The platonist certainly agrees. The test I described would only work for "accurate" maps of the territory. The platonist would consider PA to be an accurate (but incomplete) map of the actual natural numbers, while the formal system you described is not.
Ok, but what is the test which distinguishes between accurate and inaccurate maps? It seems to me that the reasoning here has become circular: It is asserted that the number 2 exists because otherwise, formal system X would be unable to prove it; and also that formal system X is a good test because the number 2 exists. I feel that at this point, you ought to abandon the formal systems and just go for straightforwardly asserting that the number 2 exists in a mystical, intuitive sense which is not open to rational disproof, but which you can use to test formal systems for accuracy.
If you would be willing, on these grounds, to assert with some confidence that the computer will never print any more pairs, why would you demure from asserting, on these same grounds, and with this same confidence, "A largest pair of twin primes exists, and this most-recently printed pair is it."?
Because this does not seem to me to match the meaning of the word 'exists'. If I say that Planet X exists, I can point to it with a telescope and I expect in principle to be able to travel there. If I say that a species of animals exists, I am asserting the possibility of shooting one and eating it. What am I asserting when I say that a number exists? If it's that a particular computer will print so many numbers and then stop, then I think this is not the same class of assertion as in the two previous examples, and it ought to have a different word.
Ok, but what is the test which distinguishes between accurate and inaccurate maps?
Both maps are accurate maps of different parts of the territory.
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.