hardness of gold as three qualities, though none of these is before or after the other, either in space or time. Every means of discrimination may be a source of plurality. Our logical notation may be used to express the rise of number. The symbol A stands for one thing or one class, and in itself must be regarded as a unit, because no difference is specified. But the combinations AB and Ab are necessarily two, because they cannot logically coalesce, and there is a mark B which distinguishes one from the other. A logical definition of the number four is given in the combinations ABC, ABc, AbС, Abc, where there is a double difference, and as Puck says "Yet but three? Come one more; Two of both kinds makes up four.' I conceive that all numbers might be represented as arising out of the combinations of the Abecedarium, more or less of each series being struck out by various logical conditions. The number three, for instance, arises from the condition that A must be either B or C, so that the combinations are ABC, ABC, AbC. Of Numerical Abstraction. There will now be little difficulty in forming a clear notion of the nature of numerical abstraction. It consists in abstracting the character of the difference from which plurality arises, retaining merely the fact. When I speak of three men I need not at once specify the marks by which each may be known from each. Those marks must exist if they are really three men and not one and the same, and in speaking of them as many I imply the existence of the requisite differences. Abstract number, then, is the empty form of difference; the abstract N number three asserts the existence of marks without specifying their kind. Numerical abstraction is then a totally different process from logical abstraction (see p. 33), for in the latter process we drop out of notice the very existence of difference and plurality. In forming the abstract notion hardness, for instance, I drop out of notice altogether the diverse circumstances in which the quality may appear. It is the concrete notion three hard objects, which asserts the existence of hardness along with sufficient other undefined qualities, to mark out three such objects. Numerical thought is indeed closely interwoven with logical thought. We cannot use a concrete term in the plural, as men, without implying that there are marks of difference. Only when we use a term in the singular and abstract sense man, do we deal with unity, unbroken by difference. The origin of the great generality of number is now apparent. Three sounds differ from three colours, or three riders from three horses; but they agree in respect of the variety of marks by which they can be discriminated. The symbols I + I + I are thus the empty marks asserting the fact of discrimination which may apply to objects wholly independently of their peculiar nature. Concrete and Abstract Numbers. The common distinction between concrete and abstract numbers can now be easily stated. In proportion as we specify the logical character of the things numbered, we render them concrete. In the abstract number three there is no statement of the points in which the three objects agree; but in three coins, three men, or three horses, not only are the variety of objects defined, but their nature is restricted. Concrete number thus implies the same consciousness of difference as abstract number, but it is mingled with a groundwork of similarity expressed in the logical terms. There is similarity or identity so far as logical terms enter; difference so far as the terms are merely numerical. The reason of the important Law of Homogeneity will now be apparent. This law asserts that in every arithmetical calculation the logical nature of the things numbered must remain unaltered. The specified logical agreement of the things numbered must not be affected by the unspecified numerical differences. A calculation would be palpably absurd which, after commencing with length, gave a result in hours. It is in reality equally absurd in a purely arithmetical point of view to deduce areas from the calculation of lengths, masses from the combination of volume and density, or momenta from mass and velocity. It must remain for subsequent consideration in what sense we may truly say that two linear feet multiplied by two linear feet give four superficial feet, but arithmetically it is absurd, because there is a change of unit. As a general rule we treat in each calculation only objects of one nature. We do not, and cannot properly add, in the same sum yards of cloth and pounds of sugar. We cannot even conceive the result of adding area to velocity, or length to density, or weight to value. The unit numbered and added must have a basis of homogeneity, or must be reducible to some common denominator. Nevertheless it is quite possible, and in fact common, to treat in one complex calculation the most heterogeneous quantities, on the condition that each kind of object is kept distinct, and treated numerically only in conjunction with its own kind. Different units, so far as their logical differences are specified, must never be substituted one for the other. Chemists continually use equations which assert the equivalence of groups of atoms. Ordinary fermentation is represented by the formula CH12 06 2C2H6O + 2 CO2. = Three kinds of units, the atoms respectively of Carbon, Hydrogen, and Oxygen, are here intermingled, but there is really a separate equation in regard to each kind. Mathematicians also employ compound equations of the same kind; for in a + b ✅ − 1 = c + d -1, it is impossible by ordinary addition to add a to b-1. Hence we really have the separate equations a = c, and b=de. Similarly an equation between two quaternions is equivalent to four equations between ordinary quantities, whence indeed the origin of the name quaternion. Analogy of Logical and Numerical Terms. If my assertion is correct that number arises out of logical conditions, we ought to find number obeying all the laws and conditions of logic. It is almost superfluous to point out that this is the case with the fundamental laws of identity and difference, and it only remains for me to show that mathematical symbols do really obey the special conditions of logical symbols which were formerly pointed out (p. 39). Thus the Law of Commutativeness, is equally true of quality and quantity. As in logic we have AB = BA, so in mathematics it is familiarly known that 2 × 3 = 3 × 2, or x x Y = y xx. The properties of space, in short, are as indifferent in pure multiplication as we found them in pure logical thought. c De Morgan's 'Trigonometry and Double Algebra,' p. 126. Similarly, just as in logic triangle or square square or triangle, or generally so in quantity = A+B=BA, 2+3=3+ 2, The symbol is not identical with+, but it is so far analogous. How far, now, is it true that mathematical symbols obey the law of simplicity expressed in the form or the example AA = A, Round round round? = Apparently there are but two numbers which obey this law; for it is certain that X× X = X is true only in the two cases when x = 1 or 0. In reality all numbers obey the law, for 2 x 2 = 2 is not really analogous to AAA. According to the definition of a unit already given, each unit is discriminated from each other in the same problem, so that in 2' x 2', the first two involves a different discrimination from the second two. I get four kinds of things, for instance, if I first discriminate 'heavy and light' and then cubical and spherical,' for we now have the following classes heavy, cubical. heavy, spherical. light, cubical. light, spherical. But suppose that my two classes are in both cases discriminated by the same difference of light and heavy, then we have In short, twice two is two unless we take care that the second two has a different meaning from the first. But |