[FOM] Kreisel on Non-Standard Functional Analysis

[rtragesser@mac.com]

	Did Kreisel overstate the significance of (or mis-state) a  
(supposed) theorem [see below] of NonStandard Functional Analysis,  
viz., that the theorem shows that A. Robinson's "infinitesimals" can  
have dramatic explanatory powers?

	 [[Philosophical interest: If the theorem Kreisel indicated [see  
below] does exist, then our mathematical and philosophical view of  
"infinitesimals" should be altered exactly by their being found to  
have dramatic explanatory powers, for that would take them beyond  
being only "(even highly) convenient logical fictions" [that Robinson  
said they were].  Some readers of Gödel unfortunately construe him as  
"a Platonist of easy virtue." For example, Fensted asserts that Gödel  
would have denied that those "infinitesimals" were merely "logical  
fictions" (even if highly convenient logical fictions).  But there  
was about Gödel's Platonism something of the German Idealist  
"coherence conception of truth/reality" - which is baldly present  
(but oddly little appreciated) in Cantor's philosophical remarks.  In  
particular, while "being (even very) convenient logical fictions"  
would not suffice to compel the acceptance of their reality by a  
Platonist of more muscular virtue (of the sort Gödel was), their  
having dramatic explanatory powers would compel the acceptance of  
their reality.]]

	In Kreisel's for the most part highly illuminating 100 page  
'Mathematical Logic' (in T.L.Saaty, LECTURES ON MODERN MATHEMATICS  
Vol.III, John Wiley & Sons, New York, 1965, 1967; pp.95-195), in the  
very brief section (less than one page) 1.74 'Remark on Model Theory  
and Recursion Theory' (pp.113-14), Kreisel reports,-

	[Kreisel's statement:]""In analysis, nonstandard Hilbert spaces  
(infinitesimals [59]) explain the occurrence of a "point" spectrum  
inside continuous spectra in the theory of operators; not unlike the  
use of the complex plane explains the behavior of power series of the  
real axis...""

	On the face of it, this would be a highly significant theorem,  
suggesting that, beyond providing slick and less encumbered ways of  
discovery and proof, Robinson's "infinitesimals" can also have  
EXPLANATORY VALUE, in the way that circles of convergence in the  
complex plane explain otherwise anomalous-appearing convergences cum  
failures of convergence of real power series.

	In the literature touting Abraham Robinson's non-standard  
mathematics, I haven't noticed any dramatic underscoring of the  
explanatory powers of Robinson's "infinitesimals".   If there is such  
a theorem as Kreisel indicates, it deserves to be widely known, and  
it (and theorems of its ilk, if there are any) should alter our  
mathematical and philosophical appreciation of non-standard  
mathematics based on non-archimedean models (in the spirit of Robinson).
	ALTER PHILOSOPHICAL APPRECIATION OF NON-STANDARD ANALYSIS?-- In his  
introductory note to Gödel's "Remark on non-standard analysis" [in  
Gödel CWII, 307-11], J.E.Fensted points out that Robinson regarded  
those "infinitesimals" as convenient logical fictions, but (Fensted)  
asserts that Gödel would have taken "them" to be real.  But it is  
hard to believe that Gödel would have done so without further ado;  
Gödel was so-to-say NOT A PLATONIST OF EASY VIRTUE.   To repeat what  
was said above: There was about Gödel's Platonism something of the  
German Idealist "coherence conception of truth/reality" - which is  
baldly present in Cantor's philosophical remarks.  In particular,  
while "being (even very) convenient logical fictions" would not  
suffice to compel the acceptance of their reality by a Platonist of a  
more muscular virtue (of the sort Gödel was), their having dramatic  
explanatory powers would compel the acceptance of their reality.

	CAN ANYONE HELP?  But at least I have found it difficult to track  
down the theorem Kreisel refers to.  The reference Kreisel gives,  
"[59]", is to A. Robinson, "On generalized limits and linear  
functionals," PacJMath 14 (1964) 269-283 [readily available on-line];  
but if the theorem Kreisel is thinking about is there, it must  
require some expert reading between the lines to see it.  Worse,  
people I've asked have not seen how to translate Kreisel's figurative  
""the occurrence of a 'point' spectrum inside continuous spectra""  
into language sufficiently literal that the problem can be located in  
the classical literature, not to mention the literature on non- 
standard functional analysis.
	DOES SUCH A THEOREM EXIST, AND DOES IT HAVE THE SIGNIFICANCE  
KREISEL'S ANALOGY WITH THE EXPLANATORY USE OF CIRCLES OF CONVERGENCE  
IN THE COMPLEX PLANE GIVES TO IT?

	One can't help but suspect that this "theorem" is one Kreisel heard  
about rather than something he saw in print or pre-print.  If Kreisel  
heard about it, might it have been from Gödel (who would likely have  
emphasized its significance)? (Or possibly Dana Scott or....?)

Robert Tragesser

Robert Tragesser

email: rtragesser at mac.com
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