A Preferred Treatment of Mill's Methods: Some Misinterpretations by Modern Textbooks

A number of modern logic books give a misrepresentation of Mill's Methods as originally conceived by Mill. In this paper, I point out what I believe is a better presentation of Mill's Methods. This treatment is not only historically more accurate, but it also represents a better conceptual introduction to Mill's Methods in general

Abstract A New-Millenium Attack on the Busy Beaver Problem ∗

Various computationally-based approaches to making progress on Rado’s Σ function (the Turing-unsolvable “Busy Beaver ” Problem: BBP) have been taken: brute force searches, genetic algorithms, heuristic behaviour analyses, etc. As a result, candidate BBP champions have been reported with a high degree of confidence for both the quintuple and quadruple formulations of the problem. However, what these previous research efforts lack is a definitive proof which explicitly confirms that these candidate machines are in fact Busy Beavers. The present paper proposes and explores the merits of combining treenormalization search techniques with specific nonhalt detection routines to explicity confirm BBP for small values of n. The start of our planned multi-year attack on BBP, this work establishes a foundation for exploiting a form of distributed computation used previously at our institution on the twin-prime problem and provides a fertile testbed for exploring both “visual ” reasoning and possible super-Turing computation. 1 The “Busy Beaver ” Problem Rado’s (1963) Σ (“Busy Beaver”) function has become a classic focus of study in the theory of computing. Although certainly directly related to the halting problem (Church 1936), BBP poses an alternative formulation of the concept of non-computability for Turing machines: given a fixed-size alphabet and a limited number of internal states, create the most “productive ” Turing machine that halts when run on ∗ We are indebted to.... Part of this research is being suppore

A new Gödelian argument for hypercomputing minds based on the busy beaver problem

9.9.05 1245am NY time Do human persons hypercompute? Or, as the doctrine of computationalism holds, are they information processors at or below the Turing Limit? If the former, given the essence of hypercomputation, persons must in some real way be capable of infinitary information processing. Using as a springboard Gödel’s little-known assertion that the human mind has a power “converging to infinity, ” and as an anchoring problem Rado’s (1963) Turing-uncomputable “busy beaver ” (or Σ) function, we present in this short paper a new argument that, in fact, human persons can hypercompute. The argument is intended to be formidable, not conclusive: it brings Gödel’s intuition to a greater level of precision, and places it within a sensible case against computationalism.