Re: Re:Is the universe computable?

From: Eric Cavalcanti <eric.domain.name.hidden>
Date: Thu, 15 Jan 2004 11:41:13 -0200

----- Original Message -----
From: "David Barrett-Lennard" <dbl.domain.name.hidden>

>>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.
Received on Thu Jan 15 2004 - 08:45:55 PST