## CHAPTER THREE

# NUMBER, MAGNITUDE, QUANTITY

## 1. NUMBERS IN THE LANGUAGE

Numbers are linguistic expressions. They play the role of subject-terms, parts of subjects, parts of predicates and quantifiers. It is just predicateterms that numbers cannot be. Examples: (1) the number 'ten' in the statement 'The number of types of elementary particles is greater that ten' is a subject; (2) the numbers 'one' and 'two' in the statement 'The molecule of water consists of one oxygen atom and two hydrogen atoms' are parts of subjects; (3) the number '300,000' in the statement 'Light propagates with a velocity about 300,000km/sec' is a part of the predicate; (4) the number 'three' in the statement 'For all three types of the airplanes, it is true that they reach supersonic velocity' is a quantifier. Analysis of the logical properties of numbers leads to the analysis of positive integers through which all other kinds of numbers are defined The definition of the logical properties of positive intergers is the subject of formal arithmetic which actually became a part of logic. And there is justification to it: the construction of formal arithmetic does not imply the existence of any developed mathematics, but it does imply explicitly formulated logic and operators exclusively by logical methods. Below we shall give a presentation of a formal arithmetic that is essentially different from the constructions of this kind known to us and agrees with our conception of logic. In establishing the properties of numbers we shall make use of the predicates of superiority and identity in order. At the same time this will serve as the definition of the above predicates for number-terms and as the ordering of the class of numbers. We shall omit description of the method of establishing order, under the following assumptions: (1) the order of numbers is established by convention with number theory, in particular, by that system of formal arithmetic which will be discussed below; (2) the order of numbers is established by our arbitrary willed decision. Thus, the expressions la is greater than V and 'a is equal to V for numbers, in our presentation, will be identical with the expressions 'a is superior in order to V and 'a is identical in order to b\ respectively. We do not distinguish here between both forms of negation for numbers, in the sense that the following assertion ~(a>b)\-(a~\>b) holds. This means that it is up to us to establish complete order in a given class of numbers and to exclude cases of uncertainty. Because of this assertion, the following statements will be valid: (a = b) -j | (a > b) A ~ (b > a), and in what follows we shall limit ourselves to the logical rules for the classical case.

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