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From: Big Red Jeff Rubard on 9 Aug 2010 15:45 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 HÃ¶lderlin, âThe Oldest System Program of German Idealismâ Now, a logical word about Freudianism. Recently I cast doubt on the ability of relatively recent âcomputational theories of mindâ to adequately account for mindedness, but one need not âstay currentâ 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 âProject for a Scientific Psychologyâ 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 Freudâs 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 âtheory of mindâ 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 âradical interpretationâ that do not transgress reasonâs modesty about its epistemic abilities and pragmatic powers. So this âscientific mythologyâ 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 (âmodes of presentationâ) are included in the intensional logic based on Russellâs 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 âextensionsâ 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 thereâs at least a little confusion about why this is so, and so Iâd like to talk about some elements of GÃ¶delâs proof which are less commonly dwelled on. A lot of people are familiar with the idea of GÃ¶del numbering, which assigns a logical statement a unique number by encoding each of its symbolic constituents: figuring out the point of GÃ¶del 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 âbase casesâ 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 GÃ¶del number and the predicate âis the GÃ¶del number of a theorem of Lâ 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 âfixed-point theoremâ stating that any theorem mutually implied the number-theoretic representation of the proof of the theorem. Suppose we devise a statement saying âI am unprovableâ (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 Ï-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 Î¼-operator, which searches for the smallest natural number with a certain property: this is equivalent to the âwhileâ or âforâ 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 âsemantic descentâ 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 âmythology of reasonâ? |