1.
As for Frege's logicism, I want to draw attention to his article
U"ber formale Theorien der Arithmetik.
Sitzungsber. Jenaische Ges.Mediz.Nat. 1885 , 94-104
reprinted in "Kleine Schriften", ed. I.Angelelli, Hildesheim 1967.
This may to be one of the first texts in which Frege states his logicist
thesis explicitly:
Erstens: es ist keine scharfe Grenze zwischen Logik und Arithmetik zu
ziehen ... Meine zweite Folgerung ist, dass es keine eigentu"mlich
arithmetischen Schlussweisen gibt, welche sich nicht auf die
allgemeinen der Logik zuru"ckfu"hren lassen ... Meine dritte Folgerung
bezieht sich auf die Definitionen wie die zweite auf die Schlussweisen
... Daraus ergibt sich die Forderung, alles Arithmetische durch
Definitionen auf das Logische zuru"ckzufu"hren.
[First: no sharp boundary can be drawn between logic and arithmetic ...
My second conclusion is that there are no particularly arithmetical
modes of argument which cannot be reduced to the more general ones of
logic ... My third conclusion concerns the definitions as the second
did concern the modes of argument ... From this there results the
request to reduce, by definitions, everything arithmetical to the
logical. ]
2.
As for the various 'programs' and 'theses', it is an important lesson to
be learned from Mr.Detlefsen's article that one should distinguish between
them according to their different authors, and sometimes even by different
years of publication.
In particular, it may be useful to distinguish between Hilbert's
"formalist program" of presenting mathematics in logical formalisms, as
foundational enterprise in its own right, and Hilbert's "logical program"
of finding (relative) consistency proofs. Go"del's results, of course,
only concern the latter. But even there, attention can be directed at what
Bernays and Schu"tte called the "extended Hilbert [logical] program" and
the classification of theories by their proof theoretical ordinals.
W.F.