Quality and Quantity: Another Look at Bennett and Bibi (Column 400)

As part of my desperate race for ratings, this time I decided to ride the winning wave of the previous column and write another piece with a philosophical–halakhic bent. If you’ve gotten at least this far, then apparently the title I chose did its job. Still, I’m sure you’ll agree that it wouldn’t be fitting to leave the festive Column 400 (who said they’d organize a party for this column?) without readers.

A First Look at Quality and Quantity[1]

In everyday speech, the terms “quality” and “quantity” are used quite frequently, and they are usually taken to describe two kinds of magnitudes that require two different modes of measurement. Some properties describe the qualities of things, and others describe quantitative characteristics. Of course, in halakhah the distinction between quality and quantity also surfaces in several contexts—namely, there is an assumption that these terms are well-defined and can be discussed.[2]

One such discussion is held regarding disagreement among judges in a court (beit din). According to halakhah, a regular beit din seats three judges. The ruling of the court is determined by the majority opinion, as learned from the verse “after the majority to incline.” Not all judges need to possess the same level of expertise and wisdom. At least in monetary cases, there can be a beit din composed of one learned scholar-judge (a gmir, a talmid ḥakham) alongside two lay judges (who at least understand what is explained to them). What do we do when they disagree? Some halakhic authorities (see for example Sefer HaChinuch, commandment 78, following Rav Hai Gaon) hold that the “majority” here is not the numerical majority of judges but the majority of wisdom and expertise; therefore, when the learned judge disagrees with the two laymen, the ruling should follow him. By contrast, Ramban in his novellae to tractate Yoma holds that the determining majority is always the numerical majority of judges.[3] Some later authorities formulate the sides of this dispute as a debate over whether what decides is the quantity or the quality of the judges. According to the Sefer HaChinuch, quality decides, whereas according to Ramban, the decision follows the quantitative majority.[4]

To the Crux of the Difficulty

Looking more deeply, this description is not so clear-cut. Seemingly, one could say that even the Sefer HaChinuch requires a quantitative majority rather than a qualitative one—except that, on his view, the relevant majority is the quantity of wisdom, not the quantity of judges. From the verse we learn that, halakhically, we rule by the majority, but the question remains whether to assess that majority by counting judges or by their quality (in cruder terms: whether we count legs or heads). According to the Chinuch, the required majority is not a majority of judges but a majority of wisdom—meaning that he too requires a quantitative majority, but of a magnitude different from that required by Ramban. What we previously called the “quality of the judges” we now call the “quantity of their wisdom.”

Up to now we assumed that the concepts “quality” and “quantity” are well defined, and the difficulty lies only in their application to reality. But we can now broaden the question and claim that the concepts “quality” and “quantity” themselves are not well defined and, in fact, are not truly different. The quality of a given thing can always be described as the quantity of something else. Just as we saw that the quality of a judge is actually the quantity of his wisdom, the same maneuver can be done for any qualitative magnitude. In this formulation of the issue, we are already challenging the very separation between the concepts “quality” and “quantity,” not just their application to this or that concrete problem. Is there any distinct meaning at all to the difference between “quality” and “quantity”?

It is clear, of course, that no basic concept can be defined precisely. When we come to define a concept, we necessarily use other concepts that are more basic and familiar than it. Therefore, at the beginning of the entire conceptual structure there must be a set of basic concepts that cannot be defined in terms of concepts more fundamental than they. From here arises the possibility of claiming that the concepts “quality” and “quantity” are basic and therefore cannot be reduced to more fundamental concepts. Robert Pirsig, in his book Zen and the Art of Motorcycle Maintenance, discusses the attempt of a rhetoric lecturer named Phaedrus to define precisely and systematically what the quality of an essay is—or quality in general—and, of course, without success.[5] His conclusion is that the “wicked Greeks” embedded deep within our culture the positivist assumption that every concept must be precisely defined in order for us to use it. If we free ourselves from this draconian demand and allow our intuitive understanding of concepts to operate naturally, we spare ourselves needless complications.

But even that does not satisfactorily address the difficulty, because our problem is not only the definition of each of these two concepts on its own, but whether there is any difference between them at all. Are these even two distinct concepts? Such a difficulty certainly requires an answer. I will offer here three failed solutions, and finally a fourth solution that will cast the previous three in a new light.

