8: Aristotelian critique

[Laura Gibbs]

Space for thoughts and responses to Chapter 8: Aristotelian critique

[Laura Gibbs]

Well, folks, I think this is where I leave you. I was prepared to give Chapter 8 a good try, but the first part of the chapter was 100% alienating for me. I am not Klein's intended audience for this book, and there is no point in wasting my time here: I'm sure this is a great book for its intended audience, but it is not a book that has much to offer for people beyond that very narrow audience. Here are my comments on the opening sentences of Chapter 8. No need to reply or clarify my misunderstandings (in fact, please don't - because I'm really done with the book now); I'm just writing this down as an explanation to myself of why I am not going to finish this book; it's very rare for me to not finish a book, but this is one of those special occasions!

Meanwhile, for those of you who continue to read and re-read Klein, more power to you - every author needs an audience, and I am glad that Klein continues to have such avid readers who find his ideas stimualting and useful. :-)

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The Platonic chorismos thesis has its strongest support in mathematics. ... I guess, and it's very frustrating that I should have to start out just by guessing, that the "chorismos thesis" maybe has something to do with the discussion of stasis and kinesis in the previous chapter, or maybe it has something to do with the gap between the original and the copy - but why am I having to guess? I looked in the index to see if it was listed there, so that I could get some clarification from Klein on just what this thesis entails - but it's not listed in the index of topics. Probably this is something obvious to the Plato scholars who are Klein's audience, in which case I really am wasting my time reading this book. Let's see if Google can rescue me: aha, yes, when I Google "Plato chorismos," I find all kinds of information about Platonic philosophy which Klein probably assumes is already familiar to his audience. Well, it's not familiar to me, alas.

The exemplary "mathema" character of mathematical objects, ...Why is the word mathema in scare quotes? Klein's use of these scare quotes is really maddening. With or without the scare quotes, I am not persuaded about the mathema character of mathematical objects (although maybe Klein is just making a circular argument - the things I think of as numbers are probably not Klein's mathematical objects). Anyway, my learning about mathematics is very much based on experience, which would fall into the category of pathema, not mathema. For example, if I lived in a world where there were never more than five of anything, I would not learn to count past five. But I live in a world which I experience (via my many pathemata) many kinds of countable things, in many varieties of numbers, even and odd, big and small, fractional and whole, etc. The mathema and pathema seem connected to me; I have no idea how they can be disconnected - especially for the arithmoi. Imaginary numbers: sure, that's mathema. But why would the arithmoi, definite numbers for definite things, be objects of mathema, without being objects of pathema as well?

their undeniably pure noetic nature, ... Why pure? The one idea I had gotten from the reading so far is that we count things, and we also have the ideas of these numbers that allow us to count. I'm not so sure what I think about the ontology of those number ideas, but at best it seems to me that it is noetic AND somatic, a combination of mathema AND pathema - so, I don't see the PURE noetic nature here... if we started out by counting on our fingers (and by that I mean, we as little children, but also we as a people who developed the process of counting millennia ago, starting from bodily experience), how is that purely noetic?

their "indifference" with regard to objects of sense... Why does Klein keep using these scare quotes? What is the reason for "indifference" being in quotation marks here? I have no idea.

all this immediately indicates the possibility of the existenceof noetic structures ... Klein's hyperbole here is very offputting: undeniably pure? immediately indicates? Sorry: I don't understand his use of scare quotes, and I don't understand his need for this kind of adverbial blustering.

which are independent of and "detached," i.e. separated, ... So, "detached" goes in scare quotes, and the word separated does not go in scare quotes, nor does the phrase independent of. So what's the deal: are independent of, detached, and separated synonyms? Or do they have differences in meaning? And why, of the three of them, is detached the only one Klein puts in scare quotes? Maybe this is some bizarre artifact of the translation from German to English; in any case, I find it both baffling and annoying. These scare quotes are a persistent feature of his style and very unhelpful in conveying his meaning.

from all that is somatic, just as the Platonic thesis affirms. Again, I have no idea how numbers - the arithmoi in particular - can be separated, detached or independent of all that is somatic, and thus I do not see how I can proceed here. I have no idea how it would be possible for something which is so grounded in human experience to be some kind of purely noetic mathema that is separated from all that is somatic.

So, yes, I concede defeat. I can find no way over the hurdle here at the opening of Chapter 8, which is definitely a sign of what a failure - for me - the first 100 pages of this book have been. Perhaps I would find more congenial material later on in the book, but based on the first 100 pages, I would say the odds are not good. I've got much better odds on the many books on my bookshelf for which I am the intended audience... so that is where I will turn now, instead of wrestling with more of Klein. Luckily, I learned all kinds of fascinating information along the way in my efforts to cope with Klein - but I think I will pursue those interests in a more direct method, instead of doing any more wrestling. :-)

[Ed Wall]

Laura

Thanks for beginning this! Reading of Klein - as with reading of some of his contemporaries (who also avoided substantial indexing) - can result in a bit of wrestling and I certainly understand your preference to do otherwise. I, myself, make such decisions.

I will remark, bringing in again my biases, that what is being discussed here is what, I think, Freudenthal called mathematizing. This is a critical notion and the cause of a great number of problems in the elementary grades and beyond.

[Laura Gibbs]

I'm not familiar with the term mathematizing but guessing from the context of your remarks, I suspect there are probably some great connections to be made here between mathematizing and the challenge posed by literacy (written language as taught and acquired in school) in the work that I do as a teacher... I'll confess that I was hoping for more of a gateway to that question with Klein - it's my own fault for having made such an assumption, but since a lot of the great insights into the peculiar condition of literacy have come from modern anthropological studies of ancient Greece (starting with the genius work by Milman Parry on oral epic, and then later by such folks as Eric Havelock, Walter Ong, etc.), I was hoping for a similar kind of experience with Klein.

Now, when it comes to literacy studies, the philosophical tradition is itself a function of literacy - as an outgrowth of literacy, it's hard for philosophy to then provide us with any real insights into orality (an awkward word, admittedly, but literacy itself has not given us much vocabulary for discussing the world that is other than literacy - words like illiterate, preliterate, etc. are clearly problematic!). I'm guessing that a similar problem might exist when it comes to looking at "naive mathematics" (or "folk mathematics" or whatever term, like orality, tries to encompass the kind of math that lets children, say, insist that 4/4 and 5/5 are not the same thing). It seems to me that the tools we need to ask and answer these questions are the tools provided by anthropology and history, by linguistics and psychology, and not by philosophy... but Klein seems to have no interest in any of those socio-historical tools, and instead is going at the problem with a philosopher's toolkit. The results are not impressive to me - but I'm clearly coming at Klein with the wrong background and wrong assumptions; his intended audience definitely does not include someone like me.

[Ed Wall]

Laura

What you say about literacy is interesting. This has not been my focus so let me say a little more about mathematizing and see if it still works. Aristotle argues that effectively one can turn situations into mathematics. Say " Johnny has two ice cream cones and Susie gives him five more. How many ice cream cones does Johnny have?" This is, of course, nonsense as what child has seven ice cream cones and ice cream cones have nothing to do with the problem. It is simply 2 + 5 = ? and this is mathematizing. Later on it gets worse when we have two trains coming from two different directions, etc.

As far as philosophical tools goes, I perhaps have a different point of view. The tradition Klein is working in is hermeneutic phenomenology which has always seemed to be me to be heavily socio-historical. However, I am sure others would disagree with me. Nonetheless, although Klein provides, for me, some significant direction, I agree that Klein does not answer many of my pragmatic questions. I need some blending with those other areas you mention (which fall significantly short also).