Comment on The Pascal's Wager Fallacy Fallacy

Sat, 04 Jun 2011 16:57:18 GMT

1) Yes, there are countably many constructions of various kinds of numbers. The construction can presumably be written down, and strings are finite-information-content entities. Yes, they're normally understood to form a set-theoretic lattice - the integers are a subset of the gaussian integers, and the integers are a subset of the rationals, and both the gaussian and rationals are a subset of the complex plane.

However, the reals are not in any well-defined sense "the" limit of that lattice - you could create a contrived argument that they are, but you could also create an argument that the natural limit is something else, either stopping sooner, or continuing further to include infinities and infinitiesimals or (salient to mathematicians) the complex plane.

Defenders of the reals as a natural concept will use the phrase "the complete ordered field", but when you examine the definition of "complete" they are referencing, it uses a significant amount of set theory (and an ugly Dedekind cuts construction) to include everything that it wants to include, and exclude many other things that might seem to be included.

2) Yes. I think the reals are a history-laden concept; they were built in fear of set-theoretic and calculus paradoxes, and without awareness of the computational approach - information theory and Godel's incompleteness. They are useful primarily in the way that C++ is useful to C++ programmers - as a thicket or swamp of complexity that defends the residents from attack. Any mathematician doing useful work in a statistical, calculus-related, or topological field who casually uses the reals will need someone else, a numerical analyst or computer scientist, to laboriously go through their work and take out the reals, replacing them with a computable (and countable) alternative notion - different notions for different results. Often, this effort is neglected, and people use IEEE floats where the mathematician said "real", and get ridiculous results - or worse, dangerously reasonable results.

3) You're right that the finite amount of sensor data will only say it is consistent with this interval. As you point out, if there are an uncountable set within that interval, then it's entirely possible for there to be no single value that is a maximum of the probability distribution function. (That's an excellent example of some of the ridiculous circumlocutions that come from using uncountable sets, when you actually want the system to come up with one or a few best hypotheses, each of which is stateable.)

Pi is always a finite-information entity. Everything nameable is. It doesn't become an infinitely large in information content just because you consider it as an element of the reals.

Yes - if you use a probability distribution over the rationals as your prior, and the actual value is irrational, then you can get bad results. I think this is a serious problem, and we should think hard about what bayesian updating with misspecified models looks like (I know Cosma Shalizi has done some work on this), so that we have some idea what failure looks like. We should also think carefully about what we would consider to be a reasonable hypothesis, one that we might eventually come to rest on.

However, it's a false fork to argue "We wouldn't use the rationals therefore we should use the reals". As I've been trying to say, the reals are a particular, large, complicated, and deeply historical construction, and we should not expect to encounter them "out in the world".

Andrej Bauer has implemented actual real number arithmetic (not IEEE nonsense, or "computable reals" which are interesting, but not reals). Peano integers, in his (Ocaml-based) language, RZ, would probably be five or ten lines. (Commutative groups are 13 lines). In contrast, building from commutative groups up to defining reals as sequences of nested intervals takes five pages; see the appendix: http://math.andrej.com/wp-content/uploads/2007/04/rzreals.pdf

Regarding "reminding you of things you didn't think of", I think Cosma Shalizi and Andrew Gelman have convincingly argued that Bayesian philosophy/methodology is flawed - we don't just pick a prior, collect data, do an update, and believe the results. If we were magical uncomputable beasties (Solomonoff induction), possibly that is what we would do. In the real world, there are other steps, including examining the data, including the residual errors, to see if it suggests hypotheses that weren't included in the original prior. http://www.stat.columbia.edu/~gelman/research/unpublished/philosophy.pdf

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