First Proposed Solution

Solomon Maimon, in his book Givʿat HaMoreh on the Guide of the Perplexed, Part I, chapter 74, deals with defining quality and quantity and proposes the following definition: by a quantitative addition, quality does not increase. In our example, the natural tendency to relate to the judges’ wisdom as quality rather than quantity stems from the fact that if we add several judges of wisdom level X, we will still obtain (to an order of magnitude) the same overall level of wisdom. That level is lower than the wisdom level of the single learned judge; therefore, the Chinuch claims that in case of disagreement, the law follows him. This is a situation where the number of judges has changed, but their quality has not. Thus, apparently we have a definition of the difference between quality and quantity: the parameter that does not change when quantities are added (wisdom) is a qualitative parameter. The parameter that does change (number of judges) is called a quantitative parameter. Another example: adding water at a given temperature to another quantity of water at the same temperature will not change the quality of the water (their temperature), only their quantity (the total mass). Therefore, the mass of the water is a quantitative parameter, whereas the heat is a qualitative parameter.[6]

This definition seems, at first glance, clear and intuitive. At least with respect to the second question (the radical formulation that challenges the very difference between the concepts), it seems to provide a respectable answer: wisdom is quality and the number of judges is quantity. Admittedly, it does not solve the first question we raised: does the Chinuch argue for a quantitative majority of wisdom or a qualitative majority of judges? Perhaps we can say that this problem is also solved: if indeed wisdom is a qualitative parameter and not a quantitative one, then necessarily Ramban requires quality, not quantity.

The discomfort with Maimon’s definition and solution arises because, upon further consideration, the second problem has not truly been solved either. If, for example, I were to add wisdom in some way to one of the judges or to several of them (say, by putting them through a judges’ workshop) without changing their number, then it would be the number of judges that remained unchanged and the wisdom that would change. So is wisdom really a qualitative parameter? Does Maimon’s definition provide us a criterion for this distinction? In the water example, too, we can certainly consider further heating of the same amount of water. That is an addition of heat without changing the amount of water, and by the author of Givʿat HaMoreh’s definition, heat would be the quantitative parameter and the (amount of) water the qualitative parameter. The conclusion is that this does not solve the problem: what changes and what does not depends on the addition process; that process determines what will be called quality (what does not change in the addition process) and what will be called quantity (what does change in that process).

Despite all this, it is clear to all of us that “quantity” and “quality” are distinct concepts. Moreover, even in the judges example, it seems intuitively clear that wisdom is a quality and the number of judges is a quantity. The question is whether we can clearly define these two concepts and the difference between them.

Second Proposed Solution

One might suggest that the difference lies in the kind of things we measure. When we measure an abstract concept (such as wisdom or heat), we say we are dealing with quality. By contrast, when we measure concrete, tangible objects (such as judges, or liters of water), we say the result is a quantity. According to this, we cannot speak of quantities in the context of abstract concepts.

But this also fails the test of criticism. In many cases we measure abstract things, and yet our sense is that we are measuring quantities rather than qualities. For example, when we measure a period of time, that measurement is of a quantity (how many seconds, hours, or days a process lasts), even though what is being measured is abstract (time). Or when we count how many new ideas there are in an article (we simply count them), that too seems like a quantitative measurement, even though what is being measured is abstract. So this proposal also does not solve the problem.

Third Proposed Solution

I said that the number of ideas in an article is a quantitative parameter. If, by contrast, we ask about the degree of novelty of the article, here it is clearly a measurement of quality, not quantity. What is the difference? This is harder still if we remember that the degree of novelty in an article depends, among other things, on the number of new ideas it contains (though not only on that, of course; the degree of novelty of each idea also matters).

Perhaps the answer lies in the fact that we can count the number of ideas in an article (not that it is always easy), but we cannot measure quantitatively the degree of novelty it contains (i.e., assign a number that indicates its degree of novelty). From this it is tempting to claim that a qualitative parameter is one that cannot be quantified. But even this proposal, although it sounds quite logical and intuitive, does not provide us a solution to the problem. There are parameters that seem qualitative and yet a quantitative number has been attached to them. For example, a person’s intelligence is currently measured by the IQ index, which is quantitative. The relationship between a person’s intelligence and that number is indeed nontrivial, and yet there is such a number. Even if we accept the objections to the IQ metric, that does not refute the proposal—for those objections merely point out that the metric is not good, but not that better metrics cannot be produced. According to the claim presented here, there is and can be no quantitative metric for intelligence (hence we assumed above that a judge’s wisdom is a qualitative parameter). That is implausible. Moreover, we measure temperature on a well-defined numerical scale (degrees). Does that mean temperature is a quantity? Above, our intuition led us to say it is a quality (it does not change when we increase the amount of water).

