Two Types of Counting: Quantity and Quality (Column 193)
With God's help
In the previous column I mentioned the reasoning of Bahag, according to which the definition of counting requires continuity, and therefore one who does not count the counting of the Omer continuously (for example, if he missed a day) can no longer fulfill the commandment of counting. I argued there that this is a conceptual rather than a halakhic requirement; that is, this requirement does not emerge from any verse, but from the meaning of the concept of counting, which exists even in a world without a commandment, or prior to the commandment.
But this immediately raises the following question: after all, Bahag cites the verse term "innocence", so he is indeed deriving it from a verse? My claim is that the verse only teaches us what concept is required in this law, and once that is the concept, the requirement of continuity follows from the very definition of the concept and not from the law. What is that concept? How might we have understood the commandment of counting the Omer had the Torah not written "innocence"? That is what I wanted to devote the present column to.[1]
Between quantity and quality: the case of majority in a religious court
In my book Shtei Agalot this discussion was presented as part of the discussion of the relation between quantity and quality, which itself was meant to illustrate the difference between an essentialist approach and a conventionalist approach to concepts. In ordinary language, the terms 'quality' and 'quantity' are used quite frequently, and they are generally understood as describing different forms of measurement. Some properties describe qualities of the thing or things, and others are described as quantitative. In Jewish law too, the distinction between quality and quantity arises in several contexts, such as the laws of majority in a religious court (see on this column 69).
According to Jewish law, an ordinary religious court consists of three judges. Not all of them need possess the same level of expertise and wisdom. A court can consist of one gamir (a learned Torah scholar), alongside two laymen. The ruling is determined by the majority view, as derived from the verse follow the majority. Some halakhic authorities (see, for example, Sefer HaChinukh, commandment 78, following R. Hai Gaon) maintain that the majority in question is not a numerical majority of the judges but a majority in wisdom and expertise. Therefore, when the gamir disagrees with the two laymen, the ruling will follow him. By contrast, Nachmanides, in his novellae to tractate Yoma, holds that the decisive majority is always the numerical majority of the judges.[2]
At first glance, this appears to be a dispute over whether what determines the matter is the quantity or the quality of the judges. According to Sefer HaChinukh, quality is decisive, whereas according to Nachmanides one follows the quantitative majority.[3]
To the heart of the problem
When one looks more deeply, it seems that this description is not so unambiguous. Although according to the author of Sefer HaChinukh it seems, ostensibly, that the quality of the judges is what matters, and not their quantity, it also seems possible to formulate the matter differently. One might say that Sefer HaChinukh too requires a quantitative and not a qualitative majority, except that the majority he requires is of the quantity of wisdom and not of the quantity of judges. Earlier we assumed that what was required was a majority of judges, and the question was whether to assess that majority through the number of judges or through their quality. According to the alternative understanding proposed here, the required majority is not a majority of judges but a majority of wisdom. According to this definition, Sefer HaChinukh too holds that quantity is what matters, except that the magnitude whose quantity we require in order to determine the majority is different from the one proposed by Nachmanides. What we earlier called 'the quality of the judges' we now call 'the quantity of their wisdom.' If so, everyone can apparently agree that the required majority is quantitative, and the dispute is only over the question of a quantitative majority of which magnitude is required to determine the halakhic ruling.
Thus far we have assumed that the concepts 'quality' and 'quantity' are well defined, except that their application to reality is unclear. Therefore, if for some reason it is clear that the required majority is a 'majority of judges,' then it is clear that the dispute is over how to measure that majority: through the number of judges or through their quality. In such a situation, ostensibly, it is clear what quality and quantity are, and the only question is how to make sure that we are indeed dealing with this dilemma and not with the alternative formulation I proposed.
However, the question can be formulated more radically, by saying that the concepts 'quality' and 'quantity' themselves are not well defined, and in fact are not really different from one another. The quality of a given thing can always be described as the quantity of something else. The quality of a judge is always the quantity of his wisdom. In this formulation of the question there is already a challenge to the very distinction between the concepts 'quality' and 'quantity,' and not merely to their application to concrete problems. But precisely for that reason this formulation is problematic, since there is a clear intuition that when we refer to the judge, wisdom is defined as one of his qualities, and similarly when we refer to wisdom, its degree is a quantity. If so, the question does not concern only the problematic nature of applying the concepts 'quality' and 'quantity.' The more basic dilemma is this: are 'quality' and 'quantity' identical concepts whose differences are fictitious, or is there a real difference between them?
