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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Why should the balance be in a vacuum? It is not because of friction. It is because there are substances in the atmosphere that have “negative weight” such as hydrogen and helium.
- 2.
This is done by considering a semigroup of objects \(\mathcal{O} =\langle \mathcal{O},\circ ; 1\rangle\), with the distinguished element 1 called the unit, and some internal structure to generate fractions and multiples of the unit.
- 3.
There can be further structure for the map M, e.g., depending on the fact that the attribute considered is either extensive (e.g., mass) or intensive (e.g., temperature).
- 4.
This error margin in the initial speed of the proof particle of mass m means that precision in speed does not matter for this experiment.
- 5.
Let f and g be total maps with signature \(\mathbb{N} \rightarrow \mathbb{N}\). We say that f ∈ Ω(g) if there exists a constant \(k \in \mathbb{R}\) such that, for an infinite number of values of \(n \in \mathbb{N},\ f(n) > \mathit{kg}(n)\).
- 6.
The reference frame of the stars is a good inertial frame for experiments carried out on Earth.
- 7.
To Aristotle the force applied is the cause and in some way the velocity is the effect. Since uniform motion in a straight line does not need any explanation, Newton searched for the variation of uniform motion in a straight line as the required effect.
- 8.
In the Principia, Newton defined force as change of momentum, i.e., \(\mathbf{\mathit{f }} = \frac{d\mathbf{\mathit{p}}} {\mathit{dt}}.\)
- 9.
References
Beggs E, Tucker JV (2006) Embedding infinitely parallel computation in Newtonian kinematics. Appl Math Comp 178(1):25–43
Beggs E, Tucker JV (2007) Experimental computation of real numbers by Newtonian machines. Proc R Soc Ser A (Math, Phy Eng Sci) 463(2082):1541–1561
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 52–66
Beggs E, Tucker JV (2009) Computations via Newtonian and relativistic kinematic systems. Appl Math Comp 215(2009):1311–1322
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):2777–2801
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 20–30
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 2008
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): 1453–1465
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”
Beggs E, Costa JF, Tucker JV (2009c) Physical oracles. Technical Report
Beggs E, Costa JF, Tucker JV (2010) Limits to measurement in experiments governed by algorithms. Technical Report, Swansea University, submitted for publication
Campbell NR (1928) An account of the principles of measurement and calculation. Academic, London and New York
Carnap R (1966). Philosophical foundations of physics. Basic Books, New York
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. Amsterdam
Geroch R, Hartle JB (1986) Computability and physical theories. Found Phy 16(6):533–550
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): 163–172
Suppes P (1951) A set of independent axioms for extensive quantities. Portugaliae Mathematica, 10(2):163–172
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this paper
Cite this paper
Beggs, E., Costa, J.F., Tucker, J.V. (2010). Computational Models of Measurement and Hempel’s Axiomatization. In: Carsetti, A. (eds) Causality, Meaningful Complexity and Embodied Cognition. Theory and Decision Library A:, vol 46. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3529-5_9
Download citation
DOI: https://doi.org/10.1007/978-90-481-3529-5_9
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3528-8
Online ISBN: 978-90-481-3529-5
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)