> "That which can be destroyed by the truth should be."
> -- P.C. Hodgell
> Today, among logicians, Bayesian Inference seems to be the new dogma
> for all encompassing theory of rationality. But I have different
> ideas, so I'm going to present an argument suggesting an alternative
> form of reasoning. In essence, I going to start to try to bring down
> the curtain on the Bayesian dogma.
So how are you going to get around Cox's theorem?
>This is not the end, but it *is*
> ‘the beginning of the end’ (as Churchill once nicely put it). I'm a
> fan of David Bohm, the physicist who developed the 'Pilot Wave'
> Interpretation of QM (which I like). So I base my argument on his
> The genius of David Bohm was that he showed that there’s a perfectly
> consistent interpretation of quantum mechanics which completely
> reverses the normal way that physicists think about the relationship
> between particles and background forces – physicists tend to think of
> particles as real static objects moving around in a nebulous backdrop
> of force fields. Bohm turned this on its head and said why not regard
> the *background forces* as primary and view particles as simply
> temporary ‘pockets of stability’ in the background forces. This idea
> is implied by his interpretation of quantum mechanics, where there’s
> a ‘pilot wave’ (the quantum potential) which is primary and particles
> are in effect ‘epiphenomen’ (mere aspects) of the deeper pilot wave.
On the contrary, in Bohm's interpretation the particles are more like
real classical objects that have definite positions and momenta. What
you describe as Bohmian is more like quantum field theory in which
particles are just eigenstates of the momentum operator on the field.
> Now my idea as regards rationality is exactly analogous to Bohm’s idea
> as regards physics. In the standard theory of rationality, causal
> explanations (Bayesian reasoning) is primary and intuition (Analogies/
> Narratives) is merely an imperfect human-invented ‘backdrop’ or
I'd say analogies are fuzzy associations. Bayesian inference applies
equally to fuzzy associations as well as fuzzy causal relations - it's
just math. Causal relations are generally of more interest than other
relations because they point to ways in which things can be changed.
With apologies to Marx, "The object of inference is not to explain the
world but to change it."
>My theory totally reverses the conevntional view. I
> say, why not take analogies/narratives as the primary ‘stuff’ of
> thought, and causal explanations (Bayes) as merely
> ‘crystallized’ (unusually precise) analogies?
> Bayesian reasoning is exactly analogous to algebra in pure math,
> because with Bayes you are in effect trying to find correlations
> between variables, where the correlations are imprecise or
> fuzzy. .Algebra is about *relations and functions* which in effect
> maps two given sets of elements (correlate them). So I suggest that
> algebra is simply the ‘abstract ideal’ of Bayes, where the
> correlations between variables are 100% precise (think of elements of
> sets as the ‘variables’ of statistics).
> Now…. Does algebra have any limitations? Yes! Algebra cannot fully
> reason about algebra. This is the real meaning of Godel’s theorem –
> he showed that any formal system (which is in effect equivalent to an
> algebraic system) complex enough to include both multiplication and
> addition, has statements that cannot be proved within that system.
> Since algebra is exactly analogous to Bayes, we can conclude that
> Bayes cannot reason about Bayes, no system of statistical inference
> can be used to fully reason about itself.
You mean Bayesian inference is incomplete? I think that would depend
on more than just the inference rule. First order logic is complete,
so Bayesian inference without second order quantifiers would be complete.
> But is there a form of math more powerful than algebra? Yes, Category/
> Set Theory! Unlike algebra, Category/Set theory really *can* fully
> reason about itself, since Sets/categories can contain other Sets/
> Categories. Greg Cantor first explored these ideas in depth with his
> transfinite arithmetic, and in fact it was later shown that the use of
> transfinite induction can in theory bypass the Godel limitations. (See
> Gerhard Gentzen)
On the contrary Gentzen showed that transfinite induction is an
example of the incompleteness that Godel proved.
> By analogy, there’s another form of reasoning more powerful than
> Bayes, the rationalist equivalent of Set/Category theory. What could
> it be? Well, Sets/Category theory is very analogous to
> categorization, a known form of inference involving grouping concepts
> according to their degree of similarity – this is arguably the same
> thing as…analogy formation! Indeed, I’ve been using analogical
> arguments throughout this post, showing that analogical inference is
> perfectly capable of reasoning about itself. My punch-line? Bayesian
> inference is merely a special case of analogy formation.
> If all this seems hard to believe at first I suggest readers go back
> and look at the analogy I gave with Bohm’s ideas about physics.
> Remember Bohm’s ‘complete reversal’ of the normal way of thinking
> about physics turned out to be fully consistent. All I’ve done is
> performed the same trick as Bohm in the field of cognitive science.
> Just as ‘particles’ become mere epiphenomena of a ‘pilot wave’,
> ‘Bayes’ becomes a mere epiphenona of analogy formation.
Note that Bohmian quantum mechanics is essentially barren. It proved
to difficult, if not impossible, to create a relativistic version that
could account for particle production (a consequence of taking
particles as fundamental).
> Time for Bayesian logicians to fill their trousers? ;)
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