In string theory, a Level 4 parallel universe is the strangest place (and most controversial prediction) of all, because it would follow fundamentally different mathematical laws of nature than our universe. In short, any universe that physicists can get to work out on paper would exist, based on the mathematical democracy principle: Any universe that is mathematically possible has equal possibility of actually existing.
Scientists use mathematics as their primary tool to express the theories of how nature behaves. In a sense, the mathematics that represents the theory is the meat of the theory, the thing that really gives it substance.
In 1960, physicist Eugene Wigner published an article with the provocative title “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in which he pointed out that it’s kind of unreasonable that mathematics — a construct purely of the mind — would be so good at describing physical laws.
He went further than this, supposing that this effectiveness represented a deep level of connection between mathematics and physics, and that by exploring mathematics you can figure out ways to approach sciences in new and innovative ways.
But the equations that work so well to describe our universe are only one set of equations. Certainly a universe could be created, as physicists have done on paper, with only two dimensions and containing no matter, which is nothing but expanding space. There could be a vast anti-de Sitter space, contracting, right next to it.
Why, then, do we observe the specific set of equations, specific set of laws, that we do? In other words, to use the phrase of British cosmologist Stephen Hawking (from his 1988 A Brief History of Time), what is the force that “breathes fire into the equations” that govern our universe?
Theoretical physics explores cutting-edge concepts — the bosonic string theory, the various superstring theories, AdS/CFT correspondence, Randall-Sundrum models — but that clearly don’t match our own universe. Most physicists leave it at that, with the understanding that some “pure math” just doesn’t apply directly to the physical universe we live in. However, according to the principle of mathematical democracy, these universes do exist somewhere.
In this controversial view, our equations aren’t preferred, but in the multiverse, every equation that can have life breathed into it will. This makes up the Level 4 multiverse, a place so vast and strange that even the most brilliant among us can only conceptualize it with the tools of mathematics.