Aumann’s Agreement Theorem, Epistemology & Evolution

The Time Traveler was confronted by a sphinx with a faint shadow of a smile on its face.

 

These notes are my side of a discussion I had with James Knight on Aumann’s Agreement theorem. I believe he is incorporating his views on the subject in one of his books (something to look forward to!). I don’t have time to write books myself but prefer to zip around from one subject to the next logging my thoughts as fast as I can in order to cover as much ground as possible in what time is left to me; there are just too many riddles to touch on! But you never know, anyone of them might throw further inspiring light on the meaning of life, the universe and everything!

In these notes I talk about evolution. I have a rather qualified and reserved view on evolution as it is currently understood. This current understanding implies that a mathematical object I refer to as  the spongeam ( my name; see links below for more details) must exist in configuration space. I personally have doubts about that. But evolution in the weaker sense of a natural history of change, whatever the mechanism(s) that have driven this change, has occurred over long ages. Observations based on the messages sent to us via fossil laden rock sediment is strong evidence of this history of change. So, when I refer to evolution below I am in most cases thinking “natural history” rather than “spongeam”.

On the Spongeam

https://quantumnonlinearity.blogspot.com/2019/09/evolution-naked-chance.html

https://quantumnonlinearity.blogspot.com/2016/03/the-mathematics-of-spongeam.html

https://quantumnonlinearity.blogspot.com/2016/02/on-structuralism-and-spongeam.html

https://quantumnonlinearity.blogspot.com/2015/11/intelligent-designs-2001-space-odyssey.html

http://quantumnonlinearity.blogspot.com/2015/06/algorithms-searches-dualism-and_13.html

 

Introduction

Most Western people, as a matter of common sense, assume there’s an invariant ontology out there that is the depository of a common truth to be targeted and if possible converged upon. For most people this is an axiomatic trivialism. A minority of relativists, postmodernists, nihilists and cynics (“What is Truth?”) might question this axiom but most common sense people take the existence of a universal target ontology as given. Also, by and large Christians have no problem with an ontology of “truth” because they believe God is a rational creator (although rationality seems break down in the paranormal). But how to get to that truth is a much more vexed question; this is the epistemological question. 

What Aumann’s theorem doesn’t do is set out to prove that there is such a thing as a Truth to agree on. That’s not what the proof shows; that’s axiomatic as far as Aumman’s theorem is concerned, a theorem which  goes on to set a lower limit on the computational time needed for agreement assuming there is something to agree on. I see Aumann’s theorem as a department of epistemology because epistemology is about developing a concept of the world on the basis of the messages received from one’s context and that context includes messages from other people as well as the objects around us. Aumann simplifies this process using a relatively elementary model.

Actually it is possible to imagine circumstances where a successful epistemology is simply not possible; e,g, where sensory data is muddled,  chaotic or only erratically available and therefore there are then  no regular patterns to agree on (cf the paranormal). However, in these notes I’ve followed the axiomatic assumption that there is some solid common “truth” out there to be discovered and agreed upon.

If one accepts the common truth axiom then in one sense Aumann’s theorem isn’t so startling or surprising: If dispassionate observers share the same logic, share unobstructed communications and (eventually) the same “facts” then convergence toward agreement is an intuitively obvious conclusion; it all boils down to dispassionate observers ultimately sharing the same set of “facts” (for presumably the logic needed to massage those facts into a theoretical narrative is universal).

But of course human beings are far more advanced and complex affairs than simple dispassionate machines trying to share a limited set of facts in order to arrive at the same conclusions, as per Aumann’s theorem. Moreover, the world human beings find themselves in doesn’t present simple collections of facts on a plate that can then be shared willy-nilly. Fact gathering itself can be a hard graft; the cosmos doesn’t give up its facts that easily let alone hand them to us embedded in readymade theories.

Further; human beings are complex adaptive systems evolved to have a very complex suite of motivations and cognitive processes that fit them for a community life. That community life mediates both facts and theories; this immediately puts an entirely different complexion on the matter, way in advance of Aumann’s simple computer simulation, a  simulation not programmed with the trade-offs forced on us by social interactions and community belonging.

The world of Aumann’s theorem is a kind of toy-town cosmos not unlike the Ptolemaic cosmos – it’s a start, but its only start. I have in the past likened Aumann’s theorem to Olber’s paradox, a conclusion about starlight which proved to be wrong, but because it was wrong it pointed to the need to question the fundamental cosmological assumptions from which the paradox was derived. Olber’s paradox was therefore a profound result. Aumann’s theorem is also a profound result if it is seen in the right light.

The full copy of these notes can be found here