Meta Wishes

Neil L. is a leprechaun. He has been visiting me or Ken once every year since we started GLL. We had never seen a leprechaun before we began the blog—there must be some connection.

Today we want to share the experience we each had with him this morning of St. Patrick’s Day.

That’s right—Neil visited both of us separately. Four years ago, when I had major heart surgery on St. Patrick’s Eve, Neil came to my previous New York apartment, grabbed a page of my notes, and took it to Ken. The past two years, Ken was on Zoom with me when Neil appeared. This time, Neil had a bone to pick with Ken—after posing a problem to me.

Neil’s Problem

Kathryn, my dear wife, and I are still in our new midtown Manhattan apartment after my procedure two weeks ago. She went to sleep in the bedroom, but I stayed up awaiting Neil’s arrival. I must have dozed off, but jolted awake to his laugh and the waft of his pipe’s green smoke.

Neil said hi and explained right away that he had a problem. “A classic problem that leprechauns have faced forever.” I nodded to him and rubbed my eyes to be awake enough to listen. Neil continued:

The other day I was minding me business and I fell into a trap. Nasty fall. The trapper was an old foe of mine. She told me she had three wishes. And I was bound to grant them as usual. Sae much, sae normal. But then she breached the rules.

No More Wishes

Now I knew the rules of wishes, so Neil did not have to tell me: Must be about something contingent, not love or death, no self-transmogrification, and most of all—well, let’s hear it from the genie in Disney’s Aladdin:

“Three wishes, three. Uno, dos, tres, no substitutions, exchanges, or refunds, and ixnay on the wishing for more wishes.”

Those were fair and I showed I could handle them forthwith. But then she asked for:

‘Please ten chirag oil lamps, each with a genie who can grant three wishes.’

This was terrible. It sent me to the lore. Ye cannot ask your servant for more wishes. Ye cannot involve another leprechaun. But lamps of themselves are just objects. That a chirag lamp has a genie is like an oyster has a pearl. Even if half the lamps be duds, that still compounds the wishes—and she wished for ten that were not duds.”

Math and Myth

Neil puffed a few more times, as if really expecting an answer from me. He even prompted, “What do ye think?”

That sent me into befuddlement. Usually I try to engage Neil on a math problem, to trick him into telling the answer to Riemann or factoring or P=NP. But despite the “” aspect, this wasn’t math—it was more about myth. Math I can take seriously, but with weighty matters personal and worldwide, I did not want to play “meta” games.

Neil read my mind: “Nay, I assure ye—it be really about math. Even the Riemann—”

That made me think: if Neil knew that it had real math content, he must know the answer already. No sense groping for it groggily. I just retorted: “I don’t want to guess the answer. Please just tell me.”

“Ye know there be only one way I ever tell ye answers…”

“Sure. OK.” I really wanted to join Kathryn for some sleep.

I blinked as Neil vanished in green light. Only his pipe smoke remained, and it curled around into a shape I couldn’t place at first. It took awhile to become sharp enough that I could tell what it was:

Myth and Math

I, Ken, shall tell the rest. Ordinarily, I would have been eager to engage Neil again about the NCAA basketball tournaments. But I too am too touched by the same weighty matters, and quite forgot the day.

My unawareness did not matter because Neil apparated and instantly started confronting me: “What gave ye the mickey to write of mathematics being ’emergent’? Ye dinna even define it.”

I had to concede I’d not only been vague but had analogized it too to two senses of “emergent” in philosophy. I started to explain: “It’s like Albert Einstein’s famous question, ‘Did God have a choice?’ among physical laws. Now about math…”

Neil cut me off: “When Einstein said ‘God,’ he meant nature—but when you say it, you mean leprechauns. You are asking: ‘Do we leprechauns have the power to change mathematical truths in advance of your learning them?’ That’s our turf—!”

I stammered that I had a right to pose the question and had not meant to insinuate. But Neil kept on: “If we really could change outcomes then what would maths rest on? On myths just as well. But where ye speak of law, we have lore. Established rules. And they suffice.”

This was waxing cryptic and I thought bringing up Kurt Gödel would only make it more so. Neil sensed I was adrift and went on: “I shall inform ye via the same story I told Dick.”

Neil’s Solution

Neil unfurled his story. At his same pause, one perception dawned on me: “Neil, did the lady’s wish imply a recursion—meaning the genies would be asked for a wish generator in the same proportion?”

Neil nodded: “The lore says yes—the lore on hearing what is said. That is what I consulted it for. The rest I could work out on paper.”

The paper part was easy up to a point. The wishes implied an infinite summation: the original , then more of the ten genies, with the third wishes to each bringing at the next level, for more wishes there, and so on. In sum,

wishes. Clearly the number of wishes is bounded by the series

Which equals . Without asking Neil how he’d computed that, I went to an online zeta calculator and obtained a bounding total of

wishes. Neil puffed and chortled and finished his story:

“So the unlucky lass had wished herself into wishing a grand total of less than one-fiftieth of a wish. Since only whole wishes can be honoured, this sprang me from the trap and kept me both me pots o’ gold to boot. I gave her one coin as a peace offering.”

I joined the laughter for just a moment. This followed mathematical rules but outlandishly, and I groped for the point. But Neil supplied it directly:

“Some of your fellow travelers have felt reaching answers by non-constructive methods to be less outlandish only in degree not kind. If ye later apprehend the answer by calculation—which is what your post styled as “emergent” knowledge—then it must be exactly the same object previously described non-constructively. Thus in the realms ye ascribed to us wee folk, we are the guardians and gatekeepers of truth, not the forgers of it.”

Neil tipped his hat and simply faded out—no green smoke for me.

Open Problems

Can Neil’s zeta-function calculations insulate against any attempt at recursively gaining infinite wishes? The zeta function whips up and down in tune with the Bernoulli numbers, but the leprechauns do have freedom to choose a bounding series. Or is there an infinite series scheme that rises above 3 wishes even so? We hope this has given some St. Patrick’s Day diversion.