1: Introduction

[Laura Gibbs]

Space for thoughts and responses to Chapter 1: Introduction.

[Laura Gibbs]

Well, here goes! Since I am not a physicist or a mathematician or a philosopher, I'm watching what Klein does here from the sidelines, and am very dependent on him to set the pace - I cannot leap ahead here, based on my own knowledge of physics or math or philosophy (all very limited). However, I did find VERY sympathetic the historical principle that he is enunciating here in the Introduction:

Hence our object is not to evaluate the revival of Greek mathematics in the sixteenth century in terms of its results retrospectively, but to rehearse the actual course of its genesis prospectively.

The attempt to recover the history of ideas as it unfolds, looking forward, has always been for me a much more rewarding perspective than beginning with the modern age as our final destination and looking backward for our knowledge of the past. As someone who works in Classics, understanding the past through its future outcomes and survivals is a real straitjacket, very confining.

I could also really connect with the geography of the flow of mathematical ideas because it matches very much the flow of fables which is my own object of study: there are fables from ancient Greece, and also strong fable traditions in ancient India (the striking similarities between the ancient fables of Aesop and the ancient jataka tales attributed to the Buddha in India have lead to all kinds of speculation about just who was borrowing from whom)... then the Arabic tradition draws on both the Greek and Indian materials, and those Arabic materials reach Europe again later, creating an amazing confluence of storytelling in the 13th through the 16th centuries, that same period in which Klein describes Europe as "absorbing the Arabic science of algebra in the form of a theory of equations, probably itself derived from Indian as well as from Greek sources"... much the same can be said for the Arabic and Syrian story collections that came to Europe at more or less the same time! (As, just to take one example, in the fables attributed to "Syntipas" - a Greek form of the name "Sinbad" - you can see those fables here, if you are curious.)

Also, while I will confess to finding philosophical questions to be really opaque and hard for me to understand, I am really excited about the way this project will give me a chance to explore Greek vocabulary in a new way, and I hope that my lack of philosophical insight into the text can be justified by my philological contributions! For example, what most attracted my notice here in the Introduction was the importance to be played by the words εἴδη (eide), "kinds" and ὕλη (hule) "material."

In English, we get all kinds of words that derive ultimately from the Greek philosophical vocabulary of εἴδη such as, of course, idea. Very closely connected is - surprise, surprise - the word idol. The Greek term εἴδωλον (eidolon) is one of real interest to me, because - far more than εἶδος (singular of εἴδη), the word εἴδωλον - "idol" - is not a word which was as narrowly philosophical and instead included all kinds of fascinating objects, such as GHOSTS (!) and other kinds of fascinating insubstantial images (like reflections in water, etc. - here's the Greek-English dictionary entry).

And with the word ὕλη (hule) ... wow! ... it is much harder here for me to see that word and think of it philosophically! I'll be curious what the other Greek readers say about this, but for me ὕλη (hule) doesn't even feel like a philosophical word at all - it means woods, woodland, etc. You can read a detailed definition in the standard scholarly Greek-English dictionary online here. Aesop's fables often take place in the ὕλη - and it's going to be hard to make myself get used to this word as being a philosophical abstraction, when to me it is a very concrete wood. I look out my window at this very moment and what do I see: ὕλη ...! Woods! So while I know that ὕλη does become part of Greek philosophical vocabulary, it's definitely something that still feels kind of odd to me!

And how did we end up with the word "material" in English, the standard word used to translate Greek ὕλη...? Well, it's because in Latin the original meaning of materia was also wood, in the sense of timber, woodstuff... fascinating, eh? Here's materia in the Latin-English dictionary.... and yes, that's no mistake: it IS related to the word for mother, mater.

So, while I will try hard to focus on the philosophical matters at hand, I will also try to do something that Klein himself is NOT doing in this book: he says he does not way to read the end back into the beginning of things... well, if you assume the highly abstract philosophical meaning of a word like ὕλη, you are letting the classical and post-classical meanings of the word determine your understanding, at the expense of the archaic, pre-classical Greek world, which did not have Platonic philosophy, did not have Aristotle, etc., and where ὕλη was the stuff you brought out of the woods... Which is not to say it does not later take on a philosophical meaning in classical and later culture - but my take on such things is that it never leaves its old meaning fully behind.

:-)

[Barry Robbins]

Laura,

Thanks for getting us started. I’ll try to keep the logos rolling with an anecdote.

When I first read Klein’s book, I happened to be “between schools,” hanging out in Cambridge, Mass. One day while walking around in the neighborhood of MIT, I happened to strike up a conversation with a physics graduate student. At some point I asked him if he had studied Aristotle’s Physics and what he thought of it. His response was that Aristotle’s work had nothing to do with physics but was a work of philosophy.

