The Mathematical Universe

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and Godel incompleteness. I hypothesize that only computable and decidable (in Godel's sense) structures exist, which alleviates the cosmological measure problem and help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.Comment: Replaced to match accepted Found. Phys. version, 31 pages, 5 figs; more details at http://space.mit.edu/home/tegmark/toe.htm

The effects of longitudinal chromatic aberration and a shift in the peak of the middle-wavelength sensitive cone fundamental on cone contrast.

Longitudinal chromatic aberration is a well-known imperfection of visual optics, but the consequences in natural conditions, and for the evolution of receptor spectral sensitivities are less well understood. This paper examines how chromatic aberration affects image quality in the middle-wavelength sensitive (M-) cones, viewing broad-band spectra, over a range of spatial frequencies and focal planes. We also model the effects on M-cone contrast of moving the M-cone fundamental relative to the long- and middle-wavelength (L- and M-cone) fundamentals, while the eye is accommodated at different focal planes or at a focal plane that maximizes luminance contrast. When the focal plane shifts towards longer (650 nm) or shorter wavelengths (420 nm) the effects on M-cone contrast are large: longitudinal chromatic aberration causes total loss of M-cone contrast above 10-20 c/d. In comparison, the shift of the M-cone fundamental causes smaller effects on M-cone contrast. At 10 c/d a shift in the peak of the M-cone spectrum from 560 to 460 nm decreases M-cone contrast by 30%, while a 10 nm blue-shift causes only a minor loss of contrast. However, a noticeable loss of contrast may be seen if the eye is focused at focal planes other than that which maximizes luminance contrast. The presence of separate long- and middle-wavelength sensitive cones therefore has a small, but not insignificant cost to the retinal image via longitudinal chromatic aberration. This aberration may therefore be a factor limiting evolution of visual pigments and trichromatic color vision