CULTURELAB

46 | NewScientist | 2 November 2013

The Outer Limits of Reason: What science, mathematics, and logic cannot tell us by Noson S. Yanofsky, MIT Press, £20.95/$29.95

Reviewed by Richard Webb

“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 shouldn’t worry too much if it doesn’t 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 Russell’s 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 infinity comes in several different sizes.

Reason is even good enough to tell us there are things reason can’t 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 isn’t 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 Microsoft’s 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 humankind’s attempts to reason about the world. Themes emerge, such as the consistent sticking point of

self-reference. The sentence that doesn’t know whether it is true or not, Russell’s set that doesn’t know whether it contains itself or not, or the computer that doesn’t 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 what’s 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. n

“Our act of observing seems to change the observed... Is this the ultimate problem of self-reference?”

The shortest route between cities must be crunchable, right? Wrong

Life in an unreasonable worldReason has very deep problems, and they affect our minds too

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