(This post is still undergoing correction and enhancement)
 |
| Different perspectives implies a likelihood of different experience sets and conflicting probability estimates, therefore setting the scene for potential disagreement. |
I was rather intrigued by James Knight's use of Aumann's agreement theorem in a blog post of his that can be found here:
The Philosophical Muser: Why Christians Disagree So Much
James' post was a response to a challenge put to him that If Christianity is true, why are there such a varied set of Christians who disagree and squabble about so much? Towards the end of his article we read this (My emphases)....
Given the state of
humanity, I’d no more expect Christians to agree on everything than I would
mathematicians to agree about politics, or opera singers to agree about
economics. But, I do wish they would – and as I often argue – Christians SHOULD
agree more, especially on objective things – and two Christians of any sex,
ethnicity, denomination, should converge on more and more consensus if they
were to sit by the fire, Aumann’s Agreement-style, and honestly, rigorously
seek the truth together, like people who care about what is true."
The exact "science" of Mathematics is a domain of knowledge incommensurable with politics and economics and no like-for-like comparison can be made. Mathematics is an activity, in fact a form of model building, which depends on very strictly agreed symbolic conventions and algorithmic procedures being followed. If in mathematics every one keeps to the same agreed conventions and procedures disagreement can't arise. The progress of mathematics bares this out; I'm not aware that mathematicians frequently and fundamentally disagree (except perhaps about un-proven conjectures). This of course is not so of politico-economics; disagreements about best economic policy and its political implementation abound. This why economics is a breeding ground for politicians and their political passions & power seeking; after all the only way to implement a particular contentious economic policy is to get political power first (Hopefully by democratic means). But let's not think any better of those sanguine mathematicians over and against those battling politico-economists - the latter are dealing with very complex and epistemically tricky material which as we will see provides one reason among others why politico-economics breaks the assumptions of Aumann's agreement theorem and promotes the sharp divisions of power politics.....and that's before we consider those ever present very human psycho-sociological factors which one expects of complex adaptive systems like human beings.
***
Aumann's agreement theorem assumes we have a set of interlocutors who start with a common information base ("common priors") but then these interlocutors bring to the discussion table differing levels of knowledge in the form of conditional probabilities that all interlocutor believe to be a trustworthy contribution to the discussion (So-called "common knowledge" as opposed to "common information"). The interlocutors update their probabilities by mutual cooperative sharing of their differing conditional probabilities (*1). They assume one another to be rational honest agents and that they can trust one another's probability estimates as they share them. According to the agreement theorem they will eventually converge on the same information set. See the following link for more on the agreement theorem: Aumann Agreement - LessWrong
Before I go any further let's get one thing straight. Most common sense people (which includes myself and people who believe there is such a thing as a single truth out there which stands above the slippery slopes of cultural relativism and critical theory) have an intuitive grasp of Aumann's theorem; that is, they understand that in an epistemically transparent world where evidence acquisition is not a problem and interlocutors are rational and honest, then agreement about truth will inevitably emerge. So the agreement theorem proves what most common sense people already believe (Of course critical theorists and cultural relativists are likely to make heavy whether of this common sense truth). Aumann's theorem is a nice confirmation of what all reasonable people already know intuitively. But the article on the theorem that I have linked to above ends with this warning:
The fact that disagreements on questions of simple fact are so common amongst humans, and that people seem to think this is normal, is an observation that should strike fear into the heart of every aspiring rationalist.
So, given this agreement theorem which is undoubtedly mathematically correct why is disagreement between humans so widespread? In this connection I made the following comment on James blog entry. As a rule my comments never get past the Philosophical Muser's approval process and are therefore cancelled. So, rather than let my comments disappear into oblivion I thought this matter to be so important that it needs airing. What follows in the next section is based on the comment I added to James post...
***
I think I agree with the general drift of your argument here but not in one or two of the details; especially, may I say, you are missing the crucial point of the agreement theorem and vastly underestimating the epistemic issues that impact attempts to get agreement
That cozy fireside talk seldom if never arises. For a start whilst our interlocutors are locked in by the fireside they are not updating their experience or accumulating any further experience. They have to try and get agreement on the basis of the information they already have. (in the form of priors and conditional probabilities) If this pool of information contains contradictions and they insist of sticking to their scripts they won’t necessarily reach full agreement even if they are rational.
