6: Concept of arithmos

[Laura Gibbs]

Space for thoughts and responses to Chapter 6: Concept of arithmos

[Ed Wall]

Nicely put and I agree Klein is very much grounded in ordinary phenomena and needs to be read that way.

[Barry Robbins]

I just read your post, Laura. I didn't get a chance to read it sooner since I was in transit to St. Petersburg, Russia, where I recently arrived and will vacation for the next week. And while I'm pretty exhausted, I do want to try to say a few things in response to your post. First of all, your use of the word "abstraction" is very interesting, especially since it is precisely the word (or at least the Latin form of it) that Aristotle uses to explain how we come to be familiar with the "pure" units as well as all other "noeta." He will argue that we "draw" such noeta "from" the things that are perceived by us, and I think it's fair to say that it is this view that ultimately has led to your immediate use of the term "abstract" and also, I suspect, colors your perception of Plato, who never uses the term to explain how we come to know the "noeta." (Klein will discuss this in great depth in chapter 7).

The phenomenon that Klein is asking us to start with is this: right now you have available to you many numbers, so that if I tell you that I just bought five doughnuts, you know what I mean by five and you would have no difficulty looking at the doughnuts I just bought and determining if, indeed, I do have five. Now, since every number consists of a number of "definite things," ask yourself the question, "How would I characterize those "definite" things in the five that I know?" What is the five that you have available to you, the five that you know, five of? Klein stresses the significance of asking this type of question (cf., 49-50).

In much the same way (to go back to an earlier post) that the vase that you have in mind when I mention to you that I have a new "vase" allows you to identify the new vase on my coffee table even though the vase you have in mind does not itself having any perceptible characteristics (which would limit it to only those vases with those specific characteristics - e.g., green color) so too, if I were to tell you that I bought five new vases, the five that you "know" must consist of units that do not limit the sort of objects that can be five; otherwise, some numbers of five objects wouldn't seem to you to be five.

As for the temporal relationship between the "noeta" and the "aestheta," that's a different question.

Sorry, if this is not clear, or simply mistaken, but I'm not all that alert right now. I'll try to write more after I've gotten some rest.

[Laura Gibbs]

WOW, Barry, St. Petersburg - how fantastic!!! Do you travel often to Russia, or is this your first time? I used to live in Poland (back in the 1980s), and I also studied Russian in school... but I never had a chance to travel there. How absolutely wonderful! I bet June is a great time to be there, too. ENJOY!!!!

About numbers: figuring out the right forms and case endings for numerical expressions is really tricky in the Slavic languages, and I remember when we were learning that in school, my teacher told us that Krushchev was so bad at being able to read numbers correctly out loud that in his speeches (he was notoriously uneducated) that his handlers had to write the words out for him, instead of just the numbers, ha ha. Now, I don't know if this is really true or not, or if it is a legend that our teacher shared with us to make us feel better about how darn confusing it is to know what form of the nouns to use with the numbers in Russian, but it is something that made an impression on me all those years ago.

It sounds like I will get to learn more about the "noeta" in Chapter 7 which I will start reading on Sunday - great! The reason I wrote up these notes about the history of counting words in Indo-European generally, and also in ancient Greek, is because it seems to me that we discount how many of our assumptions about counting do not acknowledge the way that counting is something learned... I was really struck by how the process of counting really is something that evolved in time, re-purposing words that were not originally number names at all, but instead words used for something else that then got adapted for counting purposes as time went by. Counting is now so basic to us that words like "one-two-three" seem like they are purely numbers - but the history of the words shows otherwise. So, I'm looking forward to seeing if I can find a way to reconcile my anthropological-historical take on those numbers to the philosophy that unfolds in Chapter 7! :-)

[Laura Gibbs]

Apologies in advance for presenting what might seem like asides on the chapter's main idea about arithmos - but here's another something that kept coming to mind as I read this chapter - something I think is very interesting, and actually helps to elucidate some of Klein's argument here, even though he does not get into linguistic issues.

And here it is: in Greek (as in the Indo-European languages generally), numbers are adjectives. They are not nouns. Now, admittedly, adjectives and nouns do have a lot in common, and it is because they have a lot in common that we do not immediately realize that numbers are adjectives rather than nouns, but there are a few little things that give us clues about that.

Numbers cover all three genders, and the choice of gender is determined by the noun with which the number agrees. This is seen most clearly in the smaller numbers, where the gender has different morphological forms: unus, unum and una in Latin and, even more interestingly in Greek, εἷς, ἕν, μία - "heis, hen, mia" (if anyone knows something about the etymological history of these words in Greek, I would love to know more about that - if I remember correctly, the word-initial aspiration in the Greek would mean they are related to the same root we seem in "semel" in Latin, but I do not remember the details at all).