In column 69 I cited Pirsig, who in his book Zen and the Art of Motorcycle Maintenance speaks about an attempt to define what quality is, without success. His conclusion is that it was the Greeks who deeply implanted within our culture the assumption that every concept must be defined precisely. If one frees oneself from this demand and allows the intuitive understanding of concepts to operate without analytical guidance and criticism, one reaches better insights.[4] But that is making life easy for ourselves. We still ought to strive for a definition where possible, or at least for clarification. Note that here we are not dealing with a mere question but with a difficulty. The difference between quality and quantity is not only unclear; ostensibly it does not exist. Such a thing certainly requires clarification.
First proposal: the effect of addition
Shlomo Maimon, in his book Giv'at HaMoreh, on Moreh Nevukhim, Part I, chapter 74, discusses this issue and writes that a quantitative addition does not increase quality. In the context of our example, the natural tendency to treat the wisdom of judges as a quality and not as a quantity stems from the fact that if we add several judges with wisdom level X, we still obtain, roughly speaking, the same level of wisdom. That level is lower than the wisdom of the single gamir. This is a situation in which the quantity of the judges changed and their quality did not. If so, we apparently have a definition of the difference between quality and quantity: the parameter that does not change when quantities are added (wisdom) is called a qualitative parameter. The parameter that does change (the number of judges) is called a quantitative parameter.
As another example, adding water at a certain temperature to more water at the same temperature will not change the quality of the water (its temperature), but only its quantity (the total mass).[5]
This definition appears, at first glance, intuitively clear. Clearly it does not resolve the first question we raised, concerning the way the question should be posed with respect to judges: is the dispute over what quantity is required (wisdom or judges), or is the dispute over whether what is required is a qualitative or quantitative majority of judges. But with respect to the second question (the radical formulation that challenges the very difference between the concepts), an adequate answer seems, at least apparently, to have been given here.
And yet, examining this definition with the requisite care shows that the problem has still not been resolved. If, for example, I were somehow to add wisdom rather than judges, then it would be the number of judges that did not change, while wisdom would increase. Admittedly, with judges it is difficult to imagine a realistic situation in which to do this, but in the case of water one can certainly imagine further heating of the same quantity of water. That is an addition of heat without a change in the quantity of water. According to the definition of the author of Giv'at HaMoreh, such a description implies that heat is specifically the quantitative parameter, and the water the qualitative parameter. With judges too one can imagine a situation in which we teach them, thereby making them wiser and more skilled, without changing their number (their quantity). If so, even according to the definition given above for the concepts 'quality' and 'quantity,' there is no clear distinction between quality and quantity. Everything depends on the question of what the process of addition is. This process is what determines what will be called quality (what does not change in the process of addition), and what will be called quantity (what does change in that process).
Yet despite all that, it seems entirely obvious that 'quantity' and 'quality' are understood by every rational person as two different concepts. It seems clear, intuitively at least, that wisdom is a quality, while the number of judges is a quantity. The question is whether these concepts can be clearly defined.
Second proposal: a distinction between types of things
Some would say that the difference lies in the type of things we measure. When we measure an abstract concept (wisdom), we say that this is quality. By contrast, when we measure concrete objects (judges), we say that the result is quantity. According to this, one cannot speak of quantities in the context of abstract concepts.
But now the question has moved to a more basic plane: what is an abstract concept, as opposed to a concrete object, for this purpose? Ostensibly there is no essential difference between abstract entities (attributes, or concepts in general) and ordinary physical objects. Is an abstract concept simply an object without mass? If so, then if we count photons in physics, is that quality and not quantity? And if we count a stretch of time (such as seconds or minutes)? Clearly one can speak of quantities of abstract concepts as well, for example by counting how many new ideas there are in an article, or how many seconds the race took.
