Theory of Measurement by Johann Pfanzagl
My rating: 2 of 5 stars

Only Fiction Can Save Science

Pfanzagl’s 1971 book is a classic text in the theory of measurement. It is a highly technical work which is read, I presume, only by specialists in the field called metrology. It is probably ‘inaccessible’ and consequently ignored by everyone else. It is a prime example therefore of what C P Snow called the Two Cultures, one of science, the other of the humanities, each speaking a mutually incomprehensible language about issues which have no apparent connection. I want to use this review to make a minor bridge between those two cultures, using several pieces of modern fiction to criticize Pfangl’s theory.

According to Pfanzagl, “The subjects of measurement are properties. Weight, color, intelligence are typical examples to illustrate the sense in which the word ‘property’ will be used here.” He then goes on to say, “Although we always start from relations between objects, it is the properties which are the concern of measurement, and not the objects themselves.” This is an apparently common-sensical proposition which conforms to what most of us would approve of. It isn’t, however, and we shouldn’t. It is an example of one culture leading the other astray through definitional sleight of hand.

The fundamental idea that Pfanzagl is promoting is that properties, particularly measured properties, are “manifestations” of the objects they are associated with. This is an insidious idea which infects thinking in a variety of unhelpful ways. Most importantly, it ‘naturalises’ the establishment of what are nothing more than arbitrary choices about these phenomena. Pfanzagl sees the standardization of measurement as a breakthrough, for example, “The development of a coherent system of universally accepted scales is one of the first signs of transition to a scientific stage of the society.” This ‘universal acceptability’ in fact makes the choice of scale invisible and therefore unconscious and immune to further choice or criticism.

I have written elsewhere about the logical and practical difficulties this concept of measurement causes (see for example https://www.goodreads.com/review/show...). Here, however, I would like to try a more intuitive explanation of the problem using the Yōko Ogawa’s The Housekeeper and the Professor and Nicole Krauss’s Man Walks Into a Room as a sort of dialectic to explore the issue (see: https://www.goodreads.com/review/show... for more detail)

The Housekeeper and the Professor contains a great deal about what is called Number Theory. Many people have reported in their reviews and comments that much of this theory seems incomprehensible. I think that part of the reason for this is that it appears to be entirely abstract.

But in fact Number Theory is very important in everyday life. Among other things, it is the foundation of measurement and what measurement actually means. Without understanding this we can get vary confused about what gets reported in things like economics and social sciences generally.

Numbers define what is called a metric; they never exist on their own but only with some other group of numbers. This group constitutes a metric - a rule for ordering things. For example, let's simply define a scale from, say 1 t0 10. The choice of the units in which to express things on this scale - pounds, dollars, votes etc. - can be left aside for the moment because the importance of the units pales into insignificance compared with the choice of metric, that is the sequence in which objects or events will be placed..

When we measure something, we place whatever it is on a location on the metric, say at $5.50 as the dollar value of something. What we don't do is to assign the value on the metric to the thing being measured. In other words the thing being measured becomes a property of the metric. The metric does not become a property of the thing. The way to lie with numbers is to pretend that numbers are a 'property' of anything. They are not. When we measure we assign a property to a metric.

This distinction is important because it prevents us from thinking that the thing being measured has some objective value that is inherent to itself. Measurement is always about the value of something: weight, height, temperature, price, importance, etc. Politicians and economists, and indeed con men of all sorts, like to make believe that value is somehow inherent in what they are proposing or opposing. It isn't. Value lies in the choice of metric.

For example, in Britain the proposed High Speed Line from London to the Midlands looks like a winner on a government metric of travel time reduction; but it's a dead loss if the metric used is anything to do with destruction of countryside-beauty. And there is no way to compare these two metrics without constructing another one ad infinitum. The political game is one of establishing your metric as definitive, not because it is the best one but because it gets what you want. A few billion pounds off one way or the other in estimates are are marginal significance.

Number Theory may be understood as thinking about alternative metrics, that is, alternative scales of value. A scale say running in equal increments from 1 to 10 has very different implications than one that proceeds from 1 cubed (=1) to 10 cubed (=1000). When something is placed at 5.50 on either of these scales, the resulting relative values are vastly different. Any error in the choice of a metric is orders of magnitude more important than an error made on a metric (see the graphic example at the end of this review). This is why statistical analysis, which is the analysis of error on a metric, is so often either misleading or entirely irrelevant.*

There are two types of measurement typically discussed in textbooks like Pfanzagl’s: Cardinal measurement and Ordinal measurement. The latter is simply the placing of things in their correct order on a metric. Ordinal metrics are, despite this simplicity, infinitely precise. There is always room to put a measured object between any two existing objects. All measurement of value is ordinal measurement.

Cardinal measurement is what many erroneously think of as 'scientific measurement.' Pfanzagl certainly does. However it is neither scientific, nor is it really measurement in any practical sense. Cardinal measurement is actually pure mathematics. It considers numbers and their relations solely to each other at any degree of precision desired. Cardinal measurement is concerned mainly not with the numbers themselves but with the intervals between them. When scientists , social or physical, then claim or imply that these intervals exist with similar significance in the world outside of mathematics, they are, as noted above, assigning numbers to things, a fundamental and duplicitous No No (Economics recognises this implicitly, grudgingly on the part of its Aristotelians, with its concept of declining marginal utility).

In my review above of Ogawa's book I allude to these two types of measurement when I make the distinction between Aristotelian and Platonist thinking. Platonists, because they hold with the independent existence of numbers, and their metrics, they wouldn't think of degrading numbers by allowing them to be properties of anything. They are themselves, and themselves alone. They are after all God-like. As the Professor told his Housekeeper, "The mathematical order is beautiful precisely because it has no effect on the real world."

