Daily Archives: March 17, 2014

Reflections on Noson Yanofsky’s, “The Outer Limits of Reason”

But as for certain truth, no man has known it, nor will he know it; neither of the gods, nor yet of all the things of which I speak. And even if by chance he were to utter the final truth, he would himself not know it: for all is but a woven web of guesses. ~ Xenophanes


Just finished reading Noson Yanofsky’s, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us.  He covers paradoxes, conundrums, puzzles, impossibilities, limitations, perplexities and obstructions in philosophy, mathematics, computer science, physics, and logic. Topics include: self-referential paradoxes, the ship of Theseus, Zeno paradoxes, different sizes of infinity, the halting problem, the Monty Hall problem, chaos theory, the travelling salesman problem, quantum mechanics, relativity theory, and more. Much of the book is tough going, although Yanofsky does make difficult topics understandable.

Of particular interest to a philosopher is his conclusion. After an entire book on reason’s limitations, he meditates the definition of reason. While many definitions have been offered, Yanofsky settles on: “Reason is the set of processes or methodologies that do not lead to contradictions and falsehoods.1 Processes that do not lead to contradictions and falsehoods are reasonable ones; whereas if we derive a falsehood or contradiction we overstepped the bounds of reason.

Of course determining whether some idea or process is reasonable or steps over the limits of reason is hard to determine. At one time scientists believed in ether, phlogiston, phrenology and spontaneous generation. Often the problem lies in relying on intuition, which sometimes misleads. The earth may seem stationary and flat, but it is not; motion may seem independent of speed, but it is not. But what of dark energy, dark matter, string theory, supersummetry and the like? For now we don’t know if some theories have overstepped the bounds of reason, but as we learn more we will hopefully find out.


Yanofsky is at his best extolling the virtues of reason. “Jonas Salk did not find a cure for polio using intuition … Imagination was not used to get humans to the moon … World hunger will not be solved by feelings of love and warmth…”2 Moreover to go beyond reason is dangerous, even if the answers to our problem are out there. Yes there is a shortest route for the travelling salesman, but we cannot know it; and we cannot prove that Godel’s sentence—this logical statement is not provable—is true, even though it is. In short, there is knowledge out there that we can’t access.

Why not then just speculate? Because there is nothing intelligent we can say when we go beyond reason, especially if we are trying to avoid contradictions or falsehoods. To go past reason leads inevitably to mistakes, and is characterized by guesses and conjectures. He counsels accepting our limitations lest we fall into error. Yet there is a bright side; we live, we love, we have aesthetic and moral experiences. As for reason, it both powerful and limited, but it is not all we have. In this we should take comfort.


1. Noson Yanofsky’s, The Outer Limits of Reason: What Science, Mathematics, And Logic Cannot Tell Us, 345.

2. Noson Yanofsky’s, The Outer Limits of Reason: What Science, Mathematics, And Logic Cannot Tell Us, 349.