Worldly imprecision
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Worldly imprecision Michael E. Miller1
Accepted: 16 November 2020 Ó Springer Nature B.V. 2020
Abstract Physical theories often characterize their observables with real number precision. Many non-fundamental theories do so needlessly: they are more precise than they need to be to capture the physical matters of fact about their observables. A natural expectation is that a truly fundamental theory will require its full precision in order to exhaustively capture all of the fundamental physical matters of fact. I argue against this expectation and I show that we do not have good reason to expect that the standard of precision set by successful theories, or even by a truly fundamental theory, will match the granularity of the physical facts. Keywords Precision Quantities Empirical success Fundamentality
1 Introduction Suppose you have been tasked with measuring the value of my height at some particular instant in time. You might proceed by asking me to stand up straight against a wall at that instant, making a mark just above the top of my head with a very fine tipped pen, and measuring the distance between the mark and the floor with a meter stick. Were you to do this you would likely find that the mark on the wall falls between two millimeter markings on your meter stick, say the seventh and eighth millimeter markings between the 95th and 96th centimeter markings. Having already measured one full length of the meter stick, you would come to the conclusion that I am 1.957 ± 0.001 m tall. & Michael E. Miller [email protected] 1
Department of Philosophy, University of Toronto, Jackman Humanities Building, 4th Floor, 170 St. George Street, Toronto, ON M5R 2M8, Canada
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M. E. Miller
The precision of this measurement can obviously be improved. If only one additional decimal place of precision is required you could simply obtain a rule with finer markings. With the aid of an electron microscope you could determine the value with nine or ten decimal places of precision. In fact, on first inspection it seems that the only limit to the precision with which my height can be accurately determined is the resolution provided by currently available technology. In order for the only limit on the precision to come from such pragmatic factors, there must be a physical fact of the matter not just about the tenth decimal place of my height but also about the nth decimal place for any arbitrary n. If there is some level of precision beyond which there is no longer such a physical matter of fact, that marks a principled, not merely pragmatic, limit on the precision with which my height can be measured. It turns out that there is such a principled limit, at least if we impose plausible conditions on the semantics for the ordinary language term ‘‘height’’. By my height we plausibly mean the distance between the floor and the highest point on my head, and so a measurement of my height is a measurement of that distance. Consider the determination of the position of the highest point on my head.1 Suppose we can all agree w
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