6: Concept of arithmos

[Laura Gibbs]

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

[Laura Gibbs]

Wow. Lots of thoughts on reading this chapter, but one theme that recurred throughout the chapter which I would really like some help with - because it is ALWAYS a problem for me in reading Platonic material - is just how the primacy of abstraction is demonstrated. Just to take one statement of many, here is p. 51, emphasis is Klein's: "Everyone is able to see - if only it has been emphatically enough pointed out to him - that his ability to count and to calculate presupposes the existence of 'nonsensual' units." Such statements about the primacy and priority of the ABSTRACT (nonsensual, noetic) objects occur again and again in Klein's paraphrases of Plato - and I think that Barry has said some things like this also in preparing us for Chapters 6 and 7. So: please help! This is something I have never been able to grasp about Plato at all.

I don't disagree that it is possible to create these abstract objects of knowledge... but what I don't understand is why those abstract objects of knowledge are presumed to come BEFORE the real-world objects, the sensual objects. It seems to me just the opposite: most people most of the time are perfectly happy to live their lives in the realm of sensual objects, and the quest for the abstract and the nonsensual is a secondary, derivative intellectual construction built UPON the everyday sensual world... although for some reason the secondary, derivative intellectual construction then makes claims - strange ones, to my way of thinking - about being the true primary object.

I laughed out loud at that part in Klein about the nonsensual object being "emphatically enough pointed out to him" - it really conjures up the teacher bullying the classroom about some abstraction which everyone in the class knows they did perfectly well WITHOUT... until they started going to school and were led to think in some new way - which pretends to be the way they were thinking all along, even if they didn't know it.

Here's an example of just how abstract thinking is inculcated in schools, from Walter Ong's Orality and Literacy, who is providing a summary of research reported in the Soviet sociologist A.R. Luria's Cognitive Development: Its Cultural and Social Foundations. Luria conducted research by interviewing illiterate and somewhat literate persons in Uzbekistan and Kirghizia in the early 1930s. It's absolutely fascinating stuff, and here is one item in particular that I kept being reminded of in reading this chapter in Klein:

Illiterate subjects identified geometrical figures by assigning them the names of objects, never abstractly as circles, squares, etc. A circle would be called a plate, sieve, bucket, watch or moon; a square would be called a mirror, door, house, apricot drying-board. Luria's subjects identified the designs as representations of real things they knew. They never dealt with abstract circles or squares but rather with concrete objects. Teachers' school students, on the other hand, moderately literate, identified geometrical figures by categorical geometric names: circles, squares, triangles, and so on. They had been trained to give school-room answers, not real-life responses.

So, what do you see here? Since we've all been to school, we see a circle. We cannot help but see a circle. But if we were one of the Uzbek peasants interviewed by Luria, we could not see a circle at all - we would see a moon... or a plate...

[Ian Carmichael]

I may be giving a simplistic answer here, Laura, but I'd have thought that Plato's view of reality is the eternal and immutable logically, if not also temporally, prior to the flickering fitful stuff we misconceive as 'reality'. That's, at any rate my view of his 'cave' image. My schoolboy understanding of Platonic Forms is that their existence shapes the world stuff of things that we manipulate, or fall over. So, for example, the (Platonic) ideal Chair exists and by existing allows temporal chairs to be sat upon, rearranged and fallen over in all of life's mundanity.

No constructivist is Plato, but top-down!

On the other hand, Aristotle's forms are abstractions shaped from the here and now: constructivist he would be - because we have built, painted, thrown and fallen over chairs we can abstract an idea of chairness, but Aristotle's ideal is (just) a mental summary rather than an ontological control.
Hence the superb teaching wrapped up in Raphael's School of Athens

Plato pointing up to the source of reality - the home of the Forms, and Aristotle, hand sprad out flat, feet firmly anchored on the ground, locating a very material, solid, present reality with which philosophers have to do.

[Barry Robbins]

Thanks, Ian, for calling our attention to Raphael's wonderful painting. As I recall, Plato and Aristotle are the only members of the School of Athens who are depicted as engaged in dialogue and as looking directly at each other. As I also recall, all other relationships are depicted as that of instructor/student with a few isolated thinkers.

