Wittgenstein (e.g. in his Lectures on the Foundations of Mathematics, Cambridge 1939) criticised Principia on various grounds, such as:

It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and Principia, this would be treated as evidence of an error in Principia (e.g. that Principia did not characterize numbers or addition correctly), not as evidence of an error in everyday counting.

The calculating methods in Principia can only be used in practice with very small numbers. To calculate using large numbers (e.g. billions), the formulae would become too long, and some short-cut method would have to be used, which would no doubt rely on everyday techniques such as counting (or else on non-fundamental – and hence questionable – methods such as induction). So again Principia depends on everyday techniques, not vice versa.

Wittgenstein did, however, concede that Principia may nonetheless make some aspects of everyday arithmetic clearer.

2012.05.01 Tuesday ACHK