On Wed, Aug 7, 2013 at 5:36 PM, Jurriaan Bendien <[email protected]> wrote:
> Thanks for your extensive elaboration. My understanding of “mapping” in > mathematics is, that it has a more specific meaning than you use. What more specific meaning? > I get your idea about a fixed point, but it is only one way of looking at > it, and not the only logically compelling or most obvious one. I'm not saying it is the only way. But, given what we know, it is a very general and extremely *powerful* way of looking at it. In mathematics, fixed point theory is a whole field. So, if you can frame a process -- physical, social, etc. -- as a mapping with rather basic properties so that one or more fixed points are ensured, then you may invoke the whole fixed-point machinery. I have only a limited knowledge of the field and its applications, but even with that, I can see clearly that virtually *any* process (in nature or society) whose qualitative stability (in the face of quantitative changes) you're interested in highlighting and analyzing can -- in principle -- be framed in these terms. I'll just give you an illustration I'm a bit familiar with: Since Newton and Leibniz, some systems of differential equations have yielded solutions, but the methods to solve them have been idiosyncratic, more or less based on the skill and intuition of the mathematician. Solving them was akin to performing a musical piece, you either had the magic or not, with little awareness of the procedure. (Differential equations are equations where the "law of motion" or rates of change of a variable or variable vector is known, as well as some boundary -- e.g. initial or terminal or transversality -- conditions, and you try to solve or "integrate" for the location or level or "function" of the variable vector. It's as if they give you data on where the rocket was at first, its pattern of speed/acceleration changes over time/space, and you then have to mentally reverse engineer all this to pin down the exact location of the rocket now.) However, when whole systems of these equations are interpreted as mappings (or relationships or correspondences or functions) endowed with the right properties (basically, continuity, though if you add other properties, you can narrow things down even more), and pow! -- these equations turn immediately into sweet, edible pieces of cake. I mean -- not really, you (or the computer) still need to do involved computations (which may require you to impose even more structure, "linearize," etc.), but the *structure* of the problem appears entirely manageable to even a regular brain like mine. The trick then consists of "reducing" (ha!) the particular case to the universal analog in your brain: a continuous mapping, etc. Once you begin to think in these terms, a lot of stuff makes a whole lot of new sense. > The concept of aggregation is essentially an empiricist concept, whereby we > arrive at a generalization by adding up and distinguishing groups of sense > data (observations). An empiricist generalization is certainly useful, but > has its limits. Aggregation is not "empiricist." I mean, scratch that -- everything in mathematics is "empiricist" in that all abstract mathematical structures result from the generalization of concrete procedures. But the principle of aggregation or "averaging" is a very general mathematical structure: a sum or "generalized" sum (e.g. a convex, linear, or even nonlinear transformation applied to a vector of "data"). So, I'm not sure what you're trying to say. Generalizations have limits? Of course. But compared to what? Particular procedures? And here's another analogy for you to reflect on: Generalized mathematical structures are to scale economies as particular "empiricist" procedures are to each worker operating in its own individual means of production. > Firstly, there are generalizations from experience which are > not reducible to particular experiences but go beyond them (reductionism has > its pitfalls). Secondly, in dialectical theory we can envisage the > particular and the general as existing together and related to each other as > part of a totality. A “social force” would then for example presuppose the > actions of individuals, but also be semi-autonomous from them, in such a way > that they mutually influence each other. Not sure what you mean. _______________________________________________ pen-l mailing list [email protected] https://lists.csuchico.edu/mailman/listinfo/pen-l
