From: Big Red Jeff Rubard on
Mythologies of Reason [NO TALGO GO GO GO - JR]

First I will speak here of an idea which, as far as I know, has not
occurred to anyone. We need a new mythology, however, this mythology
must be at the service of the ideas, it must be a mythology of reason.

Hegel or Schelling or Hölderlin, “The Oldest System Program of German

Now, a logical word about Freudianism. Recently I cast doubt on the
ability of relatively recent “computational theories of mind” to
adequately account for mindedness, but one need not “stay current” to
see why this might be so: we find this dialectic already in Freud. As
is well-known, Freud began his work as a neurologist studying with
Charcot, and his 1895 “Project for a Scientific Psychology” outlines a
proto-neuroscientific theory of the mind. However, shortly thereafter
he left the nerves behind and began working in the metier we know him
in: the scientific hermeneut, whose theories of drives and the
symbolic essences of psychopathology are at least almost as
interesting for the literary theorist as for the practicing clinician.

Obviously the neurosciences have improved since the days of Ramon y
Cajal staining brain cells with silver chromate; why would we think
that Freud’s compensation for their inadequacy then would hold any
interest today? Well, perhaps the Freudian orbit that cannot be
escaped is his modernist project of a scientific mythology, a “theory
of mind” that adequates mental states to potential scientific reality,
to what can reasonably be conjectured to be the case. In mapping
psychological distress onto biological states and personal histories
very widely construed, Freud traced the inside of reason, through
elaborating processes of “radical interpretation” that do not
transgress reason’s modesty about its epistemic abilities and
pragmatic powers.

So this “scientific mythology” is not only scientific, it is a
mythology: an attempt to map the entire space of signification, to
include all possible configurations of sense. Now, in formal logic,
all possible configurations of sense (“modes of presentation”) are
included in the intensional logic based on Russell’s theory of types;
I offered a very rudimentary elaboration of this in my discussion of
Montague Grammar, and do not want to elaborate the whys and wherefores
much further at this point except to note the presence in symbolic
descriptions of the hierarchy of types by other writers of the lower-
case a to denote the “extensions” senses are mapped onto.

The theory of types, like any other logic powerful enough to generate
Peano Arithmetic, is incomplete: there cannot be a finite axiom set
capable of deriving all its truths. But there’s at least a little
confusion about why this is so, and so I’d like to talk about some
elements of Gödel’s proof which are less commonly dwelled on. A lot of
people are familiar with the idea of Gödel numbering, which assigns a
logical statement a unique number by encoding each of its symbolic
constituents: figuring out the point of Gödel numbering is a little
harder. Complete logics have a primitive recursive axiom set: that
means that the theorems which are provable depend on the theorems that
have already been proven, going back to “base cases” instantiating a
finite set of axioms.

Primitive recursion is a concept in the theory of computation: the
primitive recursive number-theoretic functions are a subset of the
computable functions. If we give each sentence a Gödel number and the
predicate “is the Gödel number of a theorem of L” is primitively
recursively definable, the logic is complete. The first incompleteness
theorem shows this is not possible. If that predicate was primitively
recursively definable, there could be a “fixed-point theorem” stating
that any theorem mutually implied the number-theoretic representation
of the proof of the theorem. Suppose we devise a statement saying “I
am unprovable” (that there is no primitively recursive number-
theoretic representation of a proof of the statement).

If the logic is consistent this statement will be unprovable, but if
the logic is ω-consistent (a statement true of every natural number is
true) the statement will be provable (there will be no primitively
recursive number-theoretic representation of its proof among the
natural numbers). This is not to say that a logic powerful enough to
develop Peano arithmetic is necessarily inconsistent, and maybe it is
not even to say that its axiom set is not recursive tout court:
general recursion includes an additional device, the μ-operator, which
searches for the smallest natural number with a certain property: this
is equivalent to the “while” or “for” loops in programming languages,
but perhaps it is also equivalent to simply adding a new axiom when we
find a mathematical truth that cannot be proven using the axioms we
already have.

The point of all this as regards the Freudian import of the theory of
types is that there is not a computationally effective procedure for
“semantic descent” for defining higher-order abstractions in terms of
the more concrete objects they quantify over: but that is not to say
that psychological or other abstractions are necessarily inconsistent.
We need to work through our symbolic hierarchies and choose wisely in
terms of ordering principles, and what would that be but a “mythology
of reason”?