Anyway, I think this notion of numbers as adjectives fits in VERY nicely with Klein's insistence that we should think of the arithmos only in relation to the objects of counting, linked as adjective and noun. Numbers were indeed used to count THINGS - and the gender of the numbers was determined by the gender of those things.

Consistently in Indo-European languages, the numbers five and larger do not decline (in Latin, four also is indeclinable, but in Greek, it declines). This break between four and five seems to suggest something about the evolution of counting together with the evolution of the language itself. As the proto-Indo-European people(s) were learning how to count more things, the adjectival nature of the numbers themselves was shifting - from "normal" adjectives, to a different kind of adjective, one which was indeclinable. Some scholars have even argued that we should consider the numbers five through ten not to be indeclinable adjectives at all, but to be indeclinable nouns - perhaps even the names of the fingers themselves. It's a bit of a moot point, of course - either way, they are indeclinable.

So, while those "thinkers" of the proto-Indo-European past did not leave any written records of their thought processes, as Plato and the Greek philosophers did, there is clearly some kind of development in their thinking about numbers which we can see reflected in the words used in Indo-European to speak about the counting phenomena which later become an object of study for the philosophers.

In the absence of written records, we can only speculate, but some anthropological linguists have argued that the Indo-European root in "one" is a demonstrative adjective, "this, this here" - while the root in "two" is also a demonstrative adjective, "that, that there" - so before they were counting in the sense of incremental units, what the earliest Indo-Europeans were doing was saying "this (thing)," and then "that (thing)" in a series. For two things which were both "this" (and not "this" and "that"), the Indo-Europeans had a special grammatical number, the dual number, which still survives in some frozen case forms - body parts are still regularly declined with the dual number in Polish, for example! (The dual is also a grammatical number found in other world languages, as in Semitic; the dual is a fully functioning feature of modern Arabic even today.)

With the etymology of "three" in its Indo-European form, it may be from the same root that gives Latin "trans" - in the sense of something over, through, across, beyond - that is, not this, not even that - but beyond that. The evolution of "three" in the history of Indo-European is clearly different from that of "one" and "two" because, unlike "one" and "two," both the cardinal AND ordinal number "three" come from the same stem, which is not the case for "one" and "two." That suggests that by the time the word "three" came into use, the systematic relationship between cardinal and ordinal numbers had already been recognized, and an ordinal number "three" (third) was formed systematically from the cardinal number. Four is even more mysterious - as far as I know, the etymology is still up for grabs (with all kinds of tantalizing possibilities). Then, beyond four, we enter the realm of the undeclinables, and what clearly must be a later development of Indo-European (although still proto-Indo-European, since those indeclinable numbers up to ten are shared across the Indo-European languages as we know them).

This all seems important to me for many reasons, but I guess the most important is that in this chapter the Greek thinkers whom Klein is discussing all seem to take for granted that we all can number from one to infinity, with an ever increasing plethos of monads... well, the development of the Indo-European languages, including Greek, show that human expressions of counting and number have a distinct history of their own - the basic unlimited notion of arithmetical counting is not something timeless and universal. I am guessing that the Platonists want to argue for an unchanging and timeless reality that precedes this complicated and in some ways still very mysterious historical unfolding. I'm not sure what I think about that (see my previous comment)... but I think it is very interesting that none of the mathematicians discussed by Klein so far seem intrigued by the very words which are being used to discuss mathematics, even though those words themselves can give us all kinds of clues about the shifting systems with which people "thought" about numbers. :-)

[Barry Robbins]

As I hope you know by now, Laura, I am in awe of your intellectual capacity and your "philosophia." And, no, I don't think that we are getting nowhere. However, I do sense an extreme reluctance on our parts to turn toward those very things that Klein (and Plato) invite us to consider - namely, the phenomenon of "our own" experiences. If you grant that those numbers that you are so familiar with and which allow you to identify and make calculations with numbers of everyday objects themselves must consist of some kind of units, then why is it so difficult to ask the question "What are these numbers numbers of?" (i.e., what sort of units are these numbers numbers of). You could ask the same question, by the way, about anything that you have in mind when you say a word. If I tell you that I saw an amazing cathedral here in St. Petersburg, you immediately have some "idea" of what I'm talking about. Now I grant that we are not in the habit of asking ourselves questions like "Gee, what is that 'cathedral' that I immediately have in mind when Barry tells me about a particular cathedral he saw in St. Petersburg." And yet all that Klein (and Plato) are inviting us to do is to raise such a question. Again, I find it extremely curious that we are so reluctant to take "our own" experiences seriously and to reflect on them. Why, for example, do you raise the possibility of someone else having a very different "perception" of numbers rather than reflect on your own experience. Very curious indeed.

