The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us
Noson S. Yanofsky
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Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes.
Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known.
Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there.
natural number. If one were to say that subset D corresponds to number d0, then see if the number d0 is in D. d0 is in D if and only if d0 is not in the subset that corresponds to d0. That is, d0 is in D if and only if d0 is not in D. This is a contradiction. We conclude that the subset D is different from the subset corresponding to number d0. In fact, D is different from any subset in the proposed correspondence. Hence, our correspondence is missing at least one subset. The natural numbers
person who stands in the doorway of a room is both in the room and not in the room. • How many hairs does a man have to lose in order to be considered bald? Depending on which way the wind blows, he is sometimes considered bald and sometimes considered not bald. • Is 42 a small or large number? Human beings use vague ideas all the time. Our mindset and our concomitant human language are full of vague statements: • Sometimes we say people in a doorway are in the room and sometimes we say they
the sphere to that point. Saying that the directions have or do not have spin is like assigning 1s or 0s to the points of the sphere. We have the following conditions on assigning the 1s and 0s: 1. If a particle is spinning in one direction then it must also be spinning in the opposite (antipodal) direction. So if a 1 is assigned to a point on the sphere, then a 1 must be assigned to the opposite point because it is the same direction. Similarly, if a 0 is assigned to one point, it must be
analysis of a pun. Let us move on.) I close this introduction with a few questions about the nature of reason and its limitations. Read the book with these questions in mind. I return to these issues in the last chapter and perhaps get closer to the answers using some of the ideas presented in the book. I would be remiss in writing a book titled The Outer Limits of Reason without giving a definition of reason. After all, how can we say something is beyond the limits of reason if we do not
These instrumentalists are only concerned that the equations work and make the correct predictions. They see no reason to pay any attention to why the equations work or what the underlying reality of the physical universe is. Their motto is “Shut up and calculate!” They believe that one should not waste time pondering what is going on “under the hood” and question whether a deeper reality even exists. To them, the underlying nature of the real world is either beyond us or is not worthy of