THE OUTER LIMITS OF REASON: What science, mathematics, and logic cannot tell us by Noson S. Yanofsky, MIT Press, Reviewed by Richard Webb Reason has very deep problems, and they affect our minds too Some things you will never know There are inherent limits to logic that cannot be resolved, and they affect our minds too, says Noson Yanofsky in The Outer Limits of Reason THIS sentence is false. This sentence is also where the problems start. If true, it is false; if false, it is true. Extracting its true truth is like ironing out a Möbius strip. Things in the world we experience, however, tend to be distinctly one thing or the other. Language is a messy, human construct, so perhaps we shouldnt worry too much if it doesnt always map one-to-one with reality. But in The Outer Limits of Reason, Noson Yanofsky, an information scientist at the City University of New York, shows that our problems with reasoning about the world go much deeper than that. Mathematics is pure reason in symbolic form. Set theory, the underpinning of all modern mathematics, has an equivalent to that unreasonable sentence above in the form of Bertrand Russells famous paradox: consider a set containing all sets that do not contain themselves. Does that set contain itself Such logical limitations are systemic. Kurt Gödel and others showed that no set of fundamental mathematical axioms can be used to prove itself true. The logical axioms that underlie everyday things like arithmetic depend on us accepting as reasonable the notion that in several different sizes. Reason is even good enough to tell us there are things reason cant tell us. In the notoriously hard travelling salesman problem, there is always a shortest route connecting very many cities – but even the remorseless logic of a computer the size of the universe is never going to be able to crunch through the possibilities to tell us what it is. It is a problem logistics firms wrestle with every day. Uncomputability isnt the half of it. Three-quarters of a century ago, Alan Turing asked if an idealised computer, given any algorithm and its input, would be able to predict whether it will halt on a given output, or go into a never-ending loop. The answer to this halting problem is no: computer self-analysis is logically fundamentally undecidable. Next time you are inclined to scream at Microsofts blue screen of death, be charitable to Bill Gates. Yanofsky provides an entertaining and informative whirlwind trip through limits on reason in language, formal logic, mathematics – and in science, the culmination of humankinds attempts to reason about the world. Themes emerge, such as the consistent sticking point of self-reference. The sentence that doesnt know whether it is true or not, Russells set that doesnt know whether it contains itself or not, or the computer that doesnt know whether it is about to loop the eternal loop: these are all entities asked to decide logically something about themselves. The same stumbling block might mean we can only take science so far. Quantum mechanics is our most successful theory of reality, bar none, and yet we find its predictions of particles that are in two places at once, or cats that are both dead and alive, unreasonable. It is a challenge to our classically schooled logic. But we cannot observe these predictions directly because, in quantum experiments, our act of observing something seems to change whats observed – we are ourselves part of the experiment. Is this the ultimate problem of self-reference, one that suggests a limit to how much we can ever reason about the world The problem of human consciousness looms large, not just in the quantum problem. In thinking about thinking we have to use thought. Our brains are computational machines like any other, and so presumably subject to the same fundamental limits on their ability to reason. So what allows the human mind to establish that there are limits beyond which it cannot think Yanofsky wisely and humbly declines to speculate on the answer. But a reader of this book will more readily understand what the question is. And that sentence is true.
Posted on: Mon, 11 Nov 2013 17:33:03 +0000
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