Among the topics are a new critique of theological interpretations of physical cosmology, Graham Oppy on the Kalam cosmological argument, finitism
and the beginning of the universe, the Grim Reaper Kalam argument: from temporal and causal finitism
to God, and endless future: a persistent thorn in the Kalam cosmological argument.
, and the Foundations of Mathematics.
Of particular note are Chapter 5's discussion of the applicability of mathematics, Chapter 6's discussion of how to ground proof in the concrete, Chapter 7's discussion of idealization, which is intended to allow Weir to both reject strict finitism
and meaningfully criticize finitism
concerning proof, and Chapter 8's discussion of logic and how to recapture most of everyday mathematics within Weir's framework.
Second, if singulars, then finitism
in its particularized form, and we should wonder what sense could make the idea of the infinite that Christianity had taken great pains to imbue with cognitive value.
In Grundlagen the word "abyss" occurs only once in a polemic passage mocking Kronecker's finitism
as an overreaction to the alleged danger of falling into the "abyss of the transfinite." Ges.
When we talk of Hilbert's finitism
, we ordinarily mean his finitist proof theory or metamathematics rather than what he takes to be the finitist portion of a formalized mathematical theory.
sums up in the principle that infinite difference is finite" (156), that is, in the principle of "absolute finitism
." For F., however, unlike D., such an absolute finitism
("thinking what is") involves "privileging synthesis with respect to disjunction" (158).
Then a careful analysis is given of the similarities and dissimilarities between Wittgenstein's new conception of proofs by induction and those of the intuitionists on quantification (thereby setting the stage for a proper discussion of Wittgenstein's finitism
Perhaps the triumph owes more to the grip of finitism
among leading figures in proof theory of that period than to the philosophical and logical merits of first-order, finitistic, formulations of theories, for these merits seem rarely to have been argued for explicitly (cf.
The mention of 'finitism
' may conjure up Hilbertian associations but Hilbert's finitism
was restricted to metamathematics; mathematics itself was free to employ infinitary and other ideal elements provided such use could be shown finitistically to be free from contradiction.
In this early period Godel is especially interested in finding ways to establish the consistency of arithmetic and other parts of mathematics on grounds that are constructive but that extend beyond Hilbert's finitism
. (It is clear, especially in 1933o, that he already appreciated the difference between finitism
and intuitionism.) The "Lecture at Zilsel's" (1938a) gives a fascinating glimpse into this effort and prefigures a number of important ideas and results.
Yet it not clear that he undermines the alternatives that he offers (including here "meaning finitism
" [Chapter 15]) or that some alternative theory, in particular the Mead/Dewey theory, might be preferable.