He was, of course, quite right on several scores. Aristotle’s Physics has nothing to do with what Klein refers to as modern “mathematical physics.” And yet in the Physics Aristotle certainly makes an attempt to explain what appears to be most characteristic of the natural world that we are all familiar with – namely, “change.” (By the way, it’s in the Physics that Aristotle develops his technical use of the word hyle [How were you able to print the Greek, Laura].) So it isn’t that Aristotle is not trying to understand the natural world; however, there is nothing mathematical about his presentation, and this, of course, was at the heart of the physics graduate student’s objection. Also embedded in the physicist’s objection was an opinion that philosophy has nothing to do with any attempt to understand the physical universe.

The first point that Klein stresses in his introduction is “that it is impossible … to grasp the meaning of what we nowadays call physics independently of its mathematical form.” (This, of course, was at the heart of the physicist’s objection to taking Aristotle’s Physics seriously.)

Klein then suggests what I find most interesting and would like to understand better, namely “That the intimate connection between the formal mathematical language with the content of mathematical physics stems from a special kind of conceptualization….” (p. 4)

The question I would raise is – what is this “special kind of conceptualization”? Since it characterizes what Klein refers to as “the modern mode of thinking” (cf. Author’s note), it certainly behooves us to understand it better, especially if this modern mode is defective.

Now, while Klein does not clarify this special kind of conceptualization in the introduction, he does, I think, say a few things that help. First of all he says that it involves a “conceptional transformation … which constitutes the indispensable condition of modern mathematical symbolism.” (p. 5) So, the degree to which we are comfortable with “symbolic” mathematics is, perhaps, an indication of how far removed we are from the ancient mode of thinking. And while Klein does not say much more about the modern mode in the introduction, he does say quite a bit about the ancient mode of thinking. What then is this ancient mode of thinking? This is what Klein goes on to describe on pages 8 & 9.

So it probably would be a good idea for us to try to state clearly just what is involved in this “ancient” way of forming concepts as it is described by Klein on pages 7 & 8.

Also, I think that in ending Part I with Aristotle, Klein is suggesting that the development that occurs in Part II depends upon Aristotle’s break from Plato’s conception of numbers and in particular his different conception of the unit. We can keep this in mind when we get to section 8 of Part I.

[Laura Gibbs]

It is so going to be so nice to have people familiar with Plato and Aristotle to guide us here, Barry! I have a vague kind of sense of what words take on special philosophical meaning in Greek, but I have no specific sense at all of just which works are crucial for understanding the evolution of one term or another.

For including the Greek here, I used a little online Greek word processor that one of my students built for an online Greek course I was teaching: TypeGreek.com. You type in Beta Code (he has a note there about how the Beta Code transliteration system works), and out comes the Greek. The one fluke I've noticed is that at least using Firefox on a Mac, you cannot use Control-C to copy and paste, and instead have to right-mouse click and use the edit menu Copy option, or use the edit menu in the browser bar itself.

I too am very interested in the question of symbolism, and in particular a question that I am guessing Klein is not going to help us to answer, but which I will try to research some myself and share with the group - specifically, it is about the symbolic written characters used to represent numbers. I know that, following in the Semitic alphabetic tradition, the Greeks used letters to represent numbers - and I'll confess that I've never gotten very good at reading those kinds of numbers (it's kind of embarrassing, actually, when sometimes modern critical editions of Greek text will use the old Greek numbering system; I get so confused!). The Romans, of course, used a "tally" system, something like the way we make four little stick lines and then use a slanting diagonal line to mark off each set of 5 - the Roman works in the same way: IIII and then V which represents the set of 5 little lines, and X which is two sets of five, one on top of the other. Latin students often assume that these symbols I, V and X are letters, which in fact they are not letters as all, but non-alphabetic symbolic characters. I have a fun game with Roman numerals here, and how the Romans did multiplication of 5-10 on their fingers: Roman Finger Multiplication.

So, as we plunge into the philosophical questions of "conceptualization" as we proceed, I'll keep on asking myself some much more basic questions about the symbolic representation of Greek numbers in written form. Even if we don't learn about that directly from Klein (I cannot tell if we will or not), I will use this group as an excuse to learn something about that, since it is something I an enormously curious about myself!

[Ed Wall]

There is a sense in which Aristotle thinks that 'physics' has nothing to do with mathematics.

