The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [214]
12. A more refined discussion of computable and noncomputable functions would also include limit computable functions. These are functions for which there is a finite algorithm that evaluates them to ever greater precision. For instance, such is the case for producing the digits of ψ: a computer can produce each successive digit of ψ, even though it will never reach the end of the computation. So, while ψ is strictly speaking noncomputable, it is limit computable. Most real numbers, however, are not like ψ. They are not just noncomputable, they are also not limit computable.
When we consider “successful” simulations, we should include those based on limit computable functions. In principle, a convincing reality could be generated by the partial output of a computer evaluating limit computable functions.
For the laws of physics to be computable, or even limit computable, the traditional reliance on real numbers would have to be abandoned. This would apply not just to space and time, usually described using coordinates whose values can range over the real numbers, but also for all other mathematical ingredients the laws use. The strength of an electromagnetic field, for example, could not vary over real numbers, but only over a discrete set of values. Similarly for the probability that an electron is here or there. Schmidhuber has emphasized that all calculations that physicists have ever carried out have involved the manipulation of discrete symbols (written on paper, on a blackboard, or input to a computer). And so, even though this body of scientific work has always been viewed as invoking the real numbers, in practice it doesn’t. Similarly for all quantities ever measured. No device has infinite accuracy and so our measurements always involve discrete numerical outputs. In that sense, all the successes of physics can be read as successes for a digital paradigm. Perhaps, then, the true laws themselves are, in fact, computable (or limit computable).
There are many different perspectives on the possibility of “digital physics.” See, for example, Stephen Wolfram’s A New Kind of Science (Champaign, Ill.: Wolfram Media, 2002) and Seth Lloyd’s Programming the Universe (New York: Alfred A. Knopf, 2006). The mathematician Roger Penrose believes that the human mind is based on noncomputable processes and hence the universe we inhabit must involve noncomputable mathematical functions. From this perspective, our universe does not fall into the digital paradigm. See, for instance, The Emperor’s New Mind (New York: Oxford University Press, 1989) and Shadows of the Mind (New York: Oxford University Press, 1994).
Chapter 11: The Limits of Inquiry
1. Steven Weinberg, The First Three Minutes (New York: Basic Books, 1973), p. 131.
Suggestions for Further Reading
The subject of parallel universes draws on a broad range of scientific material. There is a growing literature that focuses on various aspects of such material, mostly intended for the nonexpert, but often well-suited for those with more background. In addition to the references called out in the notes, here is a collection of books, from the many wonderful ones that have been written, through which the reader can continue exploring topics discussed in The Hidden Reality.
Albert, David. Quantum Mechanics and Experience. Cambridge, Mass.: Harvard University Press, 1994.
Alexander, H. G. The Leibniz-Clarke Correspondence. Manchester: Manchester University Press, 1956.
Barrow, John. Pi in the Sky. Boston: Little, Brown, 1992.
_____. The World Within the World. Oxford: Clarendon Press, 1988.
Barrow, John, and Frank Tipler. The Anthropic Cosmological Principle. Oxford: Oxford University Press, 1986.
Bartusiak, Marcia. The Day We Found the Universe. New York: Vintage, 2010.
Bell, John. Speakable and Unspeakable in Quantum Mechanics. Cambridge, Eng.: Cambridge University Press, 1993.
Bronowski, Jacob. The Ascent of Man. Boston: Little, Brown, 1973.
Byrne, Peter. The Many Worlds of Hugh Everett III. New York: Oxford University Press, 2010.
Callender, Craig, and Nick