NUMBER IS DIFFERENT FROM QUANTITY
This difference is basic for any sort of theorizing behavioral science, for any sort of imagining of what goes on between organisms or inside organisms as part of their processes of thought.
Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three, there is a jump. In the case of quantity, there is no such jump; and because jump is missing in the world of quantity, it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate.
Even when number and quantity are clearly discriminated, there is another concept that must be recognized and distinguished from both number and quantity. For this other concept, there is, I think, no English word, so we have to be content with remembering that there is a subset of patterns whose members are commonly called "numbers." Not all numbers are the products of counting. Indeed, it is the smaller, and therefore commoner, numbers that are often not counted but recognized as patterns at a single glance. Card-players do not stop to count the pips in the eight of spades and can even recognize the characteristic patterning of pips up to "ten."
In other words, number is of the world of pattern, gestalt, and digital computation; quantity is of the world of analogic and probabilistic computation.
Some birds can somehow distinguish number up to seven. But whether this is done by counting or by pattern recognition is not known. The experiment that came closest to testing this difference between the two methods was performed by Otto Koehler with a jackdaw. The bird was trained to the following routine: A number of small cups with lids are set out. In these cups, small pieces of meat are placed. Some cups have one piece of meat, some have two or three, and some cups have none. Separate from the cups, there is a plate on which there is a number of pieces of meat greater that the total number of pieces in the cups. The jackdaw learns to open each cup. Taking off the lid, and then eats any pieces of meat that are in the cup. Finally, when he has eaten all the meat in the cups, he may go to the plate and there eat the same number of pieces of meat that he got form the cups. The bird is punished if he eats more meat from the plate than was in the cups. This routine he is able to learn.
Now, the question is: is the jackdaw counting the pieces of meat, or is he using some alternative method of identifying the number of pieces? The experiment has been carefully designed to push the bird toward counting. His actions are interrupted by his
having to lift the lids, and the sequence has been further confused by having some cups contain more than one piece of meat and some contain none. By these devices, the experimenter has tried to make it impossible for the jackdaw to create some sort of pattern or rhythm by which to recognize the number of pieces of meat. The bird is thus forced, so far as the experimenter could force the matter, to count the pieces of meat.
It is still conceivable, of course, that the taking of the meat from the cups becomes some sort of rhythmic dance and that this rhythm is in some way repeated when the bird takes the meat from the plate. The matter is still conceivably in doubt, but on the whole, the experiment is rather convincing in favor of the hypothesis that the jackdaw is counting the pieces of meat rather than recognizing a pattern either of pieces or of his own actions.
It is interesting to look at the biological world in terms of this question: Should the various instances in which number is exhibited by regarded as instances of gestalt, of counted number, or of mere quantity? There is a rather conspicuous difference between, for example, the statement "This single rose has five petals, and it has five sepals, and indeed its symmetry is of a pentad pattern" and the statement "This rose has one hundred and twelve stamens, and that other has ninety-seven, and this has only sixty-four." The process which controls the number of stamens is surely different from the process that controls the number of petals or sepals. And, interestingly, in the double rose, what seems to have happened is that some of the stamens have been converted into petals, so that the process for determining how many petals to make has now become, not the normal process delimiting petals to a pattern of five, but more like the process determining the quantity of stamens. We may say that petals are normally "five" in the single rose but that stamens are "many" where "many" is a quantity that will vary from one rose to another.
With this difference in mind, we can look at the biological world and ask what is the largest number that the processes of growth can handle as a fixed pattern, beyond which the matter is handled as quantity. So far as I know, the "numbers" two, three, four, and five are the common ones in symmetry of plants and animals, particularly in radial symmetry.
The reader may find pleasure in collecting cases of rigidly controlled or patterned numbers in nature. For some reason, the larger numbers seem to be confined to linear series of segments, such as the vertebrae of mammals, the abdominal segments of insects, and the anterior segmentation of earthworms. (At the front end, the segmentation is rather rigidly controlled down to the segments bearing genital organs. The numbers vary with the species but may reach fifteen. After that, the tail has "many" segments.) An interesting addition to these observations is the common circumstance that an organism, having chosen a number for the radial symmetry of some set of parts, will repeat that number in other parts. A lily has three sepals and then three petals and then six stamens and a trilocular ovary.
