The role of mathematics in the concept of repeatability and replicability.

The previous part of this project was devoted to an inquiry into the dependence of replicability on models, in particular mathematical ones. Metrology makes extensive use of mathematical models (abstract, idealized representations capturing the relation between an object, its instrument, the instrument indicator and the measurement outcome) which are based on theoretical and statistical assumptions. This means that the stability and thus the replicability of a measurement depends on the (conventional) statistical assumptions of the model, where statistical hypotheses refer to the distribution of indications and the properties of background noise.

In this part, the P.I. will inquire into the dependence of replicability on the mathematical language in which the scientific theory and measurements are described. The role that mathematics has played in the concept of replicability has been crucial since the very beginning. The struggle for building a reliable and objective science upon replicable measurements was certainly an issue for the first experimentalists. For them, indeed, it was almost impossible to replicate other scientists’ experiments for lack of transparent communication and the unavailability of any unambiguous description of the experiments. Not only was it impossible to keep record of all the details of experimental practice but also there was no universal, precise language to communicate quantities. For instance, the description of measurement involving a glass of a liquid could have meant different quantities to different scientists working in different countries or with different tools. It was also very difficult to imagine and recreate the experimental devices, and while most of the time a remedy was to provide drawings, it was certainly difficult to provide drawings so accurate as to capture all the features of those instruments.

The impossibility of achieving a complete and transparent transmission of experimental practice was mainly due to the lack sufficiently sophisticated mathematical language and apparatus. Indeed, the problem was thought to be overcome with the introduction of mathematical quantities, units of measure, and their classification. In this regard, mathematics played a major part in creating the formal conditions for repeatable measurements.

Another very important role that mathematics plays in the concept of the repeatability of measurement is that it provides a powerful tool to establish whether and to what degree measurements are indeed repeatable. Indeed, the case that repeating the same procedure leads to the very same result is very rare, as there are always some conditions of the previous measurement that cannot be perfectly met. One way to evaluate the repeatability of measurements is to calculate the standard deviation of the mean after performing a series of identical measurements. This procedure allows the scientists to check whether the results are indeed ‘the same’, notwithstanding that they actually differ.

The present project will mostly be concerned with the third and most important role of mathematics in the concept of replicability, which is to shape and determine the concept itself. The working hypothesis of the P.I. is that it is sometimes the case that some measurements are considered nonrepeatable because of the mathematical language in which they are described, not only for physical reasons. An interesting example supporting this working hypothesis comes from the literature on repeatable measurements in quantum mechanics. In quantum mechanics, as in classical mechanics, an ideal achievement would be to have measurements that detect and seize a particular property or quantity independently of the measurement process. These measurements would satisfy the repeatability condition formulated by Neumann (1955), according to which if a measurement result is , the immediate repetition of a measurement would with certainty yield the same result as the initial measurement. In order to ensure this, Neumann introduced the collapse or projection postulate which opposes an instantaneous, irreversible and non-deterministic state transition to the continuous, deterministic state dynamics governed by the Schrödinger equation. It is then commonly believed that in conventional approaches the repeatability condition is satisfied thanks to the Neumann collapse postulate. However, the collapse postulate can apply only to the measurement of discrete variables (the measurement of observables with discrete spectrum) while it is inapplicable to continuous spectrum observables, since their eigenvectors cannot be normalized (Busch et al. 1996). For instance, while a harmonic oscillator allows for repeatable measurements, given that its Hamiltonian has a discrete spectrum, the system of an unconfined particle does not allow for repeatable measurement given that its Hamiltonian does not have a discrete energy spectrum. The problem is that as long as it is not possible to generalize the collapse postulate to continuous spectra observables, quantum theory may be regarded as unsatisfactory as its most important measurements (of position or momentum) are nonrepeatable. However, this topic is controversial. While some physicists regard it as a mere technical problem that can be simply resolved by generalizing the projection postulate to all kinds of measurements (Srinivas 1980), other physicists argue that even in the case of discrete spectra, there are no repeatable measurements available (Cini 1990). In this regard, it seems that Neumann’s projection postulate is not enough to guarantee the repeatability of measurements. But this, according to them, is not a problem.

The P.I. will be concerned with investigating the role of mathematics within this discussion. Do nonrepeatable quantum measurements mainly constitute a technical problem? Would it be possible to design repeatable measurements for observables with continuous spectra? Would a change of mathematical tools render some currently nonrepeatable measurements repeatable or is the non-repeatability of position measurements a physical fact? Mathematics is normally conceived just as a tool for describing physical phenomena. Different mathematical tools can be used at need. If mathematics does shape the concept of repeatability, and different mathematical tools would shape this concept differently, in which cases and to what extent is repeatability a mathematical rather than a physical concept? Would the conventional choice of the kind of mathematical tools employed in the description of physical theory render the concept of repeatability a conventional concept as well?