Magnificent, Alamo-Girl. I haven't seen this article before. Thanks for the link!
I took your advice and revisited the excellent Tegmark article on multiverses this afternoon.
Hank, you want to declare that mathematics is unreal, that pi is unreal, yet at the same time declare perfect isosceles triangles are "ubiquitous." I do find this confusing. But as to the point of whether mathematics is real or not, here's Tegmark's view of the matter:
"A hint that a [Level IV] multiverse might not be just some beer-fueled speculation is the tight correspondence between the worlds of abstract reasoning and of observed reality. Equations, and more generally mathematical structures such as numbers, vectors and geometric objects describe the world with remarkable verisimilitude. In a famous 1959 lecture, physicist Eugene Wigner argued that 'the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious.' Conversely, mathematical structures have an eerily real feel to them. They satisfy a central criterion of objective existence: they are the same no matter who studies them. A theorem is true regardless of whether it is proved by a human, a computer or an intelligent dolphin."
I do agree with Tegmark that "a mathematical structure is an abstract, immutable entity existing outside of space and time."
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.
Why do you do that? When did I suggest mathematics was not real?
Mathematics is a very powerful tool for dealing with those aspects of existence which are countable and measurable, and can even be stretched for use with some things which, in the strictest sense, cannot be counted or measured, such as the ratio of the circumference of a circle to its radius (pi) or the ratio of the length of either leg of an isosceles right triangle to the hypotenuse. I find it strange that those who understand mathematics not only fails to find an absolute value for pi and cannot find a unit of measure that can measure both a leg and hypotenuse of an isosceles right triangle, (and these are just very common limitations of mathematics) believe mathematics is some super-metaphysical power dictating the nature of the world. It's a very useful intellectual tool, but has not other special significance, except to the superstitious who are always in awe of what they do not understand.
My comment about isosceles right triangles being ubiquitous illustrates exactly what I mean. One of the most common geomatic forms in our world cannot be measure by mathematics. I would say mathematics is quite limited, as useful as it is.
Hank
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,
The Equivalence Principle as Symmetry