Still, there is a sense that each of the three proposals captures something essentially true about the matter, including this third one. The “qualitativeness” of a parameter seems connected, in some way, to the difficulty and complexity of the quantitative metrics that represent it. I now contend that we can use the three earlier proposals to formulate a sharper and clearer definition of the difference between quality and quantity. After much pondering about quality and quantity, I arrived at the conclusion that the difference between them is related to the mathematical distinction between counting (cardinal) and ordinal numbers, and to the physical distinction between extensive and intensive magnitudes. As noted, this proposal will shed light on the three earlier proposals, each of which seems connected to the quality–quantity distinction, even if none is exhaustive.

A Halakhic Appetizer

We can perhaps begin the explanation with a halakhic question (I heard it many years ago in a short halakhic mini-lesson that a rabbi gave on Friday night at the synagogue in Yeruḥam). There is a prohibition to measure various things on Shabbat. The common explanation is that measurement is an activity associated with commerce and business dealings. And yet, that rabbi cited halakhic authorities who hold that one may measure temperature with a thermometer on Shabbat.[7] He wondered about the basis of this leniency, since all measurement is forbidden. He proposed that heat is an abstract concept, and therefore measuring it is not considered a prohibited measurement on Shabbat. We must remember that the basis of the prohibition to measure on Shabbat is the weighing of merchandise, which is the measurement of concrete physical objects. But he himself rejected that explanation, since time is no less abstract, and yet the authorities forbid measuring time on Shabbat.[8] During that lesson the penny dropped for me; I thought I understood the answer, and it immediately connected in my mind to the distinction between quality and quantity.

Ordinal Numbers and Counting Numbers

In mathematics we distinguish between two different roles of the number system: counting and ordering.[9] When we count objects and say that five objects lie before us, we mean to describe the number of objects—more than four and fewer than six. We can say that here the numbers function as “counting numbers.” Each object counted adds one to the total. An object removed from the total reduces the result by one. In such counting, each unit has significance, and the sum of all units gives the overall count. By contrast, when we measure intelligence, as in IQ tests, the number representing intelligence is ordinal rather than counting. That number is not produced by adding discrete units until we reach the total. One cannot point to the eighty-third IQ “unit” out of one hundred, because there is no such unit. Therefore one cannot take an IQ unit and add it to the tally accumulated thus far, nor subtract it. In this case the numbers are “ordinal numbers.” That is, the overall number represents the person’s IQ level and places him somewhere on a scale that orders levels of intelligence one above another. The overall number has (ordering) significance, but it does not represent how many discrete units are present in our collection. This is precisely the meaning of an ordering operation, as opposed to counting. When we say that someone has an IQ of 100, we do not mean he has one hundred discrete IQ units; rather, that his IQ level sits on the general scale in a position above 99 and below 101. He is at the 100th place on the scale. The number orders him within a given ordered scale, but it does not count anything. Of course, one can teach a person and improve his performance so that on the next measurement he will stand higher on the scale (105 instead of 100). But there are no five distinct units that were added to him. After that process he advanced along the scale and his rank in the ordering changed. Some would say that his innate intelligence cannot be changed, and that study merely improved his ability to succeed on the test. The conclusion is that the test is not a good measure for determining a person’s innate, natural ability. For our purposes, all this only indicates that measuring a person’s ability is not a counting operation but an ordering one.

What causes people to miss the distinction between these two roles of numbers is that, in most cases, there is a simple mapping between them. Suppose I have an ordinal system (like IQ) that enables me to arrange objects according to that order. Once I have a group of people arranged by intelligence, I can count all those who come before Reuven and thus determine his place in the series. That is a counting action that gives us his ordinal position; indirectly, it performs an ordering action. And of course one can also order things according to their quantitative order and thus count by means of ordering. But that is only a technical resemblance. The functioning of numbers in these two contexts is completely different.[10]

Back to Quality and Quantity

We can now infer in general that “quality” is a parameter measured by ordinal numbers, and “quantity” is a parameter measured by counting numbers. The question of majority of wisdom versus majority of headcount regarding judges is now more understandable. Wisdom parallels intelligence; as we have seen, it is at best subject to ordering, not counting. Therefore wisdom is quality, while the number of judges is quantity (for here we simply count—counting that tallies).

