From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on polar.synack.me X-Spam-Level: X-Spam-Status: No, score=-1.3 required=5.0 tests=BAYES_00,INVALID_MSGID autolearn=no autolearn_force=no version=3.4.4 X-Google-Language: ENGLISH,ASCII-7-bit X-Google-Thread: 109fba,baaf5f793d03d420 X-Google-Attributes: gid109fba,public X-Google-Thread: 1014db,6154de2e240de72a X-Google-Attributes: gid1014db,public X-Google-Thread: 103376,97188312486d4578 X-Google-Attributes: gid103376,public X-Google-Thread: fc89c,97188312486d4578 X-Google-Attributes: gidfc89c,public From: jsa@alexandria (Jon S Anthony) Subject: Re: What's the best language to start with? [was: Re: Should I learn C or Pascal?] Date: 1996/09/18 Message-ID: #1/1 X-Deja-AN: 181292931 sender: news@organon.com (news) references: <01bb8df1$2e19d420$87ee6fce@timpent.airshields.com> organization: Organon Motives, Inc. newsgroups: comp.lang.c,comp.lang.c++,comp.unix.programmer,comp.lang.ada Date: 1996-09-18T00:00:00+00:00 List-Id: In article <51knhg$j61@dub-news-svc-8.compuserve.com> grs@liyorkrd.li.co.uk (George) writes: > jsa@alexandria (Jon S Anthony) wrote: > > > In article <01bb9f26$36c870e0$87ee6fce@timpent.a-sis.com> "Tim Behrendsen" writes: > > > > It is similiar to the difference between summation and integration; > > > one consists of individual sums, the other of an infinite number > > > of sums. However, both are fundamentally adding. > > > Well, that is one option. But as "everyone" knows, the FTC allows you > > to compute definite integrals without taking the limits of sums or > > using summations at all. Incidentally, none of the standard > > definitions (Riemann Sum or something) uses "an infinite number of > > sums". Can't - infinity is not part of the real numbers... > > > Surely the definition of integration contains the phrase "...tends to > infinity", i.e. it's _as if_ there was an infinite number of sums. Nothing so sloppy as that. You can't have "real definitions" that hand wave stuff like "as if", "tends to infinity", etc. Also, "integration" is an activity - the definitions in question involve the notions of "definite integral" and "indefinite integral" of a function (over some interval). Integration then involves various techniques for finding such things. In _standard_ analysis the definitions for definite integral (the sort of "summing stuff" first mentioned) all involve the _formal_ notion of a limit (you remember that - all that epsilon and delta stuff with universal and existential quantification). Infinity (as in a completed concept, i.e., transfinite number) does not enter into it. If you are really interested, I can give you the details in email - or you can just pull out that first year calculus book... /Jon -- Jon Anthony Organon Motives, Inc. 1 Williston Road, Suite 4 Belmont, MA 02178 617.484.3383 jsa@organon.com