Aumann's Agreement Theorem and Its Discontents

 

In quantum physics, there are mathematical equations, such as Schrödinger's equation, upon which nearly every educated person agrees.  Not only do they agree on the math, they also agree on the evidence.  What they disagree upon is how to interpret this math.  Thus we have De Broglie-Bohm "pilot wave" interpretation, the Copenhagen interpretation, the "Many Worlds" interpretation, QBism (quantum Bayesianism), and so on.  In a sense, we might say that for the varying schools of interpretation, they all agree on the math, but they disagree on how to "put it into words," to translate the truth of the mathematical equations into a natural language such as English.  But this is not merely a linguistic phenomenon, a mere dispute over semantics.  It might be better to say that there's a dispute over how to translate the truth of the math into human intuition.  Each of the different interpretations of quantum physics is nestled into a different set of intuitions about the universe, our place in it, and how to understand it.  

Of course, another option is simply to choose, resolutely, not to interpret the math.  (This is similar to, but, I would insist, not the same as, the Copenhagen interpretation.)  I imagine a scientist filling a chalkboard with hundreds of lines of complex mathematical formulae, and deriving one particular equation at the very end, which he proudly shows off to his colleagues.  They ask, "What does it mean?" and he replies, equally proudly, "I don't know."  For instance, one might reasonably reflect upon the fact that humans evolved for survival, mostly likely in the African Savannah, tens or hundreds of thousands of years ago, and there's no reason to think that, under such circumstances, an organism would evolve an intuition for quantum physics.  Thus, a person might conclude, it is mere hubris to suppose that anyone ever can or will develop a real intuitive understanding of how our universe works.  (I actually met a person like this once.  I happened to take a class with a graduate student in physics when I was an undergraduate, and he was very, very intelligent, especially when it came to math, and got good grades, but he once told me that deep down, he never felt that the theoretical physics he was working on had anything to do with reality - to him, it just seemed like a game he was playing with numbers.  I don't know whether this was a deep-seated philosophical conviction on his part, or if he just had a case of imposter syndrome.)

But for many of this, this is not enough.  We want to feel smug.  We want to feel confident.  We want to feel that we really understand physics.  And we have faith that such a thing is possible.  Is it merely our own sense of pride, our striving for honor, that motivates us to feel this way?  I can imagine one of the advocates of the interpretations of quantum physics (Copenhagen, many-worlds, etc., etc.) accosting the person who refuses to interpret it at all: Where's your spirit?  The spirit of inquiry, the spirit of intellectual adventure, the human spirit - how can you just abandon it like that?  How can you be indifferent as to the nature of reality?  If that's your attitude, why bother studying physics at all?  Or science at all?  Why bother using rationality?  

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There's a rough analogy between quantum physics and Aumann's Agreement Theorem.  It's not a perfect analogy, but I hope it will help me clarify my position on the theorem.   I don't dispute the mathematical argument at its center - the math seems solid to me.  But I dispute how to interpret this math, or, better put, I want to demonstrate that more than one interpretation of the math is possible, and I very strenuously insist that, quite often, people take the consequences of this argument far further than the math can support.  People take Aumann's theorem to mean that two people cannot agree to disagree, which is, of course, obviously, patently untrue: people agree to disagree all the time.  It has, in fact, happened to me twice today.  Sadly, this faulty interpretation of Aumann's theorem begins with Robert Aumann himself, who unfortunately and unhelpfully titled his 1976 paper "Agreeing to Disagree".  I suspect that Aumann gave it this title provocatively, and perhaps even a bit tongue-in-cheek.  But ever since, people have been saying that it has been "proven" that is impossible that two people cannot agree to disagree.  If you push someone on it, they may admit that yes, of course people sometimes agree to disagree, but really they shouldn't - which is interesting.  It shows a hidden moral normative dimension to their supposedly purely logical and mathematical argument.

In reality, Aumann's theorem does not prove that people cannot agree to disagree, nor that they should not.  What does it, then, prove?