[Diane Bouvier]

Looking at this as a student of mathematics, all I see is a bunch of words I don't understand. It is going to be very dangerous for me to put my mathematical concepts and symbols on top of these words and assume that they mean what I already know. Since I suspect that the whole point of the book is that we have thrown away the type of mathematical thinking that the ancients did and replaced it with modern symbolic thinking, here I go in a circle.

So I would find it very helpful to get a good grip on the words, especially eidos and hyle, arithmos and logos.

Also wish we would choose a style for spelling, since I am going to get confused if people are using different spellings.

Is it possible to put a wiki on here?

[Ian Carmichael]

Hi Diane,
Good points - in the absence of the book for me, how do you take Klein's use of these key words?

[Laura Gibbs]

Hi Diane, I'm guessing the whole book is an exploration of the meaning of arithmos and logos, and it would take many more books besides to explore the meanings of these words, and the same would be true of eidos and hyle. So, I'm not sure if they are words we can ever really feel like we can get a good grip on, in the sense of perfectly comprehending them in their entirety (especially as English speakers) - although I'll defer to the philosophers on that one; maybe they are more confident about that than I am... Logos is the only one of these words which comes up often in my work, and just on that basis it seems to me that this is a word which the speakers of ancient Greek used for many different purposes which can never be embraced by a single word, or even just a few words, in English.

Anyway, while I hope I'm not putting my ideas in a wrong way on the ideas Klein is discussing, I'm not going to be shy about describing the ways in which those ideas do resonate with me - learning proceeds by making connections, at least it does for me, so even though this is a subject matter that is indeed very alien to me, it's been fun trying to make connections where I can. I don't think you should be shy at all about sharing your ideas here: my goal is not to have a "Klein mind" exactly, but just to see where I can make some connections between his interests in the ancient world and my own. :-)

I've been trying to use the same spelling conventions used in the book; the one mistake I might make is through the use of English alphabet "u" instead of "y" for which I apologize in advance; I'll try to be very careful and use the "y" which is the convention that is being used in the Klein book (I unfortunately tend to use the "u" option unless I'm careful and really think about it!). It's not that one is really either right or wrong; you see both styles used for transliterating the Greek letter in question, which does indeed look like a "u" in Greek, upsilon.

Wiki: If you want to explore ideas in a wiki, I'll be glad to set one up for us to use although wikis are usually not very good for discussion, which is what I am very much hoping for here, just having people share ideas based on wherever they are coming from. Since it is possible to create links from one post to another here at Fireside (each post has its own unique address you can link to), and since we can create as many discussion areas as we want, I think trying to work in this space would be a good idea. Would it help if I created discussion areas for the words arithmos, logos, hyle and eidos? I'm guessing that could prove very useful and might give some of the space you are looking for - and of course you can create any discussion areas here that you want; I've got some tips on how to create additional discussion areas here.

The next time I post something here where I discuss a specific Greek word or term, I'll create a discussion area to go with it, which we can use something like a wiki page entry. Since the same terms are going to be recurring throughout the book, I'm sure that will be quite useful! Meanwhile, if you encounter a term that you'd like to explore, please create a discussion area for it, and I promise I will join in. I find all aspects of ancient Greek vocabulary EXTREMELY interesting to explore... :-)

[Barry Robbins]

Hi, Diane,

"eidos," most simply put, is the "look" of something. For example, as I type this letter, I am sitting in my living room. I can see a coffee table in front of me and on it I can also see a number of objects that I have no trouble identifying - two books, a coffee cup, a vase, etc. What allows me to identify each of these objects as I have done is its eidos. It's similar to when someone asks us what something seen at a distance is and we respond by saying, "That looks like a dog [for example]." The eidos is "the look" that something has and which enables us to identify it as such and such a thing.

But each of the objects on my coffee table also was made out of some material or "hyle."

But let's take eidos a little further. Let's say I invite you over to my house and I tell you that I have a very interesting vase on the coffee table in my living room and then ask you to take a look at it and to tell me what you think of it. Let's say that you then walk into my living room and take a look at the coffee table and see the vase on it and then tell me what you think.

Of course, you had no trouble identifying the vase. It didn't matter what its individual color or size or shape was. In fact, it could have been any one of a thousand different vases and you still would have easily identified it. Now, what allowed you to do this was something that you recognized and had in your possession the moment I spoke the word "vase." And what you recognized (without having to reflect on it) had no individual shape, size or color, all of which are aspects that appear only in particular vases. What you somehow recognized when I used the word "vase" was the eidos of vase. And, curiously, this eidos, which allows you to identify all visible vases as vases, is itself not visible. (It couldn't be, for then it would limit vases only to those that had its particular visible characteristics.) Now the ontological status of this eidos (i.e., in what way it actually exists) will become a source of a huge disagreement between Plato and Aristotle, and will be discussed by Klein later in the book.

