Thinking About Infinity. Check My Math.

I have been thinking about infinity.

I was never good at math in school but I have always been fascinated by numbers. Here is what I have been running through my thoughts. Check my math.

infinity + 1 = infinity, which makes it seem like that 1 is a zero – no effect.

What about infinity minus 1? It has to be less than infinity. Right? So, what is the answer?

infinity + infinity = infinity

But infinity – infinity = 0

Two things inspired this infinitely frustrating thought experiment. First, I watched the film A Trip to Infinity (on Netflix). This 2022 documentary explores the concept of infinity through interviews with mathematicians and physicists.

The second inspiration was the much lighter sitcom Young Sheldon. In a recent episode, the precocious and young genius Sheldon comes to doubt the existence of zero. He is tutoring his not-very-bright neighbor Billy in math. During the session, Billy naively asks how zero can simultaneously exist as something but be nothing. The question causes Sheldon to have a kind of existential crisis. He turns to the two professors he works with and they can’t really answer the question and have some mathematical doubts too. It’s not unlike the physicist and mathematicians in the infinity film who have answers about defining infinity but don’t really agree or even seem very confident.

Sheldon rejects religion and God which is very important to his very Christian mother. Somewhat incongruously, when Sheldon talks with Billy again, Billy suggests they just pretend zero exists. Sheldon interprets this as an act of faith and that restores him.

It’s not that you can’t find a definition of “infinity.” It is that which is boundless, endless, or larger than any natural number. The ancient Greeks discussed the philosophical nature of infinity. In the 17th century, we get the infinity symbol and infinitesimal calculus. Working in the foundations of calculus, it was unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done.

By the end of the 19th century, people were studying infinite sets and infinite numbers, and infinity was clearly a mathematical concept. In physics and cosmology, whether the Universe is infinite is still an open question.

There is a section of the film that I rewatched and it still doesn’t make sense. One physicist says that if you place an apple in a box it will decay into mush and then dust. Then, it becomes microscopic particles and then it becomes one with the universe. Whoa. Give it enough time, and it will become an apple again. What?

I think the connection between the film and the TV episode is the futility of wrestling with paradoxes. You probably will end up accepting that with all of our knowledge we will likely never explain or comprehend the greater existential realities of the universe.

Aristotle said that the more you know, the more you realize how much you don’t know. Not that we shouldn’t think about these things. Just don’t expect an answer.

Zero Is Not Nothing

“If you look at zero you see nothing;
but look through it and you will see the world.” – Robert Kaplan

“an O without a figure” – William Shakespeare, King Lear

“Where did you go?”
“Nowhere.”
“Then why are you late?”
– exchange between a father and son
on a Sumerian clay tablet from 5000 years ago

I remember vividly when one of my sons at age 5 realized that zero was very important. He has a mathematical mind (he ended up in the finance world) and it hit him that nothing in math works without zero. He would discover in the years to come that it plays a role in many other things too.

It is a symbol of what is not there. Add a zero with any number and it does not change. Add a zero to the end of any number and it increases. A bit of a paradox.

This concept was invented (or is the proper word “discovered”?) in pre-Arab Sumer. It got a symbolic form in ancient India.

It obviously is mathematical but it figures then into other areas as large as the universe itself.  Mathematician Robert Kaplan follows this symbol’s history in his book The Nothing That Is: A Natural History of Zero. It is partially a cultural tale, part scientific discovery, part law of nature and it has a few strokes of Romance.

The Sumerians counted by 1s and 10s but also by 60s. That’s not so foreign to us if you consider our 60 minutes in an hour and that 6 × 60 = 360 for the degrees in a circle. Kaplan reminds us hat we have a bunch of number systems: 12 for months in a year, 7 for days in a week, 24 for hours in a day and 16 for ounces in a pound or a pint. All these systems were our ways of trying to make sense of the universe. and gigantic concepts like Time.

Our ancestors pre-zero were clearly handicapped in dealing with large sums. (Kaplan says to try multiplying CLXIV by XXIV.) The mathematically astute Greeks didn’t really have zero. It was Indian mathematicians who treated zero like any other number, instead of as a symbol.

I find math fascinating but have always been math-phobic and always struggled with math courses once we got past the arithmetic and into algebra and beyond. So, I enjoyed the story of zero. For example, how in the Middle Ages, zero comes to western Europe via Arab traders.

