Mathematical existence is a slippery concept, and I don't think anyone knows exactly what its nature is. But I do think most of us strongly believe that the number '3' has a kind of existence. Even if there were no people to count them, wouldn't there still be a three-ness existing about three rocks together? I know that the planets orbited in our Solar System before humans used math, but to stay in those orbits, they needed to follow mathematical laws, laws which necessarily involve PI. PI preceded life.
Champernowne's constant is also a number, which has its own existence just like the number '3.' Champernowne's constant is the number constructed by listing all of the integers in order -- 1,2,3, ... 145,146, ... forever, after a decimal point, creating an irrational number like PI (3.14159...) which is a decimal expansion that also goes on forever. So, Champernowne's constant = 0.123456789101112131415161
718192021... (never-ending). PI is also a never-ending, never-repeating decimal, and its existence, and Champernowne's constant's existence as numbers are certainly no less real than the number '3.'
What is amazing about Champernowne's constant that
every finite sequence of numbers appears somewhere in its decimal expansion. '7181' -- you can find that in the small portion of the decimal expansion which I gave just above. It turns out that we can prove that
every sequence of numbers of any length will exist at some point in this number.
Now, consider digital representations of reality. Take a JPG:
This is a file which a computer stores as only a sequence of numbers. Every point (pixel) in the image is coded as a number representing the color, and the width and height are also coded in the file as numbers. The JPG above exists as a number sequence somewhere within Champernowne's constant. Video sequences are also able to be represented as a string of numbers, one image after another. Add sound -- no problem. All your mp3's are just strings of numbers. Somewhere in this number is a high-definition (as high-def as you want) copy of the movie Avatar and every other movie ever made, yet to be made, or ever possible to make. Add a third dimension, rather than a flat image, and a whole room or galaxy and its motion and change over time can also be coded as a string of numbers. They all exist at a specific location in Champernowne's constant just as '7181' does. Our brains have a physical configuration and electric potentials that are just as easily represented as numbers, so our thoughts are in there too.
The dramatic implication of this is that my life (in 3D video, sound, smells, and all other senses), your life, and everyone's life, and the details of the whole universe (or any conceivable universe) over any time-frame can be found represented to any desired accuracy somewhere within Champernowne's constant.
As numbers exist, everything that ever has, ever will, or ever could exist will always exist.
We cannot completely cease to exist as long as the number '3' cannot either. Mathematical existence seems like a real kind of existence to me, but perhaps that is as much illusion as our material existence is also probably illusion.
Fun stuff.
