Just a quick note: This short article is densely written.  I couldn’t really find a way around that, other than not to post it, but I really do hope somebody will take enough interest to say whether they think it’s valid or not.  So, if you venture inside (and I hope you do), please take your time, follow the links, and have a look at the supporting material, especially the article referenced in the first paragraph (after this one), where I first tried to tell you what I’d been thinking about.  So, without further ado, the article itself:

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Not long ago, I came across new (to me) material that seems to support my speculation that there are trans-Euclidean geometrical models possible, which make positive and negative infinity the same. My previous article depends on the idea that an infinitely large circle could be equivalent to a straight line, and thus that the number line, with which most of us have been familiar since first grade or earlier, could be looked at as an infinite circle.

A couple of months ago, I was participating in an online version of a college bull-session, when I was referred to two different books about the study of infinity throughout the ages: The Infinite Book: A Short Guide to the Boundless, Timeless and Endless by John D. Barrow and A Brief History of Infinity: The Quest to Think the Unthinkable by Brian Clegg. I found both books fascinating, and both mentioned an early Renaissance thinker I hadn’t known about, known to Latin-friendly English-speakers as Nicholas of Cusa.

This thinker, in a philosophical work finished in 1440, called On Learned Ignorance, talked about the equivalence of an infinite straight line and an infinite circle. He also equated a triangle having an infinite base with a straight line, and did a number of other neat geometrical tricks, all in the name of thinking about God. An article I found online, called Nicholas of Cusa and the Infinite by Thomas J. McFarlane, summarizes his views on infinity along with the history of the study of the infinite before and since.

In short, I think that Nicholas of Cusa would like my idea. On Learned Ignorance has a very Zen feel to it – the reader is asked to embrace contradictory ideas, and ideas that on their own don’t seem to make sense. But, instead of a Zen Koan to shock the mind out of logical thinking, Nicholas invokes the idea of infinity to make the impossible seem possible, and the absurd seem plausible. The idea that a triangle with an infinite base is a line, for example, is used as an illustration of the Trinity – how three persons could equal one God.

Given these Zen-like goals, what could be more opposite than positive and negative infinity? This is the coincidence of opposites carried to its logical extreme. I couldn’t find any reference to a number line in On Learned Ignorance, but this isn’t too surprising, since René Descartes, of Cartesian Coordinate fame, wasn’t even born for another 150 years.

The reason I’m writing this, though, is not to assert that Nicholas of Cusa would like my argument, but to demonstrate that there’s a precedent for thinking of an infinite circle being equivalent to a straight line. I’ve looked at numerous references to this, from works cited above to little, one-paragraph “Doesn’t this just blow your mind?” blurbs, and I haven’t seen any mention of it being contradicted or refuted since it was written. Now, mathematics has progressed quite a bit in the last 570-odd years, so I can’t say that On Learned Ignorance validates my idea. But at least I have a philosopher, still known and respected, who wrote about a major basis for my idea.

Maybe having a real philosopher to reference will interest someone in commenting on the idea. I still haven’t found anyone to give an opinion about whether the idea is valid or not. Maybe the name of Nicholas of Cusa will stimulate discussion that so far has gone unstimulated. Can the infinite, properly analyzed, be equated with its own opposite?

Part of the explanation was that I was intentionally playing dumb. Those financial matters, who got paid what, were clearly none of her business. And it’s not as if she can exactly keep a secret. She may be getting older and more mature, but my favorite little girl in the world still lets pretty much everything that goes into her ear come back out of her mouth. You can’t set your watch by it, but you can essentially count on it coming out eventually. So, anything I want kept from the world at large has to be kept from her, too.

On the other hand, I really was having trouble understanding the question, and that played an equally strong role in why I wasn’t answering it. I think she was after something like salary independent of expenses, but the way to express that in the language of a seven-year-old is elusive, and I just wasn’t getting it. But, rather than blame the limitations of the linguistic tools at her disposal, she blamed the man trying to listen.

“A stupid daddy, even dumber than you, would get it!” she declared.

“Does that mean I’m even dumber than a guy who’s dumber than me?”

“YES!”

Now, normally, I’d believe I’d caught her in a contradiction, but this is a girl who’s been wrestling with the idea of the infinite for some time. For any finite level of stupidity, it’s not possible for me to be dumber than a daddy who’s dumber than I am. But, for the infinitely stupid, such a thing is possible.

This may also be one of those times when intuition trumps logic. Even in the real world, it’s possible for the “dumber” of two people, whose intelligence is measured by any criteria you please, to understand something that the “smarter” one fails to grasp. The old Far Side cartoon comes to mind, with a student pushing with all his might on the front door of a school for the gifted. On the door is a sign that says “Pull”.

But that’s really just a matter of two different criteria for intelligence being used. The apparent contradiction isn’t real. But it also somehow makes intuitive sense to say that that someone is so dumb that, not only is it impossible for anyone else to be dumber, it’s even impossible for someone else who is dumber to be dumber – or even quite that dumb. Again, it doesn’t make sense logically at all, so you have to banish logic from your intuition for it to make sense. Really, it’s just a long way of saying someone is impossibly dumb.

So, I think that’s what was going on. She invoked infinite regression to declare that I’m impossibly dumb.

Well, I’ve been called worse.