Therefore our interlocutors are going to have to get off
their backsides, get out on their bikes and find a set of consistent priors and conditionals. But
that brings us to the main problem: This information can only come from statistics which result of a wide and long term experience of the cosmos. Moreover, any
mature engagement with that cosmos requires thousands if not millions of bits of information.
Single interlocutors, therefor can’t survey the whole lot; ergo, their experience
therefore is very liable to being skewed. So unfortunately our interlocutors,
on top of all their other very human traits, have to face the epistemic problem
of systematic and random sampling errors.
The agreement theorem simply sets a lower limit on agreement
time. That is:
Agreement time >= Aumann agreement time:
As I found out with my own AI Thinknet project AI systems also suffer from similar epistemic problems relating to sampling bias. After all, I think the YEC organization AiG have implemented their own YEC AI interlocutor presumably by training it with a bias on YEC texts.
As I’ve said before because of these fundamental
epistemic limitations tribal identification & group think where one
outsources epistemic help to the experience of a large group of minds is an
adaptive trait and this factor shouldn’t be underestimated in terms of its
potential epistemic utility. So what James refers to as “incentives, needs and agendas”
have the potential to be adaptive whether we like it or not.
So, even without factoring those many awkward human social foibles (which potentially may have adaptive value), epistemic challenges alone are very likely to lead to agreement failure.
My guess is that disagreement due to epistemic issues is the biggest factor in disagreement.
The only antidote I see for this is epistemic humility. But the trouble with
this is that when faced with utterly convinced group-think such as we find in AiG &
Trumpite brands of Christianity epistemic humility is read as weakness. Hence, a certain
amount of vehemence is demanded in the heat of argument.
***
Olber's paradox was a famous theorem in astrophysics. This paradox shows that under plausible assumptions the night sky should not be black but a continuum of bright star light; the fact that this isn't the case pointed to the need to revisit the underlying assumptions; it was a profound piece of theoretical thinking which lead the way to our understanding of an expanding universe. I contend that likewise Aumann's theorem prompts us to think a bit deeper as to why it's not a real world model; in particular it urges us to think about both our epistemological limitations and the complexities of socio-psychology which strongly influence the acquisition of knowledge. With respect to the latter we are prompted to investigate the adaptive value of group think & group belonging along with its potential downsides and tradeoffs. Because Aumann and his successors are making us think harder about human affairs then like Olber's paradox its pedological value can't be underestimated.
The upside of group think is that it widens the number of experiencing agents contributing to the conversation and this increases the amount of incoming evidence. It's true, however, that the instincts behind group think have a big potential downside as group can to lock in error such as we see amommg cultists and fundamentalists who exploit the adaptive instincts'; in this context the survival of the group identity takes precedence over further evidential updates. But Aumann's theorem prompts us to study the trade-offs and the costs vs benefits balance entailed by joining an epistemic group with a strong sense of collective identity.
***
I would want to rewrite the quote at the beginning of this post which I took from Philosophical Muser along these lines....
Christians are expected to make heavy weather of agreement, even about on objective things – and two Christians of any sex, ethnicity, denomination will not necessarily converge to a consensus if they were to sit by the fire, Aumann’s Agreement-style attempting to get convergence; that's even if they honestly, rigorously and rationally seek the truth together, like people who care about what is true."
The Agreement theorem tells us that in principle agreement is possible if we get our priors and evidences right, but therein lies the epistemic challenge of gathering huge amounts of data some of which may present accessibility problems. That this can make agreement problematical should always be at the back of our minds and therefore our difficulty in to conforming to Aumann's theorem SHOULD be the basis of an attitude of epistemic humility rather than allowing Aumann's theorem to promote epistemic arrogance, which in my books classifies as an abuse of the theorem. Agreeing to disagree until more information comes to light should not make us shudder.
But let me repeat this.....
The only antidote I see for inevitable disagreement is epistemic humility. But the trouble with this is that when faced with convinced group-think such as we find in AiG & Trumpite brands of Christianity, humility is read as weakness. Hence, a certain amount of vehemence is demanded in the heat of argument.
Footnotes
*1 A conditional probability has the form "The probability of A given evidence B is P"; formally expressed as P(A/B). Here B is the evidence relevant to the truth of A.
