I was rummaging around in my bookcases this morning, and (as I often do these days) ran across a book I couldn’t remember reading. Pulling it off the shelf, I found something I had highlighted (apparently I have read the book) that I rather like and wanted to share. But first I need to set it up a bit.
Various paradoxes have been much discussed over the past 100 years or so in philosophy and in mathematics. I won’t attempt a full exegisis here, but just to give you a taste, here are a couple of the bestknown:

1. The (male) barber of Seville, who lives in Seville, shaves everyone in Seville who doesn’t shave himself. Who shaves the barber?
2. This statement is false.
Here’s one that’s less well known, called the Berry paradox. Quotes are from “Infinity and the Mind: The Science and Philosophy of the Infinite” by Rudy Rucker.
The Berry paradox “. . . [has] to do with the impossiblity of ever explaining exactly how we use language.” I won’t fully discuss it–Rucker does a fine job if you want to dig into it more deeply–but in abbreviated form, consider really really BIG numbers. Whole numbers–we’re restricting the discussion to whole numbers. Imagine a specific HUGE number, so huge that it requires over a billion words just to describe it.
OK, now (given that numbers are infinite), it should be possible to imagine the existence of numbers so large that they cannot be described to a person in words at all–what with lifetimes being limited. Now, since we’re talking about whole numbers, “. . . Assume that there are indeed numbers that connot be described to a person in words in the space of a lifetime; and assume that there is indeed a least such number, which we may as well call u. Now, it looks as if I have just described a particular natural number called u. But u is supposed to be the first number that cannot be described in words.”
This paradox “. . . is a version of the problem of how we can talk about things that we cannot talk about.”
Contemplating such paradoxes may be more than idle entertainment; they can take us into deep places. Here are the bits I wanted to share:
The very existence of a paradox such as this can be used to derive some interesting facts about the relationship between the mind and the universe. No one has made such a derivation as boldly as Borges
“We (the undivided divinity operating within us) have dreamt the world. We have dreamt it as firm, mysterious, visible, ubiquitous in space and durable in time; but in its architecture we have allowed tenuous and external crevices of unreason which tell us it is false.” *
Actually, since the paradoxes inhere in the very nature of rational thought, I don’t think that “we” could have chosen to dream a world free of paradoxes. Rather than saying the the paradoxes indicate that the rational world is “false,” I would say that they indicate that it is incomplete—that there is more to reality than meets the eye.
I like this way of thinking about the limits on human minds and understanding. I like the openness implied–it leaves room for the unknown, and for wonder. We don’t simply give up on rational thought–we just recognize its limits and open ourselves to whatever there may be in the Universe that’s outside the confines of rationality.
Later in the book, after considerable more discussion of such paradoxes, Rucker concludes:
No finite system can generate arbitrarily complex patterns.
No finite system can understand everything.
No finite system can define truth.
This raises all kinds of interesting questions. On the assumption that we are products of the Universe, and that however remarkably complex we are we are nonetheless finite systems– what is the relationship between rational thought (as a product of our minds) and the Universe from which our minds have sprung? If there is a God, presumably (or anyway possibly) God is not a finite system–so does the mind of God thereby escape such paradoxes? And if so–what would that mean to us, since regardless we are finite systems?
I will now leave you to your own contemplation of paradoxes, wherever they may take you. Enjoy.
* Jorge Luis Borges, “Avatars of the Tortoise,” in Labyrinths, (New York: New Directions, 1962), p. 208.
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