Piaget, quantification, and ratio scale

by Pierre Moessinger

I would like to get back to a discussion that took place in a Jean Piaget Society (JPS) forum in December 2015. It started with a question from Leslie Smith, who was wondering why there is not more use of ratio scales in measuring children’s cognitive abilities. This question can be extended to a more general one, and thus a more interesting one, namely why there not more use of ratio scale in measuring psychological attributes. Of course, this discussion was turning around Piaget and his focus on the distinction between intensive and extensive quantities. In a way, Piaget answered Leslie Smith when he said:

« For example, [if] a subject retains 5 words out of 15 in a memory test, or 4 out of 6 sectors on a spatial journey, we have no way of knowing whether these words or sectors are equivalent as between each other, nor do we know how to compare the memory of words with that of a journey . . . The difficulty in creating systems of units may thus spring from the nature of the biological, mental, or biological-mental structures themselves . . . » (Piaget, 1970, p. 45).

Namely, a word remembered is not equal to another one, we cannot compare the memory of distances, nor can we compare remembering a word to remembering a distance. Note that Piaget is mentioning a measurement problem, not a scale problem. Were he thinking of scales, he would probably have mentioned the comparison between 15/5 and 6/2… but that was not his concern. As to the problem of « creating units », Piaget refers to the « nature » of the biological and the mental. Although he does not clarify, he probably thinks of the complexity, the loosely integrated systems, be they organic or mental.

In the JPS forum, I started to try to clarify the difference between the interval scale and the ratio scale. It is important to note that Piaget was not clearly aware of this problem, which explains his absence of concern for the problem of the zero and his ignoring of the problem of interpersonal comparisons. Briefly said, with an interval scale, measures are defined in any linear transformation. For example, the Celsius scale is a linear transformation of the Fahrenheit scale (with a different zero). Technically, it means that the ratio of differences are scale independent. For example, the ratio (in Fahrenheit) between, say, 20-10F/10-5F is the same for corresponding temperatures in Celsius. Now, with ratio measures, like weight or height, you can also use another scale, like centimeters instead of inches, you can add them or subtract them, and the zero would be the same in all scales, as it will have the same meaning (usually the absence of the property measured).

As, in the JPS forum, the discussion was turning around the ratio scale (and not around Piaget’s quantification), I tried to clarify this point first:

« Hi Les,

You cannot use the ratio scale for psychological attributes, say attributes dealing with thinking and feeling. The basic point deals with the absence of a « fixed » or a « real » zero. With an interval scale, i.e., with no fixed zero, you cannot say that 40/20 = 60/30, or that 30-15 = 3(35-30) for example. The reason why there is no zero for psychological attributes would take some space and ink, and would lead to discuss ontological problems; surely, it could be due to the current state of knowledge, or rather lack of it, but it rests mainly, I think, on the famous « impossibility of interpersonal comparisons » problem. Indeed, it’s a philosophical problem, and psychologists have little interest for it, yet the discussion you raised shows that it is difficult to circumvent it. We can, of course measure psycho-physiological properties, or abilities, on a ratio scale because we can identify a point under which it makes sense to say that the property we are measuring does not exist. And therefore, we can attempt to compare 2 individuals. But take the case of preference, for example: it does not make sense to say that there is a zero degree for your desire to drink a beer, it only makes sense to express your desire in relative terms. (Incidentally, this problem has stormed economic brains for more than 2 centuries.) In the same way, there is no real zero in the IQ scale, for example, and in other epistemic attributes. »

Yet my argument did not convince my interlocutor, and I must admit I may not have been very clear, so I tried another approach:

« Hi Les,

I do appreciate your interest for Piaget, I should even say your fascination for his work. I tend to think that Piaget was not as interested in the details of his own work as you are. First and foremost, he was interested in children’s responses, trying to make sense of them, and correcting what he had said earlier if these responses led him to. Of course he was interested in philosophy when he was young, but, at some point, stopped reading philosophy to concentrate on studying children. Some of his philosophical preoccupations remained, but they were reduced to genetic epistemology problems (example: ontological problems). And, as you know, in the early 60ies, he started to read some philosophy again, as he was writing Insights. His focus on studying children led him to neglect other problems such as concept clarification, problems you are well aware of and which became particularly clear when Battro was preparing his Dictionnaire d’épistémologie génétique; nor was he particularly interested in the coherence of his theory, adjusting it only when it became inevitable (as in the case of the horizontal décalages).

