Thoughts on the Riemann hypothesis

The simultaneous appearance in May 2003 of four books on the Riemann hypothesis (RH) provoked these reflections. We briefly discuss whether the RH should be added as a new axiom, or whether a proof of the RH might involve the notion of randomness

Does priming really put the gloss on performance?

Priming has recently emerged in the literature as offering advantages in the preparation for skilled performance. Accordingly, the current study tested the efficacy of imagery against a priming paradigm as a means of enhancing motor performance: in essence, contrasting a preparation technique primarily under the conscious control of the performer to an unconscious technique promoting automaticity. The imagery intervention was guided by the PETTLEP model, while the priming intervention took the form of a scrambled sentence task. Eighteen skilled field-hockey players performed a dribbling task under imagery, priming, skill-focus, and control conditions. Results revealed a significant improvement in speed and technical accuracy for the imagery condition as opposed to the skill-focus, control, and priming conditions. In addition, there were no significant differences in performance times or technical accuracy between the priming and control conditions. The study provides further support for the efficacy of imagery to elicit enhanced motor skill performance but questions the emerging emphasis on priming as an effective tool in preparation for physical tasks

Computability and human symbolic output

This paper concerns “human symbolic output,” or strings of characters produced by humans in our various symbolic systems; e.g., sentences in a natural language, mathematical propositions, and so on. One can form a set that consists of all of the strings of characters that have been produced by at least one human up to any given moment in human history. We argue that at any particular moment in human history, even at moments in the distant future, this set is finite. But then, given fundamental results in recursion theory, the set will also be recursive, recursively enumerable, axiomatizable, and could be the output of a Turing machine. We then argue that it is impossible to produce a string of symbols that humans could possibly produce but no Turing machine could. Moreover, we show that any given string of symbols that we could produce could also be the output of a Turing machine. Our arguments have implications for Hilbert’s sixth problem and the possibility of axiomatizing particular sciences, they undermine at least two distinct arguments against the possibility of Artificial Intelligence, and they entail that expert systems that are the equals of human experts are possible, and so at least one of the goals of Artificial Intelligence can be realized, at least in principle