We can also now explain why measuring time differs from measuring heat. When we measure time, we are actually counting (tallying) the quantity of time units (say, seconds). Therefore it is quantitative counting, even though time is abstract. Counting examines how many units we have in the total quantity. In this case each unit exists in its own right and the counting merely sums the number of units; thus it is counting, not ordering. By contrast, heat is measured in degrees. A single degree has no independent significance. We do not add a degree to a given amount of degrees so that now we have one more. We also cannot point to the 21st degree out of the 40 “we have.” Heating does not mean adding one or two degrees to the system and thereby increasing the total number of degrees, but rather bringing the system to the next temperature in the hierarchy. The conclusion is that the numbers that measure degrees are ordinal, not counting. Degrees are represented by numbers that order the levels of heat in a hierarchical sequence, not by numbers that count how many discrete degrees we have in total. Therefore, when we measure heat, it is an ordering operation, not a counting one. If so, even if it is forbidden on Shabbat to measure abstract magnitudes like time, the prohibition pertains to counting measurement—for that is what happens in commerce. The conclusion is that a measurement whose essence is ordering rather than counting is permitted on Shabbat.[11]

Extensive and Intensive in Physics

Despite what was said above about measuring temperature, measuring heat in units of energy (calories or joules) is indeed a counting measurement rather than an ordering one. I count how many units of energy an object contains, and each unit of energy is an addition. If you eat another food, it will add to your body another calorie or several calories, depending on the food’s caloric content (of course one must multiply by absorption, and this is not the place to elaborate). Thus we have two measures for a body’s heat, one counting (energy) and the other ordinal (temperature). There is a relationship between them, and still it is correct to say the one is counting and the other is ordinal.

One indication of this difference is what physicists call a physical magnitude being extensive or intensive. For a magnitude like temperature there is no way to define temperature per unit of substance (per gram) or per unit of volume (cc). Temperature is a physical magnitude that does not operate against volumes and quantities. It characterizes the whole, not its parts. By contrast, energy can certainly be defined per unit volume or mass, and the total energy of the body can be obtained by summing the energies of each unit of mass or volume separately.[12] For our purposes, intensive magnitudes represent quality and extensive magnitudes represent quantity.

Connection to the Three Previous Proposals

It is now easy to see the connection to the three definitions presented earlier for the difference between quality and quantity. If we increase the amount of matter, the intensive magnitudes do not change; therefore such magnitudes are qualitative. By contrast, extensive magnitudes are indeed proportional to the amount of matter and therefore express quantities rather than qualities. From here Solomon Maimon’s definition (the first proposal) arises naturally. True, we saw it is not sufficient, but once the full explanation is presented it becomes clear that it too captures an essential element of the distinction between qualitative and quantitative parameters.

The second proposal is now also clear: qualities measure abstract things. We rejected that proposal by noting that sometimes there are quantitative measures for abstract things, yet the link between abstractness and quality remains. There are quantitative measurements of qualities (such as IQ or temperature), but there are no qualitative measurements of quantities. Quantities can be measured straightforwardly, and there is no reason to invent complex qualitative metrics for them. With qualitative magnitudes, one may perhaps propose quantitative metrics (like temperature or IQ), but the relationship between the metric and the thing measured is very complex. For example, if we wish to measure, with a numerical metric, the degree of novelty of an article, we would have to define a very complex metric—and it certainly would not be agreed upon (cf. IQ). Therefore, this is a measurement of quality. But if we wish to represent the number of novel ideas it contains, there is a simple and natural practical procedure (count them). In such a case, there is no reason to define complex qualitative metrics.

In discussing the third proposal above, we claimed that a qualitative magnitude cannot be quantified. We can now slightly update this: the numerical metrics of qualitative magnitudes are defined in a much more complex way. They have no simple relation to what is being measured. The reason is that these are ordinal metrics rather than counting metrics, and arranging levels of a complex parameter is not as simple as counting units. In counting, you tally how many units you have, and the overall number is the number of units; this is a very simple process, and the numerical representation of the result is trivial. The result is a quantity of units (hence it is a quantitative magnitude). By contrast, defining a numerical metric such as IQ or temperature is much more complex. It cannot be built as a simple count of quantities. In such cases, we must seek a numerical representation for a given level of quality and define an ordered axis along which all levels can be placed hierarchically. No wonder there are many debates about such metrics.[13]

Is This a Definition?

