Information Processing Finitism, Part II

In my previous post, I explored information processing finitism (IPF), the idea that nothing can essentially causally depend on an infinite amount of information about contingent things.

Since a real-valued parameter, such as mass or coordinate position, contains an infinite amount of information, a dynamics that fits with IPF needs some non-trivial work. One idea is to encode a real-valued parameter r as a countable sequence of more fundamental discrete parameters r₁, r₂, ... where r_i takes its value in some finite set R_i, and then hope that we can make the dynamics be such that each discrete parameter depends only on a finite number of discrete parameters at earlier times.

In the previous post, I noted that if we encode real numbers as Cauchy sequences of rationals with a certain prescribed convergence rate, then we can do something like this, at least for a toy dynamics involving continuous functions on between 0 and 1 inclusive. However, an unhappy feature of the Cauchy encoding is that it’s not unique: a given real number can have multiple Cauchy encodings. This means that on such an account of physical reality, physical reality has more information in it than is expressed in the real numbers that are observable—for the encodings are themselves a part of reality, and not just the real numbers they encode.

So I’ve been wondering if there is some clever encoding method where each real number, at least between 0 and 1, can be uniquely encoded as a countable sequence of discrete parameters such that for every continuous function f from [0,1] to [0,1], the value of each parameter discrete parameter corresponding to of f(x) depends only on a finite number of discrete parameters corresponding to x.

Sadly, the answer is negative. Here’s why.

Lemma. For any nonempty proper subset A of [0,1], there are uncountably many sets of the form f⁻¹[A] where f is a continuous function from [0,1] to [0,1].

Given the lemma, without loss of generality suppose all the parameters are binary. For the ith parameter, let B_i be the subset of [0,1] where the parameter equals 1. Let F be the algebra of subsets of [0,1] generated by the B_i. This is countable. Any information that can be encoded by a finite number of parameters corresponds to a member of F. Suppose that whether f(x) ∈ A for some A ∈ F depends on a finite number of parameters. Then there is a C ∈ F such that x ∈ C iff f(x) ∈ A. Thus, C = f⁻¹[A]. Thus, F is uncountable by the lemma, a contradiction.

Quick sketch of proof of lemma: The easier case is where either A or its complement is non-dense in [0,1]—then piecewise linear f will do the job. If A and its complement are dense, let (a_n) and (b_n) be a sequence decreasing to 0 such that both a_n and b_n are within 1/2^n + 2 of 1/2ⁿ, but a_n ∈ A and b_n ∉ A. Then for any set U of positive integers, there will be a strictly increasing continuous function f_U such that f_U(a_n) = a_n if n ∈ U and f_U(b_n) = a_n if n ∉ U. Note that f_U⁻¹[A] contains a_n if and only if n ∈ A and contains b_n if and only if n ∉ A. So for different sets U, f_U⁻¹[A] is different, so there are continuum-many sets of the form f_U⁻¹[A].

Entropy

If p is a discrete probability measure, then the Shannon entropy of p is H(p) =  − ∑_xp({x})log p({x}). I’ve never had any intuitive feeling for Shannon entropy until I noticed the well-known fact that H(p) is the expected value of the logarithmic inaccuracy score of p by the lights of p. Since I’ve spent a long time thinking about inaccuracy scores, I now get some intuitions about entropy for free.

Entropy is a measure of the randomness of p. But now I am thinking that there are other measures: For any strictly proper inaccuracy scoring rule s, we can take E_ps(p) to be some sort of a measure of the randomness of p. These won’t have the nice connections with information theory, though.

Heaven and materialism: The swollen head problem

Suppose, as Christian materialists believe, that materialism is true and yet some people have eternal life in heaven. Good experiences happen daily in heaven, and bad things never do. It is a bad thing to fail to remember a good experience. So in heaven people will have more and more good experiences that they remember. But it is plausible that there is a maximum information density in our brains, and given materialism, all the information in memory is stored in the brain. Thus, it follows that those who will be in heaven will have their heads swell without bound. Humans will eventually have heads that are millions of light-years in diameter, just to hold all the good experiences that have happened to them. But a life with such big heads just doesn't seem to be the life of human fulfillment.

Objection 1: Perhaps there are patterns to the good experiences in heaven such that the total information content in the infinite future of good experiences is finite.

Response: If the total information content is finite, then it seems likely that one will eventually get bored. Moreover, plausibly, human flourishing involves continual growth in knowledge, and it would not be fitting for heaven if this growth were to slow down eventually in order to ensure an upper bound on the total information content.

Objection 2: The laws of nature will be different in heaven, and while there is maximum information density in our current brains, heavenly brains will be made of a different kind of matter, a matter that either has infinitely many particles in any finite volume or that is infinitely subdivisible. After all, the Christian tradition does hold that we will function differently--there is speculation that we may be able to go through solid walls as Jesus apparently did after the resurrection, move really fast, see really far, etc.

Response: This seems to me to be the best materialist response. But given that on materialism the brain is central to the kinds of beings we are, there is a worry that such a radical reworking of its structure into a different kind of matter would create beings that aren't human. The dualist can allow for a more radical change in the physical aspects of the body while allowing that we still have the same kind of being, since the kind of being could be defined by the soul (this is clearest in the hylomorphic theory).

Objection 3: The dualists face the same problem given that we have good reason to think that memories are stored in the brain.

Response: Maybe memories are not entirely stored in the brain. And see the response to Objection 2: the finer-matter response is more defensible in the case of the dualist.