> On Tue, Jul 19, 2011 at 10:23 PM, Joseph Green > <[email protected]> wrote: > > But no, there is *no* reasonable way to map a two or more dimensional vector > > space onto an ordinary numerical scale without losing essential information. > > Period. That's the end of the story. > > > > It isn't a matter that this is a complicated problem. It isn't a matter that > > one can use fancy methods of linear programming to do this. Or shadow > > prices. > > Or Lagrange multipliers. Or non-linear functions. Or whatever. It simply > > isn't possible. A two-dimensional vector space, for example, obeys different > > laws than a simple numerical scale. That's no way around this. > > > > Or, to put it another way, if one reduces a two-dimensional vector to a > > single number, one loses essential information. > > > You are wading into deep mathematical waters here. The famous Cantor > diagonalization argument does precisely what you claim to be > impossible: reduce an order pair of numbers to a single number in a > one-to-one fashion. > http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument > > -raghu.
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. For example, you could find a function that would combine this information about means of production: 3 tons of steel] 5 labor-hours of work into a single number. Let's say the number was 17. But ths same function, applied to the following means of production 3 tons and one-ten thousandth of an ounce of steel 5 hours and one-ten thousandth of a second of work might give the number 243. Or conversely, the number 17.000001 might correspond to 2 ounces of steel and five million years of work So a change of only .000001, from 17 to 17.000001, would correspond to a change of tons of steel and million of years of work. Such functions are useless for economics. You couldn't really get the information about the means of production from the single number, because the slightest change in that number, from 17 to 17.000001, would cause catastrophic changes in what it stood for. And you couldn't really calculate the number 17 from the means of production unless you knew the means of production EXACTLY, with NO DEVIATION whatsoever, not 1% of deviation, not .000000001% of deviation, NO APPROXIMATION allowed whatsoever. And even then, you might find that the mapping itself, while it existed theoretically, was, for most combinations of numbers, impossible in practice to calculate. Such mapping functions exist in set-theory, and may even be continuous. It is important for set-theory that these functions exists. But such functions are chaotic, non-smooth, non-intuitive, and impossible to use in economic calculations. -- Joseph Green [email protected] _______________________________________________ pen-l mailing list [email protected] https://lists.csuchico.edu/mailman/listinfo/pen-l
