The branching many worlds interpretation of quantum mechanics famously faces the problem of why it is appropriate to interpret the weights in the wavefunction as probabilities. The many minds interpretation is designed to solve this problem. Infinitely many minds traverse the branching multiverse, and move in accordance with objective chances defined by the wavefunction. It sounds like everything is just fine probabilistically: each mind's movements are nicely stochastic.
Yes, the movements are, but what about the initial setup of the minds? Presumably minds come on the scene when in some branch of the multiverse matter is so arranged as to form something like a brain. And infinitely minds come into existence then. But much earlier than evolution has managed to produce brains on earth, the branching multiverse will have branches with Boltzmann brains: brains that come into existence randomly out of quantum fluctuations in little bubbles of order. Some of these Boltzmann brains will have brain states exactly like ours. And each Boltzmann brain, on the many minds interpretation, will get infinitely many minds. So am I associated with a real brain or a Boltzmann brain? There are infinitely many minds with states like mind associated with a real brain and infinitely many minds with states like mine associated with a Boltzmann brain. The infinities are, presumably, the same. So how is it that I know that I have two hands?
Here's one move that a defender of many minds could make. Assume a pre-existing infinite bucket of minds, existing unconsciously apart from the physical universe. Consider all the brain-coming-into-existence events throughout the multiverse and across time. Assign probabilities to these brain-coming-into-existence events in proportion to the weight assigned the event by the wavefunction. Now add this dynamics: Whenever a brain-coming-into-existence event happens, each mind in the infinite bucket has the indicated probability of getting pulled out of the bucket and connected with this brain. Then as long as we're confident that, roughly, the number of Boltzmann brains weighted by the probabilities is less than the number of normal brains weighted by the probabilities, it seems all should be well.
There are potential technical problems with the normalization of the probabilities. Also, there is the issue that the metaphysics now seems excessively dualistic, in that we are supposing that bucket of minds independent of the physics from which the minds are pulled. Maybe one could just suppose a bucket of mind-haecceities? I am not sure.
This post is inspired by remarks Rob Koons made about retrospective probabilities in the many worlds interpretation. Essentially the point of this post is that that problem isn't solved by the many minds interpretation.