Networks with Real Weights

Abstract

This chapter describes the network in its full richness: the internal weights may assume any real number. One may argue that such networks are, for all practical purposes, useless since systems with infinitely precise constants cannot be built. However, the real weights are appealing for the mathematical modeling of analog computation that occurs in nature, as discussed in Chapter 2. In nature, the fact that the constants are not known to us, or cannot even be measured, is irrelevant for the true evolution of the system. For example, the planets revolve according to the exact values of G, π, and their masses, regardless of our inability to gauge these values. Although one could replace these constants with rational numbers in any finite time interval, and observe the same qualitative behavior, the long-term infinite-time characteristics of the system depend on the precise real values.