version of this book
Discretized finite difference equation models have two serious problems. To scientists and mathematicians who think they are working with continuous or completed infinite structures these models seem ugly and arbitrary. This is a matter of taste and familiarity. Most applaud the digital revolution when it brings flawless audio recordings, high definition television or cheap and powerful computing. However many mathematicians and scientists prefer the illusion that the work they do transcends the `simple minded' structures of discrete bits. One theme of this book is how limiting that illusion is. Such things change slowly. Tastes will change if the approaches I advocate lead to the results I expect but that will happen only long after those results have been digested.
A far more important problem is the difficulty in deducing detailed macroscopic predictions from these models. The intellectual thinking based research model that has evolved in the last few centuries does not consider mere ideas or intuitive arguments to support them as an adequate basis for hard science. New science starts out that way but it must be converted to a mathematical theory with testable predictions before it is considered proper science. In recents years the physics community has relaxed half of this requirement but it is the wrong half. Much research in foundations in such as quantum gravity is far from producing models with experimentally testable consequences. That is no longer seen as essential as long as the models are mathematically sophisticated. This puts the existing mathematical formulation ahead of experiments by implying we can base new theory on mathematical extensions of the existing theory without experimental tests. This is nonsense. It is an example of pushing an intellectual approach to problem solving way past the point that it is productive. The models in this chapter are the reverse. They have experimentally testable consequences but these predictions are based on simple mathematics and intuitive arguments that are far from providing a substantial mathematical development of the model.
Nature is under no obligation to be based on laws that are discoverable by any particular research model. Indeed if nature is simple at the most fundamental level then those simple laws are almost certainly such that it is extremely difficult to develop a complete theory from them. If the laws were simple and the derivation of a theory straightforward then the laws would have been discovered and the theory developed long ago.
The theory presented here is in a primitive state of development. It is important because it shows how a more complete local theory is possible. This can motivate effective tests of Bell's inequality and help to define what constitutes an effective test. If and when such tests show is false, there will be an intense interest in developing a more complete and correct theory. I do not think the existing research model will be able to produce such a theory. We will have to work for a longer time and collectively with ideas like those presented here that are still at a primarily intuitive state of development. If my models are the correct approach we will need to create new branches of mathematics to describe the large scale properties of discretized finite difference equations. We must do this with primarily intuitive arguments for why that mathematics is likely to be important. Both the development of this mathematics and experimental data from test of Bell's inequality and follow on experiments may be essential to creating a detailed mathematical theory describing macroscopic consequences. This development is likely to be more far more difficult than the development of .
version of this book