What does Wittgenstein mean when he says "there are no numbers in logic"?

Question

From the Tractatus:

5.453 All numbers in logic stand in need of justification. Or rather, it must become evident that there are no numbers in logic. There are no pre-eminent numbers.

What does Wittgenstein mean by saying that there are no numbers in logic? To some extent he was an adherent of Frege and of Russell who both gave logcial constructions of numbers. And elsewhere in the Tracatus he gives statements which touch on the logical construction of numbers, eg:

6.03 The general form of an integer is [0, ξ, ξ +1].

So what does Wittgenstein mean in 5.453?

Answer 1

In the Tractatus, Wittgenstein rejected the Logicist program of Frege and Russell to define the concept of number based on logical notions only (included the extension of concepts, for Frege, and the theory of classes for Russell).

We can see :

4.1272 [...] one cannot say, for example, ‘There are objects’, as one might say, ‘There are books’. And it is just as impossible to say, ‘There are 100 objects’, or, ‘There are ℵ0 objects’. And it is nonsensical to speak of the total number of objects.

for the rejection of Russell's Axiom of Infinity, necessary for the foundational project developed in the Principia.

And also 4.1273, for Wittgenstein's critique of Frege and Russell's definition of successor.

For Wittgenstein, numbers are not "logical objects":

4.128 Logical forms are without number. Hence there are no pre-eminent numbers in logic [...]

And 5.453 All numbers in logic stand in need of justification. Or rather, it must become evident that there are no numbers in logic.

Number is a sort of "primitive" concept:

6.021 A number is the exponent of an operation.

And :

6.031 The theory of classes is completely superfluous in mathematics.

In the Tractatus, mathematics (better : arithmetic) is essentially calculations, i.e. an activity based in signs manipulation (see 6.2 and on).

This point of view will be developed by later Wittgenstein in the theory of language games based on rules.

Answer 2

In order to help distinguish Wittgenstein's view of number from that of Frege and Russell, G. E. M. Anscombe described Frege and Russell's view of number as a genuine concept rather than as a formal concept:

For Frege and Russell (natural) number was not a formal concept, but a genuine concept that applied to some but not all objects (Frege) or to some but not all classes of classes (Russell); those objects, or classes, to which the concept number applied were picked out from others of their logical type as being 0 and the successors of 0. (page 126)

For Wittgenstein number was a formal concept defined (6.02) by its use as the exponents in any formal series. Formal series contain operations that

can be iterated - 'the result of an operation can be the base of that very operation' (5.251) (page 124)

Numbers answer the question

which term it is, which performance of the generating operation the term results from. (page 126)

As a formal concept, numbers are not objects among other objects in logic. Hence, for Wittgenstein there are no numbers and no pre-eminent numbers such as "0" in logic. However, numbers are expressed "only by the way we apply the corresponding sign" (page 123). The way the sign for a number is applied is as the exponent in a formal series.

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