----- Original Message ----- From: "David Barrett-Lennard" <[EMAIL PROTECTED]>

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>>0xf2f75022aa10b5ef6c69f2f59f34b03e26cb5bdb467eec82780 >> didn't exist in this universe (with a very high probability, it being a >> 512 bit number, generated from physical system noise) before I've >> generated it. Now it exists (currently, as a hex string (not necessarily >> ASCII) on many systems (...) > You admit a base 16 notation for numbers - which means you allow numbers > to be written down that aren't "physically realized" by the > corresponding number of pebbles etc. So much for talking about pebbles > in your previous emails! I think that it doesn't matter what base you choose to write down the number. It is an integer, therefore it is physically realizable *in principle*. If you write '1aa3' in base 16, it means '6893' in base 10, which corresponds to a given number of pebbles. We may think that there is somehow more "reality" in 6893 in comparison to 1aa3, but they are both in the same footing, except that we are more used to the first representation. Why would one claim that the corresponding decimal representation of Eugen's 512-bit number has any more reality that the hexadecimal one? This shows well how we take for granted the connection between a number's representation in digits and the physical representation in pebbles. But to take from any representation to any other, some operation is necessary. What 6893 means is "take 3 pebbles, sum those with 9x10 pebbles, then sum 8x100, then 6x1000 and you will have the number of pebbles represented by 6893" This operation uses implicitly the concepts of sum and multiplication, and of the physical representation of the first 10 digits (or maybe we could argue that even those are actually the representation of successive sums of units). It tooks us years in primary school to master these concepts and operations until we thought they are "natural". An interesting fact is that it is very easy to represent integer numbers that cannot be physically realizable in pebbles or in atoms, not even using all of the atoms in the universe. 10^(56^579), for example. I believe that this representation is as good as the corresponding decimal or hexadecimal one, since any of them requires some operation to be converted in pebbles. But before one argues that this is an argument for arithmetical realism, it is not *necessarily* the case that 10^(56^579) exists independently of *this* representation either. I have no formed opinion on arithmetical realism, even though I tend to accept that there is some external reality to the integers. But is the "reality" that is assigned to numbers of the same kind that is assigned to their physical representation? Are we not discussing just words without any meaning? -Eric.