Contemplating the binary and the infinite

Perhaps my favorite sequence in the world is this one; we’ll call it “S”:

01101001100101101001011001101001…

It is a sequence permanently trying to reconcile one with zero, something with nothing, existence with nonexistence, trying to find the perfect balance between the two, realizing that there is none, and spinning off into infinity. For if you start by mindlessly setting zeros and ones side by side, you will realize that it is horribly off balance:

0101010101010101010101010101010101…

For each one might be thought to balance out the preceding zero; but then, what will balance out the sequence “01”? The whole sequence is in fact a mindless repetition of this pair, and so there is no balance whatsoever. But the sequence I have written seeks to rectify this by responding at every turn. Responding to 0, it gives 1; responding to 01, it gives 10; responding to 0110, it gives 1001; and so on.

Actually, the sequence has an amazingly simple interpretation. Write down all the numbers, starting with zero, in binary:

0, 1, 10, 11, 100, 101, 110, 111…

Now write down the sum of their digits, again in binary:

0, 1, 1, 10, 1, 10, 10, 11…

Now take only the last digit:

0, 1, 1, 0, 1, 0, 0, 1…

And there you have the sequence S. In other words, the interpretation of S is this: for each counting number written as a binary expansion, we assign a 1 if the number of 1s is odd and a 0 if the number of 1s is even.

When one meditates on the binary number system, one senses there is something so basic, so extraordinarily fundamental about it that one is about to touch the essence of reality itself. Zero or One, Yes or No, Exists or Does Not Exist, True or False. It is not simply that these are binary choices we live by; it’s that without even such a binary choice, there would be no choice at all! That is, if one does not even have the option between existence and non-existence, how can anything be said to exist? Or how can it be said to not exist?

If such binary choices are necessary, they are indeed also sufficient. One can describe every quantity using them. Once we have used up “0” and “1,” then it suffices to string them together: “1+1=10.” It is then natural to think that one more than 10 is just 11, and so then comes 100. In general, 10…0 is simply 1 more than 1...1. And at each iteration, all the preceding numbers merely repeat themselves, but with a leading “1” attached. So it is natural for the sequence S to consist of mirror images, constantly trying to achieve “balance,” only to be continually thrown off balance because in fact there is always another place for a leading 1 to be added on.

Mathematics has a deceptive way of reducing down infinite sequences and sets to a list of abstract principles. It makes one feel in control of the whole thing. Yet the actual experience of it is quite difference. That sequence never really ends. There is no balance. Existence and non-existence do not combine, nor are they ever confused; they eternally remain opposites, and all of reality derives its existence from the endless iterations of this simple truth.

And so, as I gaze off into infinity, contemplating the structure of this mysterious sequence, I recognize that there is an enormous difference between rational understanding and contemplation. The former simplifies reality and puts it under my control; the latter magnifies reality and makes me desire more and more. The heart longs to actually find the end of the sequence, to find a resolution to this grand and mysterious dance between Zero and One; but it will never come. Yet in some sense it already has, thanks to rational understanding…

And so on…