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: fc89c,97188312486d4578 X-Google-Attributes: gidfc89c,public X-Google-Thread: 103376,97188312486d4578 X-Google-Attributes: gid103376,public X-Google-Thread: 1014db,6154de2e240de72a X-Google-Attributes: gid1014db,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: 1996/09/26 Message-ID: #1/1 X-Deja-AN: 185508020 sender: news@organon.com (news) references: <01bb8df1$2e19d420$87ee6fce@timpent.airshields.com> followup-to: comp.lang.c,comp.lang.c++,comp.unix.programmer,comp.lang.ada organization: Organon Motives, Inc. newsgroups: comp.lang.c,comp.lang.c++,comp.unix.programmer,comp.lang.ada Date: 1996-09-26T00:00:00+00:00 List-Id: In article <1996Sep24.133312.9745@ocsystems.com> jvl@ocsystems.com (Joel VanLaven) writes: > George (grs@liyorkrd.li.co.uk) wrote: > : Surely the definition of integration contains the phrase "...tends to > : infinity", i.e. it's _as if_ there was an infinite number of sums. > > : G. > Actually, not all functions are integrable. The most complete and > irrefutable definition of the integral of a function to my knowledge is : > > Given a partition P of [a,b] > (P is a finite subset of [a,b] including a and b) > P={x0,x1,x2,...xn} such that a=x0, b=xn, and x(j+1)>xj > call Mj the lub(f([x(j-1),xj]) (least upper bound) > call mj the glb(f([x(j-1),xj]) (greatest lower bound) > > The number Uf(P)=SUM(Mj(xj-x(j-1))) 1<=j<=n > is called the P upper sum for f > > The number Lf(P)=SUM(mj(xj-x(j-1))) 1<=j<=n > is called the P lower sum for f > > The unique number I that satisfies the inequality > Lf(P)<=I<=Uf(P) for all possible P of [a,b] > is called the definite integral of f from a to b. *IF* it exists. If f is not "nice" it won't. When it does, it is also the LUB of the set of all lower sums and the GLB of set of all the upper sums. Actually, I've always been partial to the upper-and-lower sums definition for the definite integral. However, it is _not_ any more "complete and irrefutable" than Riemann Sums. > So, you could talk about an infinite (aleph 2)! number of sums, but it Actually, when people use the side-ways "8" notation, the typical intent is simply that of indicating "arbitrarily large" or (somewhat less so) denumerably infinite (size of naturals - aka Aleph0). Your "Aleph2" is a denotation for the size of sets the size of the power set of R (the reals) where Aleph1 denotes the size of sets equinumerous with R. In general you have this whole "backbone" of transfinite numbers constructed (typically) via the power set operation. > is ridiculus. We are talking about more uncountable that uncountable. > No one integrates this way. It is a mathematical abstraction. This is Well, Archimedes _did_! :-) (Just one of the reasons he is often regarded among the top 3 mathematicians of all time). But you only need countably infinite sums. > the definition that we then prove is equivalent to some simpler method > for certain "nice" functions. Riemann sums only work for uniformly > continuous functions. The RS definition is _equivalent_ to the ULS definition. This is reasonably straightforward to prove, but it is definitely NON-trivial. > mathematics there are REAL infinities but a good mathematician NEVER > actually calculates an infinite anything, as that is impossible! they Depends on what you mean by "calculate". If you mean "construct" (ala' Poincare or Intuitionism) then what you say is accurate. OTOH, letting P be the power set of operation, then P(R) is a perfectly nice entity (IMO) and the cardinality (size) of P(R) is a perfectly nice transfinite number. > While a hand-wavy infinite sums explanation usually > satisfies the non-mathematicians (like say engineers), the truth is at > once more complicated, simple, and beautiful. Hey, you really are a mathematician! :-) /Jon -- Jon Anthony Organon Motives, Inc. 1 Williston Road, Suite 4 Belmont, MA 02178 617.484.3383 jsa@organon.com