Happy St. Patrick’s Day—Again Again.
Leprechauns and proofs
Neil L. is a Leprechaun.
Today I want to relay my experience with him this morning of St. Patrick’s day.
Neil L. came to see me today, he visited me the last two years, and fell asleep on my sofa waiting for me. It is a very comfortable sofa, one that I have used for “power” naps many times. Seeing him asleep I grabbed him. As he awoke he said: “The day, the day what a wonderful day. Please release me and let me go my way.” I said why should I? He remarked that if I let him go he would grant me three wishes. Any. I released him, and he pulled out a pipe and began to smoke. Green smoke came out, which had a surprisingly pleasant smell.
He then added they could not be “any” wishes. Leprechauns have magical powers, but he explained there are limits to my wishes. He said that over the years there have been some abuses of wishes and now there were rules among Leprechauns. The wishes had to be requests that required only information—I could not ask for money, or some strange power, or try to ask for more wishes. The rules went on for quite a while, and I started to become impatient. I wanted to make my first one. At last he nodded with a smile, “please ask away, on such a great day.”
Question and Answer I
The question I thought I had to ask is of course: Give me a proof that
What else? Well I might have asked for a proof that , but I decided to go with conventional wisdom in the spirit of the day.
Neil L. smiled at my request. And after he blew a large green cloud of smoke from his pipe, there appeared in his small hand a single sheet of green paper. He said with a twinkle in his eye: “Here be a proof that . Really a gem of a proof. You should be happy, hope so, hope so.” More green smoke.
The sheet had this written on it:
This is a proof that . So suppose it is false, that is
But this implies that . For large this is clearly wrong and we have reached a contradiction.
I screamed this was nonsense—that is not a proof that is not equal to . No way. He answered, “but the sheet has a proof—perhaps not the one that you were looking for—but a proof.”
I calmed down a bit and saw his point. He did give me a “proof,” of course a proof that was silly. Proofs can be incorrect and proofs can be correct. Okay I think I will have to restate my next wish more carefully. At least I had two more.
Question and Answer II
“I see your game Neil L., and I will be more careful with my next question.” He just smiled and took another puff on his pipe. This time I asked him: I want a proof that a professional mathematician would accept as a correct proof that shows the separation of the complexity classes and . Oh and make the proof in English and in a normal sized font. I thought he just might give me a unreadable proof. Ha. I thought this time I had him.
He said, “Clever. You could have gotten a proof by a great troll who knows her maths well, and only knows a long dead language.” Again after a puff of green smoke he handed me a book—it must have been over a 500 pages. He said with a twinkle in his eye: “Here be a proof that . Really a gem of a proof. You should be happy, hope so, hope so.” More green smoke.
I opened the book to the first page. Then scanned another page, and then jumped to the middle. It was mathematics. He had not cheated me there. But the proof was unreadable to me. Here are the opening sentences:
This is a proof that . As is well known  we need only show that every loose co-double sheaf over the -adic structure of the polynomials of locally bounded degree, is either final in the sense of Williams  or is near-final and is quasi-closed in the natural topology induced by the Fortnow-Tao ring .
The proof went on for hundred of pages like this. I quickly turned to the references. They were there, but some were to papers that were published in years ahead—one in the year 2033, for example. What is the “Fortnow-Tao ring”?—it had not been invented yet.
He had gotten me again. It was a professional proof, and might even be right, but I would never understand it.
Question and Answer III
I was now sorry I had let him go. He sat there looking at me with his pipe and puffing out green smoke from time to time. He seemed very happy with himself. He said, “You have one request left, what be your pleasure?”
I finally figured it out. I asked him
He waved good-bye and in a final puff of green smoke was gone.
Happy St. Patrick’s Day. What did I ask him? What would you have asked?
There is one more puzzle today—Neil L. has a secret, can you see what it is?