I don't expect laws to be formally defined, I just expect them to be few, sensible, and transparent. Reasonable people shouldn't have a hard time inferring whether a specific activity is illegal.
Right now, our body of law is vast, complex, opaque and uncertain. Because of this, as I said above, most people likely break the law every single day. This gives rise to tyranny by selective enforcement (http://en.wikipedia.org/wiki/Selective_enforcement), burdened economies and reduced investments due to untested and protectionist regulatory regimes, and an overall reduction in freedom and civil liberties.
By contrast, yes, the knowability and certitude afforded by a hypothetical-but-impossible legal system based on mathematics would be preferable. This is really not a complicated idea.
“Corruptissima republica plurimae leges. [The more numerous the laws, the more corrupt the state.]” — Tacitus, the Annals ca. AD 69
So's our society. Roman law was quite a bit simpler, but justice in ancient Rome was extremely unpredictable.
By contrast, yes, the knowability and certitude afforded by a hypothetical-but-impossible legal system based on mathematics would be preferable. This is really not a complicated idea.
As HL Mencken said, 'For every complex problem there is an answer that is clear, simple, and wrong.' Godel's theorem tells us that in any sufficiently expressive formal system (such as mathematics), you can have consistency or completeness, but not both. A hypothetical mathematically-precise legal system would thus yield cases that either couldn't be judged because there was no prescription for such a situation (a div zero problem, if you will), or else would yield contradictory results on the same facts at different times.
People are emotional and subjective by their very nature. It is not possible to be objective - one can only tend towards objectivity by applying constant introspection and reassessing one's world view. Nonetheless, an Ape is not capable of being a Vulcan. It is, frankly, a travesty that people's fates are being decided by other people's stomachs.
I'd argue that you really can't say anything with such certainty. If true friendly AI is ever possible (and that is up in the air), such a system would be a better one. Obviously, this is all speculation.
So, you're saying that's more dystopian than the thought of secret laws for 'national security', on top of a body of laws that already spans over 200,000 pages, whereby even with non-poorly-written laws, you're still likely breaking one every single day?
Because that's what we have now. So if those are my options, then shit yes, I will take mathematical certainty of the legality of my actions any day.