The audacity of thought is not to repeat 'to the limit' that which is already entirely retained within the situation which the limit limits; the audacity of thought consists in crossing a space where nothing is given. We must learn once more how to succeed.

In the ordinal view, number is thought as a link in a chain, it is an element of a total order. In the cardinal view, it is rather the mark of a 'pure quantity' obtained through the abstraction of domains of objects having 'the same quantity'. The ordinal number is thought according to the schema of a sequence, the cardinal number, according to that of a measurement.

4.23..If 'thought' means: instance of the subject in a truth-procedure, then there is no thought of this thought, because it contains no knowledge.

The 'economy of number' proposed by Peano is an economy of signs whose paradigm is algebraic, whose transparency is consensual, and whose operational effectiveness is therefore not in doubt. He thus participates forcefully in that movement of thought, victorious today, that wrests mathematics from its antique philosophical pedestal and represents it to us as a grammar of signs where all that matters is the making explicit of the code.

We do not want to count; we want to think the count.