APPENDIX I HUSSERL'S DESCRIPTION OF THE ORIGIN OF A SYMBOLISM FOR NUMBERS.l According to the Philosophy of A rithmetic, the concept of number differs from that of multiplicity in that it specifies the precise form of multi­ plicity involved in a given case. When faced with a group of objects, it is enough to know that there is a multitude there to have the concept of multiplicity apply. In order to apply the concept of number, we must be able to say how many objects there are; that is, we must specify which of the forms of multiplicity is present in this case. When we are limited to authentically given numbers, we have only a handful of such forms to deal with. Only a few numbers are actually given in direct consciousness. It would be easy to decide on a few names or signs to represent them symbolically. Now that symbolically given multitudes are introduced, however, the picture changes. We know that a specific number corresponds to each multitude that can be symbolized, but now there is no limit to the diversity of forms of multitudes that can be involved. The weakness of our imagination and memory and the limits of our ability to perform actual collective liaisons do not matter any more. We know that every number, every form of multiplicity, can be changed into another form simply by the addition of one number, and that this process can, theoretically, be carried out indefinitely. The only limitation on it is that imposed by our mental powers of collecting, and since we are no longer dealing with actually collected multitudes, this element of factual limitation falls away. Husser! says, "So the re­ strictions on the conception of number concepts fall away, together with the restrictions that check our presentation of multitudes. In a symbolical, but in a very determined way, we can speak of numbers even where an authentic presentation is forever denied us, and on this level we are even in a position to determine the ideal infinity of the realm of numbers." 2. Thus we are now faced with a limitless variety of forms of multitudes or numbers; what sort of symbolism are we to devise for them, so that, given the symbol, we will know exactly which form of multiplicity we are dealing with? The first problem is to devise an appropriate system of symbols. What materials do we have to work with? We have the names of numbers which are authentically given to us. Let us say they go as far 1 To § 10, Chapter I. a "So sind wirklich mit den Schranken, die unser Vorstellen von Mengen hemmt, auch diejenigen fiir die Konzeption der Zahlbegriffe gefallen. In symbolischem aber ganz bestimmtem Sinne konnen wir von Zahlen sprechen, wo dert'n eigentliche Vor­ stellung uns fiir immer versagt ist, und wir sind auf dieser Stufe sogar in der Lage, die ideelle Unendlichkeit des Zahlenreiches festzustellen." Phil. Al'uh., pp. 251-52. APPENDIX I 225 as ten or twelve. We have the operations of addition and subtraction, and the relationship of more and less.1 We have the principle that each number, each form of multiplicity, differs from its immediate prede· cessor and successor by one unit each; it is one more than its predecessor and one less than its successor. One way of building a series of signs to symbolize all possible numbers, the first method studied by Hussed, would be to do so in an unsystematic way, through "systemless symbolization of numbers." 2 This could be done, he says, by taking the symbols for authentic numbers and adding them together to form complex symbols for new numbers which are not given in direct encounter. Thus "10 + 5," "9 + 6 + 8," "7 + 10 + 5," etc., could each be complex signs for numbers greater than those we encounter directly. The difficulties in such an unsystematic method are apparent. The complex symbols become very long and unwieldy once the numbers become large, and the strain on memory and imagination becomes very great. Also, it becomes difficult to determine whether two such complex symbols stand for the same number or not, and if they do not, it is hard to determine which of the two represents the larger or the smaller number. Thus the main purpose of numeration, to indicate which form of multiplicity we are dealing with in a given case, is not achieved by this method. This purpose can best be attained, says Husserl, by use of a systematic method of forming symbols. 3 The systematic method Hussed adopts is developed as follows. Let our symbols be defined in a sequence running parallel to the sequence of number concepts. This means that each symbol will be introduced as being equal to its preceding symbol plus one, just as each number is equal to its predecessor plus one unit. This results in a sequence of symbols as follows: I; 2 = I + I; 3 = 2 + I; ••. 9 = 8 + I. This process can be continued indefinitely, and will generate a system of signs each of which corresponds to a given form of multiplicity, to a given number, since each number is also generated from its predecessor by the addition of one. However, it should be noted that nothing we have said so far, none of the rules we have introduced, justifies our repeating the cycle of symbols, "I," "2," "3," etc., after we have passed beyond the number 9. Strictly speaking, we must find altogether new symbols for all the succeeding numbers. "Every new step in the symbolical formation of numbers demands a new step in naming them." 4 This would result in an unlimited 1 Husserl explains the relationship of more-less after he treats the origins of au­ thentic numbers, and uses this relationship to explain addition and subtraction, which he considers the two fundamental operations of arithmetic, the two to which all other operations can be reduced (ibid., pp. 203-II). The explanation of the re­ lationship more-less does not involve any principles different from those presented in Husserl's treatment of the origins of groups. The relationship arises, once again, as a result of psychic acts of relating, but in this case the things related are groups. In forming this relationship, we form a higher order group, we form a group of groups (p. 99). Once we relate two such groups, we can see, says Husserl, which is greater than the other. This is the origin of the relationship more-less. I "Die systemlose Zahlsymbolisierungen." ibid., p. 252. 8 Cf. ibid., p. 254. , "Jeder neue Schritt der symbolischen Zahlbildung erfordert einen neuen Schritt der Bennenung." ibid., p. 256. 226 APPENDIX I number of symbols, each different from the others, generated by our series. The series would continue: 9; x = 9 + I; Y = X + I; Z = Y + 1. ••. We would have as many different symbols as there are numbers. This result will not do, since the limits of memory and imagination once again make such a series of symbols useless. The difficulty can be resolved by carrying on the sequence for a given distance, then repeating the same sequence over again, but indicating somehow that a repetition is being carried on. Thus we build up the first sequence, I, 2, 3, 4, 5, 6, 7, 8, 9. Then we repeat it, but add a "I" to each repeated symbol to indicate that we are now in the second phase of the cycle: 10, II, 12, 13, 14, IS, 16, 17, 18, 19. When we repeat the sequence another time, we add "2" to each symbol: 20, 21, 22, 23, 24 ..• and so forth, with appropriate changes made for hundreds, thousands, etc.l This system of symbols satisfies all the requirements for a good set of signs for the number system. Each symbol differs from its predecessor by one unit, so the system of symbols is parallel to the system of numbers. Furthermore, we can tell at a glance which of two symbols represents the larger of two numbers, and how far apart they are in the number series. Thus these symbols serve to identify clearly which number, which form of multiplicity, we are dealing with in any given case. If we accept the principle of a recurrent cycle of symbols for our number system, the question then arises, how long should each cycle be ? How many units should it include? The only requirements imposed are (I) the cycle should not be too short, or else the symbols quickly become too complex and too long. If we repeat the cycle after only two or three units, for example, we would soon have very complicated symbols to deal with. (2) It should not be too long, or else we would have too many different basic symbols in the first cycle. For example, if the cycle ex­ tends over seventy-five different units before repeating itself, we would have to handle seventy-five different basic signs. Both of these re­ quirements are imposed, once again, by limitations of our memory and imagination, perceptual discrimination, and other such psychological factors. 2 The fact is that our number symbols repeat in cycles of ten units. There is no apriori necessity that dictates this. The cycle could just as conveniently have extended over, for example, twelve units before re­ peating, and Hussed observes that mathematically a number system based on twelve units would be more advantageous.s Hussed suggests a historical explanation for the choice of ten units in the basic cycle, re­ ferring to the fact that primitively ten fingers are used as the most common means of counting.4 1 Ct.