The prohibition of measuring on the Sabbath: the problem
After a long time of wondering about the issue of quality and quantity, whose main elements have been described thus far, I believe that I nevertheless succeeded in clarifying for myself the difference between them. The solution to the problem that I will present here is based on a mathematical distinction between two different functions of the number system. The penny dropped for me on the Sabbath about twenty years ago, when I heard a lecture in the synagogue about the prohibition of measuring on the Sabbath. As is well known, there is a prohibition against measuring various things on the Sabbath. Some halakhic authorities hold that one may measure temperature with a thermometer on the Sabbath. Ostensibly, the basis of this permission lies in the fact that temperature is an abstract concept, and therefore measuring it is not considered prohibited measuring on the Sabbath. It should be noted that the basis of the prohibition against measuring on the Sabbath is the weighing of merchandise, which is a measurement of concrete physical objects. In that lecture, the rabbi wondered why time, which is no less abstract a concept, is forbidden to measure on the Sabbath. When I thought about this question, the distinction between quality and quantity suddenly became clear to me.
Ordinal and cardinal numbers
In mathematics, two different operations of counting are defined, or perhaps more accurately: two different kinds of use of numbers. When we count objects and say that five objects lie before us, we mean the quantity of the objects, which is greater than four and less than six. One may say that here the numbers function as 'cardinal numbers.'[6] Every additional object that is counted adds one to the total count. An object removed from the total count lowers its value by one. Each unit has significance, and the sum of all the units yields the total count. By contrast, when we measure intelligence, by means of IQ tests for example, the number that represents the level of intelligence is not created by adding units. One cannot take an IQ unit and add it to the count accumulated thus far, or subtract it from it. The reason is that in this case the numbers are 'ordinal numbers.' That is, the total number representing the IQ level represents a certain level, and the units within it have no distinct significance. IQ 100 is an intelligence level lower than IQ 110 and higher than IQ 90. One cannot add 10 IQ units in order to change the level of intelligence. Even if someone becomes smarter, in one way or another, we say that he reached the level of IQ 110, not that 10 IQ units were added to him. No one can point distinctly to those units that were added to him.
Thus, in this example each number represents a certain level of intelligence, and the function of hierarchical numbering is to arrange the levels one after another in some order, not to count the quantity of units that have accumulated in total. The numbers that represent intelligence are a scale (whose units are arbitrary in certain respects) meant to give a quantitative measure to qualities, and to arrange and classify them according to that measure. Therefore these are ordinal numbers and not cardinal ones. They do not count units, but rather arrange levels in a hierarchical order.[7]
The prohibition of measuring on the Sabbath: the solution
If we use this distinction for our purposes, we will understand that time is counted by cardinal numbers. We count how many seconds have passed, where each second is a unit added to the total count. Temperature, by contrast, is measured by ordinal numbers. The total number is not an aggregate of individual units that together make up a total count. The number has a holistic significance, and not the significance of an aggregate of units. A temperature of 50 degrees is higher than 40 degrees and lower than 60 degrees.[8] There is no meaning to adding one degree as a unit in order to obtain a higher temperature reading.
We thus learn that although time and temperature are both abstract parameters, there is a difference in the way numbers are used in relation to them. A measurement forbidden on the Sabbath is a measurement whose result is expressed in cardinal numbers, such as measuring a quantity of merchandise or measuring time, and not a measurement of a parameter that is measured by ordinal numbers, such as temperature or intelligence.
Third proposal: quality is measured by ordinal numbers and quantity by cardinal numbers
From this one may infer more generally that 'quality' is a parameter measured by ordinal numbers, and 'quantity' is a parameter measured by cardinal numbers. The question of a majority of wisdom versus a numerical majority with respect to judges is now more understandable as well. Wisdom is parallel to intelligence, and as we have seen it can at most be ordered and not counted. Therefore wisdom is a quality, whereas the number of judges is a quantity.
This distinction is connected to the earlier proposals. Concrete things can always be counted in the manner of cardinal counting. We count how many concrete objects there are in a collection. Among abstract concepts, there are some that can be counted (the number of ideas in an article, or an amount of time), and there are others that admit only qualitative ordering. That is, the intuition is correct that a qualitative parameter is always an abstract concept, but it is not correct that every measurement of an abstract concept is necessarily a measurement of qualities.
I would only note that even this explanation does not place the concepts 'quality' and 'quantity' on an entirely distinct and fully defined footing. We are now left to clarify what characterizes those abstract magnitudes that can only be ordered and not counted. My determination above concerning the impossibility of defining basic concepts remains in force. Phaedrus, the hero of Pirsig's book, will have to continue his ceaseless (and apparently futile) chase after a definition of quality. But even if there is no definition here, there is certainly clarification.
