Epideictics II: “All true theorems have corollaries”

 

CHIEFTAIN — Conan! What is worst in life?
CONAN THE BARBARIAN — To help your friends, see them parade in front of you and hear the exultation of their women.

In the Aristotelian canon the method of logic is discussed in full continuity with the methods of rhetoric and even aspects of persuasion. In this older view, logic stands in relation to the λέγω (I speak, I arrange, I reckon, I gather) as mid-20th century science-and-technology to the nature of things. Just as engineers hoped to organize (i.e. formalize, shape, make intelligible) our useful interactions with the world in terms of their technical (infrastructural) reliability, old logicians hoped to give organize true arguments in terms of their form.

Note that here logic is not really concerned about truth — it takes truth as a given and hopes to generate technical tools that make it reliable. This changes only with the emergence of abstract formal languages that are far-removed enough from ordinary argument that logic appears to internalize the μάθημα (plainly, the true knowledge-ness) of mathematics. Because translation of plain arguments to the abstract language is awkward, it would seem that their truth emerges from the abstract language itself.

Of course, mathematicians see this slightly differently already: beyond the mere power to preemptively embarrass bullshit vendors, the abstract language works its magic through its aesthetic effects — although no one can stop from formalizing each and every special pleading as an additional axiom, the standard mathematics is beautiful, economical and filled with astonishing symmetries and self-similarities. Indeed, once in a while genuine mystical experiences can be had in a smaller math classroom. How wonderful is the fact that we somehow have found the correct true-knowledge, the one that enables you to think about arguments far larger than your raw cognitive powers?

But mathematicians themselves have been severely embarrassed by this mystical worship of mathematics. Naïve reasoning about arguments far larger than anyone’s individual powers of reasoning have produced incorrect answers (Russell’s paradox being a painful sticking point) that, while mostly inconsequential to applications and even absent from discussion in general mathematical discourse, threaten the integrity of the mystery of math. Mathematics is magic and also logical; it’s only natural that the core task of logic would be to protect mathematics. A good name for this outcome would be the Tarski emergency. In light of it, logic finds itself entangled not with truth-as-given (maybe Hume’s truth-as-a-feeling, maybe our very own ambient conditions), but with a special kind of internal truth — logical truth — that emerges from a special kind of argument — deduction.

Now, the Tarski emergency is a legitimate concern if there ever was one. But the cool aura and the sheer power of mathematics has led to a lot of cargo-culting and fetishization (of the emergency itself; any book on linear algebra you can find is okay and anything you can express in linear algebra is potentially very powerful — but note also the second order potentia here). Here, for once, you don’t have to trust me: witness the great logician Jean-Yves Girard urinating on analytical philosophy from a great altitude. In a twist cosmic irony, computing has provided an alternate role (an alternate emergency, really) for logic — but that’s not what “rationalists” are talking about, right?

What’s hilarious about all of this epochal conflation of both truth and logic with logical truth is that it dissipates under the weight of jokes. I forget where I saw the opening quotation; in it, Conan the Barbarian makes a mockery of logical negation and negative arguments. Of course there’s worse in life than helping your friends — starvation, torture, psychosis. And what if we negate this again?  Then the best in life is to hinder your enemies, see them route around and parade elsewhere, and hear nothing from their women. Some people even make great use of this drifting negation as formalized in Greimas squares.

(Source — Youth mode: a report on freedom)

Indeed, important and interesting arguments can be made out of Greimas squares — they’re not alogical (even if the drifting negation is not a specially formal procedure) or illogical (they’re often true!) — they’re just not deductive; their ultimate source of truth might be coded in jeans, not in axiomatics.

The group that produced the argument summarized above is not facing an emergency (at least not one with the civilizational import of the Tarski emergency) — why would they be more axiomatic than needed? Their answers to the aporias encountered along the way are mostly extralogical (dressing blank) but converge on grand themes (“maximizing the opportunity for strategic misinterpretation”) that radically clash with, say, the dharma of the Tarski litanies. As it turns out, the dharma of logicism (a kind of worship for logic) assumes away too easily that representations can be accurately and cheaply obtained — which holds in the magical realm of mathematics but fails dramatically  in business, military action, serial dating or anything where “strategy” means anything.

This is also the radical distinction between the time of tempo and the time of kairos (for clickbait’s sake, the time of revolution).  In tempo, Tarski litanies stab at the wind, haphazardly, and never let go after hitting something, anything really; but in kairos (which is nothing but ambiguous signs and choppy water) logical reference is rudderless and ineffective. This is why it takes something utterly uninterpretable (coded in that movie as the madman, the Joker) to trigger change.