Thats not what mathematical realism is supposed that mean. If non contradict is the criterion of mathematical truth, MR suggests that every non contradictory entity exists somewhere , but doesn't require it to exist everywhere ... or need your input. So if it's non contradictory to eat an enchilada, one of your counterparts already is ...but there was is no reason for you to be, since not eating an enchilada is also non contradictory.
But that's still not how MR works: existence is the criterion of truth, not truth of existence. If it's not in Plato's heaven, it's not true even if it's non contradictory.
Are the beings immortal? How do they reproduce, if each of them lives in a separate reality? Are they born in an empty universe? How do they learn anything?
How likely are they to accidentally kill themselves by summoning e.g. a huge ball of fire? Especially if they had no previous experience with fire. Or a lake of acid. What about radioactive materials, black holes, etc. It is possible to imagine an interesting thing that would kill you in reality. Though, do they have bodies? Are those bodies fragile? Do they have a metabolism? Can they be killed? Hurt?
Living in a world where anything can be created by thinking also means living in a world where the only things that exist are those that you can imagine.
If you conjure a thing, does it remain changeless, or can it change? Can things break? Will ice melt? Changes of things and interactions among things can be unpredictable.
If things remain perfect and changeless, you could conjure a Turing machine and let it run various computations.
Initially the beings are pure minds existing in a empty universe, so there's no risk of dying or killing yourself, but plenty of driving yourself mad. If they want a body, they have to imagine it into existence like anything else. They reproduce by imagining other beings into existence. I'm not really sure where the first one came from or how it learned anything, but at this point they have a thriving society and a culture for training new minds how to exist in harmony with the others. One of the chief concerns of the beings is maintaining the norms of this culture with the worst possible punishment being ostracism for people who don't play by the rules.
Objects imagined into existence follow the laws of physics you imagine along with them, so you could have ice that melts or a perpetual motion machine if you want that instead.
It's also possible to create "planes" with more restrictive rules (sort of like spinning up a VM in a computer).
I am writing a novel in which the protagonist meets a visitor from a "dimension" where mathematical realism is literally true. Namely, anything that the visitor can logically conceive of he can also "will" into existence.
Naturally at first the protagonist is pretty jealous of the visitor. After all, who wouldn't want to be able to summon a steaming hot burrito whenever they feel hungry. But now I need the visitor to convince the protagonist that his life isn't so bad after all.
So I am collecting a list of reasons why mathematical realism might be bad. For the sake of this article, assume that there is a society of "freely interacting" beings in the mathematical realism world, so you can only "wish" things into existence in my world if I allow you to. Or we could create a shared world where things can only be created according to a shared set of mutually agreed upon rules.
Here are some of the "bad" things I've come up with so far, but I'd like to collect more.
One obvious downside of being able to wish anything into existence is that you might (intentionally or not) wish something really awful into existence.
A favorite topic of rationalists. If you can wish for anything, there's a real chance of repeatedly wishing for things that gratify your short-term reward system. This results in either an endless loop of obsessively doing the same behaviors over and over again, or "burning out" and being unable to take pleasure from those types of rewards.
The real world is fascinating and filled with a seemingly endless set of things to explore or discover. Living in a world where anything can be created by thinking also means living in a world where the only things that exist are those that you can imagine. A sufficiently clever person might get around this by (for example) simulating the big-bang and recreating the entire universe. But someone who isn't sufficiently clever might just find themselves surrounded by the small set of things they can easily imagine, unable to ever experience surprise.
Similar to playing a game with all of the cheat-codes, living in a world where you can solve any problem simply by wishing might feel dull and unrewarding. Again, a sufficiently clever person can simply create restrictions and challenges for themselves, but the temptation to "cheat" will always be there.
Even if you are clever and virtuous enough to create a decent life for yourself, there's a chance that everyone that you know might be wireheading or too scared by the eldritch abominations they have created to be your friends.
If you're clever, you can simply "make" friends, but these relationships may feel shallow or unreal for the same reason that "cheating" takes away the sense of satisfaction of overcoming a challenge. If someone is literally created to be your friend, how can you ever know if the "really" like you or if you're just forcing them to be friendly.
Even if other beings cannot specifically create things in "your" world, there still may be other ways for them to harass you. Suppose there is some method that all beings use to communicate with one-another, they could span or troll you using this medium.
I hope to edit this post as I think of other ideas, but please post in the comments if you have any ideas I haven't mentioned.