One obvious problem is posed by what it would mean for the world to be “made of” mathematical structures. The notion of mathematical structure is well developed, so that’s not the issue. A structure, strictly speaking, is a property or a group of mathematical objects that attach themselves to a given set. For instance, the set of real numbers has a number of structures, including an order (with any given number being either less or more than another number), a metric (measuring the distance between points in the set), an algebraic structure (the operations of addition and multiplication), and so on.
In Many Worlds in One: The Search for Other Universes, critic Alex Vilenkin says that “the number of mathematical structures [in the multiverse] increases with increasing complexity, suggesting that ‘typical’ structures should be horrendously large and cumbersome. This seems to be in conflict with the beauty and simplicity of the theories describing our world.” In order to get around that problem, Tegmark assigns lower weights to more complex structures, but since this is done without a priori justification, it is an ad hoc move, which of course violates Occam’s razor. So, as much as I enjoyed our conversation with Max, for the time being I remain skeptical of the MUH and related hypotheses. Maybe we just need to wait for the appearance of an infinitely intelligent mathematician.
[This just in from Max Tegmark himself!]
Thanks Massimo for the fun conversation during the interview and for raising these important questions! They are excellent ones, and a key reason why I spent three years writing this book is because I wanted to make sure to finally answer them all properly. Needless to say, I couldn't do justice to them in our short interview, so I'm very much look forward to hear what you think about my detailed answers in chapters 6, 10, 11 and 12. I think you'll find that our viewpoints are closer than your post suggest - for example, your statement "Tegmark assigns lower weights to more complex structures" is not something you'll find in the book. Rather, I describe how the measure problem is a terrible embarrassment for modern cosmology (regardless of whether the MUH is true or not) that we need to solve, and that our untested assumption that truly infinite things exist in nature are my prime suspect: we've never measured anything to better than 17 decimal places, have only 10^89 particles in our universe, and manage to do all our publishable physics simulations with computers that have finite resources, so even though my physics courses at MIT use infinity as a convenient tool, I respectfully object to your "OPS" argument that we somehow have experimental evidence for infinity in physics. Without infinity, there are, as you say, no Gödel issues in our physics.
I look forward to continuing this interesting conversation! ;-)