Katabasis
No—the reason paradoxes trouble us is because their absurd conclusions make us rethink all of our premises. A paradox is like a staircase, in which each step leads inexorably to the destination. But you get to the top, and the destination is impossible; you’ve stepped off into empty air.
Katabasis
My description of the book goes as follows: “An allegory of a fanfic that contains allegories to real life.”
It takes the structure of “Dante’s Inferno” and adds a power system called “magick” which can only be a direct mapping to what we consider “linguistics”, specifically those that study “symbolic systems” or the grammar of language, including math, of course.
The plot itself is good, but I only felt like I got through it because of the random neuron-activating references scattered throughout the book. And the point of me writing this post was to talk about the philosophical paradoxes/spells and the resolutions we have to them.
Paradoxes
M.C. Escher
The book cover of Katabasis is a reference to the work of M.C. Escher. I enjoy this because the rest of the paradoxes are mainly logical or linguistic ones, which do involve the “brain” in interpretation. But what is so special about Escher’s works is the notion that humans indeed carry a visual grammar and cognition that leads to paradoxes such as this.
The paradox is resolved by recognizing how our visual systems are, in fact, processing the image. There is this paradoxical effect because our visual language uses the junctions to infer about movement. The problem only arises when we want to integrate or combine the junctions and the deduced movement, which can be crafted to be a contradiction.
The important thing to realize is that Escher is not playing with our reflective systems, rather he’s playing with something “pre-reflective”, sort of our “visual-language” system.
Sorites’ Paradox
1,000,000 grains of sand are a heap. Take away a single spec of sand, and it is still a heap. Repeat and you will end up with 0 specs and a heap.
The paradox arises from and is resolved by asking for the definition of a “heap” or any predicate like “tall” or “bald”.
There is a simple solution: defining the predicate quantitatively. A heap is only a heap when there are at least X grains. The intuitive problem is that we don’t necessarily have an “objective” definition of a heap, but who cares for that here.
Zeno’s Paradox
This paradox is described with Tortoise & Hare interlocutors, but it’s pretty simple to give the idea of it. If it rains, it will be wet outside. It rained. So it is logical that it will be wet outside. But do I believe in the conclusion?
And this is where an interlocuter creates the paradox by saying If the first premise is true and the second premise is true, then the conclusion is true. But do I believe in that new premise?
The resolution here is that the grammatical rules of a system are not defined within the system. When people do formal logic, “modus ponens” isn’t in the “object language” of formal logic; it’s just in plain English, where we might say something about logical equivalence and semantic entailments.
Curry’s and Russell’s Paradox
I grouped these two together because I think they are mechanically the same. These two paradoxes arise from the use of “names” versus the actual content of the “name.”
Curry’s paradox is something like this. S is the sentence: “If S is true, then X is true.” By assuming the sentence S and because of the self-reference, anything you put in the implication is true.
Actually, looking at the Wikipedia for Curry’s, it also shows how it can be made to be a variant of Russell’s Paradox. Russell’s Paradox in natural language is this: “Does the set of all sets that do not contain themselves, contain themselves?” or R = { x: x not in x}.
The question here is how the name R is resolved. Does it resolve to contain itself or not? This is what causes the paradox.
The resolution to these problems is actually not too bad; some grammar rules just need to be added about how names are to be resolved and how they can’t be treated in a self-referential way.
Liar’s Paradox
This sentence is false.
or
The next sentence is false. The previous sentence is true.
There really isn’t a resolution besides choosing a different system or banning it, but it has spun off into theorizing about the use of language as a big deal.
Godel’s 2nd Incompleteness and Halting
I’m starting to move towards things that aren’t considered “paradoxes”, but more so unintuitive results that weirdly use semantics.
The first is good old Gödel and his ability to make Russell slip into a depression with a theorem. It says that whenever you can find a self-reference, the formal system can not be pure or have both properties of “completeness” and “consistency”.
The problem here is that the application of a function predicate or even an algorithmic predicate runs into a sort of “metaphysical” problem when it is asked to “interpret” itself.
Essentially, if you can encode the “liar’s paradox” sentence by way of syntactical encoding, you can also construct a predicate that can’t do the work you would like it to do.
Gabriel’s Horn and Painter’s Paradox
This is the kind of paradox, where it is a paradox because the definitions and results are unintuitive, but it is fully valid if the math is done. There really isn’t a “paradox” here, rather a confusion about what infinity and finiteness mean.