## Numbers: The Abstract Approach

The Cardinal and Ordinal approaches may satisfy philosophers, but are likely to irritate mathematicians, who seldom think of numbers as closely connected with the rest of our conceptual structure, reasonably referred to by words, ‘Nought’, ‘One’, ‘Twelve’, ‘First’, ‘Second’, ‘Fifth’, but as abstract entities, presented symbolically, ‘0’, ‘1’, ‘12’, etc. (where we read ‘0’ as ‘O’, like telephonists). A purely symbolic account is, we shall see, as unsatisfactory as a purely cardinal or purely ordinal one. In the end we shall be led to the conclusion that there are three distinct, but interlocking facets to our concept of natural number: 1. a cardinal one, as answers to the question Quot?, How many?,

where the answers can be Nought, One, Two,…etc. 2. an ordinal one, either as adjectives, First, Second, Third,…etc.,

or as intransitive counting numbers, One, Two, Three,…etc., where the numbers form a queue of order-type w.