We can now ask whether what we have found is a definition of quality and quantity and the difference between them. Phaedrus claimed that quality cannot be defined, and it would seem that here I have offered a definition of it. But I think a closer look at the book shows that Phaedrus sought a numerical metric for quality, not a definition of the concept itself. Beyond that, what I have offered here is more a description than a definition. I do not presume to claim that with what is presented here you will always be able to determine unambiguously whether we are dealing with a quantity or a quality. This is an illumination, not a definition. As a thought experiment, consider a being who does not possess within him the understanding of the difference between quality and quantity. I very much doubt that such a column could explain it to him. The description here illuminates the distinction for one who already has a potential understanding of it within, but who has not succeeded in articulating it.

One Last Note: Matter and Form

There is an ancient distinction between matter and form, or between essence and accident. In the second gate of my book Two Wagons I elaborated on it and argued, among other things, that quantity pertains to matter/essence and quality to form/accident. Every description of an object or concept is part of its form. Everything I can say about a person, an object, or a concept is a description of it. And what about the statement that it exists? Or the statement that it is one, or two? Such statements relate to the essence itself, not to its attributes and form. Thus there I refuted Anselm’s ontological argument (see also the first conversation in The First Existent), arguing that God’s existence is not a description of Him and therefore not part of His perfection—nor is His oneness. I will not go into the details here, but only point out that in this sense as well, statements about quantity differ from statements about quality.

[1] The source of these ideas is a discussion I conducted in note 9 of my book Two Wagons.

[2] See, for example, R. Yosef Engel’s book Lekach Tov, rule 15, where he discusses whether quantity is preferable to quality or not.

[3] See on this topic the commentary Minḥat Ḥinukh to Sefer HaChinuch there, and the book Shaʿar HaMishpat on Shulḥan Arukh Ḥoshen Mishpat, §18, and what he cites there from Shevut Yaʿakov.

[4] R. Yosef Engel, who discusses the question of quantity versus quality in halakhah (see note 2 above), for some reason does not address this issue at all. Some early and later authorities explain in this way the fundamental dispute between Beit Shammai and Beit Hillel. See Minḥat Ḥinukh there, explaining that the root of the dispute is in Yevamot 14a; and Tosafot there in Yevamot and in Bava Metzia 59 write that this is the root of the Beit Shammai/Beit Hillel dispute, and more.

[5] See there from p. 147; the formulation of the problem is on p. 150.

[6] See further discussion and examples in Rabbi Menachem Mendel Kasher’s MeFaʿne’aḥ Tzefunot, ch. 11, §9.

[7] See Shemirat Shabbat Ke-hilchatah, ch. 29 §9 (as well as ch. 28 §30 and ch. 40 §2).

[8] I do not know whether there is a particular decisor who forbids measuring time and permits measuring heat, but he assumed that the prohibition to measure time is agreed upon.

[9] In set theory, the terms “cardinal numbers” and “ordinal numbers” are used in a somewhat different sense, with respect to infinite cardinals (see, for example, here). To my understanding, there is a connection between the mathematical distinction and the distinction I make here, and this is not the place to elaborate.

[10] Indeed, in contexts of infinite numbers (cardinals and ordinals) the resemblance between the two systems is not complete. Therefore, mathematicians use this distinction only with regard to infinite numbers.

[11] I do not mean to say that the decisors were aware of the distinction between the two roles of numbers. But intuitively, it is certainly possible that this stood behind their ruling. I note again that I do not know of a particular decisor who forbids measuring time and permits measuring heat. In light of what I have said here, however, there is no principled impediment for there to be such a view.

[12] Even in measuring a system’s energy, one cannot point to the 8th unit out of 41 units. In that sense, energy looks like an ordinal, not a counting magnitude. But it is an extensive magnitude; that is, one can distinctly indicate the energy of “the third unit of matter from the left.” Questions still remain here in light of quantum physics, which casts doubt on this claim of mine as well, and this is not the place. For our purposes, what matters is the definition, not its applications in physics.

[13] Not regarding temperature itself, to be sure. But anyone familiar with statistical mechanics knows how complex that metric is, and how global in essence it is—unsusceptible to computation as an extensive sum over the system’s parts.