Foundations
20 June 2011
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.
Politicians. Twitter.
2 June 2011
I enjoy interacting with organisations and politicians on twitter. It allows access to quick, pertinent information that I don’t think would be possible through a telephone or email. As with any mode of communication, success depends on both ends knowing how the medium works. @brisbanecityqld is a good example of a productive organisational account. I’ve also heard good things about Premier Bligh’s @TheQLDPremier. These two work because someone has decided to take twitter seriously as a medium to communicate with the public, and has been adequately resourced to do so.
On the other end of the spectrum, the account of one politician recently couldn’t articulate a policy position, because their ideas ‘won’t fit into 140 characters’. Here, a well run campaign, or just a politician who understand the web, could defer to a medium which does not hold such a limit. But this didn’t happen, so there was a breakdown.
If we do operate in a world where politicians are professionals in the narrow field of appealing to voters, taxpayers or business, then political parties need to ensure that they are not left unsupported in domains that they don’t understand (here, the internet). Perhaps this is ridiculous idealism, but I would like for the people running the council, state or country to have a broader expertise and be able to respond when questioned by a citizen, because they are intelligent and capable individuals in their own right. Maybe it is fair to insist that agents of the state are literate in modern communication platforms, consider all that they demand of us.
Nashi Frangipane Tart
11 May 2011
The other week I chanced to bring home some nashi fruit. As the first whispers of winter’s approach reach us, it was nice to taste something so bright and sweet. I decided to make some sort of nashi pastry, mostly to give myself an excuse to buy a whole heap. I mostly followed this recipe, though I made the pâte sablée from Michel Roux’s pastry book.
I expected the high water content of nashi pears to be an issue, so I was careful to dry them thoroughly after poaching. Still, the tart was quite wet, so I drew some of the liquid out with a towel. I didn’t foresee the size of the pears to be a problem, but looking at it now, they could have been sliced a bit more elegantly. Between these two things and my probably under-creamed frangipane, the centre of the tart cooked much more slowly than that around the rim. Luckily, it’s still delicious to my unrefined palate, served with a drizzle of the poaching liquid, reduced to a syrup.