21 April, 2016

Anti-Perfect Objects

Epistemic Status: likely; I probably didn't make obvious mistakes, but I doubt this concept is anything more than an encapsulation of something trivial.


To me, Cantor's diagonal argument is pretty good as an intro to what antiperfection is all about. If you hang out in the geek aisle long enough, you'll see it. Given that you're reading a blog as obscure as this one, I'm sure you've probably already seen it. I hope, because I'm writing a cliff notes version and it's going to be mangled to heck.

Seriously, look it up, if you need to.

Anyway, Cantor worked with a notion called 'cardinality', and applied to infinite sets. He discovered that most infinities we know about are really the same infinity, albeit with fake glasses and a mustache.

For instance, an infinite set like 0,1,2,3,4... (naturals) is equivalent to itself sans one element, e.g. 1,2,3,4,5... (counting numbers) Then, by induction, we see an infinite set is equivalent to itself sans infinity elements, for instance 2,4,6,8... (evens).

It also means it's equivalent to itself plus infinite elements, so ...-1,0,1... (integers) are equivalent as well.

It keeps going. An infinite set is equivalent to itself times infinity elements. The .25, .5, ,75 rationals are equivalent as well! The proof is outside the scope of this article, mostly because I forgot bits of it and it'll take cognitive effort to recover them.

Ok, so what's not equivalent to the natural numbers?

The reals, as it happens.

Here's the crux of it's relevance: the diagonal arguments says, suppose there reals were denumerable (read: countable). Then we list them out in some ordering. The reals are infinite sequences of converging rationals, so we have a list a items with infinite elements each. For convenience, suppose we just convert that to a binary representation in its stead.

So: just take the first digit of the first sequence, and flip that bit. Add it to the sequence we're constructing. Now, take the nth bit of the nth sequence, and flip that bit. Append that to the sequence.

For obvious reasons, this sequence can't be on the list. QED

you can stop skimming now i mean: II.

Now, I consider this a good enough example of a antiperfect construction. The essential insight is that of the special case which defeats the general theorem. The antiperfect lies close enough to the normal that it prevents regularities of the normal from becoming universally true. It kills theorems just by existing.

This doesn't mean antiperfect objects are objectively well-defined, or that they are a minority in a set. An antiperfect object is always antiperfect relative to some fledgling conjecture. And from the Diagonal Argument above, you can see the antiperfect objects good just as easily be the majority in a set.

But in a sense, the diagonal argument itself is kinda antiperfect, because most prototype of my minds category of 'antiperfections' have antiperfections are special cases that are unique in that the generalization they refute is more or less true for the rest of the set.

Consider that famous proof of the Halting Problem's uncomputability. If we did have a program which solves the halting problem, then we could feed a bizarro implementation of that program which halts iff the given program doesn't, and vice versa.

Now we ask ourselves what the output will be when we feed this program its own source code, and are forced to admit a contradiction.

What it's interesting about this is that the proof just says the problem is generally undecidable, because a very specific input can't be decided by any would-be implementation. You very well could implement a program which decides on most programs, but it just isn't universal. Its deserved titled would be stolen by the antiperfect programs.


I'm probably overstepping the bounds of my theory here, but I also think antiperfection is a way to think about uncanny valley effect.
At least, a new perspective. Uncanny valleys are antiperfect in the sense that the uncanny faces are close enough to ideal to be recognizably human, but far from it enough to be disturbing.

There's a good chance I'm missing something though, because the uncanny valley doesn't really effect me that much. The above image is pretty common in popular treatments of this valley, and honestly, I don't get what's so creepy about it. I'm just weird.

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