paradox as function of both time and place
My friend Richard says I love paradox.
Part of my process in these thoughts has now been developed by circumstance (which I find delightful). On one of my daily walks I passed a free library.* I always look in every new free library I pass and to my surprise there was a copy of Jonathan Strange and Mr. Norrell, a novel I had been thinking about just a few weeks before. I had to look. (I was even considering taking it, but it was a large paperback, and I didn’t want to carry it.) As I looked, I saw beside it a much smaller volume, one whose spine design I immediately recognized as a Harper Torchbook. Which meant it would be a good book. This is my process of thinking slowed down almost past conceptualization, but I want to set out what occurred methodically. I picked it up and discovered it was W. V. O. Quine’s from a logical point of view.
This struck like a thunderbolt. I read Quine as an undergraduate studying logic, and had found him to be very thorough in his thought, which was usually, clear not opaque, if not always transparent.
I almost walked away without it, but I went back and took the book. In so doing, I decided it was a sign that these thoughts of mine are worth the effort. Which brings me to paradox. First, a preliminary distinction. My finding the copy of Jonathan Strange and Mr. Norrell was a coincidence; my finding Quine’s from a logical point of view was a synchronicity. This is a necessary distinction as there may indeed be a synonymy between synchronicity and paradox. Now, for Quine’s influence on this work of mine, which thereby is changing the course of my plan of thinking. I’m going to phrase this next part a la Quine: I’d suggest the statement “a synonymy between synchronicity and paradox“, can be restated thus: “there may be a state of meaning that is equal to another state of meaning that may or may not be both equal to the first state of meaning and the opposite of the first state of meaning.” I imagine a good logician would complain that my initial statement (synonymy between synchronicity and paradox) by using the word ”paradox” as my predicate pretty well guarantees logically it’s true. To which I say, yes, how odd.
Let’s go back to Quine for another moment. In his essay Two Dogmas of Empiricism, he proposes a concept of science: “total science is like a field of force whose boundary conditions are experience”. He continues that the interior of the field becomes more abstract the further it gets from experience (empirical evidence). It’s delightful, very clever, and like all metaphors, problematic when it poses as a definition of science, or more in the spirit of Quine’s meaning (I think/hope), a description (evocation?) of the scientific process that accounts for science’s adaptability. For instance, a new point of empirical data may affect the science of logic, or other sciences, or lots of sciences together, but Quine’s theory is the field will hold. Which I’m willing to accept; it’s the “boundary conditions” that interest me as a very ready source of paradox. First, Quine couldn’t have accounted for fractal geometry back in 1951 when he wrote his essay. Which is a good thing because one of the things that led Mandelbrot to conceive fractals was he was hired to map the coast of England. Which led him to realise as a cartographer, he was going to have to impose an idea of coastal outline because the closer he looked at where the outline of the coast was, the more it fell apart. There was no clean line. Waves, rocks, tidal shoals, you name it, they all did not work for the idea of “coastal outline”. This was part of the process he identified as leading to his conceptualization of fractals.
Now, consider fractals as the boundary conditions of the field of science (as proposed by Quine). First, there is no boundary condition, there is an infinity of boundaries, in fact, an infinity within each point of a proposed boundary. The fractal boundary of science would imply an infinite number of boundaries within the finite boundary. Which sounds rather problematic, as I suspect Prof. Quine believed the line of demarcation of empirical evidence would be a simple one, a Y/N proposition, or at least, definable. I’d suggest not only is the boundary of the field of science fractal, it’s paradoxical. Paradoxical as it would mean, as one of many meanings, if the boundary is infinite, the boundary is not a boundary. It cannot possibly be a boundary.
Let’s go back to coincidence and synchronicity as I experienced it via the free book library. If I had walked away from the Quine book, left it behind, then none of this would have happened. I would have seen the book, recognized it, and declined the opportunity. That would have been a coincidence. But I didn’t. Instead, I went back, took the book, intent upon reading it as I felt it would inform this work. Which it has done (perhaps not as Prof. Quine might have liked). Again, if I didn’t read the book, and it didn’t inform me, then the situation would have remained coincidental. But when I followed through on my intention of reading the book, in order to inform my own writing, then a) I created a synchronicity, a moment of alignment or interaction of previously unrelated things/moments, and b) the book triggered the idea that a fractal boundary of science is paradoxical, in that a fractal boundary is possible and it is not. I’d like to suggest that sounds like a probability collapse, much like Schrödinger’s cat example, but a parallel process, which according to current research is very quantum mechanic-al like, but I’m not a physicist nor mathematician. Which leads me to my final observation: it’s not the establishment of empirical data that demarcates science, it’s the establishment that something is not empirical data that is definable. We don’t say what the boundary condition of science is, we say what it is not. I’d like to add, consideration of what is paradox got me here, and I’d like to suggest the idea that what is not empirical data is what defines empirical data is kind of paradoxical, and on another level, also analytic, if a Möbius strip can be analytic. Which throws Schrödinger’s cat back amongst the pigeons.
Finally, as an undergraduate, I read Quine’s “The ways of paradox, and other essays’, which may or may not have inspired this piece through buried recollection. I’ll have to re-read it to find out, and I don’t want to kill that poor cat!
Me: A paradox is the set known as paradox, which contains only truths, the only member of the set being the statement: I am not part of this set.
Berry paradox (person, not a fruit) “The smallest positive integer not definable in under sixty letters.”
Since there are only twenty-six letters in the English alphabet, there are finitely many phrases of under sixty letters, and hence finitely many positive integers that are defined by phrases of under sixty letters. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under sixty letters. If there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property “not definable in under sixty letters”. This is the integer to which the above expression refers. But the above expression is only fifty-seven letters long, therefore it is definable in under sixty letters, and is not the smallest positive integer not definable in under sixty letters, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is definable in under sixty letters), there cannot be any integer defined by it.
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