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.

Walking with Einstein and Gödel

I picked up the book When Einstein Walked with Gödel this past week at the library because of the title and the photo on the cover of the two mathematicians walking across a campus in Princeton, New Jersey.

I was disappointed that the entire book was not about the two of them, but is instead a collection of essay by Jim Holt. The title essay is one I really like as it deals with one of my favorite topics – our changing notions of time. It comes from a friendship between Albert Einstein and Kurt Gödel when they were both working in Princeton in the 1930s. Einstein had shaken the physical world with his work, and Gödel had shaken mathematics. They ended up taking almost daily walks to their offices at the Institute for Advanced Study.

Gödel would have looked pretty fancy (he liked white linen suits) and Einstein would have looked like the absent-minded genius that we know with his crazy hair and too-big pants.

But what really interests me in reading the essay today was the walking. Today was a very nice spring day that was warmer than it has been. I took the covers of the deck furniture and sat outside with my lunch and coffee. And I went for a walk.

I love walking and I am a firm believer in the power of walking to spark creativity and thought. (More on that tomorrow) Of course, it would be great to have the content of those walking conversations between Al and Kurt. I imagine that the conversations went beyond math and physics, though I’m sure math and science were the main themes.

I have so far only skimmed a few of the other essays in the book, but each could be a walking conversation. Did you know that the word “scientist” was only coined in 1833? It was a philosopher, William Whewell, who used it in his efforts to “professionalize” science and separate it from philosophy. Holt quotes Freeman Dyson (another person at the Institute who I actually got to meet and talk with briefly when he gave a talk at NJIT) as saying that “Science grew to a dominant position in public life, and philosophy shrank. Philosophy shrank even further when it became detached from religion and from literature.”

I certainly couldn’t keep up with Einstein and Godel on the science of time, but I would love to put in my own ideas and get some feedback from the boys.

Some of Holt’s questions that he attempts to answer in the essay are also intriguing ideas for a walking conversation. Does time exist? What is infinity? Why do mirrors reverse left and right but not up and down? And the biographical sketches of famous and not-so-famous thinkers makes me want to go on walks with them too – Emmy Noether, Alan Turing, Benoit Mandelbrot, Ada Lovelace and others.

The Answer to Life Is 137

When Douglas Adams wrote The Hitchhiker’s Guide to the Galaxy, he wrote that “The answer to the ultimate question of life, the universe and everything is 42.”  He was joking, but I wonder if the answer really might be 137.

Take a look at one thing about 137 in mathematics: Using two radii to divide a circle according to the golden ratio yields sectors of approximately 137° (the golden angle) and 222°.

In physics, 137 is the approximate denominator of the fine-structure constant. Being a dimensionless physical constant, it is approximately 1/137 and has the same numerical value in all systems of units.

Physicists have postulated for more than a hundred years that 137 might be at the center of a grand unified theory, relating theories of electromagnetism, quantum mechanics, and, especially, gravity. It’s the DNA of an atom.

As the inverse of the fine-structure constant, it is related to the probability that an electron will emit or absorb a photon (Feynman’s conjecture).

Some physicists has suggested that if the number that unified the relationship between all these concepts turned out to be 1 or 3 or a multiple of pi, that would make more “sense.” But why 137?

Leon Lederman thought that because the number 137 “shows up naked all over the place,” that means that scientists on any planet in the universe using whatever units they have for charge or speed, and whatever their version of Planck’s constant may be, will all come up with 137, because it is a pure number.

But it shows up frequently outside of math and physics.

In mysticism, the Hebrew word קבלה (Kabbalah) has a Gematria (numerical value) of 137.  It describes the “corresponding loops” which clasped together enjoin the two sections of the Tabernacle’s ceiling. These loops divided the Holy Place and the Holy of Holies – the physical dimension and the spiritual dimension – and at the boundary line of the physical world, the number 137 emerges.

Moses’ Tabernacle, the earthly dwelling place of God, was 13.7 meters long. NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) has taken the best measurement of the age of the Universe to date. and ”scientists now have the best estimate yet on the age of the Universe: 13.7 billion years.”

