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AlephNeil comments on Shock Level 5: Big Worlds and Modal Realism - Less Wrong
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The idea is to abolish the distinction between 'mathematical existence' and 'physical existence' or, if you like, between 'possibility' and 'actuality'. Of course all mathematical structures exist as mathematical structures. But it's not obvious (to say the least!) that all mathematical structures exist in the same sense that the physical universe exists.
If the physical universe were a purely mathematical structure - just part of the set of all ideas, implied by some rules of mathematics, but not existing in any way that 2+2=4 does not exist - then how would we, as part of the answer to a math problem, know the difference between that and 'really existing'?
For a start, we'd want to abandon the idea that mathematical structures are merely "ideas". A mathematician can have an idea of a structure, but the same abstract structure can often be conceived of in many different ways, and some structures are too complicated to be conceived of at all (e.g. a non-principal ultrafilter).
A structure (like the set of natural numbers together with its arithmetical operations) is not the same thing as a proposition (like "2+2=4" or "addition of natural numbers is commutative"). Structures satisfy propositions, and it may or may not be possible to systematically investigate the propositions satisfied by a structure by setting out 'axioms' and 'rules of inference' (both of which I suppose you'd call "rules of mathematics").
Better to say "not existing in any way that the numbers themselves don't exist".
The real question here is "how is it possible for a mathematical structure to contain an intelligent observer?" Once you have an intelligent observer they can in principle teach themselves logic and mathematics, which will entail finding out about mathematical structures other than the one they're inhabiting.
In the New Scientist version of Tegmark's mathematical universes paper he writes "every mathematical structure ... has physical existence." But what does "physical" add? When we learn the word "physical" as children we are referring to objects we see, feel, hear, etc., and to the laws of nature that describe them. But clearly a radically different mathematical structure, i.e. different from our laws of nature, is not on the same page, so to speak.
Consider ghosts. Suppose that ghosts exist pretty much as Hollywood depicts them, and also suppose that ghost behaviors and abilities follow (highly complex) mathematical laws, albeit radically different laws from QM and relativity. (Have I just supposed two contradictory things? I'm pretty sure I haven't.) Would ghosts then merit the label "physical"? I think they'd still be paradigms of the nonphysical, and the radical difference of the correct descriptions of ghosts versus particles would be the dead giveaway.
If we remove the (apparently unmerited) label "physical" and just assert that mathematical structures exist, there won't be much disagreement.
I understand that that is the intuitive idea. But how is the hypothesis to be formulated in such a way that we could evaluate its probability, even in principle?
I wrote this some years ago: Sink the Tegmark!. As you can see, I share your skepticism as to whether there's enough sense to be made of Tegmark's theory that we can derive empirical predictions from it.
Even so, you should definitely read Tegmark's original papers - he does address this question somewhat.