Re: Numeric comparison with nil - Math masochists only!!

Everett, even in the act of adding two numbers, the computer is following
our instructions & algorithms.  The computer doesn't know the difference
between 'add' or 'subtract' or 'exponentiate' or 'do this cool MMX function
thing'.  The entire process is an abstraction, so while I agree that it IS
impossible to *literally* do addition or multiplication with infinite
strings of numbers (unless you have some cool sci-fi infinite-computer
thing) you can *figuratively* do multiplication on infinite stings of
numbers.  One example as when you use ruby's rational class for numbers like
1/3 or 17/9.  Neither of those numbers can be *completely* represented in
the computer as a floating point number, as they go on indefinitely.  So
instead they are represented as a fraction and the math on an infinite
string of digits is done by changing the way the computer sees & acts with
the number.

Now, one thing I do know, is that figuring out how to do the whole 'surreal
multiplication with infinite numbers' thing is going to be a pain in my
arse!!  I've got some ideas but I'm not sure my programming-fu is up to the
task just yet. But hey, gotta set goals high right?

  hex


On Thu, Dec 30, 2010 at 7:19 AM, Everett L Williams II
<[e-mail]>wrote:

> Xavier Noria wrote:
>
>> On Wed, Dec 29, 2010 at 8:06 PM, Everett L Williams II
>> <[e-mail]>  wrote:
>>
>>
>>
>>> You are confusing computer logic and meta-data manipulation. No computer
>>> can
>>> natively deal with the representation of, much less the calculation of
>>> anything that involves infinity, either negative or positive. You
>>> certainly
>>> can define a set of rules and attempt to create a program that models
>>> those
>>> rules, but you cannot naatively do any such calculation. Computers are,
>>> by
>>> definition, finitie and deterministic, and there is not room here to
>>> explain
>>> exactly what that means, but there is plenty of information spread all
>>> over
>>> the internet on the subject. Let me take a small stab at an example.
>>> Given
>>> to finitie numbers whose sum is within the capacity of the instructions
>>> of a
>>> computer, I can add those two numbers and get a third number. Anything
>>> beyond that is modeled and entirely dependent on my logic rather than the
>>> logic of the computer. So, you can declare that infinity plus 6 has
>>> meaning
>>> and you can declare what that meaning is, providing a routine that will
>>> decode your expression of infinity and then follow your instructions for
>>> creating whatever you have defined as infinity plus 6, but there is no
>>> native instruction, even in floating point, that can impinge on the
>>> correctness or the calculation of the answer. It is entirely dependent on
>>> the meta-logic and meta-data that you have provided. Even extended
>>> precision
>>> math libraries can break a large number down into segments and then use
>>> the
>>> native facilities of the computer in a logically and mathematically valid
>>> process that leads to arithmetically correct answers, but infinity cannot
>>> be
>>> represented in any nat8ive form within a computer.
>>>
>>> If you look up infinity on the wiki, you will find pages upon pages of
>>> various means of manipulating infinities, and yours may be the latest.
>>> When
>>> I have the time and energy, I will look, but it is hard to get excited
>>> about
>>> the umpty-unth attempt.
>>>
>>>
>> My reply addressed a couple of points of your post:
>>
>> 1. If we are programming symbolic mathematics, we are doing
>> mathematics. The convenience or lack thereof of such and such concept
>> for scientists doesn't matter in discussing whether something can be
>> given a well-defined formal meaning.
>>
>> 2. Computations in computers: from a formal point of view I disagree,
>> but do not want to enter into that. If by metadata you mean eg
>> programs versus CPU registers, and if you agree that we can represent
>> something infinite like the set of quadratic polynomials in on
>> variable in Z, then we agree in this point. Not its members, but the
>> set and its rules, akin to how we represent Z in C.
>>
>>
>>
>>
> I'd go a bit past registers to the total logic of the computer. No
> instruction in any computer can deal with infinity in any form, either
> logically or physically. Of course, programs for symbolic manipulation can
> and have been written, but there is no enforcement or checking related to
> the logic or hardware of the computer. Unless something is physically wrong,
> computers, adding binary 1 to binary 1 will get binary 10 every time. Your
> program, consisting of meta-data and meta-logic is a construct entirely
> dependent on your definition of all parts of the program. By the way, if you
> are programming symbolic mathematics, the computer is merely following your
> algorithm. All elements and properties of that algorithm are external to the
> computer.
>
>
> Everett L.(Rett) Williams II
>
>