Similarly, Piaget was not interested in the problem of scales, which is our focus here. He was interested in children’s behavior of quantifying (intensively or extensively), but that is a different matter. You mention that for Piaget, « children’s quantification is based on ratio scaling ». We have to distinguish measurement and scale. We measure children’s quantification abilities on a scale, and children’s understanding of quantification goes itself through stages which cannot be translated into a ratio scale.

Piaget will not be of much help in our discussion of the difference between the interval scale and the ratio scale as he was unaware of the problem of the fixed zero, which, as I argued in my last mail, is the central characteristic of the ratio scale.

Let me mention an anecdote to illustrate this point. When Routledge asked me to correct a translation they had obtained for Sociological Studies, a book you know well, I called Piaget to tell him that I would have to mention in the English version, at least in a note, that he presupposed a ratio scale in his interpersonal equilibrium, thus ignoring the problem interpersonal comparisons (in other words, ignoring the problem a fixed zero was raising). Maybe I was a bit abrupt in saying this, anyway he made it clear he did not like my remark… the result of this discussion was that I did not correct the manuscript for Routledge. You know, I suppose, the rest of the story.

Now, what about this zero problem, on which I seem to be the only one to insist? In Sociological Studies, Piaget speculates about interpersonal balance, considering, for example that the action of A towards B is equal to the satisfaction B gets out of A’s action, and he writes something like a – s = 0. Here, we are comparing differences, i.e., we are on a ratio scale. Such a scale has a fixed zero, which gives meaning to interpersonal comparisons. But the zero has no real meaning here (for efforts and satisfactions), and interpersonal comparisons raise an insuperable philosophical problem. Had he read enough Pareto or Walras, he would have been aware or the problem of the zero. As to the question of the impossibility (or the absence of meaning) of interpersonal comparisons, it was treated mainly by phenomenologists, a category of authors Piaget had little interest for. »

After some people joined the discussion, in particular Trevor Bond, I felt I had to summarize:

« Let me remind that Piaget was focusing on the distinction between extensive and intensive measures, which, for him summarized the difference between mathematics and logic. Talking about units, was for him, talking about extensive quantities, it was talking about the possibility of mathematical operations. Now he mentions the “difficulty”, not the impossibility, of creating systems of units to measure the development of knowledge. That seems to leave open the door for new knowledge to measure – say — judgments, thoughts, opinions, etc., quantitatively. He then adds that this endeavor depends on the nature of “biological, mental, or biological-mental structures”, an assertion, as I mentioned, that remains a bit vague.

This makes me think that Piaget was hoping that his stages be, at some point in the future, found in the brain, i.e., that they would correspond to some neural networks. He did not insist on this point though, as he probably understood that such a position smacked reductionism. Also, Piaget always gave the impression that mathematical variables are more respectable than qualitative ones. Useless to say that he believed that scientific progress goes hand in hand with mathematization of science.

Of course quantification made big steps in psychology, as well as in the social sciences, in the 20th century, and there is no reason to think that this movement will stop. But it also led to big failures. For example, game theory or decision theory, which both come with a sophisticated mathematical apparatus, later appeared as unreliable and useless. And I am not talking of conventional micro-economics, with the central notion of utility, which is apriori and apsychological (there is no way to know the value of the utility of a particular individual).

The problem with psychological variables is that they are ambiguous, because they emerge out of many systems, particularly because they are both physiological and psychological, “biological and mental”, as Piaget says, i.e., they stride 2 levels. And the upper level cannot be simply reduced to the lower level. As we know, thoughts cannot be reduced to neural networks, the basic reason being that the same thought can correspond to different sets of fired neurons; in other words, a particular thought cannot be reduced to a particular brain state (see Sperry, Bunge, and others). Because psychological properties are complex, we can only approach them by means of indicators, and indicators are always perfectible. The problem of the measurement of intelligence (via IQ tests) illustrates this point. Thus, there are reasons to be less confident than Piaget in the possible quantification of children’s understanding of the world. He knew, there is a real difficulty in “creating systems of units” for measuring – say — psychological variables. Today, we better understand the limits of such creation. »

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