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. 1

The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930’s working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal credit. This is incorrect. It was Turing alone who achieved the characterization, in the opinion of Gödel. We also explore Turing’s oracle machine and its analogous properties in analysis. Keywords: Turing a-machine, computability, Church-Turing Thesis, Kurt Gödel, Alan Turing, Turing o-machine, computable approximations,

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This is a critical analysis of the first part of Gödel’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Gödel’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. Either... the human mind... infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems.

Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl’s third claim is wrong, by his own standards. To explain this thesis, let me first introduce the term ‘revisionism’. It is understood here, following Crispin Wright, as the term that applies to ‘any philosophical standpoint which reserves the potential right to sanction or modify pure mathematical practice ’ [Wright 1980, p.117]. I want to make a distinction between weak and strong revisionism. The point of reference is the actual practice of mathematics. Weak revisionism then potentially sanctions a subset of this practice, while strong revisionism potentially not only limits but extends it, in different directions. In strong revisionism, certain combinations of limitation and extension may lead to a mathematics that is no longer compatible with the unrevised one. ‘May lead’, not ‘necessarily leads’: it is all a matter of reserving rights; whether there is occasion to exercise them is a further question. To illustrate these categories, let me give examples of non-revisionism, weak revisionism, and strong revisionism. Non-revisionism can be found in Wittgenstein’s Philosophische Untersuchungen, where philosophy can neither change nor ground mathematics: Die Philosophie darf den tatsächlichen Gebrauch der Sprache in keiner Weise antasten, sie kann ihn am Ende also nur beschreiben. Denn sie kann ihn auch nicht begründen. Sie läßt alles wie es ist. Sie läßt auch die Mathematik wie sie ist, und keine mathematische

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