Back to counting the Omer
In light of this distinction, we can return and explain Bahag's derivation from the word "innocence". My claim is that the Torah teaches us here that the counting of the Omer must be cardinal counting and not ordinal counting (and on this Tosafot disagreed with him). From this point onward, cardinal counting implies that there must be continuity, because when one counts one must pass through all the units being counted. In ordinal counting there is no need for continuity, for one need only check at which stage one currently stands, and there is no necessity to position it in relation to the previous stages. From this it also follows that even a minor who counted under the educational obligation can continue counting with a blessing even according to Bahag, since as a matter of fact he counted the entire sequence.
Incidentally, in the previous column I cited the explanation of the author of Dvar Avraham to the difficulty of why, outside the Land of Israel, we do not count two days out of doubt. He explains that counting requires certainty, and therefore counting out of doubt is not considered counting. I said that here too this is a rule learned from analysis of the concept of counting and not a legal requirement. Straightforwardly, there is room for such a requirement even if the counting is ordinal (as Tosafot hold), for you still have to know where in the hierarchy you are.[9]
[1] Its main point is taken from note 9 in my book Shtei Agalot.
[2] On this topic, see the commentary Minchat Chinukh on Sefer HaChinukh there, and the book Sha'ar HaMishpat on Shulchan Arukh, Choshen Mishpat, section 18, and what he cites there from Shevut Ya'akov.
[3] R. Yosef Engel, in his book Lekach Tov, section 15, discusses the question of quantity versus quality in Jewish law in various contexts, and it is noteworthy that he does not address this issue at all.
[4] See on this, from another angle, column 143.
[5] See further discussion and additional examples in the book Mefa'ne'ach Tzefunot by Rabbi Menachem Mendel Kasher, chapter 11, section 9.
[6] In set theory the terms 'cardinal numbers' and 'ordinal numbers' are used, in a somewhat different sense, with respect to infinite cardinals. To the best of my understanding, there is a connection between these two uses, and this is not the place to elaborate.
[7] Of course, one can arrange quantities in a hierarchical order according to quantity, in which case cardinal numbers become ordinal ones.
[8] Heat as energy, which is measured in calories, is a cardinal and not an ordinal magnitude. And for those in the know I will add: there is a connection between this distinction and the physical distinction between intensive and extensive quantities, although they are not entirely congruent.
[9] One might perhaps have distinguished and said that precisely in cardinal counting this is an obvious requirement, because you are counting how many units (days) you have, and if you count two different days you do not know how much has accumulated. But in ordinal counting, if you count the two days then, either way, you have arranged the count in the correct order even if it is not clear to you. But this seems forced.
Discussion
HaChinukh is not speaking about quantification. Still, even without precise quantification it is clear that wisdom is an ordinal quality and not a cardinal count. One person is wiser than another. We make such determinations every day.
As for the question of the added value of the number of judges, I discussed that in TOD 145-+.
By the way, to illustrate the point: 20 degrees Celsius is ostensibly a temperature that is double the quantity of 10 degrees Celsius.
But if we convert them to Kelvin, we get 293.15 degrees as against 283.15 degrees. Not really double. Precisely for the reason mentioned above.
Regarding the counting of the Omer, I would say exactly the opposite – if the counting is cardinal, one could count everything all at once (just as when counting trees, it is enough to glance and say there are 3 trees here, and there is no need to count them one by one); by contrast, ordinal counting can only be done across the whole span of time: now one day has passed, now two days have passed (as with measuring temperature. When the thermometer reads 50, there is no accumulation here of 50 degrees; rather, this is degree number 50).
Very true.
In principle you are right, but when I am on the 23rd day, I do not now have all the days before me. In order to count them I have to go through them one by one as they occur, and that is done throughout the entire counting. Still, this requires further consideration.
Another note regarding temperature and IQ – ostensibly the reason the counting is ordinal and not cardinal stems from a technical matter: we use an indirect metric and therefore it is not quantified. But theoretically one could measure temperature, say, as the average speed of molecular motion times the number of molecules per unit volume, or measure wisdom, say, as a function of the number of interneuronal connections, and obtain a quantified result. So why say that the method of measurement we chose is a substantive difference that turns temperature or wisdom into a quality (as opposed to a quantity)?
