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      Notice what the scale does not provide. First, it does not provide a unique number to each mineral. The scratch test cannot deliver verdicts for minerals that do not scratch each other except that they are of the same hardness. Second, and more importantly, it only delivers ordinal information, i.e., information about the location of the mineral on the scale. It does not, however, let you infer that if one mineral has hardness 2 and the other hardness 4, that the latter is twice as hard as the former. Diamonds are not, in any meaningful way, ten times as hard as talc. Finally, it does not even tell you that if M is 4, another sample N is 6, and still another one P, is 8, that P is harder than N by the same amount that N is harder than M. The Mohs scale does not provide this information, and it would be a mistake to believe that it does.

This latter sort of information can be gleaned from interval scales, however. Those are used, for example, for measuring temperature. The difference between 90 and 100 degrees F is the same as the difference between 80 and 90 degrees. With an interval scale, we not only rank temperatures on an ordinal scale, we can also measure temperature differences. What we still can’t do, however, is make magnitude comparisons between temperatures. For example, something with a temperature of 60 degrees F is not twice as hot as something with a temperature of 30 degrees F. This is so because the Fahrenheit scale has no natural zero point. 0 degrees F does not mean the absence of temperature altogether (as it does with the Kelvin scale).

      Contrast this with ratio scales, such as the centimeter scale. A measurement of 0 centimeters means the absence of extension (in the direction measured; never mind how this could be done). Thus, if you measure something to be 200 centimeters long, you can infer that it is twice as long as the thing which I measure to be 100 centimeters long. Therefore, the use of ratio scales provides more information than that of interval scales, and much more than we can get from ordinal scales: you can rank different objects by length ordinally, and you can measure differences in length. In addition, you can measure length ratios (thus the name). The interesting point here is that what information you can glean from the world is not only dependent on the physical characteristics of the measuring instruments, but also on the scale you choose to implement. Such choices are often determined by practical concerns.

      Given that ratio scales deliver much more information than, say, ordinal scales, the question arises why we don’t use ratio scales for measuring any property. The example of the property of hardness makes this implausible. What would it mean to measure hardness on a ratio scale? First, we would need an absolute zero point – but what material would exhibit the property of the absence of hardness? Second, even interval scales seem to be inapplicable to hardness. In what exact sense could differences in hardness be exactly same for different pairs of materials? What the implausibility of developing ratio (and interval) scales for some properties tells us about these properties is unclear. One possibility is that being hard is simply not a fundamental property of the universe, and that only fundamental properties are amenable to being measured on a ratio scale. However, it would be quite surprising if it turned out that the notion of a fundamental property corresponds to what scales are appropriate for its measurement.

      2.3.2 Operationalism

An entirely different answer to the puzzle about hardness (and related ones) is that the measurement procedure actually defines our scientific terms in the first place. This view is called operationalism. According to it, measurement procedures give meaning to otherwise ill-defined predicates (those are the terms used to refer to properties). A famous example is the predicate “intelligent.” An operationalist would claim that the term “intelligent,” as ordinarily used, is unsuited for scientific purposes. We simply don’t know whether or not it applies to the behavior of a person, because we don’t have sufficient agreement between researchers as to what property is singled out by the term. Thus, we define “intelligent” more precisely as standing for a property which is causally responsible for the performance of a person in a standardized intelligence test. Such tests deliver numbers on a scale. The view is that those numbers may or may not measure any property we were interested in before we started to theorize, but rather single out a property as corresponding to a precise predicate that is scientifically useful. On this view, hardness may be a pretheoretical property, corresponding to different human experiences of resistance to pressure, but the “scientific hardness” of a sample of material is its location on the Mohs scale. End of story.

      A form of operationalism was influential for Swiss physicist Alfred Einstein’s development of the special theory of relativity; he also insisted that legitimate scientific concepts have to be defined by appropriate measurement operations. Thus, the meaning of “space” is given by the operations we apply for measuring length, and that of “time” by the operations for measuring duration. We will revisit this issue in Chapter 11. However, as a general theory of the meaning of scientific terms, operationalism is inadequate, as can be shown by the following consideration.

      First, if methods of measurement indeed fix the meaning of a predicate, then there can be no such thing as an incorrect method of measurement, because there is simply no room for a mismatch between a property and how it is being measured – the former is defined in terms of the latter! For example, if intelligence is defined in terms of the standardized test with which we measure it, the test cannot fail to measure intelligence correctly. The measurement methods are correct as a matter of convention. But, second, it seems that it is possible to develop methods for measuring some quantity that are wildly inadequate. For example, measuring hardness by shining lights of various colors onto minerals seems clearly wrong, even if we supply a “scale” (looking prettiest under color 1 (red) = hardness 1). Seeing that this “measurement” of hardness is just crazy clearly shows that there is a well-(enough)-understood notion of hardness even before we try to measure it.

This seems to support a sort of realism about the properties we try to measure, in the sense that measurement is the estimation of the magnitude of an independently existing property. In particular with respect to properties such as length, weight, and duration, for example, a realist stance seems most plausible. That is, there are real, observer-independent features of the world. It certainly seems to be the case that objects have a determinate length independently from our attempts to measure it, and processes have an equally determinate duration. But as we will see later, things are actually not all that clear cut. It turns out that what length we measure depends on the motion of the object relative to the observer. This phenomenon is called length (or Lorenz) contraction, and we discuss it in some detail in Chapter 11. However, even with such simple properties such as mass, we can already notice that its measurement is at least in some cases heavily influenced by already existing background theories. Let’s have a brief look.

      2.3.3 Theory-Ladenness of Measurement

      Suppose I want to determine the mass of my bowling ball. To that end, I might put it onto an ordinary balance and compare it with a known mass, such as a number of metal cubes each weighing 1, 10, or 100 grams. Things get more – much more – complicated if I want to know the mass of a distant star. Obviously, I can’t put it onto any balance. So how do I measure it? The details are actually quite complicated, but we can roughly say that measuring the mass of a star involves various background theories. For example, if we want to measure the mass of a binary star, we first determine a center of mass between the two stars, then their distance from that center which we can then use, together with a value for the period (the time it takes to complete one orbit around each other) and a certain instance of Kepler’s Third Law, to calculate the mass. In other words, in order to “measure” the star mass, we measure other quantities and use those values, together with certain equations, to calculate the mass. Obviously, the correctness of such a “measurement” not only depends on the correctness of other measurements (in this case, at least that of the period), but also on the correctness of certain background assumptions, such as Kepler’s Laws. Measurement is not a simple and unmediated estimation of independently existing properties, but often a determination of certain magnitudes before the background of a number of

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