Some of us trained in the theoretical dark arts instinctively move toward the dual of a problem. To move toward the light, we must understand darkness in a deep way. Most great epics are set up as the tension between a force and its dual. In one way, moving toward a dual is a protocol. A proven "one weird trick" that ends up being generative.
And so we have come to the question: What is the dual of the Protocol?
A protocol is a creature of high decoupling. It is a system of agents blind to the whole, driven by local state interacting with immediate neighbors. Earlier in a rant, I characterized this as Stateless (as in 'global' stateless).
A protocol (following that rant's somewhat mushroom-based logic) is a state of joy, rather than a state of pain. A system that is simple in its nature, which in most cases yields an outcome that is unreasonable in magnitude to what we put in. A protocol is almost like a lever that provides computational leverage that we can apply toward nasty boulders in our way.
So the dual must be a thing of global, painful state. My candidate dual of a protocol is a Mathematical Proof.
Proofs have all the markers: Globality, Painfulness, and Statefulness.
A proof is a rigid chain where no link can be broken. Step two demands step one. If it were loosely coupled, Step 90 might require only step 89. But as it happens, in proofs, Step 90 will require all the steps 1 to 89 in exactly the same order. Proofs are singular, indivisible entities akin to diamonds, which have a crystalline structure (and are some of the hardest[1] things around). Usually, a single crack gets the whole thing to crash, unlike protocol systems which are highly resilient against single or even a bunch of point failures.
[1] Sidequest: Regarding hardness as an outcome provided by certain protocols, notice there is a proof system that is doing the heavy lifting under the covers. We will come to this interplay in a minute, although let me say that this is somewhat expected in dual systems, so this is not a contradiction but something that should count as more evidence.
It is hard but we need to take a momentary look at "painful." Proofs leave behind a toll of ruin. The history of the Great Provers is a history of madness and early graves. My theory is that the human mind, a biological machine rated for twenty watts, is not wired to supply the voltage required to hold the global state of proof. Force a low-energy protocol engine to sustain the weight of the mountain, and the engine burns out spectacularly.
And boy, does it burn. One of my first high school math coaches saw the amount of interest I took in number theory; she placed her palm on my shoulder, looked me in the eye, and said something that shook me. She said, "Have fun with this, but don't indulge too much. It is known to have ruined lives." Up until that point, no one had talked to me about math in a negative light. So this is something that stuck in my head forever.
You just have to take a look at the wreckage.
(I collected some examples, but when I read it again it was so sad and painful to read. We already have enough trauma in the world as of now, so I moved that section to a separate gist. It can be skipped without missing the general argument here)
https://gist.github.com/why-not/6c0c4fc36eaa85cae1fd70127bf949e1
But here lies hope. For the first time, we have a partner not limited by biology. We have an entity not capped at twenty watts. With AI, we can handle the voltage. It appears to me that the mountain can be scaled without minds lost.
To understand globality, we must understand that one weird special case of proof by protocol: The Induction Method.
Proof by induction acts like a protocol. It merely asks to show that if f(n) is true, then it implies f(n+1 is also true. It (the protocol) verifies the first step and lets the truth dominoes fall in a single motion. It is a local-state mechanism. While useful, induction (and its cousin, contradiction) are "last mile" logistics. They are protocols for the last mile, but they do not operate at the high altitudes and the cold, thin mountain air source of truth that is Deduction.
Induction and contradiction fail to teach anything because of this. They are narrow protocols, more like support crew than Proof. Much like oracles, they answer the narrow question in a specific, cryptic way, and go silent for all eternity. Induction doesn't explain. Without explanations, a system fails to be generative. It is hearsay knowledge that there is a mountain over there with terrible beasts guarding the foothills. But no clear strategy formation happens of how to cross or go around this mountain.
Proofs are mountains, and protocols are rivers.
I framed it this way to also show that mountains are where rivers arise. A protocol is made of water, a fluid simple procedure that floods the landscape with unreasonable effectiveness. The proof is the mountain. It is a difficult, immovable procedure that defines the boundaries of that landscape. The river moves the land, the mountain shapes the river. Duals.
A formal theory of protocols will run downhill from a mountain landscape made by explanatory proofs. So one must scale the mountains. Or in the words of the great philosopher Mediocrates, "Stand around at the foothills and talk about scaling it with a hot cuppa tea."
We must look at the architecture of the Great Proofs.
Abel's proofs of the limits of algebra are said to have inspired group theory by Galois later. Both died super young.
Abel's proof that there is no closed solution to the qunitic polynomial equations and beyond is the Mountain. And Newton Ralphson method is the protocol river that shows how to get the roots of the quintic to any desired precision. This is the dual in action.
Also, Turing's proof showing the connection between memory and compute seems to have had a real generative impact.
I forgot some of this, it has been ages since I looked at them, so I can't claim yet that they were all explanatory proofs rather than inductive ones. I am not entirely sure of how much value looking at Great Proofs will yield, but I am sure it will yield a ton of fun. Like most of us, I am bankrupt for time. But I still think a slow burn thread where we metabolize some of these proofs will be useful.