davidad

We Are In The Digits Of Pi And Live Forever

I ran across this odd article today (from which I stole the above title). There are several reasons I consider it odd.

1. It has not been proven that the digits of pi contain all finite sequences of numbers. It has not even been proven that the digits of pi contain an infinite number of “9”s. Past some unknown point, it might consist of an infinite nonrepeating sequence of digits between 0 and 8. Even the binary representation of pi could conceivably include no occurrences of the sequence “11” past some point - instead, all occurrences of “1” could be replaced by “10”, so that “11” would read as “1010”, and “01101” would read as “01010010”. If an infinite, non-repeating sequence contained all finite-length sequences, this transformation would preserve its infinite non-repetition, but it would no longer contain any sequences with two consecutive “1”s. “Pish posh,” you may say, “but surely the sequences are still there in some sense.” This is true, for the simple example I gave. I’ll return to this issue further down.
2. Given these issues, why pi? Why not simply use the Champernowne constant, defined as that number whose expansion is the concatenation of all finite sequences of numbers, in lexicographical order? (0.123456789101112131415…). Perhaps it’s because pi is considered to be some sort of fundamental constant and thus exists in a more deep way than such a construction. Whether this argument makes sense or not, there exist other, more “fundamental” constants such as the Stoneham numbers that have been proven to contain all finite sequences.
3. Why use any single mathematical constant at all? If the constants can be said to exist independent of the universe, then the finite sequences (countable numbers) certainly do; finite numbers are probably more likely than infinite ones to exist, assuming a reasonable definition of existence.
4. The only “alternate universes” which can be said to exist because they have an encoding as a countable number or finite sequence of bits are those which are logically consistent. No universe exists in which something both happens and doesn’t happen (although two universes may exist, one in which it happens and the other in which it doesn’t). Thus, one example given in the article, that if the one you love rejected you, then somewhere in the digits of pi you marry her, then the person in that alternate universe must not really be the same person.

This leads to an interesting question: what alternate universes do exist? Not all of the ones we can imagine could actually have existed. In particular, it’s not very likely that there’s an alternate universe in which Saddam Hussein won the Iraq war and reduced the USA to a pile of smoldering rubble. Such a scenario would not be logically consistent. However, it is more conceivable that there is an alternate universe in which World War II was lost by the Allies, and even more conceivable that there is an alternate universe in which Al Gore won the 2000 election. If you believe in a quantum multiverse, then perhaps those universes which exist within our multiverse are those in which the events which differ were caused by a quantum wavefunction collapsing differently, perhaps feeding into a chaotic system such as weather, and then feeding into the chaotic system of human society.

But is a universe somewhere which is just like our own except that 15 billion, 240 million, 862 thousand and 719 years, 8 months, 4 days, and 38 minutes after the Big Bang, a number of quantum shifts suddenly occur, transmuting a volume of air 1 meter above my laptop into a solid lead brick? However improbable such an event might be, did it occur in some universe? Is there some universe in which tomorrow, everyone in the world wakes up blind? After all, given a program which accurately simulates the dynamics of our universe, how hard would it be to add a little bit of code which introduces such a shift at a specific time?

Oops: I intended to write a very insightful post, but now I am out of my intellectual depth, and I forgot what point I was trying to make in the first place.

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