Frege's ‘series of natural numbers’

Abstract

We all know that one of Frege's achievements was to define the series of natural numbers. So what could be less remarkable than for the phrase to occur forty times in the English version of the Grundlagen der Arithmetik?1 Beware: 'the series of natural numbers' is neither a series nor of the natural numbers. The first warning sign is the definite article. In ?76 Frege repeats his homily against using 'the' until there has been a proof of existence and uniqueness. Yet this is the very passage in which 'the series of natural numbers' makes its debut unaccompanied by any such thing. Then, when he does define natural number (?83), the adjective actually employed is 'finite'. This brings out the surprising fact that Frege never uses the term 'natural number' except in this one phrase 'the series of natural numbers'.2 Surely there is something here that requires explanation. There are puzzles, too, about 'the series of natural numbers' itself. For instance, it is different from 'the series of natural numbers beginning with o' (?83), yet how can it be? Although German usage sanctions some specific exceptions, the rule for phrases like 'die naturliche Zahlenreihe' is that the adjective qualifies the whole of the following compound and not its first element. That is to say, the correct rendering is not Austin's 'the series of natural numbers' but 'the natural series of numbers'. The series in question may be called natural ('durch die Natur der Sache bestimmte', ?io), because it is determined by the successor relation. The numbers, on the other hand, are the cardinal numbers, infinite as well as finite; for Frege defines the successor relation (?76) to apply to cardinal numbers in general. This is confirmed by the Grundgesetze der Arithmetik. For when the Grundlagen is recapitulated there (see ?43 ff.), 'die naturliche Zahlenreihe' becomes simply 'die Anzahlenreihe'-the series of cardinal numbers. And, reverting to the Grundlagen, there is nothing surprising about the fact that 'natural number' is not used outside the context of 'the natural series of numbers', since it is not used inside it either. Frege's idea of a series (Reihe) has nothing in common with the ordinary idea of order in a straight line. It is more like the graph of an arbitrary binary relation. Thus each relation 0 determines a Fregean series, 'the bseries', which may branch or merge or go round in circles (Begriffsschrift ?26), or may contain reflexive loops (Grundlagen ?84) or disconnected