Jim Piccarello on Mon, 16 Mar 2009 12:30:51 -0400 (EDT)

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Re: <nettime> Digital Humanities Manifesto

On Mar 14, 2009, at 10:33 PM, Evan Buswell 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  
since the only operation you have is the identify operation. I  
thought we were talking about a binary device that could change  
state. So, say, a light switch where define "0" as the light being  
off for 1 second and "1" as the light being on for one second. Then  
we need some convention for specifying when the transmitter begins  
sending numbers (The light is off for 60 seconds. Have 60 '0' been  
sent or is nothing being sent? Or has the number been sent and I  
missed receiving it.) We also need some convention to identify a  
single number (8 bits, 16 bits, or every bit transmitted) But all  
this gives is the ability to send numerals. Do we also include the  
ability to indicate operations to be performed on the numerals sent?

> Also: dichotomous (digital) states are not isomorphic with the natural
> numbers, they are isomorphic with binary numbers, i.e. the set [0, 1],
> not the set [0, 1, 2 ...]. To get the latter, you need to construct a
> system of mapping an arbitrary number to a *set* of digital states,
> of which many such systems exist and compete---see, e.g., endianness.
> To actually be isomorphic with the natural numbers, you would need
> an infinitely large set of states, effectively canceling the digital
> nature of the supposed device, as each state would be infinitely close
> to (in practice, indistinguishable from) another state.

> But then, when we actually deal with the natural numbers, as a whole, we
> deal more with natural numberness than with each discrete number.

  I don't understand what you mean by "numberness."

> This is something a digital system is perfectly capable of representing.
> I guess it's less that (countable) numbers are isomorphic to digital
> states than (countable) numberness is isomorphic with digitality.

I'm not sure what you mean by digitality if we decide we cannot  
represent all of the natural numbers.

> But this is getting into pretty ill-defined territory.

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.

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