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.
My description of the book goes as this: “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, this is 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, that do involve the “brain” in interpretation. But, what is so special about Escher’s works is the notion that human’s 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 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’s Paradox
1,000,000 grains of sand is a heap. Take away a single spec of sand 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 quantitavely. A heap is only a heap when there are at least X amount of grains. The intuitive problem is that we don’t neccessarily have an “objective” definition of a heap, but who cares for that here.
Zeno’s Paradox
This paradox is described with Tortoise & Hare interlocuters, but it’s pretty simple to give the idea of it. If it rains implies it will be wet outside. It rained. So it is logically so 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 is 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 mechanically are the same. These two paradox arises 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? Is what causes the paradox.
The resolution to these problems is actually not too bad, some grammar rules just needs 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 of theorizing about the use of language a big deal.
Godel’s 2nd Incompleteness and Halting
I’m starting to move towards things that aren’t considered “paradoxes”, but moreso unintuive results that use semantics in a weird way.
The first is good old Godel and his ability to make Russell slip into a depression with a theorem. It says that whenever you can find self-reference, the formal system can not be pure or having both properties of “completeness” and “consistency”.
The problem here being, the application of a function predicate or even an algorithmitic 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 is fully valid if the math is done. There really isn’t a “paradox” here rather a confusion about what infinity and finite means.