Extreme Philosophy: On the Limits of Self-Referential Truth: Why Paradox Has Been Binned By Naive Category Theory

1. Here are two papers of EXTREME PHILOSOPHICAL SIGNIFICANCE: [1] Lawvere, F. William, "Diagonal arguments and cartesian closed categories with Author Commentary,"  Lecture Notes in Mathematics, 92 (1969), 134-145, available at:   http://www.tac.mta.ca/tac/reprints/articles/15/tr15.pdf [2] Yanosky, Noson (2003) "A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points,"  available at: http://arxiv.org/pdf/math/0305282v1 2. Unless you've studied a bit of category theory, i.e., read Lawvere and Schanuel (2008, 2nd edition) and Lawvere and Roseburgh (2003), Lawvere [1] will be very obscure even with Lawvere's commentary. But take a look and get a feel. Then, look at Yanosky [2] which explains in a more breezy (but precise) way what the genius Lawvere was up to, and even more cleverly in order to reach a "wider audience", dropped category theory altogether and explains Lawvere's discoveries in easy enough "set and function" language. 3. I realize that category theory is not for everybody (yet) and recently, in the literature, there is a push-back accusing category theory of making "foundational claims" that are unjustified. For example, that the entirety of mathematics can be put on a category theory footing and replace set theory as the fundamental theory which all other theories must bow down to. But I don't think category theory as it is practiced sets out to make any really big claims like these--that would be the job of propogandists. Rather it "solves" some rather apparent fundamental problems by "resolving" the problems into a diagrammatic logic. If you buy the diagrams as BEING DENOTIVE then you might also see how category theory IS linked to Aristotle's great work On Categories. Mac Lane in a footnote joked about how the title "category theory" came from "purloining words from the philosophoers, Aristotle and Kant" [pp. 29-30 of Categories for the Working Mathematician]. He doesn't say anything more about this jokey link. But if you read and understand Aristotle's motive in his Categories, you can see immediately that Aristotle set up foundational problems so they can be resolved. He analysed knowledged into what might be called "said-of" and "thing-in" and asked what are those abstractions that are primary, that is, what are those properties that are extended and therefore, must be. He listed 10 categories [what they are appears arbitrary] and he showed how you can use these primary categories to categorize everything else, that is, that which is not so extended and universal. Now, this mental-conceptual move to abstraction in order to solve a particular problem is a natural function. Lawvere & Schanuel in Conceptual Mathematics explain this movement in terms of isomorphisms: e.g. think of how you can understand what's happening in a film even after walking into the cinema late. In media res, you know Humphrey Bogart is playing a particular character and Audrey Hepburn is playing another character, and when you sort out who's who in the film, suddenly, you can follow the plot in the film with the actors as playing their roles. Similarly, being born in the middle of things, we open our eyes, stretch our arms and legs, and explore the universe, fully confident that we will be able to sort EVERYTHING out. This confidence comes from something pretty powerful within ourselves that enables us to gain knowledge. And the point here is that knowledge isn't at its rock bottom paradoxical. It is in all likelihood isomorphic. 4. Lawvere [1] takes a swipe at the propogandists who have been using some of the great theoretical work of theorists (such as Russell, Cantor, Godel, Tarski) and turned them into very general claims about the nature of paradox at the heart of knowledge. To put this into a general philosophical context, Aristotle's optimism was founded on his discovery of a general scientific method which if simply re-iterated, would eventually uncover all the mysteries of the universe. It was based on observing that which is and translating those into propositions which could be understood. If at the heart of heart of "proposition making" we have paradox, then this whole enterprise is doomed to failure. So, burdened with the prospect of failure, why start the programme of knowledge? 5. The answer by Lawvere [1] and Yanofsky [2] shows why the propogandists of paradox are simply wrong. In Yanofsky's terms, Lawvere's great little paper [1] has been largely ignored by category theorists and philosophers alike because it is written in a forbidding unpopular formalism. Yanofsky translates the results of Lawvere's paper by saying the classical paradoxes of self-referential truth (e.g. Liar's paradox, Russell paradox, Godel's incompleteness and so on) are just instances of overstepping the limitations of a discourse ("discourse" is my term). There must be a way of limiting what a discourse can say about itself. This "problem" comes up in law and finance whenever they try to talk about themselves. I call it the problem of structure. That is, there is no such question in law and finance that says, "What is the structure of law? What is the structure of statements about finance?" There is no call for self-consciousness within laws or financial practice. Rather, the call for such professional consciousness comes from without. But there is a way of understanding such questions about professional discourses from a category theory perspective. And not only do the questions about the structure of law and finance make sense, they actually direct in some fashion a resolution to answers about the structure of law and finance. For example, one of the things I have been harping on in this blog is that there is a fundamental structure to law and finance in the forms of an individual unit which I have dubbed the "financial contract" and the "great cycle of default invariance." From these structures, we can explain a lot of current practice at the individual-to-individual level of financial transactions on up to historical and contemporary nausea of continuously impending financial catastrophes. It's all a matter of "mapping" and translating apparent limitations within the discourse of law and finance into a notation which allows for mental journeys and conceptual calculations. By the way, one of the virtues of seeing how paradoxes are slain in [1] and [2] is that we can recover a sense of optimism that Aristotle once had in the unity of science. Again, I say, judge the value of a theory by its contribution to the unity of science.