The question Are there infinitely many perfect numbers? is a classic old unsolved problem. However, we keep finding perfect numbers (via Mersenne primes) and produce a lot of knowledge on perfect numbers, e.g. distribution and more. And when I am reading about this topic perfect numbers are often treated (maybe it is just my perception) as if they were infinitely many. And to be honest: After learning about perfect numbers etc. my intuition now tells me that there have to be infinitely many of them, no?.
I just realized this personal bias and want ask those who are experts in this field: Isn't it more likely that there are infinitely many perfect number? Are there some mathematical statements that make the existence of infinitely many perfect numbers more likely? And: Does it even make sense to ask this question?
Remark 1: In the book The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike the authors point out various aspects from different mathematical fields that all point to the validity of the Riemann hypothesis. They concluded that it is quite unlikely (whatever this means) for the RH to be false (they emphasized: not because of a lot of empirical data, but because of some other mathematical connections that make the validity more likely). This motivated my questions: So are there some mathematical facts that make the existence of infinitely many perfect numbers more likely?
Remark 2: Sorry for such a provocative title :)