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I’m so glad you enjoyed that article and the re-read of the Tegmark article!!! I see we are once again in agreement:

Mathematics "lives" in a timeless realm (hello Level IV and #5D!), yet is also "inside" space and time -- because it is the language of intelligent creatures.

I would add that our familiar 4D universe is itself a manifestation of mathematical structures. (dimensionality)

It appears that Hank Kerchief has determined that irrational numbers such as Pythagoras's Constant refute mathematical Platonism in his worldview.

I however find irrational numbers, Gödel's Incompleteness Theorem and Turing machines that do not halt --- all to be part and parcel of the mathematical structure of ”all that there is.” After all, why we think the mathematical structures must be integers, real numbers, true/false decisions, finite, etc.?

Penrose points to the Mandelbrot set as an example of Platonism – and it continues. I agree, but I believe geometry is even more obvious evidence for mathematical Platonism. For example,

Swarzschild Geometry

Riemannian Geometry

Lorentz Transformation

The Equivalence Principle as Symmetry

Brane New World

For Lurkers: mathematical Platonism says that the structure (such as pi) already exists and the mathematician comes along and discovers it.

How does this type of Platonism handle things like Beethoven's musical output? Is it claimed that the symphonies "existed" somewhere and Beethoven only discovered them (a rather Zen-like posture.)

I would suggest the mathematics is invented more along the lines of an artistic creation than it is discovered. Mathematics is generally invented to describe something in the "real world" so different people do get similar results.

As an example, I invented (for what seemed a good purpose at the time) the following sequences of numbers:
Pick a prime, (2,3,5,7...,etc); then first write the integers using that prime as a base; (1,2,3,4,5.... become 1,10,11,100,101 in base 2 or 1,2,10,11,12 in base 3 respectively); then "reflect" the number about the "decimal" point; (1,10,11,100 become .1, .01, .11 ,.001 or 1/2, 1/4, 3/4, 1/8, etc.); next take the resulting fraction and replace the numerator by the number such that numerator*replacement is congruent to 1 modulo the base; use the resulting fractions as the sequence: 1/2 => 1/2, 1/4 => 1/4, 3/16 => 11/16, etc.) I'm not sure what it would mean to say that this sequence "existed" prior to my building it. (Unfortunately, the sequence didn't have the properties I wanted.)