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"Whenever a part of reality behaves in a way that conforms to the number-axioms - for example, if putting apples into a bowl obeys rules, like no apple spontaneously appearing or vanishing, which yields the high-level behavior of numbers - then all the mathematical theorems we proved valid in the universe of numbers can be imported back into reality. The conclusion isn't absolutely certain, because it's not absolutely certain that nobody will sneak in and steal an apple and change the physical bowl's behavior so that it doesn't match the axioms any more. But so long as the premises are true, the conclusions are true; the conclusion can't fail unless a premise also failed. You get four apples in reality, because those apples behaving numerically isn't something you assume, it's something that's physically true. When two clouds collide and form a bigger cloud, on the other hand, they aren't behaving like integers, whether you assume they are or not."

This is exactly what I argued and grounded back in this article.

Specifically, that the two premises:

1) rocks behave isomorphically to numbers, and
2) under the axioms of numbers, 2+2 = 4

jointly imply that adding two rocks to two rocks gets four rocks. (See the cute diagram.)

And yet the response on that article (which had an array of other implications and reconciliations) was pretty negative. What gives?

Furthermore, in discussions about this in person, Eliezer_Yudkowsky has (IIRC and I'm pretty sure I do) invoked the "hey, adding two apples to two apples gets four apples" argument to justify the truth of 2+2=4, in direct contradiction of the above point. What gives on that?