[EMAIL PROTECTED] to hpm > [re. the existence of non-computable real-valued observers] > That's a bit like saying there is some truth to 1+1=3 just because we > can argue about it
[EMAIL PROTECTED] to GLevy > [Re. Dubito ergo cogito] > Many things are doubtful. 2+2=4 isn't. There you go again. But being sure isn't the same as being right. Despite the intuitively compelling nature of arithmetic as we know it, it is really quite arbitrary. It is compelling only because we evolved in a world that provided some survival advantage to brains that interpreted sense experience that way, by way of major approximations and conflations. But its formalizations, like the Peano axioms and the inference mechanism that produces theorems like 1+1=2 really are just arbitrary system of rewriting rules. Its perfectly easy to construct equally pretty systems where 1+1 = 3 or 1+1 = 1, starting with different initial strings or using different rewrite rules. And you can build universes in such systems, where the arithmetic you find so correct never rears it misshapen head. What's more, there are situations in our own neighborhood where alternate arithmetics are more natural than everyday arithmetic. For instance, in a lasing medium, if you introduce one photon in a particular quantum state, and then add another photon in the same state, it is likely that you will find three photons in that state (then more and more - Boson statistics: the probability of a new recruit to a state occupied by n particles is proportional to n/(n+1)). Photons in the same state are in principle indistinguishable from one another, so occupancy of a quantum state is a purer model of counting than the everyday one: when you count pebbles, thay remain quite distinguishable from one another, and it takes an arbitrary high-handed act of abstraction to say that THIS pebble, with its unique shape, color and scratch pattern is somehow the same as this other, completely different pebble. The quantum world in general, with its superpositions, entanglements and ephemeral virtual particles is probably poorly served by bimodal Aristotelian logic, never mind mathematical frameworks idealized from grouping pebbles. But because you are so exclusively wedded to these parochial ways of thinking, you feel you can just reject out of hand the existence (among many other things) of beings able to store, compute and communicate reals, even though many of their properties can be puzzled out. PAH!
