Although traditionally taken for granted by many theoretical physicists, the notion that the physical world (specifically, the Level III multiverse) is a mathematical structure is deep and far-reaching. It means that mathematical equations describe not merely some limited aspects of the physical world, but all aspects of it, leaving no freedom for, say, miracles or free will in the traditional sense. Thus there is some mathematical structure that is isomorphic (and hence equivalent) to our physical world, with each physical entity having a unique counterpart in the mathematical structure and vice versa.

Let us consider some examples. A century ago, when classical physics still reigned supreme, many scientists believed that physical space was isomor-phic to the three-dimensional Euclidean space R3. Moreover, some thought that all forms of matter in our universe corresponded to various classical fields: the electric field, the magnetic field and perhaps a few undiscovered ones, mathematically corresponding to functions on R3. In this view (later proven incorrect), dense clumps of matter such as atoms were simply regions in space where some fields were strong. These fields evolved deterministically according to some partial differential equations, and observers perceived this as things moving around and events taking place. However, fields in 3-dimensional space cannot be the mathematical structure corresponding to our universe, because a mathematical structure is an abstract, immutable entity existing outside of space and time. Our familiar perspective of a 3-dimensional space, where events unfold, is equivalent to a 4-dimensional spacetime, so the mathematical structure must be fields in 4-dimensional space. In other words, if history were a movie, the mathematical structure would not correspond to a single frame of it, but to the entire videotape.

Given a mathematical structure, we will say that it has physical existence if any self-aware substructure (SAS) within it subjectively perceives itself as living in a physically real world. In the above classical physics example, an SAS would be a tube through spacetime, a thick version of its world-line. Within the tube, the fields would exhibit certain complex behaviour, corresponding to storing and processing information about the field-values in the surroundings, and at each position along the tube these processes would give rise to the familiar but mysterious sensation of self-awareness. The SAS would perceive this 1-dimensional string of perceptions along the tube as passage of time.

Although this example illustrates how our physical world can be a mathematical structure, this particular structure (fields in four-dimensional space)

is now known to be the wrong one. After realizing that spacetime could be curved, Einstein searched for a unified field theory where the universe was a four-dimensional pseudo-Riemannian manifold with tensor fields. However, this failed to account for the observed behaviour of atoms. According to quantum field theory, the modern synthesis of special relativity theory and quantum theory, our universe (in this case the Level III multiverse) is a mathematical structure with an algebra of operator-valued fields. Here the question of what constitutes an SAS is more subtle [30]. However, this fails to describe black hole evaporation, the first instance of the big bang and other quantum gravity phenomena. So the true mathematical structure isomorphic to our universe, if it exists, has not yet been found.

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