(All rights belong to their owners. Images used here for review purposes only.)

**The Godelian Puzzle Book: Puzzles, Paradoxes and Proofs, by Raymond Smullyan.**

I first encountered the works of Raymond Smullyan in the book **What is the Name of this Book?** He started as a magician, then turned to mathematics with a degree from the University of Chicago in 1955, and a Ph.D. from Princeton in 1959. Several of his books are designed to teach the math principles of logic, and as such are great primers for this subject. The problem, as I see it, is that many of the readers that did get his puzzle books never actually learned anything from them, and were turned off by the apparent repetition of the puzzle set-ups. I can see that, having read two of his books back to back in the mid-80’s, and then not continuing on with his other titles. On the other hand, I liked his sense of humor, and I think he was a funny writer. Unfortunately, he passed away in February at age 97, and now I’ll never have a chance to meet him in person.

The current title was released in 2013, and was written as an introduction to Kurt Godel’s incompleteness theorems. The first few chapters set up the concepts of logic, infinity and paradoxes, many of which mirror the related chapters in the Martin Gardner **Colossal Math Book**. In fact, Gardner mentions Raymond in relation to Newcomb’s Paradox, and Raymond specifically talks about Martin’s interest in Newcomb’s paradox in this book. Again, everything is connected, and I love it

There’s one particular joke here that I want to repeat. “A certain man was in quest of immortality. He read many occult books on the subject, but none of them gave him any practical advice on how to become immortal. Then he heard of a certain great sage of the East who knew the true secret of immortality. It took him twelve years to find the sage, and when he did, he asked, “Is it really possible to become immortal?” The sage replied, “It is really quite easy, if you just do two things.” “And what are they?” the man asked quite eagerly. “First of all,” replied the sage, “from now on, you must always tell the truth. You must never make a false statement. That’s a small price to pay for immortality, isn’t it?” “Of course!” was the reply. “Secondly,” continued the sage, “just say ‘I will repeat this sentence tomorrow.’ If you do these two things, I guarantee you will live forever.” After thinking for a few minutes, the man said, “Of course if today I truthfully say that I will repeat this sentence tomorrow, then I will indeed say it again tomorrow, hence again the next day, and the next and the next and so on, but your solution is not very practical. How can I be sure of truthfully saying that I will repeat this sentence tomorrow if I don’t know for sure that I will be alive tomorrow? Your advice, though interesting, is simply not practical.”

“Oh,” said the sage, “You wanted a *practical* solution! No, I deal only in theory.”

The second half of the book is more textbook oriented, discussing the actual principles of mathematical logic theory and introducing symbolic logic. And, this is where my eyes started glazing over. If you actually want to sincerely learn the proofs, and you sit down with a pencil and paper and write out the symbols consistently to learn what they represent, coupled with the nomenclature of the proofs, then yes, this is a good, useful book. But there isn’t much in the way of real world examples to link the theory to practical application, so I got lost. I mean, I understood the basic concepts, and I could read some of the expressions in the proofs, but I’d never be able to explain the proofs to someone else at this point.

(Case in point.)

But, the main point is that in mathematics, mathematicians want to create specific types of math with clear rules that either hold, or don’t hold, and can be proven one way or another. Examples would be, in the world of natural numbers (positive integers starting with 1): that even numbers are divisible by 2; odd numbers leave a remainder of 1 when divided by 2; when adding two numbers the order doesn’t matter (2 + 3 = 3 + 2 = 5); if you add two numbers to get a third value, you can use either set when adding a third number (5 + 7 = 2 + 3 + 7 = 2 + 3 + 2 + 5 = 2 + 3 + 2 + 2 + 3); etc. So, we have these math guys trying to prove that the true statements are true, and that the false statements are false, and along comes Kurt Godel with a paper in 1931 proving that in any math system, there will be statements that you can not prove are either true OR false using the rules of that system. You CAN prove the statements one way or another if you analyze the one system using the next system up, but that becomes a meta-analysis, and that’s where logic theory steps in, with a vocabulary that groups statements, formulas, variables and sets together to generalize the system you want to look at. There is something of an example showing how this analysis works involving the Peano axioms as applied to the natural numbers, but that’s at the end of the book, is kind of short, and isn’t actually explained in simple words.

If you look at Smullyan’s puzzles, they often take the form of Knights and Knaves. The knights always tell the truth, the knaves always lie. In some cases, he introduces “certified,” where the status of the knight or knave has been established (or “uncertified” if it’s not established). And, we also get “magicians,” who can be either knights or knaves, but they don’t actually do anything different. Examples of these puzzles are:

1) You’re walking along a road and you get to a fork. One tine of the fork goes to the town of Knights, and the other goes to the town of Knaves. At the fork are two people, A and B, one is a Knight, the other is a Knave. What question can you ask one of them to determine which road goes where?

2) You meet three people (A, B, C), one of whom is a magician. They tell you:

A) B is not both a knave and a magician.

B) Either A is a knave or I am not a magician.

C) The magician is a knave.

Which types are all three people, and which one is the magician?

3) You find an island of certified and uncertified knights and knaves. You meet someone who says “I am not a certified knight.” Is he/she a knight or a knave, and certified or uncertified?

What I found ultimately useful in this book is that Raymond finally explains his puzzles’ tie-ins to logic theory.

1) Knights represent statements that are true.

2) Knaves represent statements that are false.

3) Certified means that the statement is provably true or false.

4) Uncertified means that the statement is undecidable. You can’t prove it is either true OR false.

5) “Magician” represents some property of the statement, e.g. – that natural whole numbers can be even, odd or prime.

So, you could have a statement regarding the natural numbers that is true about a specific property and still not be able to prove it. The rest of the book then goes on to develop the vocabulary and theorems for how to show Godel incompleteness for any system, including logic theory itself.

Overall, I like this book, and I now have a slightly better idea of what Smullyan was trying to demonstrate in his earlier puzzle books. I would have preferred more examples of unprovably true or false statements in number theory, because I do want a practical answer. Also, there are quite a few obvious typos in the book (misspellings, repeated or missing words, and the use of “proportional logic” instead of “propositional logic” on page 256) that were distracting but otherwise non-malicious. This is not a simple child-friendly introductory primer to Godelian theory. But, it could be good for math undergrads. Recommended if you like Smullyan’s other books.

—

Answer 1) If you ask A, say something like “If I asked B ‘does the left fork goes to the town of Knights?’, what would he say?”

Answer 2) A and B are both knights, C is a knave, and A is the magician.

Answer 3) Let’s call the person “A”. If A is a knave, they are lying and then HAVE to be an uncertified knight. But, knights don’t lie, so A can’t be a knave that is lying. However, if A IS a knave, certified or uncertified, then they are telling the truth in not being an uncertified knight, and knaves don’t tell the true. Alternatively, if A is a knight, who always tells the truth, then there is no option but for A to be an uncertified knight. (That is, A is a true statement that can’t be proved.)