[FOM] Kreisel on Non-Standard Functional Analysis: The ontological and the epistemic

[Haim Gaifman]

Dear Robert Tragesser,
You seem to conflate two issues: (1) Realism (that is, the position that 
certain mathematical
statements have objective truth-values, independent of our abilities to 
decide them (2) Organization of
the mathematical material in a fruitful way that yields better insights, 
suggests good questions, reveals certain
regularities, etc. The first issue is ontological, the second is 
epistemic. The question of explanatory
power is epistemic.

Ontologically, there is no more to the complex numbers than there it to 
the reals, since
the complex numbers can be construed as pairs of reals and every 
question about complex
numbers can be equivalently stated as a question about the reals. So a 
Platonist with respect
to the reals is also a Platonist with respect to the complex numbers. 
But  complex number
theory---which incorporates geometric structures and geometric 
intuitions---is a different theory,
 in terms of heuristics, guidelines, the kind of questions it asks,  
etc. Its explanatory power
with regard to the behavior of the Taylor series of 1/1+x^2 consists in 
showing that this behavior
falls under a general pattern concerning complex functions, a pattern 
that is missing when
one considers only real functions.  

Concerning non-standard arithmetic, one is ontologically committed to it 
whenever
one is committed to the portion of set theory within which one can 
construct the non-standard
model in question. If Gödel  was a Platonist with respect to that 
fragment of set theory
(as presumably he was) then he would be a Platonist with respect to that 
non-standard
model of arithmetic, or analysis, because that model can be constructed 
with the set theory
that he treats realistically.

Robinson was a skeptic with respect to actual infinity. He treated both 
the standard and
the non-standard models of arithmetic as "fictional entities" that are 
useful for getting good
finitistic results. The fact that they are useful did not imply, for 
him, that every question
that can be stated in their language has objective truth value.

Haim Gaifman

     

> 	Did Kreisel overstate the significance of (or mis-state) aco  
> (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 '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
> Ph: 845-358-4515, Cell: 860-227-7940
> Address:
> 26 DePew Avenue #1
> Nyack, NY 10960-3839
>
>
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