Landsburg writes:
To me, by far the most satisfying solution is a full-fledged Platonic acknowledgement that numbers are indeed just “out there” and that they are directly accessible to our intuitions. I embrace this view for (at least) three reasons: A. After a lifetime of thinking about numbers, it feels right to me. B. Pretty much every one else who spends his/her life thinking about numbers has come to the same conclusion. C. It seems enormouosly more plausible to me that numbes are “just out there” than that physical objects are “just out there”, partly because there is in fact a unique system of (standard) natural numbers, whereas the properties of the physical universe appear to be far more contingent and therefore unnecessary.
But the set of all possible minds is so vast that the fact that numbers feel real to us feels to my mind as extremely weak evidence that numbers are real.
I think (it's not my post) that it's supposed to be evidence that numbers are real because the set of all possible minds is vast. Because there are so many possible minds, it's unlikely that a mind chosen randomly from that set would have similar intuitions about numbers to mine. It's even more unlikely that a third mind would also have those intuitions. Yet, for some reason, this vastly unlikely thing happens anyway. This implies that there is some reason which is responsible for all the minds feeling the same way. One such reason would be "the ...
My beliefs about the integers are a little fuzzy. I believe the things that ZFC can prove about the integers, but there seems to be more than that. In particular, I intuitively believe that "my beliefs about the integers are consistent, because the integers exist". That's an uncomfortable situation to be in, because we know that a consistent theory can't assert its own consistency.
Should I conclude that my beliefs about the integers can't be covered by any single formal theory? That's a tempting line of thought, but it reminds me of all these people claiming that the human mind is uncomputable, or that humans will always be smarter than machines. It feels like being on the wrong side of history.
It's also dangerous to believe that "the integers exist" due to my having clear intuitions about them, because humans sometimes make mistakes. Before Russell's paradox, someone could be forgiven for thinking that the objects of naive set theory "exist" because they have clear intuitions about sets, but they would be wrong nonetheless.
Let's explore the other direction instead. What if there was some way to extrapolate my fuzzy beliefs about the integers? In full generality, the outcome of such a process should be a Turing machine that prints sentences about integers which I believe in. Such a machine would encode some effectively generated theory about the integers, which we know cannot assert its own consistency and be consistent at the same time.
So it seems that in the process of extracting my "consistent extrapolated beliefs", something has to give. At some point, my belief in my own consistency has to go, if I want the final result to be consistent.
But if I already know that much about the outcome, it might make sense for me to change my beliefs now, and end up with something like this: "All my beliefs about the integers follow from some specific formal theory that I don't know yet. In particular, I don't believe that my beliefs about the integers are consistent."
I'm not sure if there are gaps in the above reasoning, and I don't know if using probabilistic reflection changes the conclusions any. What do you think?