If you were asked to create captions for Plato and Aristotle, what do you think that they are saying to each other, given their gestures toward each other? (It may help to keep in mind the text that each is carrying.)

Klein will elaborate on the views of both Plato and Aristotle in chapters 7 and 8. In section B of chapter 7, in particular, Klein will discuss in great simplicity and depth the "divided line" section of the Republic, the section which immediately precedes the "cave" passage that you made mention of, Ian. This section should prove especially interesting for us, since, as you will recall, in it Socrates attempts to provide some clarity about the relationship of everything that is perceptible and everything that can be grasped by our thinking. I look forward to any light that you will be able to shed on Klein's analysis of this amazingly interesting passage.

[Laura Gibbs]

Aha, thanks as always for the heads up about what is coming, Barry - it sounds like Chapter 7 may give me an excuse to go read Plato's Cave in Greek, which is something I've always wanted to do but never had a good excuse for! :-)

[Laura Gibbs]

Hi Ian, I guess "constructivist" is definitely a good word here to express my sense of disorientation in the Platonic university. Plus "construction" fits in really well with the "hyle" of Aristotle which Barry has helped us see as "building material." Somehow I just don't connect with this idea of something that is not just before reality, but which seems to have claims to be "more real than reality" if you see what I mean, which is how the Platonic idea of the forms seems to work.

It's one thing to toy with the idea that reality itself is all an illusion - the idea of Maya from Eastern philosophies definitely makes sense to me. If a dream can seem real to me when I am dreaming it, then how do I know that what I experience as waking is not just itself a dream...

But the Platonic Forms, if they are a pre-reality, seems to me so inaccessible as to not be very useful as a concept. So, I really enjoyed all the observations that Klein was making about arithmos and counting in this chapter, but any of the comments about the idea of counting existing "before" we count still just do not click with me at all. That's why I thought the historical evolution of counting terms was so interesting: unlimited counting may seem completely obvious to us now (and thus easy to imagine as pre-existent, since it is not so obvious-seeming) - but unlimited counting was not so obvious to the proto-Indo-Europeans, who really had to work up to it with their vocabulary/thoughts.

[Ian Carmichael]

Hi Laura, again,
Perhaps another analogy - this time from the world of computer programming. One of the modern (?) breakthroughs in programming was the development of object-oriented programs - C++, Java, etc... These worked on the creation of classes, where the template and behaviours for what would be real objects were framed and held in the super-real programming construct-land.
From these classes objects were instantiated so myChair was an instance of the Chair class (Form), customer(123) was an instance of the Customer class... But the program couldn't create the instance without the class (form) existing first. To use the Java vocabulary, a constructor of the class is called to create an instance of the class, or an object. Class is prior. (Yes, then we could step back a pace and have to consider the mind of the programmer and how they abstract the class! But assuming the programmer can't be considered, then we have a fair picture of a pre-existing Reality.)

Pre-objected oriented programs built objects from the experience of having built other objects, and there wasn't a form or template available to work with until the next generation of languages hit.

I don't know that this makes the concepts any more helpful or accessible, but I doubt that was Plato's concern, either. Accessible/inaccessible, hidden/explicit would not have been controlling ideas. "This is the way it is..." he'd be saying to Aristotle as he (Plato) pointed up to the Forms. "No, mate," says Aristotle - "it's the here and now stuff we work with - look at the world around and think about that. Head out the clouds and feet on the ground please, Plato!"
(Well, Barry wanted some captions! Aristotle can be an Aussie as well!!)

[Laura Gibbs]

Ha ha, Ian - I definitely like the idea of Aussie captions (like that "Strine" translation of the Bible that was making the rounds on the Internet a while back!).

About the idea of the priority of things, it's not that I do not understand what Klein and Plato are insisting on - I just don't understand on what grounds one can make such an argument, and when anyone would want/need to assent to it.

