even are "eide" that seem to do this perfectly for all numbers. The "eide" are NOT the units that make up a number. Remember that Klein's chief point about any number is that it is a DEFINITE number of DEFINITE things. The units are the DEFINITE things, the "eide" (even and odd) are what make numbers DEFINITE. This is true regardless of whether we are talking about six apples or six "pure" units.

I hope this helps a little, Laura. I am in absolute awe of your energy and determination; however, I think that it might be wise to go a little more slowly. Chapter 7 is perhaps the most difficult passage I have ever read. I would urge you to make sure you have a fairly firm grasp of chapter 6 before tackling it.

As soon as I get settled back in the states I will try post a summary of chapter 6 that I hope will be of help. I know that I want to make sure I have a pretty fair grasp of it before I move on to chapter 7.

[Laura Gibbs]

Hi Barry, I hope you had a safe trip back! I always used to get jet lag flying to Europe, but I usually did not do too badly coming back westward, with the sun.

As for Klein, here are the factors in my mind right now:

1) The book is not very well written - that's just my opinion, but it's an opinion that becomes more and more strong as I proceed through the book (the two-page "note" or whatever that is at the beginning of Chapter 7 made me laugh out loud - I still haven't got a clue what his purpose was in proceeding in that way - was it just an afterthought? something optional? who knows...). So, in addition to being about a difficult topic, the style itself is completely off-putting to me, for all that the scholarship may be impeccable (that's something I cannot judge). When I've got several shelves of books I would like to read, I don't see what would justify spending more time on Klein than I am spending right now; it's just not the right book for me.

2) It's turned on to be a philosophy book, rather than a book about mathematics - so I'm actually learning not much of value to me from this book at all, and instead what benefit I am getting is from reading a couple of other books on the side that actually are about the history of mathematics. In particular, I'm really enjoying Burton's History of Mathematics, which is a very hands-on book about how people DID math - I've loved learning about the ways the Greeks used geometry for computational purposes; I wish Plato had not been so scornful of applied mathematics, because I think it is beautiful stuff. I'm also browsing through Heath when I have questions not answered by Klein (and that is pretty frequently, since Klein has zero interest in giving us any historical context - which is something personally I cannot do without).

3) My motivation to keep going is simply this: according to both Heath and Burton, Plato is not important for the history of math, except insofar as he was a "math booster," someone whose enthusiasm about math prompted OTHERS to do mathematical research. Now, it sounds like Klein thinks that Plato is somehow really important to the later history of Greek mathematics (although I could be wrong; Klein's apparent lack of concern with signposting as he writes is really maddening). But if in fact Klein is onto something about Plato's legacy for MATH which both Heath and Burton have missed, I'd really like to know about that. That historical question is open as far as I am concerned, because Burton simply repeats pretty much what Heath says when it comes to Plato, and Heath's book is itself very old and could be off the mark with regard to Plato's significance.

So far, all Klein has shown me is that math can be helpful in understanding Plato's philosophy, which is all well and good - but not something of any interest to me personally; on the shelves holding the books I would like to read, there's not a single philosophy book to be found. If, as Klein gets into Diophantus and Vieta, it turns out that Plato's influence was significant, well, that will be a very useful corrective to my reading of Burton and Heath, who do not reserve much of a place for Plato in the history of math. So I'm not abandoning Klein - but it's definitely what we could call a "pensive" activity, in the Roman sense of the weight of wool to be spun. :-)