Aristotelians are a lot less fussy about how they treat numbers. Krauss’s protagonist is an Aristotelian and as a consequence hopelessly damaged. His condition makes it impossible to have any real connection with the world. He has forgotten everything about his life, which is now not even a story. He is lost because the ‘properties’ of the things he most loved have disappeared with those things.

Aristotelians, like Pfanzagl casually refer to numbers as properties of things without thinking carefully about what they're doing. They act as if numbers do have a material and causal effect on the world. They casually assign these magnificent entities to the most meaningless and banal events. Aristotelians fling cardinal measurements around with gay abandon, hoping to deflect attention from the metrics they are really peddling. This is why they are employed by charlatans and crooks to convince us of unhealthy things.

Sociologically and psychologically, Aristotelian, cardinal, measurement separates and distinguishes us. It pretends to identify us by assigning numbers to us: height, weight, IQ, race (yes race is a number). It establishes what it claims is our independent existence. This is perhaps its most insidious ideological effect.

Platonist, ordinal, measurement shows how we are related to one another, how we are ordered on an infinite number of metrical scales. Each of these metrics is part of reality but not one of them is definitive, even in combination. Platonist measurement thus respects both our individuality and our interdependence.

My fourteen year-old grandson challenged me recently to demonstrate the paradoxes involved in Aristotelian cardinal measurement. I asked him to draw a straight line with a ruler on a sheet of paper, which he duly did. I then asked him to measure the length of the line. He did so and reported a length of 15 centimetres. I disagreed and said I read 15.1 centimetres. He then confirmed my re-reading of the ruler. I then challenged his confirmation and measured the line again with a compass and a more precise ruler. This technique yielded a length of 15.08 centimetres.

At this point the penny dropped for him. No matter how precisely we measured the line segment, there was always a greater level of precision possible. Greater precision would always show the 'error' of less precise measurements. The problem only came about however because we implicitly presumed that the measurement we were making was a property of the line. Demonstrably therefore this cannot be the case.

When we ordinally measure the line we assign it to the numerical scale, to say, exactly 15 centimetres. The line in a sense becomes an attribute of this scale which may have any number of other lines we are interested in assigned to it. No uncertainty then: the line is 15 centimetres, snuggled right in there between 15.1 and 14.9, in ordinal simplicity.

In the novel, the Professor attaches notes to his jacket to remind himself, or actually to inform himself anew every 80 minutes, about ideas and things in his life, like the existence of his housekeeper. Sometimes these notes fall off or are changed by either the Professor or his housekeeper.

Crucially, however, they do not constitute a part of the Professor, they are not a property of him or his identity. Rather they are icons, as it were, pointing elsewhere, recalling either his place in the order of the world or his place in the order of mathematics. They are reminders of Platonist metrics in other words. This is yet another indication of Ogawa's intention.

In short, getting a basic grasp of Number Theory is not important only for mathematicians. It's an essential skill in modern life in which numbers are used to justify or condemn almost everything from taxes to immigration. It's unlikely that informed judgments are possible without that skill. The skill is an aesthetic one. It requires taste to distinguish a better metric.

This is one reason to listen to music, especially classical music, very carefully. Music is metrical sound, numerical scales for the ear. The instruments in a Beethoven symphony become ordered in and through the score. The score itself, a combination of aural metrics, remains what it is, a way to order the instrumental parts. It does not become a property of the instruments.*

And there are better scores than others. These are metrics that include lesser metrics without erasing or denigrating their lesser brethren. Such are Beethoven's to Hayden's symphonies for example. In mathematics this is expressed in Set Theory wherein one set, a metric, is part of a larger set, a special case of the larger set, just like Newtonian Physics is a special case of Relativity Physics.

Aesthetic judgments like this are different from other sorts of judgments. But they are not arbitrary. They involve a distinctive form of logic. The American philosopher C. S. Peirce called this logic 'abductive' to distinguish it from the more well published (Aristotelian) logics of inductive and deductive reasoning. Real genius in mathematics is always shown abductively. Induction and deduction is typically only used to show this genius for what it is.

In summary, Pfanzagl’s theory of measurement is flawed not in its logic but in what is termed its ontology, it’s basic presumptions about what exists in the world, and in its epistemology, how we access what exists as human beings. For me it is an example of not scientific but scientistic thinking, the presumption that there is a fixed, universally valid method for thinking. This attitude is exemplified through the subtle elimination of alternatives at is usually the definitional stage of analysis. And it is here that the junction of the two cultures can be made most visible as I hope Ms’s Ogawa and Krauss would concur.

Rise up, you readers and writers of novels!


*For more on the analogy of measurement and music, see: https://www.goodreads.com/review/show...

For further explanation of metrics and metrology see: https://www.goodreads.com/review/show...

For more on the aesthetics of mathematics, see: https://www.goodreads.com/review/show... and https://www.goodreads.com/review/show...

See also: https://www.goodreads.com/review/show... for further discussion of the connection between aesthetics and mathematics.

Illustration of the relative importance of the choice of metric

Below is a simple graph showing a strait forward linear metric on the x axis and the natural logarithm of the values of this metric on the y axis [y = f(x) = ln(x)]. Each is a very distinct metric despite the fact that both are expressed in the same numerical scale of 1,2,3 etc. Measurements taken on one will be dramatically different from measurements taken on the other. Any error in measurement on either metric will likely be insignificant in comparison with the difference in measurements between the two metrics.

Students of economics will recognise this graph as indicating the declining marginal utility of money, an established principle of micro-economics. Yet the declining marginal utility is rarely used in analysis because it is difficult to estimate and to use in calculations. Therefore it is presumed in all of financial and risk analysis that there is constant marginal utility of wealth and income - an example of the many times that economics and other social sciences look for their keys under a lamp post simply because there is light there.



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