From: Everett L Williams II Date: 2010-12-30T21:19:23+09:00 Subject: Re: Numeric comparison with nil - Math masochists only!! --------------000203050300020603080006 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Xavier Noria wrote: > On Wed, Dec 29, 2010 at 8:06 PM, Everett L Williams II > 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 --------------000203050300020603080006--