Facing nonexistential realities

Neil L. is a Leprechaun. He has graced these pages before.

Today, the day before St. Patrick’s Day, we ponder universal riddles of existence. Ken, who will visit me this coming week, insisted on reporting what happened this morning.

I woke up early, walked alone into our living room, and was amazed to see Neil already here. He was looking out the window at dawn sunlight flooding over St. Patrick’s Cathedral and the many other landmarks we can see from our apartment in midtown Manhattan.

Spring has at last given an earnest of coming. The scene was transfixing, exhilirating, all the glory of nature except for the green-clad little man puffing his pipe with his back to me.

“Neil—?”

He turned and tipped his hat to me but then turned back to the window. I wondered if I had caught him in meditation. I stood back until he turned a second time. He had always visited me late on the evening before St. Patrick’s Day, or the morning of it.

“You’re here early.”

“Aye. Top o’ the morning to ye.”

“Why?”

He gave me a long stare but warmly, not hostile.

“By the living daylights in ye…”

Then I understood. The Romans have their Ides, the Irish have the 17th, but I got the day in between—when last year I was not here but in an operating room for long hours.

“Thanks” was all I could say. I had actually come out to look up a mathematical idea for a “part 2” post I’ve been struggling with ever since posting the “part 1” years ago. This pulled me back to math, then to how Neil has always tricked me and I’ve never been able to pin him down. I realized this might be my best opportunity.

“Neil, may I ask you a Big Question? Not directly personal.”

He simply said, “Aye.” No tricks yet.

## The Question

I had my chance—something I’ve wanted to ask him for years but not had an opening for.

“Neil, what can you tell me about female Leprechauns?”

Neil had mentioned family, cousins, even siblings—I figured they had to come from somewhere. But nothing I’d read mentioned female leprechauns. I braced for silence again, but Neil’s reply was immediate and forthright:

“They comprise the most heralded subset of our people. They are included in everything we do.”

“How so?”

“Our society agreed early on that every female leprechaun should have the highest station. All female leprechauns are sent to the best schools, with personal tutoring reserved in advanced subjects. They are our superpartners in every way.”

“How do you treat them?”

“If a lady leprechaun applies for a position, she is given full consideration. Whenever a lady leprechaun gives advice, it is harkened to. None of us has ever ‘malesplained’ or demeaned a lady leprechaun in any way.”

“Are they as short—uh, tall—as you?”

“Every adult female leprechaun is between one cubit and one-and-a-half in stature, like us menfolk. We don’t have quite the variation of your human species.”

“I would surely like to meet one.”

Neil heaved a sigh.

“Ye know the trepidation with us and your womenfolk.” Indeed, I recalled Neil’s story of what was evidently his own harrowing encounter with the wife of William Hamilton. “It is symmetrical, I’m afraid. No female leprechaun has ventured into your world.”

“But can you give me an example of a female leprechaun? Someone who has done something notable.”

## The Reality

Neil puffed on his pipe. I knew I had him cornered and moved in.

“Neil, is everything you just said about female leprechauns true?”

To my surprise, he answered quickly.

Aye. As in mathematics, each of my statements was perfectly true.”

Oh. Then I realized how empty domains are treated in mathematical logic. I looked at him sharply.

“Neil, there are no female leprechauns. All of your statements have been vacuous, bull-.”

“That’s not right or fair. That which I told ye have been important values of our society from the beginning. Many studies of female leprechauns have established their natures precisely. We have devoted vast resources to the quest for female leprechauns. Readiness—how we would integrate them—this is enshrined in all of our laws. Even your late Senator, Birch Bayh, was informed by us at Notre Dame.”

Neil’s sincerity was evident. I still felt bamboozled. He carried on.

“Ye have whole brilliant careers studying physical objects that may not exist. The analysis of them is still important in mathematics and other scientific applications. And in mathematics itself—”

I was actually relieved to turn the subject back toward mathematics, as Neil and I usually talked about.

“—your most storied advance of the past quarter century was achieved by ten years’ concerted study of the Frey-Hellegouarch curve. Which by Fermat’s Last Theorem does not exist.”

## Existence

I could not argue with that. This set me pondering. Whether mathematical objects exist in reality defines the debate over Platonism. But what, then, is the reality of nonexistent mathematical objects?

Nonexistent objects still follow rules and aid deductions about objects that do exist. So, they too exist? Moreover, we often cannot prove that those objects do not exist—for all we know they may be just as real. Polynomial-time algorithms for satisfiability and many other problems—those are bread and butter in theory. I wanted Neil to tell me more about some of them.

“Neil, have your folk worked on the projective plane of order 10?”

Aye. Not only did we long ago have all the results of this paper on them, our engineers recommended it as the thoroughfare pattern for new towns.”

“You have built them?”

“Indeed—in two new districts around Carlingford back home.”

“But they don’t exist.”

“By our laws and treaties they exist until your people prove they don’t. Then they have a 50-year sunset. Which means the towns have 20 years left.

“It’s actually a much better system than the Scots have. Their villages of this type such as Brigadoon come back every 100 years just for one day. But Brexit may put paid to our two towns sooner. We are protected under E.U. law, but those towns are just over the Northern Ireland border.

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Nay. Ye know the rules. We can nay tell that which your folk do not know, unless there is nothing else ye learn that ye could not learn without us.”

I pondered: what could I ask about computational complexity that Neil could respond to in this kind of zero-knowledge way? But Neil cut in.

“It will be much more fruitful to discuss objects that ye know do not exist—and yet ye use them anyway.”

## Mathematical Leprechauns

I thought, which could he mean? I recalled the story we told about a student getting his degree for a thesis with many new results on a strong kind of Lipschitz/Hölder function even though an examiner observed that non-constant ones do not exist. Ken heard the same story as an undergraduate, with the detail that it was an undergraduate senior thesis, not PhD. The Dirac delta function? Well, that definitely exists as a measure. I forgot an obvious example until Neil said it.

“There is the field with one element: ${\mathbb{F}_1}$.”

Of course. We posted about it.

“It is an active research subject, especially in the past dozen years. Your Fields medalist Alain Connes has co-written a whole series of papers on it. Yuri Manin also. Your blog friend Henry Cohn got a paper on it into the Monthly—think of the youngsters… Edifices are even being built upon it.”

“Can you give some less-obvious ones for our readers?”

“There is the uniform distribution on the natural numbers. This paper gives variations that do exist, but it makes the need and use of the original concept clear.”

Ken had in fact once finished a lecture on Kolmogorov complexity when short on time by appealing to it.

“There are fields F whose algebraic closures have degrees 3 and 5 over F. Those are highly useful.”

I knew by a theorem of Emil Artin and Otto Schreier that they cannot exist. But it struck me as natural that they should exist. I wondered how much one would need to tweak the rules of group theory to make them possible.

“Then there is the free complete lattice on 3 generators.”

Wait—I thought that could exist if one loosened up set theory. So I asked:

“Neil, do some of these objects exist in alternative worlds known to leprechauns in which the rules of mathematics are crazy, not like the real world?”

“What makes you think ye don’t live in one of those other worlds?”

Before I could react, Neil picked up a solid glass sphere art object that belonged to Kathryn. He pulled out a diamond-encrusted knife and slashed it as green light flashed in every direction until he stopped. Then from his hands he gave me two glass spheres of the same size.

“For the happy couple—la le Padraig suna ye-uv!

And he was gone.

## Open Problems

What is your view on the nature of mathematical reality? Do nonexistent mathematical objects exist? What are your favorite examples?