Ralph Dumain We argued about this before and reached an impasse. ^^^ CB: Yes, and I don't mean to be shitty about it. Sometimes with some time passing, it may be possible to debate it further in a friendly way and take things further. Actually, I was going over the steps on imaginary numbers in my mind again recently. So, I'll put that down below as I read what you say.
^^^^^ You read Van Heijenoort's piece as well and disagreed with him. The imaginary numbers as negation of the negation is utter nonsense to me, but you disagreed. I'm not sure what else I can say. ^^^^ CB: Over the years, in the different math classes and discussions I have had, I believe imaginary numbers are defined as numbers that when squared give a negative number as the product. When types of numbers are presented, they are presented in an order. Counting numbers, whole numbers, rational numbers, irrational numbers, real numbers, imaginary numbers ( I may have missed a detail). I can't think through the way in which the latter is a negation of a negation, but it is not immediately nonsense to me , tht there isn't a way of thinking of this progression of the development of numbers, one type out of the previous type, that would be a sense of negation and negation of negation , in the last instance. For one thing, numbers _are_ inherently "imaginary" . So, "imaginary numbers" is in some sense imaginary "squared", imaginary imaginaries. This could in some sense be a negation of a negation. Hard to see how it could be proven that it just is not a negation of a negation, no way imaginable ( sorry). There could just be a way in which Engels is thinking about it that Van Heijenoort's is not. Greater expertise in math is not always necessary an advantage in seeing a philosophical aspect of math. As to Engels being a century behind on calculus and not knowing about the theory of limits "polishing up" calculus from its beginnings in infinitestimals, I believe that there is still the Newtonian root concepts in calculus. The limits ideas makes Newton and Leibeniz more rigorous, but it doesn't overthrow their basic ideas. Engels understanding very well may be at a deeper level. I'm not willing to concede that Engels doesn't see something that these others don't, for one thing because Marxism is a deep revolution in thought , and cuts in at on a lot of areas of knowledge that even the most sophisticated experts in an area, embued with other philosophical concepts ( unconsciously) don't see how Marxism impinges on their fields. The other thing I'd like to develop is how arithmetic (counting numbers) may originate in commodity exchange, one of Engels' favorite topics. This would be a prime Engels approach, mathematics deriving from practical actiivity, not Platonic ideas. I also notice that math itself , its _ideas_, develop based on paradoxes, puzzles, problems within it. Calculus itself deals with an aspect of Zeno's paradox, in that the infinitesimal uses in an affirmative way the fact that there is an infinity between any two numbers, that we can always get a smaller number, unlimitedly, i.e. an infinitesimal. ^^^^^^^ I'm not up on philosophy of mathematics, but mathematical objects, not being material objects, nor arbitrary imaginative constructs, present a curious situation to materialists. Platonism in this case asserts the objective 'existence' of mathematical truths independently of subjective considerations, beliefs, consciousness, and psychology. But what exactly does this mean if one rejects Platonism in all other matters, and rejects psychologism as well? ^^^^^ CB: I think Engels approach to math is exactly to emphasize rejection of Platonism. Numbers are _not_ natural ideas, just in our heads by birth as homo sapiens. Recall Piaget discussed here, the pebbles on the beach, the child allegedly rediscovering counting by seeing pebbles on the beach. This is a Platonic/Robinsonade tale that Engels seeks to reject. Mathematical ideas arise socio-historically-empirically, not psychologically-Platonically. I was just thinking about the mathematical Platonist, Goedel. I'm thinking that incompleteness means that no set of axioms are enough to _deductively_ derive all the propositions known to be true in a system. This implies that there is always some proposition _inductively_, i.e. empirically derived, messing up the Platonist Goedel's Platonist system of fully deductively derived truths. At least Goedel was honest. ^^^^^^^ Mathematics historically begins in practical human activities but at some point transcends them, as Engels himself admitted. The discovery of alternative axiomatic systems, beginning with geometry, was certainly a turning point, though it turns out that Riemannian geometry was put to practical use by Einstein. A revolution was begun by the Greeks with the axiomatization of geometry. which is something different from the practical advances in mathematics issuing from several civilizations including India and China. But the question even at the beginning is, what are the cognitive operations involved in the creation and use of abstract number objects? I think Cantor's conception of making one-to-one correspondences captures even intuitively the notion of counting. Piaget investigated this matter with respect to developmental cognitive psychology. This is, however, entirely a different question than the conflation of the logical and the empirical. ^^^^^^ CB: On counting, I have been trying to see how it is in simple commodity exchange, the cognition of which is in the first Chapter of Marx's Capital. I'm thinking the concept of contradictory concept "identical individuals", the notion that any two _different_ things are the _same_ thing, is fundamental to number and to exchanging two different types of "use-values" as if they are the same thing , equivalent is the root of the concept of number. In other words, that first writing in cuneiform of commodities being exchanged in Mesopotamia might be the origin of arithmetic too, the empirical origin of counting numbers. I'm trying to remember the Russell-Whitehead short definition of numeber. Does it have to do with containment ? Don't they develop something a bit different than Cantor ? Oh yes "theory of types". This _may_ begin to fit with what I am saying, in that individual commodities are abstractly a type. An individual apples or coat is an individual example of a type, such that two individual members of a type are the "same". At 09:02 AM 3/6/2006 -0500, Charles Brown wrote: >Yet Rosa could make an even more damning case, but one which would apply to >a range of ideological phenomena in the 20th century. The suppression of >the nature of abstraction, which can already be found in Engels' conflation >of the logical and the empirical (noted by Van Heijenoort), is >characteristic of all the horrendous indoctrination perpetrated by >Marxism-Leninism. But the same phenomenon can be found across the board, >from Alfred Korzybski's General Semantics to Alan Watts' New Age >disquisitions. There needs to be a better accounting for the whole shebang, >rather than simply to fall back on formal logic and ordinary language. >Clearly something is amiss. > >^^^^ >CB; Can we focus on Engels conflating the logical and the empirical ? Also, >the critique of imaginary numbers as a negation of the negation. In general, >Engels argues that math derives from material life experiences. Is it being >said that the Platonists are correct and numbers are real ideas "out there" >? I know the Heijenoort criticism has been alluded to here before; what is >it in detail ? _______________________________________________ Marxism-Thaxis mailing list [email protected] To change your options or unsubscribe go to: http://lists.econ.utah.edu/mailman/listinfo/marxism-thaxis
