Impossibility

  • John D. Barrow
Oxford University Press: 1998. Pp.279 £18.99, $25

To those enamoured of the belief that the human spirit knows no bounds or limitations, a tour of twentieth-century science must be a rather depressing experience. If there is any common denominator running through the scientific breakthroughs of our age it is the idea that there are limits.

Starting with Einstein's speed limit of the Universe in a ray of light and Heisenberg's limits on what can be measured with certainty, through Gödel's limits on what can be known by following a set of rules and on to Arrow's famous result about the impossibility of a perfect democratic society, the history of modern science is riddled with logical and physical limits of all sorts.

In this illuminating, well-written account of Limits (with a capital L), John D. Barrow chronicles and explains the limits of science as a reality-generation mechanism — and why it matters.

As he carefully points out, limits to the ability of science to answer a given question about the world as we see it come in all sizes and shapes. There are political and ethical limits on how much latitude a society is willing to give to investigators to answer a question. There are practical limits on the amount of time or energy or money that can be spent in answering a question. And, finally, there are logical limits.

Although Barrow pays more than lip service to the former sorts of limitations on our ability to tease out the ‘scheme of things’, it's clear that his real focus is on logical limits. Rightly so, too, because ultimately these are the limits that count — at least for the scientist and scholar, if not the politician and social agitator. When one is looking for a needle in a haystack, it's of more than passing interest to know that a needle is really there to be found.

As Barrow points out, all the scientific knowledge we have about nature comes from models that we create of natural phenomena. These models, in turn, are almost always mathematical in character. It follows then that it's only a small approximation to say that the issue of logical limits to science comes down to the limits of computation, as any mathematical model can be regarded as an algorithm for processing inputs (the statement of the question and its circumstances) into an output (the answer).

This notion of scientific knowledge raises several fundamental questions. How does the mathematical model relate to the real-world phenomenon it purports to represent? Does the Turing-machine model of computation impose intrinsic limits on what we can know? What is the relationship between the computational powers of the human brain and those of our computing machines?

One question missing from the list — and the book — arises from the assumption that our models of nature must necessarily be mathematical. This question is: is there an alternative to a mathematical formulation of a model of natural phenomena?

The difficulty here is that one is left perched on the horns of a dilemma. Either you use the mathematical representation of the question of concern, and then try to justify why mathematical insolubility implies the same for the model's real-world correlate. Or you forsake mathematics altogether, and then face the difficulty of trying to create a convincing real-world substitute for the mathematical notion of proof, in order to produce a knockdown argument for why the question of concern is logically unanswerable. Impossibility evades this dilemma simply by ignoring the non-mathematical alternative.

But no matter. Taken on a whirlwind tour of the mind, Gödel's theorem, quantum theory, free will, voting paradoxes, time travel, computational intractability, sandpile models, the topology of space, nanotechnology, the forces of nature, the evolution of the Universe, complexity science, computer chess-playing, percolation theory, human consciousness, economic forecasting, black holes, the Brouwer fixed-point theorem, artificial intelligence and the Heisenberg uncertainty principle, one can only wonder how Barrow can possibly make all these things fit together into a coherent story about the limits to science. Well, contrary to all expectations, he does make them fit — and in only 250 pages!

So for about as good an account as you're going to get of where science stops, read this book. It won't tell you any final answer. But the journey is far more interesting — and important — than the destination.