Circles: vicious and otherwise

This was originally posted to the blog that I had while I was a student at University of Michigan, on September 29, 1999.

Not all circles are vicious.

EscherEveryone knows about the vicious ones. The famous “this statement is false” sentence. The slightly less famous set of all sets that do not contain themselves (does the set contain itself, or not?). And Douglas Hofstadter’s published variations, such as “This sentence contains threee erors” (it contains two spelling errors, which means that it also has the error of claiming that it has three errors, but since that is an error then it really does contain three errors, in which case the third one would not be an error and it would contain only two errors, which would mean it had three errors… and so on…).

But people tend to assume that all circles are vicious. They are not. For example: “This statement is true.”  Sure, it’s largely uninformative, but it’s not self-contradictory. Similarly, the set of all sets that do contain themselves is perfectly reasonable as a set. Clearly, it contains itself: no paradox there. A similar case of a non-vicious circle is Douglas Hofstadter’s “The whole point of this sentence is to convey what the whole point of this sentence is.” Once again, it doesn’t tell you very much… but it certainly is (if nothing else) true. It contains no paradox.

The reason I think this is important is that it comes up when discussing radical constructivism, i.e. the idea that there is no objective “truth” about the universe, but rather that our perception of the universe is completely derivative from and inexorably entwined with our own network of presuppositions, biases, and so on, that come from our very non-objective view of the world.

The standard response to radical constructivism is to claim that it is somehow self-contradictory because: “if no theory about the universe is objectively true, then aren’t you saying that your theory about the universe is also not objectively true?”  (You can almost hear the “Ah-ha, gotcha!” in their voice when you read it, can’t you?)

I think this kind of retort is based on a misunderstanding. There is a circularity going on, to be sure: There is no viciousness to this circle.

This is what I believe: “I accept that all beliefs that I have, including this one that I’m expressing right now, are the product of my personal biases — biological, experiential, and cultural — and have no basis in any so-called objective fact-of-the-matter.

This is a circle, because the belief that I am expressing is talking about all of my beliefs including the belief that I am expressing, but it is not a vicious one. It is not self-negating. I don’t have privileged or objective knowledge of constructivism. It is simply something I believe — it fits together with my other beliefs like a jig-saw puzzle. None of them is tied to any “ultimate ground” or fact.

This may bother you if you can only be happy in a universe where you feel you have to be right about things.  So, feel free to tell me that my philosophy is disturbing to you.

But it’s not self-contradictory.


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