"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.  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
ideas.

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.

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
scaffolding.  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.

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)

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.

Time for Bayesian logicians to fill their trousers? ;)

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