What does Wittgenstein mean when he says in the Tractatus that there are no preeminent numbers in logic? He makes mention of this three times in the Tractatus:

4.128 The logical forms are anumerical. Therefore there are in logic no pre-eminent numbers, and therefore there is no philosophical monism or dualism, etc.

5.453 All numbers in logic must be capable of justification. Or rather it must become plain that there are no numbers in logic. There are no pre-eminent numbers.

5.553 Russell said that there were simple relations between different numbers of things (individuals). But between what numbers? And how should this be decided—by experience? (There is no pre-eminent number.)


Although this question is similar to What does Wittgenstein mean when he says "there are no numbers in logic"? that question asked why Wittgenstein claimed there were no numbers in logic. That answer, based on G. E. M. Anscombe's interpretation, involved the difference between genuine and formal concepts. For Frege and Russell numbers were genuine concepts that could be applied to objects or classes. For Wittgenstein they were merely formal ways to represent exponents of operations.

Still even viewed as formal concepts marking which in a sequence of operations one was performing shouldn't there be a pre-eminent number, such as 0, which started this sequence? G. E. M Anscombe attempts to explain Wittgenstein's approach: (page 125)

One might thus well think that for the concept 'number' it would have been enough for Wittgenstein to say as he does at 6.022-03: 'The concept "number" is the variable number....The general form of the whole number is [0,ξ,ξ+1]'一so long as this was supplemented by some account of '0' and of the special operation '+1'. In fact Wittgenstein goes about it in quite a different way. At 6.02 he gives the following definitions:

Ω0x = x; Ωn+1x = ΩΩnx

This explains the meaning of a zero exponent of the operator 'Ω' and also the meaning of an exponent of the form 'n + 1' given the meaning of the exponent 'n'. He then defines the ordinary numerals in terms of 0 and +1, as follows:

1 = 0 + 1; 2 = 0 + 1 + 1; 3 = 0 + 1 + 1 + 1; etc.

This enables us to interpret the use of any ordinary numeral as an exponent; e.g. Ω3x = ΩΩΩx. And a number is always 'the exponent of an operation' (6.021)...

Whether this works or not, it at least minimized anything special, or pre-eminent, about '0' which represented the exponent of an operation that was not performed. The '+1' represented an operation that was performed.

Anscombe, G. E. M. An Introduction to Wittgenstein's Tratatus. 1971. St. Augustine's Press.

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