There are two errors in the
the heading of this week’s post

The theme of this week’s post is adapted from here.

When I was a boy, my grandfather claimed he owned the very axe that Thomas Jefferson used. Of course the handle had been replaced three times over the years, and the head replaced once. Later I had reason to believe that the old man had purloined that story and the axe really had never belonged to the president. A man in Hollywood claims to have part of the Olympic torch because he lit a match from the passing torch and later was able to light the pilot light in his water heater from a flame that had been passed from match to match until he got home. These are variations of the story of the ship of Theseus given at the same Web site listed above.

Turning the paradox of preservation of identity around the other way, consider the dilemma often posed in science fiction stories involving multiple copies of people being made in teleporting. Which copy is the real person? If the beaming process requires disassembling a human atom by atom and re-constructing the same arrangement at a distant location using other atoms, did the person die? Is the copy the same as the original person - including putative aspects such as having a soul? When it comes time to beam the person back home, which of the two copies gets to go? Would either of them want to go? Obviously if the process involved dying, then they might not want to go since they both were just born - or were they?

How do you know you were not just beamed here with a complete set of self-consistent memories about a non-existent past? Some creationists believe that Earth was created about 5,000 years ago by a theistic deity who saw fit to include in creation a complete history. Geologists and fossil hunters are simply studying what was in God’s mind when he created the universe. How would you refute this conjecture? Does it constitute the basis of a theory?

What happens when an unstoppable force meets an immovable object?

Puzzles and paradoxes like these are much more than simple pastimes. Thinking about them can help us better understand deep mysteries. Russell’s famous paradox of the barber can be dismissed as a joke, or it can help catalyze progress in understanding the principles of logic.

Sometimes what we normally consider to be purely secular, paradoxes become part of a religious experience. The paradox of Thomas Jefferson’s axe is mirrored in the Buddhist concept of self, but they express it differently. An example might be the statement, “You can’t step in the same river twice.”

Much has been made (often wrongly) about Gödel’s proof that in any sufficiently strong formal logical system, one can find theorems that are true, but not provable. As with the uncertainties associated with quantum mechanics, these logical and physical observations can be used in arguments supporting the existence of free will and other aspects of religion. I leave it to the reader to decide if such applications are justified.

So did you find the two errors in the heading at the top of this entry?

In response to the interest my original tutorial generated, I have completely rewritten and expanded it. Check out the tutorial availability through Lockergnome. The new version is over 100 pages long with chapters that alternate between discussion of the theoretical aspects and puzzles just for the fun of it. Puzzle lovers will be glad to know that I included an answers section that includes discussions as to why the answer is correct and how it was obtained. Most of the material has appeared in these columns, but some is new. Most of the discussions are expanded compared to what they were in the original column format.

[tags]godel’s proof, barber paradox, decision theory, thomas jefferson’s axe[/tags]