Fundamentals of a Theory of Measurement

Proposal for the establishment of the concept of size

The goal of a theory

Goal of the theory of measurement is to allow a safe acquirement and reproducibility of measuring characteristics. One shall show the necessary conditions for the cognitive requirements to make scientifically relevant measurement predictions possible. After all, one has to put an end to the unacceptable situation, that our science of measurement upon which we rely, are not always clear in what the basis and content of measurement predictions concern. Time is ripe for a presentation of the rational fundamentals of metrology, in order to allow scientists to make judgements free from intuitive principles, plausible facts, or authorities. Only possessing a rational theory one can know the reason why one knows and how sound our knowledge is.

1. The irrevocable cognitive starting position

  • 1.1. Criteria are settings of the logic and are, therefore, logical. Criteria are neither true, nor untrue. Only their degree of logic is open to discussions.
  • 1.2. Characteristics are criteria transferred upon objects, in order to appropriate them mentally. They are also neither true, nor untrue. It is also crucial, how logical and how suitable they are, in order to handle the respective objects, both mentally and practically, in the desired way.
2. The object of the theory
  • 2.1. The object of the theory are (physical) magnitudes/quantities.
  • 2.1.1. Physical magnitudes/quantities are quantitative features.
  • 2.2. In order to designate a quantity, the human intellect has first to acquire a notion (concept) of the respective quantity. The concept, therefore, will depend first of all on human perception and ability of knowing and second, on interests.

The concept of heat presupposes a sensitivity to heat, just like the concept of temporality relies upon memory (i.e. the ability to remember). Duration is thus a quantity perceptible in a temporal observation of things. Heat and duration are aspects of things perceived by us, while manipulating them. These aspects do not allow us to conclude that things can exist as objects outside human observation. The later is, however, insignificant for the treatment of quantities and for the theory of measurement. We are saying today that heat is an aspect of the molecular motion within a body, or a system (for example, a gas). The concept of heat retains, nevertheless, its meaning. The same applies to the quantity "mass": it is no thing in itself, but rather an aspect of a thing, namely the measure of its mechanical resistance during interactions. The quantity "velocity", on the other hand, depends really on the chosen distance, i.e. on the initial and final points of a path. The selected section of the path has to be always specified, in order to convey reconstructable knowledge.

3. The method

  • 3.1. Knowledge about magnitudes/quantities is acquired through measurement.
  • 3.2. The method (mean) of measurement is a comparison.
  • 3.3. The aids of measurement are the standards and the scales. The units are not measured, but rather fixed by definition.

A basic pattern of recognition is the comparison. To measure means to compare quantities. A multiplicative comparison means to compare an unknown dimension with a known one, i.e. with a unit, with the help of aids (scales). In this way, the unknown dimension will be expressed as as a multiple, or as a fraction, of the defined unit. The so obtained quantity is a number. The concept of measurement involves, therefore, two cognitively different measures: one unknown and one known, as well as their comparison by a measurer (or a measuring device). The result is knowledge. This holds for relative comparison - as in the case of hardness - where each time the harder material is chosen as a reference.

4. Implementation and significance of the theory

  • 4.1. The unit of a measurable quantity (the comparison factor = measurement unit) has to be appropriately defined and implemented through conventions.
  • 4.2. The quality of a definition of measurement units is given by the degree of mathematical representation and by the accuracy, reliability and constancy of its realizations.
  • 4.2.1. Measurements units are not a question of truths, but rather of usefulness and of validity. There is no way to determine the "real" magnitude of a measurement unit. One can, however, for the sake of usefulness, chose the smallest unit, or the most significant phenomenon related to the magnitude of interest, as basis for a scale.
  • 4.3. The irrevocability of our level of knowledge makes the above procedure for valid measurements, compulsory. Theories and statements contradicting the necessary procedure are false and have to be rejected. A theory of measurement based on the cognitive status and the corresponding metrology is an unconditional basis for measurement statements made by any (natura) science.

There cannot be useful quantitative knowledge without consistent standards being valid everywhere. These standards and a consistent reference system allow meaningful, comprehensible measuring values. Any statement regarding the fact that a body/a system "rests" or "moves" is also a measurement relative to a reference set by the observer. In terms of linguistics, this observation of "resting" or "moving" is a purely metaphorical seeing and speaking habit of the observer using terms usually used for living beings. However, for lack of an objective distinctive feature of both states, which lies in the matter itself, inanimate bodies are far beyond the properties of "resting" and "moving". For this reason, there cannot be serious mechanics of "moved systems". In contrast, Newton's classical mechanics only deal with the change of the kinetic momentum (of the acceleration), which is the consequence of an objectively acting force and, in doing so, mathematically eliminates the subjective quantity of velocity (v). Thus, Newton's mechanics will always be essential. It is indeed rightly termed "classical".

The only instance above logic is the unconscious expectation and a (possible) prejudice, which can hinder its correct application. It is our duty to make these expectations and prejudices known, in order to do justice to logic. If, however, the fundament of all experimental sciences - the theory of measurement - is not founded upon rational principles, (but, for example, on intuition), what kind of competence could such a science convey? These will lead to useful practical results through trial and error. Scientists, however, will miss clarity about: a) what is the measuring standard and what is the object of measurement, b) what happens cognitively during the measurement, and c) what exactly they measure there? As examples for missing clarity we can mention the popular conviction that time were measurable and that time and mass were things. Both are, however, physical magnitudes/quantities which we, due to our cognitive ability, transfer upon things. We grasp, therefore, only their aspects: metrologically seen, time is a measure of duration (Newton) and mass is the measure of a mechanical resistance. Their respective units, like any unit of measurement, have to be fixed by definition. Only this way are we able to measure. After all, every perception is a measurement, even if not each one can be quantified.

Standards are not measured; they are provided! The question concerns not their truths, but rather their validity.
Their units are provided to us by norms to which material standards have to obey.

I think these considerations cannot be objected; there is no place for ´if´ and ´but´, otherwise the whole theory of measurement is questionable.

Translated by Dr. George Galeczki (Cologne/Germany)
Clarification pt. 4 translated by Martha Greiner-Jetha (Gröbenzell near Munich/Germany)

© HILLE 1997-2017
Clarification pt.4 5/2017 new translated

Look also: General Foundations of Mechanics

Letter of the Chairman of the DPG (Deutsche Physikalische Gesellschaft)
When, on the occasion of my 70th birthday, the former chairman of the DPG Prof. Dr. Alexander M. Bradshaw congratulated me in form of a longer letter, I of course thanked him and used the occasion to include a copy of my paper "Grundlage einer Theorie des Messens. Vorschlag zur Begründung des Größenbegriffs (zur geplanten DIN 1313)" - my lecture during the DPG meeting in Munich in 1997. He replied to me on September, 28th 1998 in brief but profound:

Dear Dr. Hille,
thanks a lot for your letter darting July, 22nd 1998 with attachments.
I have enjoyed reading it and have sent copies to the members of our board.

A.M. Bradshaw

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Lecture delivered on the Spring-Meeting of the German Physical Society in March 1997, at Ludwig-Maximilians-University, Munich/Germany