You "know" that “even numbers can be divided into two equal parts. You "know" that twelve is twice six. These are things that are "changeless" and "timeless." Please help me understand why we are having so much difficulty acknowledging this. This is all that Klein (and Plato) are pointing out. Now in the Republic Plato will liken this situation to that of prisoners in a cave, who see only the shadows of such timeless things without any awareness of them and who have enormous difficulty when compelled to turn their heads and to look at such things.

But at this point, all that we need do is recognize that we are in constant touch with such "changeless" things without realizing it. Granted that if someone whacks you on the head, you may lose touch with such things, but then that does not mean that those things have changed. Evenness will continue to be what it is regardless of your mental condition.

Also, just as a reminder, Klein doe not write in such a way as to "share" an insight, but rather to invite us and to help us to have our own insights (this is consistent with the description of true education that Socrates speaks of in Book 7 of the Republic and which I might have made mention of in an earlier post).

I do think that it would be enormously helpful to raise specific questions about specific passages in Klein's book or to make an effort to outline the construction of each chapter.

[Laura Gibbs]

Hi Barry, Klein is very un-"outlineable," at least to my reading, because he does not seem to give any particular thought to each chapter as an exposition. Your comments are VERY helpful, because even when reading Klein very carefully, I do not see him saying quite exactly what you are saying, but I am confident that perhaps after the magical Platonic part of Chapter 7, it will come into focus. I am an inveterate outliner, and would outline Klein if I could... but I can't.

As to how much Klein wants us to value our experience of numbers, I guess I am fundamentally misunderstanding what the Platonic goal is - is it really about daily life as I think you are saying here? It seems to me just the opposite. Klein says (p. 49) that scientific (by which I guess he now means the same as his previous emphasis on the word theoretical...?) arithmetic or logistic is "no longer interested in the requirements of daily life" - and then there is the passage about the "soul's turning away from the things of daily life, its conversion" (which you also cited, and which I guess will become more clear in Chapter 7) - it sounds to me that what Klein is telling us we must do is turn away from daily life... and instead, my own personal interest is in the VARIETY of daily life.

You say that this is very curious. Well, speaking for myself, I find Klein's apparent lack of interest in the variety of human life to be very curious. It seems that Klein (following Plato) thinks that somehow we will all see the same thing when our souls turn away from daily life... a timeless, changeless, universal, permanent reality? If I see and value the variety of daily life, over time, and amongst different cultures, then why would I expect there not to be an equal or greater variety of spiritual life...? I would suspect in fact that the varieties of daily life and the varieties of spiritual life would mutually reinforce one another, multiplying them, rather than unifying them.

We can both say that a number is "odd" or "even" - but the cultural connotations of oddness and evenness can be profound. I was fascinated to find out that the Pythagoreans saw a hideous quality in the prime numbers, something appalling about the fact that the prime numbers were not made up of factors. So even if the Pythagoreans and I both agree that 7 is a prime number and 9 is not, I am personally less interested in our agreement about that, than in the apparently dramatic cultural difference between what the Pythagoreans saw/felt/thought about the prime number 7, and my own perception of it being prime (my father is an amateur scholar of prime numbers, and has been all my life, so I grew up thinking prime numbers were very cool, the opposite of hideous).

If we turn away from daily life in order to contemplate the changeless, timeless, permanent reality, what happens to what the Pythagoreans believed about the prime numbers, and my very different belief about them...?

The Greeks were interested in the monads, but they did not simply count, increasing the multitude of monads one by one - they saw that the numbers came in species, kinds, like the animals - the odds and the evens, the primes, and so on, or the square and rectangular numbers that we learned about from Theaetetus. I find all of that fascinating, which is what keeps me plugging away here.

Klein is going to insist, it seems (p. 54) on the "nature of the priority of the unit over number." I'm not so sure I'm going to want to turn away from the numbers (in all their variety and all their kinds) to ponder the monad. But we'll see. I am always looking for signposts when I read, and Klein says on p. 60 that "the Platonic definition of the monad" is what we will discover in Chapter 7. I didn't get any insight into that from the first part of Chapter 7, but that was the Pythagorean part, so hopefully I will learn more about this Platonic definition of the monad in the following section. Klein keeps saying how the Neoplatonists we met back in Chapter 2 were wrong in their conceptions... while they still seem just fine to me! In order to appreciate how wrong they were, apparently I need to understand the Platonic definition of the monad: that is what is at the top of my list for reading through the rest of Chapter 7 next.