It appears that what seemed to be a quirk or peculiarity of human operation - namely, that we occidental humans get numbers by counting or pattern recognition while we get quantities by measurement - turns out to be some sort of universal truth. Not only the
jackdaw but also the rose are constrained to show that for them, too - for the rose in its anatomy and for the jackdaw in its behavior (and, of course, in its vertebral segmentation) - there is this profound difference between numbers and quantity.
What does this mean? That question is very ancient and certainly goes back to Pythagoras, who is said to have encountered a similar regularity in the relation between harmonics.
The hexago-rectangle discussed in section 5 provides a means of posing these questions. We saw, in that case, that the components of description could be quite various. In that particular case, to attach more validity to one rather than to another way of organizing the description would be to indulge illusion. But in this matter of biological numbers and quantities, it seems that we encounter something more profound. Does this case differ from that of the hexago-rectangle? And if so, how?
I suggest that neither case is as trivial as the problems of the hexago-rectangle seemed to be at first sight. We go back to the eternal verities of Saint Augustine: "Listen to the thunder of that saint, in about A.D. 500: 7 and 3 are 10; 7 and 3 have always been 10; 7 and 3 at no time and in no way have ever been anything but 10; 7 and 3 will always be 10."
Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three, there is a jump. In the case of quantity, there is no such jump; and because jump is missing in the world of quantity, it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate.
Even when number and quantity are clearly discriminated, there is another concept that must be recognized and distinguished from both number and quantity. For this other concept, there is, I think, no English word, so we have to be content with remembering that there is a subset of patterns whose members are commonly called "numbers." Not all numbers are the products of counting. Indeed, it is the smaller, and therefore commoner, numbers that are often not counted but recognized as patterns at a single glance. Card-players do not stop to count the pips in the eight of spades and can even recognize the characteristic patterning of pips up to "ten."
In other words, number is of the world of pattern, gestalt, and digital computation; quantity is of the world of analogic and probabilistic computation.
Some birds can somehow distinguish number up to seven. But whether this is done by counting or by pattern recognition is not known. The experiment that came closest to testing this difference between the two methods was performed by Otto Koehler with a jackdaw. The bird was trained to the following routine: A number of small cups with lids are set out. In these cups, small pieces of meat are placed. Some cups have one piece of meat, some have two or three, and some cups have none. Separate from the cups, there is a plate on which there is a number of pieces of meat greater that the total number of pieces in the cups. The jackdaw learns to open each cup. Taking off the lid, and then eats any pieces of meat that are in the cup. Finally, when he has eaten all the meat in the cups, he may go to the plate and there eat the same number of pieces of meat that he got form the cups. The bird is punished if he eats more meat from the plate than was in the cups. This routine he is able to learn.
Now, the question is: is the jackdaw counting the pieces of meat, or is he using some alternative method of identifying the number of pieces? The experiment has been carefully designed to push the bird toward counting. His actions are interrupted by his
having to lift the lids, and the sequence has been further confused by having some cups contain more than one piece of meat and some contain none. By these devices, the experimenter has tried to make it impossible for the jackdaw to create some sort of pattern or rhythm by which to recognize the number of pieces of meat. The bird is thus forced, so far as the experimenter could force the matter, to count the pieces of meat.
It is still conceivable, of course, that the taking of the meat from the cups becomes some sort of rhythmic dance and that this rhythm is in some way repeated when the bird takes the meat from the plate. The matter is still conceivably in doubt, but on the whole, the experiment is rather convincing in favor of the hypothesis that the jackdaw is counting the pieces of meat rather than recognizing a pattern either of pieces or of his own actions.
It is interesting to look at the biological world in terms of this question: Should the various instances in which number is exhibited by regarded as instances of gestalt, of counted number, or of mere quantity? There is a rather conspicuous difference between, for example, the statement "This single rose has five petals, and it has five sepals, and indeed its symmetry is of a pentad pattern" and the statement "This rose has one hundred and twelve stamens, and that other has ninety-seven, and this has only sixty-four." The process which controls the number of stamens is surely different from the process that controls the number of petals or sepals. And, interestingly, in the double rose, what seems to have happened is that some of the stamens have been converted into petals, so that the process for determining how many petals to make has now become, not the normal process delimiting petals to a pattern of five, but more like the process determining the quantity of stamens. We may say that petals are normally "five" in the single rose but that stamens are "many" where "many" is a quantity that will vary from one rose to another.
