It is shown that many different problems have the same degree of unsolvability. Among these problems are: THE INDUCTIVE INFERENCE PROBLEM. Infer in the limit an index for a recursive function f presented as f(0), f(1), f(2),.... THE RECURSIVE INDEX PROBLEM. Decide in the limit if i is the index of a total recursive function. THE ZERO NONVARIANT PROBLEM. Decide in the limit if a recursive function f presented as f(0), f(1), f(2),... has value unequal to zero for infinitely many (...) arguments. Finally, it is shown that these unsolvable problems are strictly easier than the halting problem. (shrink)
This paper addresses the problem of reflexivity in modern social inquiry in general and in sociology in particular. This problem is inherited from Weber''s very conception of sociology, is transformed by phenomenology and ethnomethodology, deepened by the linguistic turn of hermeneutics and Wittgenstein''s later philosophy, and has been the central concern of the work of Alan Blum and Peter McHugh. The issues and spectres raised by reflexivity are methodological arbitrariness, the need to take responsibility for one''s own talk (and (...) the cultural assumptions embedded in talk) and, finally, the deep fear of nihilism – the sense that with regard to inquiry (along with everything else in the world) nothing matters. As such, reflexivity raises the most fundamental issue that can be raised for modern social inquiry. Through an oriented interpretation of the work of Blum and McHugh and other contemporary social theorists (particularly Gadamer and Arendt), this paper works through what a dialectical engagement with these issues look like. (shrink)
We reduce to a standard circuit-size complexity problem a relativisation of the $P = NP$ question that we believe to be connected with the same question in the model for computation over the reals defined by L. Blum, M. Shub, and S. Smale. On this occasion, we set the foundations of a general theory for computation over an arbitrary structure, extending what these three authors did in the case of rings.
Let f be a function from N to N that can not be computed in polynomial time, and let a be an element of a differential field K of characteristic 0. The problem of large powers is the set of tuples x̄ = (x 1 ,..., x n ) of K so that x 1 = a f(n) , and the problem of large roots is the set of tuples x̄ of K so that x f(n) 1 = a. These (...) are two examples of problems that the use of derivation does not allow to solve quicker. We show that the problem of large roots is not polynomial for the differential field K, even if we use a polynomial number of parameters, and that the problem of large powers is not polynomial for the differential field K, even for non-uniform complexity. The proofs use the polynomial stability (i.e., the elimination of parameters) of field of characteristic 0, shown by L. Blum. F. Cucker. M. Shub and S. Smale, and the reduction lemma, that transforms a differential polynomial in variables x̄ into a polynomial in variables x̄. and their derivatives. (shrink)
In this paper I examine the particularist attack on deductive uses of moral principles, reviewing the critiques of the uniformity of moral reasons and impartiality in ethics, looking principally at arguments from Larry Blum, Jonathan Dancy, and Margaret Walker. I defend the action-guiding-ness of moral principles themselves, but consider various ways to accommodate the objections coming fromparticularism. I conclude that one objection to the impartialist theory of value must be conceded without qualification: generalism is unable to account for the (...) unique and irreplaceable value of individual persons. I present an example which supports my view andshows that, in the context of lived experience, replaceability is contradicted. Indeed there may be few constants of value in the narrative of one’s life, as experiences overlay supposed constants with continual new shading, and create even deeper sorts of transformation in valuing. In the end, both particularized moral judgment and the articulation of fact with principle contribute to moral discernment. (shrink)