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serialhex wrote:
> 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
> <rett / classicnet.net>wrote:
>
>    
>> Xavier Noria wrote:
>>
>>      
>>> On Wed, Dec 29, 2010 at 8:06 PM, Everett L Williams II
>>> <rett / classicnet.net>   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
>>
>>
>>      
Finite addition, subtraction, multiplication, and division are native 
functions of all but the simplest computers, whether the computer is 
self-aware or not, silly as that is to even discuss. What we are talking 
about here is native instructions as opposed to algorithmic processes. 
Since I have been programming computers for 44 years, I tend to think 
that most people, at least in these parts understand the distinction. As 
they say, if you don't understand it, you can't program it. And, your 
understanding of it will be what is programmed. Most of us who have been 
programming for more than five minutes know that we must limit our use 
of an instruction to the finite limits of the hardware capability. 
Programmers must constantly be aware that all processes must be bounded, 
or programs may loop until stopped, a condition not desirable, at least 
in my experience. There is both a quantitative and a qualitative 
difference between using the basic capacities of hardware and 
instruction sets, and overlaying on those extreme sets of logic not 
amenable to any computation.

Everett L.(Rett) Williams II



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