Yes, but I don't think this uses the word 'exist' in the same way.
Granting this distinction for the moment, I still think that this points to how a platonist would answer your original question:
[T]o assert the existence of the number 2 seems to me rather vague. What would the world look like if it weren't so?
The platonist might reply: If the number 2 didn't exist in the platonic sense, then one implication would be that you would be unable to construct proofs of that number's existence within the formal systems that you're thinking of.
In other words, the platonist might argue that formal systems such as PA are accurate (if incomplete) maps of the "territory", which is the objectively real platonic realm within which abstract mathematical objects exist. If the statement that the number 2 exists were false of the territory, then, since the map is accurate, the corresponding formal statement would not hold within the map. Hence, an empirical implication of the number 2's not existing would be that you would have the subjective experience of observing the failure of attempts to prove the formal statement that the number 2 exists.
then one implication would be that you would be unable to construct proofs of that number's existence within the formal systems that you're thinking of.
This seems to me rather confused, because it is easy to construct a formal system with which we have precisely that experience. Consider this variant of the Peano Axioms:
I recognize the title could be more informative. At the same time I believe it says what is important.
I believe in a deity, I believe in mathematical entities in the same way.
The community of LessWrong (from whenceforth: LessWrong) is deeply interesting to me, appearing as a semi-organized atheist, reductionist community.
LessWrong seems very interested in promoting rationality, which I applaud. The effort does seem scattered, though, and this is the reason I post.
One has Eliezer's website with some interesting posts. The same of this community. The community links to some posts when you are coming for the first time into it, and you also have a filter for top posts. One has the blog. And recently, the center for modern rationality (in the same page as harrypoter fanfiction about rationality).
The point being there is no defined roadmap to go from AIC (average irrational chump to make an analogy to Game - which also seems to come up around quite a bit) to RA (again, rationality artist).
I write this post as to maybe generate a discussion on how the efforts could be concentrated and a new direction taken.
Should the creation of the Center for Modern Rationality envision this same concentration, this post may and should be disregard.
If it does not, then I leave it to your consideration.
Hang.