In a comment a while back, I had said that I did not understand any of the insistence on "timeless" things and Barry had replied something to the effect of timeless things being the only things we can think about. That's something I still just do not understand. I'll keep trying to understand it... but it sure is not sinking in, and it reminds me in a very unfortunate way of a religion that imposes itself on believers: woe betide the people who do not believe in this-god or that-god, even if the message of this-god or that-god has never reached them. If the limitless monads that are the Platonic arithmoi are timeless, then what exactly are we to make of our proto-Indo-European speakers who could only count to four, and didn't even exactly have numbers with which to do that - although I will give those ancient nameless people MUCH credit for having invented numbers as time went by (it seems quite a genius invention for which they deserve a great deal of credit!)... but I see that invention as unfolding in time, not as anything timeless.

[Barry Robbins]

I wanted to try to summarize the chapter, but right now I only have time to try to say a few things about the beginning.

Klein first spends time making the point that every number is always a definite number of “definite things.” Now, if we can see that this is, indeed, a characteristic of “every” number, then it, of course, follows that those numbers that we have at our disposal before we begin to perform operations with any numbers of objects must also consist of “definite” things.

Klein next attempts “to understand how the conception of ‘pure’ numbers, as opposed to the “visible” or ‘tangible’ numbers, arises out of the natural phenomenon of counting.”

It is very important to keep in mind that in Klein’s view the conception of “pure” numbers arises out of the ordinary act of counting. So any conception we have of pure numbers comes from our actual experience of dealing with visible and tangible numbers. (I say that this is very important because I have noticed a curious suggestion in our comments that Klein is not dealing enough with what we are all familiar with. Nothing, I repeat, nothing is farther from the truth. The problem is that we are all very out of touch how to begin “thinking” about those phenomena with which we are so familiar.)

As a result of “the continual practice of counting and calculation” we become familiar with numbers and their relations. Thus, we become able “to execute any operation of counting and calculation we wish.” In other words, we now have numbers at our disposal which enable us to perform such acts of counting and calculation. Again, we don’t fabricate such numbers (any more than we fabricate the “vase” that we know and which enables us to identify visible objects as vases). This, by the way, is why we play counting games with young children. Our intent is for them to come to know and to have at their disposal all the numbers.

It’s at this point that one could raise the question, “What are these numbers that we have at our disposal the numbers of?” It certainly is not necessary to raise such a question, no more than it is necessary to raise a similar question about the “vase” that we have at our disposal (e.g., what sort of a vase is it?). And this is why Klein turns to making an attempt to characterize the significance of raising this sort of a question in contrast to the ordinary way in which we make use of these numbers. He describes it as “the soul’s turning away from the things of daily life,” as a “change in the direction of its sight,” and, citing language that is used in the “cave” section of the Republic, as a “conversion” and “turning about.” It is a very significant event and one which Socrates will suggest (cf. Republic 518c) is what genuine education aims at.

As for the “timeless things” that you mentioned having trouble with, Ian, we are in touch with such things all of the time. Four is an even number, and that is something that will never change. Also, when you use a word like “table,” here too what you intend is not any particular table that is subject to change, but something that has a meaning apart from anything that exists in time. Now, we must be careful not to make anything mystical out of this – it is merely something that we are all familiar with. However, it is this phenomenon which could lead us to ask about such “timeless” things without prejudice, but simply out of curiosity. This is what, I think, Klein is trying to reveal about the ancient mode of forming a concept. It really is grounded on ordinary phenomena and an interest in gaining clarity about them.

I’m afraid that this is all I have time for right now. I’m about to go for another walk in this amazing city of St. Petersburg. I would suggest that we not move too quickly and, perhaps, raise some specific question about specific passages in the text or even about the thread of the logic. Klein is not easy; neither is Plato, but I would suggest that that they are friends who are primarily interested in inviting us to share the questions that will help us to turn toward thinking about those things that we are in touch with all of the time, but in a rather sleepy way.

[Laura Gibbs]

I am so jealous of you there in St. Petersburg - and for the longest day of the year even! How marvelous! The famous "white nights" of that northern city. Wow!

