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Incalculable? The Human Quest To Conceive Of The Largest Number

Trying to count all the way to the end...
Trying to count all the way to the end...
Etienne Ghys*

-OpEd-

LYON — Those following the news know that Google technology recently defeated one of the world's greatest Go champions. To illustrate how significant this achievement is, the media explained that the number of possible combinations on a goban (the Japanese name for the Go board) is superior to 10 to the power of 170 (in other words, 1 followed by 170 zeros). Meanwhile, the number of atoms in the universe is "only" 10 to the power of 80. These numbers are gigantic and beyond human understanding.

When children learn to count until 100, they sometimes think they're done and that 100 is the biggest number there is. It often makes their heads spin when we explain to them that numbers are infinite, that we can always add 1 to any number to find an even larger one.

But children aren't entirely wrong: Do numbers that are "too big" — those that a human being cannot even describe, conceive or understand — really "exist"?

When you move beyond a certain quantity, like "the sand of the sea, which is too great to be numbered" (Genesis 32:12), perhaps it doesn't make much sense to keep counting. The Go champion's defeat is a reminder of our limits: The human brain is finite and "only" contains 100 billion (10 to the power of 11) neurons. A human being can only appreciate a finite quantity of numbers. So if that's really the case, what's the largest of these?

Our ancestors thought the universe was finite. Archimedes even wrote a text, The Sand Reckoner, in which he wondered how many grains of sand would be necessary to fill the entire universe. His answer was 10 to the power of 63. Much later, in the modern era, the universe became infinite, thanks to Newton's physics.

Over the course of the 20th century, with Einstein's theory of relativity and the Big Bang, the observable universe was once again viewed as finite, with a radius of 13.8 billion light-years. Of course, mathematics is more abstract than physics, and the question of mathematical infinity — whether actual or potential — continues to puzzle mathematicians and philosophers alike. By what magic can we finite beings reason about what's infinite?

The 1980 edition of the Guinness World Records included the so-called Graham's number, the largest ever used to that point in a mathematical demonstration. We've made progress since then: The website Googology Wiki lists the largest numbers conceived to this day.

There was even a duel between two philosophers at Boston's Massachusetts Institute of Technology. The winner was the one who could write the largest finite number on a chalkboard. Of course, there had to be rules, so as to exclude answers such as "1 + the biggest number ever imagined by a human being" and other cheats of that kind.

We too can play a game. Let's try to describe a number with fewer than 100 characters. The definition must be accurate enough to enable us to effectively calculate the number and it must be understandable to a reasonable mathematician. For instance, it could be something like this: The power of 100 of the product of all divisors of 10 to the power of 100,000.

To the winner, I offer the sum of 1 million euros (10 to the power of 6), divided by that number, rounded down to the nearest cent.

*Etienne Ghys is a mathematician and research director at the École Normale Supérieure in Lyon, France.

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