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A B Forbes, N F Zhang, A Chunovkina, S Eichstädt, F Pavese (eds.):
Advanced Mathematical and Computational Tools in Metrology and Testing XI
Series on Advances in Mathematics for Applied Sciences, Vol. 89
© 2018 World Scientific Publishing Company (pp. 85–118)
Metrology and mathematics — Survey on a dual pair
Karl H. Ruhm
Swiss Federal Institute of Technology (ETH),
Institute of Machine Tools and Manufacturing (IWF),
Leonhard Strasse 21, LEE L219, Zurich, CH 8092, Switzerland
E-mail: [email protected] When we do usual Computation in Science, especially in Metrology, we assume that we do Mathematics. This is only partly true in spite of the fact that database handling and data processing are most important indeed. The field of Measurement plus Observation is much more entangled with Mathematics. The following survey “Metrology and Mathematics” will focus on selected ideas, concepts, rules, models, and structures, each of which is of a basic mathematical nature. When doing this, it is not surprising at all that most theoretical and applicational requirements in the field can be reduced to just a few basic logical and mathematical structures. This is also true for complex processes in fields like humanity and society. Signal and System Theory (SST) supports this claim. However, such an endeavour asks for an orderly and consistent definition of quantities and processes and of their mathematical models, called signals and systems. These models represent all kinds of issues, items, and phenomena in the real world. The term model indicates the superordinate means of mathematical description, observation, and representation in Science and Technology. The question arises, what the role of Logic and Mathematics looks like in detail. The term structure will be consistently pivotal. And, as is generally known, structures are best visualised by graphical means, here called Signal Relation Graphs (SRG).
Keywords: Metrology, Measurement, Observation, Mathematics, Stochastics, Statistics, Signal and System Theory, Model, Structure, Relation, Signal, System.
Introduction
One should think that Metrology, as Measurement Science and Technology, is somehow comparable to Pure Mathematics and Applied Mathematics. Pure Mathematics follows purely abstract goals. In principle, Measurement Science does so too, but only seldom. Most Pure Sciences, including Measurement Science, are in need of mathematical support, and rely or have to rely on Applied Mathematics: As a fact, Metrology is interested in Applied Mathematics, but very seldom in Pure Mathematics, which fosters top-down frameworks (bird perspective) and allows holistic views. On the contrary, applicational topics in everyday measurement life utilise bottom-up frameworks (frog perspective), allowing “only” highly specialised and thus limited based insight. In summary, so-called abstract needs in Measurement Science call for concepts and results of Applied Mathematics. Thus, Pure Mathematics and Measurement Science do not reside on the same level. This is all common feeling, may be considered natural, and at least comprehensible.
Measurement Science aims at descriptions of past, present and future states. It either regards unrealistically the whole, hypothetical infinitely large real world, or rather realistically the particular, bounded, tangible small real world. The sensory process together with the observation process may serve as an example of such a restricted real process. It acts as a subsection of Measurement Technology in data acquisition and data processing.
In order to succeed in such a description of real world measurement and observation processes, we have to “download” abstract concepts, methods, and procedures from superior scientific insights, which are delivered by Measurement Science. They emerge in form of terms, rules, standards, definitions, analytics, inter- and extrapolations, procedures, best practises, trade-offs, calculations, refinements, corrections, tricks, programs with coding and decoding, and so on.
It may be surprising that all these abstract and manifold artefacts are fostered by few concepts, methods and tools, presented in the next sections.
These concepts, methods, and tools are first and foremost of a logical and mathematical character indeed. This is even true, if we at times do not have enough quantitative information about structures and parameters of the processes under investigation. Nevertheless, they stand by in the background. A first informal statement concerning a relation (interaction, dependence, association, correlation, assignment, coherence) between well-defined quantities of interest is at present sufficient for subsequent, determined, quantitative activities: Such a preliminary black box mode proves truly convenient in practise. Of course, we finally have to know, which of the concepts, methods and tools in the wide field of Applied Mathematics may best serve our particular, narrow needs, a challenge to be supported beforehand by Measurement Education.
Normally, the metrological community enjoys mathematical concepts only reluctantly, unless they come in form of shortly useful “cooking recipes”. One reason for this aversion of becoming committed in mathematical fields may be lacking overview and skills concerning the superior, holistic, and systematic principles and structures, which Mathematics readily provides to everybody. Admittedly, the principal structure of cause and effect is widely accepted [2, 4], but a deeper engagement in theoretical and practical consequences and possibilities is often missing; another challenge to be met by Measurement Education. Besides, amazing analogies can be detected as to many Philosophers of Science respecting their commitment to logical and mathematical structures.
Often, direct consequences result from such reservations. Design and realisation are studied from scratch and invented anew, although solutions are ready for use from generalising background sources. An outstanding example is the important task of error correction, which is once and for all offered by the structure of an inversion procedure of a mathematical model. However, who cares?
The following sections describe and recommend some selected mathematical concepts and structures, which primarily refer to measurement and observation procedures. This is not too demanding an endeavour, since most theoretical and applicational requirements in the field may, surprisingly enough, be reduced to about half a dozen main structures according the motto “Keep it Simple”. This is also true for complex processes: Signal and System Theory (SST) supports this claim. In this presentation, the path to this concept will be gradually prepared. Moreover, the main aim will be, to create the impression that such a holistic approach serves even way beyond Natural and Technological Sciences.
The disposition of this survey starts with selected mathematical tools to describe quantities and processes in a general way. These two basic terms represent any sort of phenomena and items on the real world side: The term mathematical model, concerning these two terms, stands for the superior means of abstract description, behaviour, and representation in Science and Technology.
Above all, the most tantalising omnipresent claims “Our mathematical universe” [10] or “Why our world is mathematical” [11] remains. Such popular statements will be commented shortly in the concluding section.
1 Assumptions and Preconditions
We will consider Measurement and Observation Procedures [9] and will use their common terminology. Fortunately, after slight adaptations of terminology and corresponding statements, the presented concepts prove valid in other fields too. Additionally, we assume that all quantities to be measured and observed are well defined, in spite of the fact that there are always severe discussions about definitions and meanings of quantities.
The well-known Cause and Effect Principle allows multiple quantities as causes, considered as inputs to the process. They may even occur correlated. We may also recognise multiple quantities as effects, ...
