Russell and Whitehead, famously, tried to secure a foundation in logic for formal mathematics. The common1 view is that they failed for reasons expounded by Gödel (his incompleteness theorems). Not all mathematical truths can be captured as theorems of a formal system (like that in Principia Mathematica. This is incompleteness), because if they were an inconsistent pair of statements could be constructed (this is the hard part). Since classical logic is explosive, we resort to incompleteness to avoid triviality.
In my reading for a piece on Russell’s theory of descriptions, however, I came across a much simpler and quite disarming problem with the Fregean project which Russell and Whitehead were furthering. In short, the argument goes like this. Complex numbers are essential parts of mathematics, not just in explicitly “complex” theorems, but in real analysis and probably every subfield of maths2. The basis of complex numbers, the imaginary unit, is defined as the square root of negative one. But this is a definite description which, according to Russell, should denote uniquely. But it doesn’t, and the indistinguishability of the two roots of x2 + 1 sinks the theory. The paper, was targeted at Frege, and maybe I am mistaken for considering it an objection to Russell and Whitehead, but the idea seems valid.
Of course, as soon as I open up Complex Variables and Applications to check on the arguments validity against modern complex analysis, I find it has problems. Primarily because the imaginary unit is not defined by a definite description. Instead, statement 1.1.1 on page one describes a complex number as an ordered pair of real numbers. While solving x2 + 1 = 0 is a property of (0,1), it’s not its defining characteristic. (0,1) and (0,-1) are unambiguously different (complex) numbers. While Brandom’s paper causes problems for our pre-theoretical ideas of what i and -i are, that’s not a particularly strong case to make against modern mathematics as it is practiced.
1 I am always as nervous when claiming that a view is common as I am skeptical when reading. If you disagree, entertain the proposition for a moment and/or tell me why I’m mistaken.
2 For example the distribution of the primes depends on the Riemann hypothesis. Who knew that maths was a tangle? Everyone.