Evan Buswell on Thu, 19 Mar 2009 17:59:39 -0400 (EDT)

 Re: Digital Humanities Manifesto

```On Sun, Mar 15, 2009 at 10:34 AM, Jim Piccarello
<jpicc@blackbird-studio.org> wrote:

>>> AND the operations defined in each system mirror each other.
>>
>> Isn't this redundant? Unless of course, the system is defined in
>> such a way that it places limits on what operations are definable,
>> which isn't the case with mathematical numbers, nor (theoretically)
>> digitality. I'm pretty sure that's right, but I'd be interested to
>> hear otherwise.
> First, are we talking about a two-state device that never changes
> state? In that case isomorphism isn't redundant, it's irrelevant

I think I said this a bit too quickly and unclearly, sorry :-). What
I meant is this: isn't it the case, barring some arbitrary limit
on the number or kind of definable operations, that if there is a
one-to-one correspondence between the objects in each system, then *it
must follow that* the operations defined in each system mirror each
other, i.e. there is a one-to-one relationship between operations in
one system and operations in another? Intuitively, I say yes (provided
both systems are Turing complete), but my mathematical background is
not quite strong enough to say that definitively.

> If we decide to limit the size of number to, say, 8 bits then we
> could describe this using modular arithmetic. So 1+1 = 2 but 1+ 255
> = 0. Then we would be modeling the numbers {0.1,2,...255} So we
> would have addition, subtraction, multiplication, and division mod
> 256.

Given the widely varying possibilities of the construction of a
machine, the situation is even *more* complicated, but actually less
problematic IMHO with respect to its relationship with algebra (the
natural number system). See this from HAKMEM:

> Item 154 (Bill Gosper): The myth that any given programming language
> is machine independent is easily exploded by computing the sum of
> powers of 2. If the result loops with period = 1 with sign +, you
> are on a sign-magnitude machine. If the result loops with period =
> 1 at -1, you are on a twos-complement machine. If the result loops
> with period greater than 1, including the beginning, you are on a
> ones-complement machine. If the result loops with period greater
> than 1, not including the beginning, your machine isn't binary -
> the pattern should tell you the base. If you run out of memory, you
> are on a string or bignum system. If arithmetic overflow is a fatal
> error, some fascist pig with a read-only mind is trying to enforce
> machine independence. But the very ability to trap overflow is
> machine dependent. By this strategy, consider the universe, or, more
> precisely, algebra: Let X = the sum of many powers of 2 = ...111111
> (base 2). Now add X to itself: X + X = ...111110. Thus, 2X = X - 1,
> so X = -1. Therefore algebra is run on a machine (the universe) that
> is two's-complement.

Apart from the other things in there relevant to this conversation
(probably undermining the side I seem to have fallen on), I want to
point out that this implies that in algebra addition, subtraction,
etc are actually all addition, subtraction, etc modulo infinity.
Therefore, it is the size of the set, not the nature of the operators
that is at issue.

I guess what I was getting at with the difference between the natural
numbers and a digital system is that the cardinality of an arbitrarily
large but finite set is different than that of a countable set, as
one can create a 1:1 mapping from that finite set to the infinite
set, but not the reverse---though one may question how countable
something actually is given the finite material resources involved in
*any* computation. Now I'm thinking that maybe this was not a very
interesting thing to point out, but anyway...

Cheers,

Evan Buswell

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