# Computational Models of Measurement and Hempel’s Axiomatization

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## Abstract

We have developed a mathematical theory about using physical experiments as oracles to Turing machines. We suppose that an experiment makes measurements according to a physical theory and that the queries to the oracle allow the Turing machine to read the value being measured bit by bit. Using this theory of physical oracles, an experimenter performing an experiment can be modelled as a Turing machine governing an oracle that is the experiment. We consider this computational model of physical measurement in terms of the theory of measurement of Hempel and Carnap (see Fundamentals of Concept, Formation in Empirical Science, vol 2, International Encylopedia of Unified Science, University of chicago press, 1952; Philosophical Foundations of Physics, Basic Book, New York, 1928). We note that once a physical quantity is given a real value, Hempel’s axioms of measurement involve undecidabilities. To solve this problem, we introduce time into Hempel’s axiomatization. Focussing on a dynamical experiment for measuring mass, as in Beggs et al. (Proc R Soc Ser A 464(2098): 2777–2801, 2009; 465(2105): 1453–1465; Technical Report; Accepted for presentation in Studia, Logica International conference on logic and the foundations of physics: space, time and quanta (Trends in Logic VI), Belgium, Brussels, 11–12 December 2008; Bull Euro Assoc Theor Comp. Sci 17: 137–151, 2009), we show that the computational model of measurement satisfies our generalization of Hempel’s axioms. Our analysis also explains undecidability in measurement and that quantities are not always measurable.

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