ABSTRACT
A natural number can be used both as an ordinal and a cardinal number. The continuing philosophical interest in natural numbers and the continuum may be contrasted with the situation of negative and complex numbers. There are interesting philosophical views on the nature of arithmetic which have little to do with mathematical logic. A famous example is Kant's theory in terms of the form of the pure intuition of time. Formal systems help to get a systematic survey of relation between existence theorems and computable functions. In the cases of classical number theory and analysis, people have seen that second order systems give us more adequate characterizations of natural numbers and the continuum. The definition of mechanical procedures in terms of general recursiveness or Turing computability in mathematical logic is generally regarded as of great importance. Godel points out that the precise notion of mechanical procedures is brought out clearly by Turing machines producing partial rather than general recursive functions.