"Arithmos" is the Greek word for "number." It is cognate with the Greek verb "arithmein," which means "to count." Klein will have a great deal to say about the concept of arithmos in chapter 6; however, for now, given its relationship to counting, you can think of it as "counting number," although keep in mind that there is no other word for "number" in Greek. Klein will repeatedly stress (and we will need to try to understand this stress better) that an arithmos always means a "definite" number of "definite" objects. As such, those "numbers" that we are nowadays so comfortable talking about (e.g., rational, irrational) are not arithmoi at all.

I hope this helps.

Barry

[Laura Gibbs]

Hi Barry! This explanation of eidos is very helpful because it shows how the word has a common everyday usage, a common everyday meaning, the "look" of something... but at the same time, the philosophers, when they get hold of the word, start to do things with it that do not happen in ordinary conversation, as when you ask the question about the ontological status eidos, wondering "in what way it actually exists." Most people can go around in the world using a word like eidos and NOT worrying about the philosophical problem of "in what way it actually exists."

What's tricky for us, as we read Klein's book, is that while the word eidos literally means "look" (the English noun look, not the verb), that English word does not suggest the philosophical discourse in which the word eidos becomes embroiled, so the word eidos is going to be translated in various ways - the list in the translator's note provides these different possible English words corresponding to Greek eidos: kind, form, species, idea, figure.

In some ways, having these different English options, in addition to "looks," is helpful, because a number is not like a vase on a table. You cannot look at a number... but of course numbers seem very real to us (at least, some numbers do!) - so even though it does not have a "look" we can see with our eyes, numbers do have "eide", they have "looks" - although instead of the word "looks" being used with respect to numbers, we are going to see the word "kinds" used instead. So, for example, in the passage from Olympiodorus in chapter 2 which so intrigued me, he says that numbers have TWO "eide" - they have two "kinds" - which are even and odd (p. 13).

I hope I am getting that right! Meanwhile, I wait to be enlightened about the "hyle" of numbers, something other than their "eide" - that is the main question I will be asking myself as I read through chapter 3. I really want to understand what the "hyle" of numbers is all about and what those "hylic monads" will mean! :-)

[Barry Robbins]

Hi, Laura,

Actually, you can look at numbers. This is how we become familiar with them. To take my earlier example, if you were to ask me right now how many books are on my coffee table I would tell you that I can "see" three books on my coffee table. (I added one since my earlier post). And in a manner analogous to the earlier situation when I said the word "vase," you immediately understand what I mean by "three." Now, Klein's point will be that, just as when I said "vase," you immediately understood what I was talking about, likewise, by the word "three" you also understand what I am talking about and have "three" in your possession. Nor would it matter if I were talking about three cups, three chickens, or three sunsets. And in a manner analogous to what I earlier described re the vase, the "three" that you understand must be of such a character as to allow for recognizing three regardless of the character of any particular three objects . So the question that Klein is inviting us to ask is this - the three that you have in your possession, what is this three three of? This is a very important question, since by asking it, we are no longer interested in any "practical" application, but are interested simply in understanding. Klein will raise this question in chapter 6 of Part I. But an even more important question that Klein will raise is what makes the "three" that you have in your possession "one" number.

As for your question about "hylic monads," we can discuss that as we go through chapter 2. By the way, the term "hyle" used by Olympiodorus in attempting to explain Plato's distinction between arithmetic and logistic is one that is never used by Plato. Klein will argue that all of the Neoplatonic scholia cited fail to adequately account for Plato's distinction between arithmetic and logistic.

[Laura Gibbs]

Ah, but how do I "see" evens and odds...? And those are the two kinds (eide) of numbers, or so we have been told (and that not just by Olympiodorus, but by Socrates). Do I mentally count it off in my head "odd-even-odd-even" as I count the objects...? is that how I "see" the even kind of number and distinguish it from the odd kind of number? Since the idea that there are two "kinds" of numbers - even and odd - seems crucial to the definition of number, I'm hoping to learn more about what it means to see that, to see an "even" number or to see an "odd" number. I just made a note about that in response to something Diane had written here; as typical of a modern person, I am hypertextualized by signs, in this case by the signs of the numbers... and our system of signs for the numbers sure does make it easy to discern between even and odd, more so than in the Greek sign system for the numbers. :-)

I definitely got the feeling that poor Olympiodorus was indeed being set up to be taken down... I hope I will be able to free myself from the same misunderstanding that trapped the neo-Platonists!

[Ed Wall]

Listen, a bit like one does with poetry, can be helpful in reading an analysis like this.