Some people have connected the science, math and mysticism. 137 refers to electrons and the odds of an electron absorbing a single photon. In simple Kabbalah language, 137 is about Vessel and Light. It is about the physical body of man (Vessel) and our ability to ignite the Light in the soul.

One of the important physicists of the 20th century, Richard Feynman, wrote about the number 137:

“It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the ‘hand of God’ wrote that number, and ‘we don’t know how He pushed his pencil.”

According to the Bible, Abraham died at age 175, but when he was commanded by God to offer his son up as a sacrifice, he was 137. According to the Torah, Moses’ father lived to 137, and so did Ishmael and Levi.

Physicist Leon M. Lederman numbered his home near Fermilab 137.  He tried to unite the Ancient Greeks’ earliest scientific observations, Einstein, and the Higgs boson, which is nicknamed the God Particle.

“One hundred thirty-seven is the inverse of something called the fine-structure constant. …The most remarkable thing about this remarkable number is that it is dimension-free. …Werner Heisenberg once proclaimed that all the quandaries of quantum mechanics would shrivel up when 137 was finally explained.”
― Leon M. Lederman, The God Particle: If the Universe Is the Answer, What Is the Question?

Wolfgang Pauli, a pioneer of quantum physics, died in a hospital room numbered 137, a coincidence that disturbed him.

Physicist Pauli and psychoanalyst Carl Jung were both obsessed with the power of certain numbers, including 137. They were fascinated by the atom’s fine-structure constant and its Kabbalistic significance. They formed an unlikely friendship and began a mystical quest that led them through medieval alchemy, dream interpretation, and the Chinese Book of Changes.

They were two people who believed 137 was at the intersection of modern science with the occult, and that it was a mystical number with a meaning beyond physics.

In 137: Jung, Pauli, and the Pursuit of a Scientific Obsession by Arthur I. Miller, it is reported that Pauli once said that if the Lord allowed him to ask anything he wanted, his first question would be “Why 1/137?”

Is there a primal number at the root of the universe
that everything in the world hinges on?

The Pi of Rivers

I met pi in school. You probably met pi that way too. It is that number used to calculate the circumference of a circle. Pi is shown symbolically as:

π

Pi is the ratio of the circumference of a circle to its diameter. It is an “irrational number” which means its exact value is inherently unknowable.

Using computers, we have calculated billions of digits of pi, starting with 3.14159265358979323…   –  but no recognizable pattern emerges. So strange. The digits of pi continue to infinity. Does anyone really understand infinity?

Ancient mathematicians did not like irrationality because it didn’t work with the concept of an omniscient God.

Recently I read about another pi connection which is also strange. In 1996, the UK earth scientist Hans-Henrik Stølum published a paper announcing that pi explains the seemingly chaotic paths of rivers in a mathematically predictable pattern.

This is called a river’s sinuosity. By dividing the river’s actual meandering length by the length of the direct line drawn from source to sea.

Of course, some rivers flow pretty straight from source to mouth , so they have small meandering ratios. Some rivers wander all over the place and have high meandering ratios.

But the average meandering ratio of rivers seems to be pi. Good old 3.14.

Albert Einstein used fluid dynamics and chaos theory to show that rivers tend to bend into loops.

If a river has a curve that will generate faster currents on the outer side of the curve. Those currents will cause erosion and so a sharper bend. That will eventually make the loop tighten. I have read that then chaos will eventually cause the river to double back on itself and form a loop in the other direction.

I did some more research on this river connection and found that this claim may not be accurate.

Someone put up a website at one point to crowdsource river data. The site at PiMeARiver.com seems to be dead now. People could put in the coordinates of the mouth and the source of a river, and the length of the river (from Google Maps and Wikipedia probably) to calculate the sinuosity of a river. That study looked at 258 rivers and found an average sinuosity of an un-Pi-like 1.94.

Hmmm. Maybe it is another mathematical constant, like the golden ratio (phi) which we often find in nature. That value is 1.618. Nope.

