Re: Impredicativity in Rosennean parlance

[Torkel Franzen <***>]

Judith Rosen writes:

>I recommend the many discussions on impredicativities in Essays on 
>Life, Itself. Specifically, on page 34, continuing to page 40; pages 
>135-140; pages 166-170; pages 292-295, to name a few. On page 160, 
>there is a short one which I will type into this post:

  Reading these excerpts suggests to me that in reading Rosen, it's
better not to take too seriously his apparent references to formal
logic, but concentrate instead on his own thinking.

  The following comments are intended to amplify this view.

>In this sense, syntactic or algorithmic systems, or formalizable 
>systems, are extremely weak in entailment: almost everything about 
>them must be posited, and very little is actually entailed within such 
>a system. That is, almost every why question about what is in the 
>system cannot be answered from within the system, and thus their 
>answers take us outside the system. In this sense, the objective 
>appearance of formalizable systems, their apparent rigor, is an 
>illusion.

  Rosen suggests no justification for this description, which is on
the face of it arbitrary. Indeed it is unclear what is claimed
("very little", "almost every why question"). I would guess that
to anybody familiar with the formal systems commonly used or
studied in logic, what is striking is rather how *much*
is provable within these systems, including statements about the
systems themselves.

>It is a characteristic of strong systems of entailment that they 
>possess enough entailments to allow closed loops. In inferential 
>systems, such loops are often called impredicativities (Bertrand 
>Russell's vicious circles).

  The mention of Bertrand Russell suggests that Rosen has in mind
some connection between impredicativity, as he uses the term, and
impredicativity in the logical sense. It's unclear what connection
this is. "Allowing closed loops" has no obvious application in
connection with, for example, the impredicative comprehension
principle for second order arithmetic given earlier.

  The situation is similar with Church's Thesis. CT, as understood in
logic, is specifically the (non-mathematical) thesis that every
function or predicate computable by an algorithm is recursive, where
recursiveness has a number of equivalent definitions. The definition
in terms of Turing machines is a bit of a favorite in the logical
tradition, since this was in fact the definition used by Turing in
arguing the thesis (also known as the Church-Turing thesis).  The
statement that "the thesis came to be imported into science and
epistemology, through the idea that every model of every physical
process had to be formalizable" diassociates CT from its logical sense.

   Such metaphorical or analogical use of terms taken from technical
fields is often quite harmless, and can be helpful, but it can also be
an obstacle to clear exposition and clear thinking. When looking for
Rosen quotes elsewhere, I find the following:
  
  Within a short time, however, Goedel proved his celebrated
  Incompleteness Theorem, which in effect show that syntactic rules
  captured only an infinitesimal part of "real" mathematics--in effect
  that Church's Thesis was false, even in this realm.

Here, "Church's Thesis" has nothing to do with CT as understood in
logic. A reader of the above might be confused to later learn that
Gödel, on the contrary, emphasized the importance of CT as
establishing the significance of the incompleteness theorem when
applied to general formal systems. The statement that the
incompleteness theorem "in effect" showed that syntactic rules
captured only "an infinitesimal part" of "real" mathematics again
suffers from having no apparent justification and making no definite
claim.