Apparently the difference between quantity and quality is at the level of perception. Phenomena that are emergent and whose measurement and cardinal quantification are possible only at the deeper level (the molecular, physical level) are considered qualities, whereas phenomena that are quantified at the higher level are considered matters of quantity.
The difference is not only in the form of measurement. The form of measurement indicates that one is measuring something different. Temperature is not average speed. Average speed creates the magnitude called temperature. Just as light is not an electromagnetic wave. The wave creates the phenomenon of light when it strikes the retina. A thermometer measures temperature, not average speed. Through temperature one can derive the average speed because of the connection between them.
Maybe not entirely related – is it forbidden to measure time on Shabbat? Where is that written? How does one even do that? After all, any person can notice that time has passed; it cannot be that it is forbidden on Shabbat to notice that time has passed.
The Maharal, in the book Tiferet Yisrael, says that sometimes the addition of a large quantity creates a new quality.
It could also be that a quality can have the same value on a different scale (I have twenty apples, you have twenty oranges. Abaye is sharp and raises difficulties; Rava is measured and draws conclusions), whereas in measuring quantity all the possibilities are on the same scale – this is somewhat similar to the difference between an ordinal number and a cardinal number, but pointing in a somewhat different direction.
Noticing is not measuring. Measuring time is forbidden according to the Shulchan Arukh, Orach Chayim sec. 308, se'if 51, in the name of the Maharil.
From examining the Maharil responsa, it appears that he is uncertain about the question whether it is indeed possible to measure time. He tries to explain an existing custom, and even then he only argues that perhaps an hourglass is a measuring instrument, and a sundial is even more of a measuring instrument.
It seems to me that we were simply lucky that in his time they had not yet invented the thermometer, and the philosophical casuistry is nice in itself but does not explain the halakhah. One could have engaged in the same casuistry as a result of encountering an electric pole.
(The source: Maharil Responsa, siman 200
…they found no clear prohibition, and even so he did not state that it is permitted either, since the prohibition had become widespread… and one can say that it is a utensil whose primary function is for forbidden use, since it is made for measuring … and even though measuring time is not full-fledged measurement, still one may compare it.
And after the leading authority of the generation hesitated about it, who can come after the king who has already done it. And what you heard about one of sand, with sections, and I do not know whether he means that they show by shadow through the sections; in any case, from where do we derive a distinction, and perhaps it is even worse, for it measures the shadow itself.)
It may be that my words were written in Turkish, but at first glance I see neither any connection between your words and mine nor any logic. Perhaps some electric pole inspired you and you wrote, on that basis, an obscure poem.
So here is the translation into spoken Hebrew:
You asked how we know that it is forbidden to measure time; my answer was that the Shulchan Arukh concludes that it is forbidden. What is unclear here? Do you think that is not correct? Now, for some reason, you begin casuistic discussions about the Maharil responsa, whose purpose and relevance to us I did not understand. I was not dealing with the question whether his ruling is correct and whether it indeed derives from the Maharil. That is not our topic.
Regarding measuring temperature, several halakhic authorities wrote that it is permitted. The question is whether those authorities ignore the ruling of the Shulchan Arukh (perhaps because they understood the Maharil as you do)? I assume we will agree that they do not (halakhic authorities do not generally ignore the Shulchan Arukh). If so, the question is on what basis they rely. Necessarily, in their view there is a distinction between measuring time and measuring temperature.
I suggested an explanation for that in terms of counting and ordering.
This is the best translation into Hebrew that I can manage for my argument. I hope the next response, if there is one, will be written in prose and not in poetry. In my poverty, poetry is hard for me to understand.
And as for your remark, although as stated it does not have even the slightest connection to our discussion, there too in my opinion you are mistaken. In my view it does indeed derive from the Maharil, because his doubt stems from the question whether measuring a shadow is measuring time or not. But a modern clock, which explicitly measures time, is certainly forbidden. And indeed that is what the commentaries on the Shulchan Arukh there wrote. But, as stated, that is not our topic.