As for numbers: what I wanted to suggest with my post about the laborious history of "one-two-three-four-five" which is traced in the names of the numbers themselves, is that there is nothing that I can see which is "timeless" about numeracy. Admittedly, it is something very familiar to us, and often we do not remember that it is something we learned... but it is something that we learn as children, and in the history of human culture it was something that was acquired only gradually over time. Indo-European offers a quite fascinating window on the linguistic history of the counting words, in fact, as I tried to explain, albeit briefly, in my comments on this chapter.

For us, the words "five six seven eight nine ten" are not the names of our fingers anymore... although it seems quite likely that this was once the case. But wherever they came from, these words did not simply fall from the sky, or abide timelessly in some mental landscape - they evolved in the history of the Indo-European languages, and they evolved later and differently from the word one, and from the words two and three, and from that quite mysterious word, four.

So, anything that claims to be timeless strikes me as extremely suspicious. I still haven't gotten to the parts of Chapter 7 which you have referenced here - about some kind of shift in thinking that Plato is going to describe and which Klein is going to adduce. That sounds like something very interesting, and I am looking forward to learning about that; I'm guessing that will come up in this week's portion, since I've read through the first part of Chapter 7, about the Pythagoreans.

[Barry Robbins]

Laura, right now you have in your possession many different numbers. These numbers allow you to identify all sorts of countable collections that you come in contact with as being just so many candles, pencils, cupcakes, friends, etc.; these numbers that you have in your possession are not subject to change the way the number of pencils that you may have will change as you write with them and eventually will no longer exist. But those numbers that you have right now in your possession are not numbers of anything perceptible, of anything that exists in time. The five that you know will never be anything other than five. It's really as simple as that. Please, don't try to take it for anything more than that. It's fairly obvious, I think, that these numbers that you are so familiar with are, in fact, not subject to change. Now, how they exist is another question, and, to repeat myself, Aristotle will disagree completely with Plato as to the way in which they exist. But we really are not going to get anywhere if we don't at least try to take seriously these very simple phenomena. Klein is really right about how difficult it is for us to deal with the phenomena in forming concepts.

And, yes, this is an incredible city. The streets full of people at 12:00 midnight. Some of the finest opera in the worls (Meriinsky Theater) for next to nothing. Tomorrow the Russian museum during the day and a concert in the evening. On Thursday a performance by Vladimir Davidovich Ashkenazy for next to nothing. Simply amazing. I want to live here.

[Laura Gibbs]

Enjoy the city, Barry! It has exerted its charms on many!!! I would love to go there someday... my Russian could easily become functional again, I am sure, and what fun that would be.

As to numbers never changing, I don't know: whack me hard enough on the head and I could lose them and have to regain them. Whack me even harder and I might gain some "savant" insight into numbers. A friend of mine has autistic twins, one of whom speaks and likes to engage in conversation, and who has some of those fascinating number perceptions that the rest of us do not have. Are those numbers really the same for him as they are for me? I suspect he sees them, really sees them, and in patterns that give them a meaning entirely different from the meaning they have for me. Does that mean they are the same numbers? Or different? Just to take on example, when I told him I had gotten married he asked me the date and then instantly said in real admiration, "You got married 324 days ago" (or whatever the number happened to me; it was something slightly under a year at the time). It was eerie: Oliver Sacks wrote an essay about a pair of twins who shared prime numbers with one another, clearly somehow grasping (seeing? knowing?) those numbers in a way denied to the rest of us... and here was this boy clearly sharing in that other realm of number perception (one of many other realms of number perception)...? Intense.

So, I'm reluctant to insist on some kind of sameness about the numbers until I understand better what Klein is saying, and I honestly do not understand what he is saying yet. I haven't given up - but if you think that we are getting nowhere, believe me, it is not through my lack of trying. I would have given up on this book a few chapters ago if it were not for the faith people clearly have in the power of Klein's insights. So far, I'm still waiting, but prepared to carry on! :-)

[Ian Carmichael]

Thanks Laura,
This was the point I wanted to make in the post I failed to post. Plato's concept of number, as I read it, is not an abstraction from the countability of things, but rather the countability of things is the physical instantiation of number. (That is, the reason we can count sheep is thanks to the timeless concept of number waiting to 'land' on the sheep. But, although that's what I understand of Plato, I'm not a subscriber to his viewpoint.)