What about if you look at pi/phi? You get 1.94. Okay, that’s a strange “coincidence.”  Or something more than coincidence?

I need to be careful with all this, because I saw the film titled Pi. I saw this science fiction film when it was released in 1998. It is a difficult film to label. It is surrealist, psychological, thriller, that delves into religion, mysticism, the relationship of the universe to mathematics and number theory. It was written and directed by Darren Aronofsky in his directorial debut.

I read it as a cautionary tale. It is about a genius oddball mathematician, Max, who has been working for a decade trying to decode the numerical pattern beneath ordered chaos. The ordered chaos he studies is the stock market.

Max’s belief that there is some mathematical “code” underlying everything compares in my mind with Einstein trying to find that theory that explains it all. That quest frustrated Einstein through the end of his life.

Beware of that quest.

Coincidences

I came across a book at the library this past week quite by coincidence. Well, maybe..

The book is Fluke: The Math and Myth of Coincidence. Don’t be frightened by it being written by a mathematician, Joseph Mazur. It is about the seemingly improbable, surprising moments in our lives that seem to be coincidences. Maybe you attribute those events to serendipity. Or Fate. Look at some of the synonyms for coincidence: correspondence, agreement, accord, concurrence, consistency, conformity, fluke, harmony, compatibility. Do you attribute these kinds of events to coincidence or something else?

Others have said that “extremely improbable events are commonplace.” In 1866, the British mathematician Augustus De Morgan wrote, “Whatever can happen will happen if we make trials enough.”

What are the odds of being hit by lightning  once? More than once?  Roy Sullivan, a park ranger in Virginia who spent a lot of time outside in all kinds of weather was struck 7 times.

Enter the mathematical concepts of probability. This was one of those things that actually interested me in that rare interesting math class I was required to take.

Have you heard of the birthday paradox? What is the lowest number of people who must be in the same room to make it likely that at least two people will have the same birth day and month? Answer: 23. With 30 people in the room, the probability of a shared birthday is about 0.7 (or 70 percent).

Joseph Mazur knows that we are intrigued when someone wins the lottery four times in a row. How did you react when you learned that Abraham Lincoln had dreams that foreshadowed his own assassination? Creepy?

That statistics course you had to take may have taught you about correlation and causation. People confuse the two. Maybe cavemen believed that waking up caused the sun to appear.  You talk about a friend you haven’t talked to in years and they call you on the phone that day. Correlation does not imply causation. A correlation between two variables does not imply that one causes the other.

Some of Mazur’s examples seem to be “pure coincidence.” You find  your college copy of Moby Dick in a used bookstore in Paris on your first visit to the city? How do we explain the unlikelihood of strangers named Maria and Francisco, seeking each other in a hotel lobby, accidentally meet the wrong Francisco and the wrong Maria, another pair of strangers also looking for each other?

Mazur asserts that if there is any likelihood that something could happen, no matter how small the probability, it is bound to happen to someone at some time.

“What are the odds?” is what you might say in one of these situations. Like a déjà vu experience it might feel like some ripple just went through time, space or your universe.

In the paper, Methods for Studying Coincidences, mathematicians defined a coincidence as a “surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection.”

In The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day, David Hand says that principle “tells us that events which we regard as highly improbable occur because we got things wrong. If we can find out where we went wrong, then the improbable will become probable.”

It’s no coincidence that ukuleles are popping up in ads on Facebook and other websites this week for me, because I was searching and looking at them on Amazon.com last weekend.

There’s the joke about two guys in a Dublin pub drinking and discovering a series of amazing coincidences in their lives. Another patron listening is stunned by the coincidences. But the bartender says, “Nah, it’s just the O’Reilly twins have been drinking too much.”

More Reading

theatlantic.com/science/archive/2016/02/the-true-meaning-of-coincidences/

washingtonpost.com/opinions/the-improbability-principle…

Connecting with Coincidence: The New Science for Using Synchronicity and Serendipity in Your Life

There Are No Accidents: Synchronicity and the Stories of Our Lives