The Definition of Points
in [Logic]
|
Prev: On Ultrafinitism
Next: Modal logic example
From: RLG on 25 Mar 2007 04:52 "The Ghost In The Machine" <ewill(a)sirius.tg00suus7038.net> wrote in message news:slchd4-8v1.ln1(a)sirius.tg00suus7038.net... > > > Well, that's just it...there's no last digit. However, > were there a last digit one might run into either > > ....999999999 > ....999999999 > > which has no borrow, or > > ....999999990 > ....999999999 > > which will need one. After all, we multiplied by 10... > > Of course that's why limits need to be used anyway; my > strawman logic verges on the ridiculous. :-) Yes, on the standard real number line there are no infinitessimals. Non-standard number lines have infinitessimals, like the surreal number line, but they are not given decimal representations. > But a few paradoxes get messy without them: > > -1 = 1 + 2 + 4 + 8 + ... > since if x = 1 + 2 + 4 + 8 + ... then 2x = x-1 > ? = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ... > could be 0, 1, 1/2. Yes, divergent series can be arranged to converge to any number one likes. R
From: RLG on 25 Mar 2007 04:52 "The Ghost In The Machine" <ewill(a)sirius.tg00suus7038.net> wrote in message news:slchd4-8v1.ln1(a)sirius.tg00suus7038.net... > > > Well, that's just it...there's no last digit. However, > were there a last digit one might run into either > > ....999999999 > ....999999999 > > which has no borrow, or > > ....999999990 > ....999999999 > > which will need one. After all, we multiplied by 10... > > Of course that's why limits need to be used anyway; my > strawman logic verges on the ridiculous. :-) Yes, on the standard real number line there are no infinitessimals. Non-standard number lines have infinitessimals, like the surreal number line, but they are not given decimal representations. > But a few paradoxes get messy without them: > > -1 = 1 + 2 + 4 + 8 + ... > since if x = 1 + 2 + 4 + 8 + ... then 2x = x-1 > ? = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ... > could be 0, 1, 1/2. Yes, divergent series can be arranged to converge to any number one likes. R
From: Don Stockbauer on 25 Mar 2007 09:05 The Definition of Points "Widdle bitty itsy bitsy guys what ain't got no derminshuns in n e direction an what provide mathematicians opportunity to engage in ED (Endless Discussions) cornserning what the heck they is phil-o-soffet- vently. Amen. "An dats de truff." - Edith Ann
From: The Ghost In The Machine on 25 Mar 2007 13:06 In sci.logic, RLG <Junk(a)Goldolfo.com> wrote on Sun, 25 Mar 2007 00:52:06 -0800 <CMOdnaSEuvKrt5vbnZ2dnUVZ_s-rnZ2d(a)comcast.com>: > > "The Ghost In The Machine" <ewill(a)sirius.tg00suus7038.net> wrote in message > news:slchd4-8v1.ln1(a)sirius.tg00suus7038.net... >> >> >> Well, that's just it...there's no last digit. However, >> were there a last digit one might run into either >> >> ....999999999 >> ....999999999 >> >> which has no borrow, or >> >> ....999999990 >> ....999999999 >> >> which will need one. After all, we multiplied by 10... >> >> Of course that's why limits need to be used anyway; my >> strawman logic verges on the ridiculous. :-) > > Yes, on the standard real number line there are no infinitessimals. > Non-standard number lines have infinitessimals, like the surreal number > line, but they are not given decimal representations. I'm not up on surds, myself. Best I can do is dy/dx, which is not a number, but a concept relating to functions. ;-) > >> But a few paradoxes get messy without them: >> >> -1 = 1 + 2 + 4 + 8 + ... >> since if x = 1 + 2 + 4 + 8 + ... then 2x = x-1 >> ? = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ... >> could be 0, 1, 1/2. > > Yes, divergent series can be arranged to converge to any number one likes. > The series 1 - 1/2 + 1/3 - 1/4 + 1/5 + ... is conditionally convergent. Makes life even more interesting. :-) > > R > > -- #191, ewill3(a)earthlink.net Useless C++ Programming Idea #992381111: while(bit&BITMASK) ; -- Posted via a free Usenet account from http://www.teranews.com
From: doslong on 25 Mar 2007 20:45
On Mar 14, 1:52 am, Lester Zick <dontbot...(a)nowhere.net> wrote: > The Definition of Points > ~v~~ > > In the swansong of modern math lines are composed of points. But then > we must ask how points are defined? However I seem to recollect > intersections of lines determine points. But if so then we are left to > consider the rather peculiar proposition that lines are composed of > the intersection of lines. Now I don't claim the foregoing definitions > are circular. Only that the ratio of definitional logic to conclusions > is a transcendental somewhere in the neighborhood of 3.14159 . . . > > ~v~~ There is no point in the real world at all, so we cannot define it exactly. I mean , the concept of point is absolutely illusion of mankind at all. |