> On Wed, Jul 20, 2011 at 2:52 PM, Joseph Green > <[email protected]> wrote: > > Hi raghu! It's nice to find someone else who is interested in set > > theory and the philosophical foundations of mathematics. But what I > > said was that there was no "reasonable" way to map a two or more > > dimensional vector space onto an ordinary numerical scale with losing > > essential information. > > > > There are, as you point out, set-theoretic ways to map a > > two-dimensional space onto a single numerical scale, but they destroy > > the mathematical structures in these spaces, are not smooth functions, > > destroy any practical approximations, etc. That's why you don't see > > these mappings used in economics, and never will. > > > [raghu] Indeed. I do understand what you are saying and agree on an intuitive > level (to paraphrase, it is possible to create mappings of an ordered > pair of numbers to a single number, but there is no way to design a > calculus that is preserved over such a mapping), but it doesn't seem so > easy to make this claim precise. >
Very true, one would have to formalize what one wants in such a mapping. And that's often much harder than it seems. But it's fairly clear that any such mapping would have to be quite wild. For example, it's pretty easy to show that it can't be piecewise differentiable. Yet current economists generally assume the functions they use are differentiable (among other things, so that one can define marginal quantities). Indeed, it's pretty easy to show that such a mapping couldn't even satisfy a Lipschitz condition, and yet it seems fairly clear that it would have to satisfy a Lipschitz condition simply in order to allow one to use approximations. But a simple consideration might better illustrate what is going on intuitively. If one says there are eight tons of steel in a product, one knows precisely how much steel. If one says that are eight tons of steel and wood in a product, a single number now does double service, measuring both the amount of wood and one steel. This might seem elegant. Unfortunately, the result is that one doesn't know how much steel is in the product (other than it is less than or equal to eight tons), nor how much wood. > Anyway you seem to be making what I would call an anti-Hayekian > argument. Yes indeed! This is certainly one of the reasons that the Hayekian arguments are wrong. > Friedrich Hayek made a lot of mileage out ofthe so-called > economic calculation problem which is really a rather trivial and > obvious observation. Here's Wikipedia's summary of it: "Without money to > facilitate easy comparisons, socialism lacks any way to compare > different goods and services. Decisions made will therefore be largely > arbitrary and without sufficient knowledge, often on the whim of > bureaucrats." > > With money, it is certainly true that easy comparisons are > facilitated, but this is not rational in any sense except a circular > one. Yes, using a single numerical scale simplifies comparisons. In my article showing that the labor-hour is *not* the natural unit of economic calculation, I comment on the issue of what subordinate use might nevertheless be made of simple numerical scales, and suggest that several contradictory scales might be in use simultaneously. (See the section "approximate assessments" of part 3 of the article at www.communistvoice.org/27cLaborHour3.html.) > Your argument may suggest that having a single number (the price) > to compare two disparate goods is only slightly better than having no > numbers at all; it may convey some information, but only a meagre > amount. -raghu. Part of the problem can be illustrated as follows: If a price of something is high, is it high because the product is scare and undersupplied? because the inputs are expensive? because it is taxed? because the buyer and seller are making mistakes in their calculations? because people expect crop failures in the coming year? or a war in the coming year? or because there is a monopoly? etc. Any of these things could cause the price to be high, and hence the price being high doesn't definitely indicate any one of them. It takes knowledge aside from the price in order to know why the price is high. And yet, to know what is likely to happen and to make planning decisions, one needs to know the specifics. -- Joseph Green _______________________________________________ pen-l mailing list [email protected] https://lists.csuchico.edu/mailman/listinfo/pen-l
