# Computational Models of Measurement and Hempel’s Axiomatization

## 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.

## Keywords

Turing Machine Proof Mass Uniform Motion Bisection Method Collider Experiment## References

- Beggs E, Tucker JV (2006) Embedding infinitely parallel computation in Newtonian kinematics. Appl Math Comp 178(1):25–43CrossRefGoogle Scholar
- Beggs E, Tucker JV (2007) Experimental computation of real numbers by Newtonian machines. Proc R Soc Ser A (Math, Phy Eng Sci) 463(2082):1541–1561CrossRefGoogle Scholar
- Beggs E, Tucker JV (2008) Programming experimental procedures for Newtonian kinematic machines. In: Beckmann A, Dimitracopoulos C, Löwe B (eds) Computability in Europe, vol 5028 of Lecture notes in computer science. Springer, pp 52–66Google Scholar
- Beggs E, Tucker JV (2009) Computations via Newtonian and relativistic kinematic systems. Appl Math Comp 215(2009):1311–1322CrossRefGoogle Scholar
- Beggs E, Costa JF, Loff B, Tucker JV (2008a) Computational complexity with experiments as oracles. Proc R Soc Ser A (Math, Phy Eng Sci) 464(2098):2777–2801CrossRefGoogle Scholar
- Beggs E, Costa JF, Loff B, Tucker JV (2008b) On the complexity of measurement in classical physics. In: Agrawal M, Du D, Duan Z, Li A (eds) Theory and applications of models of computation (TAMC 2008), vol 4978 of Lecture notes in computer science. Springer, pp 20–30Google Scholar
- Beggs E, Costa JF, Tucker JV (2008c) Quanta in classical mechanics: uncertainty in space, time, energy. 2008. 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 2008Google Scholar
- Beggs E, Costa JF, Loff B, Tucker JV (2009a) Computational complexity with experiments as oracles II. Upper bounds. Proc R Soc Ser A (Math, Phy Eng Sci) 465(2105): 1453–1465CrossRefGoogle Scholar
- Beggs E, Costa JF, Tucker JV (2009b) Physical experiments as oracles. Bull Eur Assoc Theor Comp Sci 97:137–151. An invited paper for the “Natural Computing Column”Google Scholar
- Beggs E, Costa JF, Tucker JV (2009c) Physical oracles. Technical ReportGoogle Scholar
- Beggs E, Costa JF, Tucker JV (2010) Limits to measurement in experiments governed by algorithms. Technical Report, Swansea University, submitted for publicationGoogle Scholar
- Campbell NR (1928) An account of the principles of measurement and calculation. Academic, London and New YorkGoogle Scholar
- Carnap R (1966). Philosophical foundations of physics. Basic Books, New YorkGoogle Scholar
- Froda A (1959) La finitude en mécanique classique, ses axiomes et leurs implications. In: Henkin L, Suppes P, Tarski A (eds) The axiomatic method, with special reference to geometry and physics, studies in logic and the foundations of mathematics. North-Holland Publishing Company. AmsterdamGoogle Scholar
- Geroch R, Hartle JB (1986) Computability and physical theories. Found Phy 16(6):533–550CrossRefGoogle Scholar
- Hempel CG (1952) Fundamentals of concept formation in empirical science, vol 2 of International encyclopedia of unified science. University of Chicago Press, Toronto Suppes P (1951) A set of independent axioms for extensive quantities. Portugaliæ Mathematica 10(2): 163–172Google Scholar
- Suppes P (1951) A set of independent axioms for extensive quantities. Portugaliae Mathematica, 10(2):163–172Google Scholar