It was considered “dangerous Saracen magic” and associated with the Devil. But those who worked with numbers every day were not only the rare mathematicians but any merchants or money lenders.

All this leads to double-entry bookkeeping, equations, the invention of calculus, and the scientific revolution. Leap into our lifetimes and most people know that computers see everything as zeros and ones.

Kaplan’s other books were on the library shelf besides this one and they are all math-related. The other title that got my attention is The Art of the Infinite. (It has a subtitle that may seem impossible to some of us: “The Pleasures of Mathematics.”) I do recall that for me as a child it wasn’t so much zero that caught me by surprise as it was infinity.

I think I have come to understand zero. I’m not sure I have any greater grasp of infinity than I did as a child. In the same way that I now know much more about the stars and the heavens than I did as a child, I still look up at the night sky with a childlike wonder and know I will never understand it all. Others know far more than me and yet they will never understand all of it. In physics and cosmology, whether the Universe is infinite is still an open question. My lack of knowledge and my wonder are infinite.

I asked earlier if zero is discovered or invented, and it turns out that this was a question famously debated by Kurt Gödel and the Vienna Circle. Kaplan writes, “The disquieting question of whether zero is out there or a fiction will call up the perennial puzzle of whether we invent or discover the way of things, hence the yet deeper issue of where we are in the hierarchy. Are we creatures or creators, less than – or only a little less than — the angels in our power to appraise?”

Other cultures, disconnected from Mesopotamia, Greece, India, or Europe, such as the Mayans, also had to discover zero. (Ah, “discover” – and so I have revealed my answer to that question.)  As infinity began as a philosophical concept before it became mathematical, zero moved from math to philosophy.

Nothingness as a philosophical term is a huge topic of its own. Nothingness is the general state/domain/dimension of nonexistence. It is where things pass to when they cease to exist. But it can also be where they come into existence, as in some cultures where God is understood to have created the universe ex nihilo, “out of nothing.”

Nothing is infinite.

Yes, I see the paradox.

Consider Infinity

Science and atheism usually sit at the same table. They are friendly. Sometimes they sit at the agnostics table, and it’s not that they are never friendly to the believers, but they have their usual place.

One time, over drinks, one of the believers said to science, “Well, I know you believe in one miracle.”

“Oh, what’s that?” said science, laughing.

“The Big Bang. Everything from nothing. That’s a pretty big miracle.”

Which brings us to infinity. It’s a topic so vast and unimaginable for most of us to wrap our brains around.

Infinity? Forever? Wait, what came before that big bang?

Physicists have a hard enough time figuring what happened at the very first moment of the big bang. But what about before that? Did time or anything exist before it?

Theories are out there. Maybe there was a series of bangs and they keep happening.  What about that whole string theory thing?

Maybe the universe isn’t infinite. Maybe it is just really big. Sounds like a joke, but cosmologist Janna Levin uses that kind of questioning in one of her books. And she is looking at a group of Big Questions –  black holes, the big bang, extra dimensions, and dark energy – questions so big we have to sometimes laugh.

That hit me hard in the funny part of the brain in Woody Allen’s Annie Hall. The idea that the universe is expanding is scary.

In Janna Levin’s How the Universe Got Its Spots: Diary of a Finite Time in a Finite Space we can follow her through the the paradoxes of finitude. Those hot and cold spots left over from the Big Bang have a pattern that may eventually reveal the true size and shape of the cosmos.

“For a long time I believed the universe was infinite. Which is to say, I just never questioned this assumption that the universe was infinite. But if I had given the question more attention, maybe I would have realized sooner. The universe is the three-dimensional space we live in and the time we watch pass on our clocks. It is our north and south, our east and west, our up and down. Our past and future. As far as the eye can see there appears to be no bound to our three spatial dimensions and we have no expectation for an end to time. The universe is inhabited by giant clusters of galaxies, each galaxy a conglomerate of a billion or a trillion stars. The Milky Way, our galaxy, has an unfathomably dense core of millions of stars with beautiful arms, a skeleton of stars, spiraling out from this core. The earth lives out in the sparsely populated arms orbiting the sun, an ordinary star, with our planetary companions. Our humble solar system. Here we are. A small planet, an ordinary star, a huge cosmos. But we’re alive and we’re sentient. Pooling our efforts and passing our secrets from generation to generation, we’ve lifted ourselves off this blue and green water-soaked rock to throw our vision far beyond the limitations of our eyes.”