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Hypothetico-deductivism: the current state of play; the criterion of empirical significance: endgame
Any precise version of H-D needs to handle various problems, most notably, the problem of selective confirmation: Precise formulations of H-D should not have the consequence that where S confirms T, for any T', S confirms T&T'. It is the perceived failure of H-D to solve such problems that has lead John Earman to recently conclude that H-D is "very nearly a dead horse". This suggests the following state of play: H-D is an intuitively plausible idea that breaks down in the attempt to give it a precise formulation. Indeed I think that fairly captures the view among specialists in the field of confirmation theory. Here I argue that the truth about H-D is largely the reverse: H-D can be given a precise formulation that avoids the longstanding technical problems, however, it relies on a fundamentally unsound philosophical intuition. The bulk of this paper involves reviewing the problems affecting previous attempts at giving precise formulations of H-D and displaying some recent versions that can handle these problems. It then briefly explains why the basic intuition behind H-D is itself unsound, namely, because H-D involves a tacit assumption of inductive scepticism. Finally, the historical relation between H-D and the positivists' quest for a criterion of empirical significance will be reconsidered with the surprising result that having glossed H-D as fundamentally unsound it is concluded that a sound version of the criterion of empirical significance is now available. The demarcation criterion, the positivists' philosopher's stone that serves to separate claims with empirical significance from claims lacking empirical significance having finally been found, it is argued that we should regard empirical significance as just one among a variety of virtues and not follow the positivists in taking it to be a sin qua non for all meaningful statements
Tempered distributions and Fourier transform on the Heisenberg group
The final goal of the present work is to extend the Fourier transform on the
Heisenberg group \H^d, to tempered distributions. As in the Euclidean
setting, the strategy is to first show that the Fourier transform is an
isomorphism on the Schwartz space, then to define the extension by duality. The
difficulty that is here encountered is that the Fourier transform of an
integrable function on \H^dis no longer a function on \H^d : according to
the standard definition, it is a family of bounded operators on
Following our new approach in\ccite{bcdFHspace}, we here define the Fourier
transform of an integrable functionto be a mapping on the
set~\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}endowed with a suitable
distance \wh d.This viewpoint turns out to provide a user friendly
description of the range of the Schwartz space on \H^d by the Fourier
transform, which makes the extension to the whole set of tempered distributions
straightforward. As a first application, we give an explicit formula for the
Fourier transform of smooth functions on \H^d that are independent of the
vertical variable. We also provide other examples
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