The Definition of Points
in [Math]
|
Prev: Guide to presenting Lemma, Theorems and Definitions
Next: Density of the set of all zeroes of a function with givenproperties
From: stephen on 19 Apr 2007 17:27 Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: >> >> There are contexts in which '1.000...' and '0.999...' are considered >> equal, and others in which they are considered no more than equivalent, >> and in some contexts perhaps not even that. > Then, are you admitting that the cranks that complain that 1 and > 0.999... are not the same may actually have a valid point? If all they are saying is that "1" and "0.999..." are different "strings" then they have a valid point. If they are saying they represent different numbers, then they do not have a valid point. You claim to be a programmer. You should be quite familiar with the fact that "equality" depends on context. What does the following code do? int main() { char *s="cat"; char *t="bobcat"; char *u=strstr(t,s); if ( s==u ) printf("%s == %s\n",s,u); else printf("%s != %s\n",s,u); }
From: David R Tribble on 19 Apr 2007 18:30 MoeBlee wrote: >> I wouldn't put it that way, but I think you're onto the basic idea. >> Yes, bijectability (equinumeroisity) is an equivalance relation on any set. > Tony Orlow wrote: > Bijectibility creates a set of equivalence classes. It doesn't determine > equality, though. That's why words like "equinumerosity" irk me so. Well, there's "equal", and then there's "equal". All of the sets within the same equivalence class are equal in the sense that they have the same number of elements, specifically, that a bijection exists between every pair of those sets. The sets are not equal, though, in the sense of set equality, because all of them contain different elements. So the cardinalities of those sets are equal, but the sets themselves are not. Perhaps you prefer "equicardinality"?
From: David R Tribble on 19 Apr 2007 18:39 Tony Orlow wrote: >> No, but sqrt on the negatives produces imaginary numbers. Besides, sqrt >> can be defined, like + or -, geometrically, through construction. > Stephen wrote: > You cannot define the sqrt(-1) geometrically. You are never going > to draw a line with a length of i. That's not entirely true. It's all a matter of definitions. For example, suppose I define "1" to be the length of a line one cm long drawn in a north/south direction. I can further define, arbitrarily mind you, that "i" is the length of a line one cm drawn in an east/west direction. But we both have to agree on these definitions, of course, if they are going to make any sense.
From: Lester Zick on 19 Apr 2007 19:17 On 19 Apr 2007 12:03:11 -0700, David R Tribble <david(a)tribble.com> wrote: >Lester Zick wrote: >>> Tony, time for you to do a little work for yourself. I've already gone >>> through this. You describe for me the mechanics of using binary truth >>> values and I explain to you I'm interested in truth not binary truth >>> values and how to ascertain truth in mechanical terms initially and >>> not how to work with truth values mechanically once ascertained. >> > >Tony Orlow wrote: >> So, you are of the opinion that science can be performed without >> collecting any data, doing experiments or studies, and ascertaining >> truth from fact? Then you are in as much a religious limbo as the >> Cantorian ball replicators. How far has this Ivory Tower approach gotten >> you so far? If you get down the mechanics of deduction, then you can >> consider more readily what underlying principles may be causing whatever >> phenomena you're investigating. > >I've got to admit that this is one of the few guilty pleasures I >get from reading sci.math: seeing two cranks arguing with >each other over the meaning of "truth". I feel the same when I read you, David, trying to define "infinity". ~v~~
From: Lester Zick on 19 Apr 2007 19:20
On 19 Apr 2007 15:39:49 -0700, David R Tribble <david(a)tribble.com> wrote: >Tony Orlow wrote: >>> No, but sqrt on the negatives produces imaginary numbers. Besides, sqrt >>> can be defined, like + or -, geometrically, through construction. >> > >Stephen wrote: >> You cannot define the sqrt(-1) geometrically. You are never going >> to draw a line with a length of i. > >That's not entirely true. It's all a matter of definitions. For >example, suppose I define "1" to be the length of a line one cm long >drawn in a north/south direction. I can further define, arbitrarily >mind you, that "i" is the length of a line one cm drawn in an >east/west direction. But we both have to agree on these definitions, >of course, if they are going to make any sense. Well I daresay we can find crackpots most anywhere who'll agree with your definitions, David. By the way why exactly do you define "1" to be a line of length "1"? I mean surely there must be other numbers you could use? Don't be shy. Why not define "1" as the square root of 00? ~v~~ |