With God’s help, 19 Tevet 5779
It seems to me that the rationale for being lenient about measuring temperature on Shabbat is that the prohibition of measuring on Shabbat stems from the fact that measurement is generally bound up with buying and selling, where one measures the dimensions or weight of an object being sold; and measuring time too is done for the sake of hired labor, to know how much to pay – and since such measurement is done for a commercial purpose, the Sages prohibited all such measurement. From here one may discuss whether perhaps they did not prohibit measuring temperature, since there is nothing comparable to it in a commercial transaction (though one might discuss the issue from the angle of measuring the height of the mercury column in a thermometer, which is a measurement of length, and there is something comparable to that in commerce).
Regards, S.Z. Levinger
With God’s help, 19 Tevet 5779
It seems that the counting of the Omer is 'ordinal' according to everyone, for each day one counts its ordinal number among the days of the Omer. The dispute between the Behag and Tosafot is about another question: whether each day is a mitzvah unto itself, in which we count its ordinal number and thereby connect it to the starting point, or perhaps all the days are limbs of a single organism that needs to be 'complete.'
Regards, S.Z. Levinger
Perhaps the discussion regarding the wording of the count – 'la-omer' or 'ba-omer' – is also connected to this discussion. According to Tosafot, where each day on its own measures its distance from the starting point, it is appropriate to say 'today is … of the Omer'; and according to the Behag, where all the days are part of a complete organism, the wording 'today is … in the Omer' is appropriate.
And while speaking about 'la-omer' or 'ba-omer' – I should note that the wording commonly used in our communities, where people say: 'Today is such-and-such days, which are such-and-such weeks and such-and-such days of the Omer,' is a composite wording, preserving the Ashkenazi custom to mention 'ba-omer' at the end (as opposed to the Sephardi custom to say: 'Today is such-and-such days of the Omer, which are such-and-such weeks and such-and-such days'), together with the Sephardi custom to say 'la-omer' and not 'ba-omer.'
This combination was created by Rabbi Ben-Zion Meir Hai Uziel when he was the rabbi of Tel Aviv-Jaffa, and he explains it in his responsum to Rabbi Yitzhak Nissim (dated 29 Nisan 5695, Shu"t Yayin HaTov vol. 2, Orach Chayim siman 7), that mentioning the Omer at the end is preferable, because thereby it is evident that both the counting of days and the counting of weeks are of the essence of the mitzvah; and on the other hand he adopted 'la-omer' in accordance with the custom of the Ari (even though according to grammatical precision 'ba-omer' is more correct, as the Taz writes).
At the end of the responsum, Rabbi Uziel explains his principled approach: 'In general, my method is to bring closer together, as much as is permitted, the differences in wording between Sephardim and Ashkenazim in a matter that is noticeable, and thereby to prevent puzzlement and questions' (Rabbi Nissim, in his responsum in siman 8, establishes the custom of the Sephardim – to say 'la-omer' immediately after the counting of the days).
Regards, S.Z. Levinger
Paragraph 2, line 1:
This combination was created by the distinguished Rabbi…
Very nice. I did not understand why, according to your explanation of the Behag, you conclude that it would also be permissible for a bar mitzvah boy during the Omer count to continue counting with a blessing. It may be that the requirement is to count under obligation in a quantitative sense; that is a discussion seemingly unrelated to the matter, as to what kind of quantitative count is required for the mitzvah.
Possible, but there is no reason to assume that. There is a requirement for a cardinal count. You are asking whether perhaps they also want a cardinal count under obligation. Maybe. But if you want to add a requirement, you have to prove it.
With God’s help, 23 Tevet 5779
In the decision of judges according to the majority, even though they know who is greater than whom in wisdom (for they 'begin from the side' so that the lesser judge can express his opinion freely and not suppress his view before one greater than he) – nevertheless each judge has an equal vote.
It makes sense to say that a greater number of sages expresses a 'greater amount of wisdom' regarding the subject under discussion, for each of the judges adds a unique personal angle of vision, and not infrequently the lesser one sees correct aspects that escaped the greater one. The majority of opinions, therefore, reflects a majority of perspectives and directions of thought, and for that reason each judge has an equal vote, small and great alike.
Regards, S.Z. Levinger
When the author of Sefer HaChinukh relates to counting on the level of the amount of wisdom, how can one practically determine the rank of each of the judges? If this is something similar to a Markov decision process, then I don’t understand what three judges add. One is enough. And if this is an actual quantification of the trait of wisdom, then I really find it hard to grasp how that is possible.