With this difference in mind, we can look at the biological world and ask what is the largest number that the processes of growth can handle as a fixed pattern, beyond which the matter is handled as quantity. So far as I know, the "numbers" two, three, four, and five are the common ones in symmetry of plants and animals, particularly in radial symmetry.
The reader may find pleasure in collecting cases of rigidly controlled or patterned numbers in nature. For some reason, the larger numbers seem to be confined to linear series of segments, such as the vertebrae of mammals, the abdominal segments of insects, and the anterior segmentation of earthworms. (At the front end, the segmentation is rather rigidly controlled down to the segments bearing genital organs. The numbers vary with the species but may reach fifteen. After that, the tail has "many" segments.) An interesting addition to these observations is the common circumstance that an organism, having chosen a number for the radial symmetry of some set of parts, will repeat that number in other parts. A lily has three sepals and then three petals and then six stamens and a trilocular ovary.
It appears that what seemed to be a quirk or peculiarity of human operation - namely, that we occidental humans get numbers by counting or pattern recognition while we get quantities by measurement - turns out to be some sort of universal truth. Not only the
jackdaw but also the rose are constrained to show that for them, too - for the rose in its anatomy and for the jackdaw in its behavior (and, of course, in its vertebral segmentation) - there is this profound difference between numbers and quantity.
What does this mean? That question is very ancient and certainly goes back to Pythagoras, who is said to have encountered a similar regularity in the relation between harmonics.
The hexago-rectangle discussed in section 5 provides a means of posing these questions. We saw, in that case, that the components of description could be quite various. In that particular case, to attach more validity to one rather than to another way of organizing the description would be to indulge illusion. But in this matter of biological numbers and quantities, it seems that we encounter something more profound. Does this case differ from that of the hexago-rectangle? And if so, how?
I suggest that neither case is as trivial as the problems of the hexago-rectangle seemed to be at first sight. We go back to the eternal verities of Saint Augustine: "Listen to the thunder of that saint, in about A.D. 500: 7 and 3 are 10; 7 and 3 have always been 10; 7 and 3 at no time and in no way have ever been anything but 10; 7 and 3 will always be 10."
No doubt, in asserting the contrast between numbers and quantities, I am close to asserting an eternal verity, and Augustine would surely agree.
But we can replay to the saint, "Yes, very true. But is that really what you want and mean to say? It is also true, surely, that 3 and 7 are 10, and that 2 and 1 and 7 are 10, and that 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 are 10. In fact, the eternal verity that you are trying to assert is much more general and profound than the special case used by you to carry that profound message." But we can agree that the more abstract eternal verity will be difficult to state with unambiguous precision.
In other words, it is possible that many of the ways of describing my hexago-rectangle could be only different surfacings of the same more profound and more general tautology (where Euclidean geometry is viewed as a tautological system).
It is, I think, correct to say, not only that the various phrasings of the description of the hexago-rectangle ultimately agree about what the describers thought they saw but also that there is an agreement about a single more general and profound tautology in terms of which the various descriptions are organized.
In this sense, the distinction between numbers and quantities is, I believe, nontrivial and is shown to be so by the anatomy of the rose with its "5" petals and its "many" stamens, and I have put quotation marks into my description of the rose to suggest that the names of the numbers and of the quantities are the surfacing of formal ideas, immanent within the growing rose.
But we can replay to the saint, "Yes, very true. But is that really what you want and mean to say? It is also true, surely, that 3 and 7 are 10, and that 2 and 1 and 7 are 10, and that 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 are 10. In fact, the eternal verity that you are trying to assert is much more general and profound than the special case used by you to carry that profound message." But we can agree that the more abstract eternal verity will be difficult to state with unambiguous precision.
In other words, it is possible that many of the ways of describing my hexago-rectangle could be only different surfacings of the same more profound and more general tautology (where Euclidean geometry is viewed as a tautological system).
It is, I think, correct to say, not only that the various phrasings of the description of the hexago-rectangle ultimately agree about what the describers thought they saw but also that there is an agreement about a single more general and profound tautology in terms of which the various descriptions are organized.
In this sense, the distinction between numbers and quantities is, I believe, nontrivial and is shown to be so by the anatomy of the rose with its "5" petals and its "many" stamens, and I have put quotation marks into my description of the rose to suggest that the names of the numbers and of the quantities are the surfacing of formal ideas, immanent within the growing rose.
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