Galileo's Paradox
in [Math]
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From: Bob Kolker on 1 Dec 2006 13:32 Eckard Blumschein wrote: > > Any rational number has a numerical address. Just the entity of all > rational numbers is a fiction. The set of rational numbers exists in exactly the same sense that the set of integers exists. Bob Kolker
From: Lester Zick on 1 Dec 2006 14:04 On 30 Nov 2006 23:55:39 -0800, "Tonico" <Tonicopm(a)yahoo.com> wrote: > >Lester Zick ha escrito: >........................................................ >> >> >What sort of evidence? Surely not empirical evidence. Mathematics done >> >> >abstractly has no empirical content whatsoever. >> >> >> >> Except apparently for axioms and definitions. >> >****************************************************************** >> >What axioms of what part of maths have "empirical" evidence in the >> >sense Eckard is tryuing to convey?!? >> >> I don't know what sense Eckard is trying to convey. My response was in >> reply to Bob not Eckard. > >****************************************************************** >Bob was answering Eckard's ranting, and you got in the middle talking >something about axioms...and you don't know what Eckard was trying to >convey? Why should I care what Eckard was trying to convey since I was replying to what Bob was trying to convey to which he appears to have nothing to convey. >****************************************************************** > >> > For him, and for other trolls, >> >Cantor "not having evidence" for his idea (what stupid this sounds!) >> >means that he (cantor) never foiund an aleph_null under his bed, or >> >that so far no one can buy aleph_beith apples out there. >> >> Well empirical evidence would certainly be one criterion for the truth >> of what one claims. > >**************************************************************** >"Truth"?? Who gets to define what "truth" is, except religious >fundamentalists and other cuckoos of the kind? Certainly not modern mathematikers. > And what do we care of >that in maths? Indeed. > The basic questioning for a mathematical set of axioms >is, imo, whether it is a consistent such set...and whether it is >interesting to deal with, of course. If you mean this by "truth" then >fine. >**************************************************************** No it is not what I mean. >Otherwise one is forced to rely on analytical >> criteria for the the truth of infinities which no modern mathematikers >> appear willing to assert and demonstrate. > >**************************************************************** >Again: what does "the truth of infinities" in the mathematical world >mean? Produce them either empirically or prove them analytically instead of just assuming what you're talking about. In other words show the truth of what you're talking about or shut up. Shape up or ship out. > And again: I don't think any mathematician is interested at all >in assert and demonstrate whatever about "truths" (whatever the meaning >of that is for you), UNLESS you're referring to consistency of axioms, >relevance, interest....and all this has been widely done the last 130 >years or so. >*************************************************************** So you speak for all mathematics do you? >> >What "empirical evidence" are there in group theory's axioms? Or in >> >Topology? >> >> The axioms and definitions themselves are empirical. > >**************************************************************** >Perhaps it is that we don't really understand what each other means by >"empirical". For example, what empirical evidence (of what, where, >when...?) does the axiom stating the existence of a unit element in >group theory have? Or the axiom in Topology that states that the empty >set is part of the set of open sets? I agree we don't understand the term "empirical" the same way. Not my fault since I've discussed the subject at length over the past couple years here and elsewhere. Basically any tautologically undemonstrated judgment is empirical. Doesn't matter whether the judgment is sensory, perceptual, cognitive, or whatever. If you assume an axiom such as "a straight line is the shortest distance between points" the assumption is empirical until and unless demonstrated true analytically. The same applies to definitions. Most people completely misunderstand the meaning of an empirical judgment. Most think it means getting out the tape measure, scales, and so forth. The problem originated with Aristotle and his concept of syllogistic inference. Aristotle was history's first empiricist in formal terms. He found he could not establish the truth of any conclusion syllogistically except by regression to further premises whose truth he could not establish either except by further regression ad infinitum. Which meant he could establish no truth syllogistically at all without some kind of true basic premises which he set out to find in unreducible perceptual terms. Which left us epistemologically exactly where we are today in terms of all kinds of mathematical and scientific methodologies. In point of fact however empirical judgment is nothing more than input to a process of tautological regression whose ultimate goal is reduction to self contradictory alternatives. That's how the mind and brain work, tautological rather than syllogistic inference because it can produce reductions to truth in exhaustive mechanical terms. ~v~~
From: MoeBlee on 1 Dec 2006 14:14 Bob Kolker wrote: > MoeBlee wrote: > > > > > What ordering do you claim to exist on the Cartesian plane? How do you > > prove its existence? What kind of ordering is it (what are its > > properties from those I mentioned)? > > Without resorting to the axiom of choice or well ordering one can use > the lexicographical ordering (alphabetic ordering of two letter words, > so to speak) of two tuples. But this ordering has no arithmetic > significance. Right, of course we can take the lexicographic ordering on ordered pairs of reals, since the reals have an ordering (even a linear ordering). But in this case, what are the ordering properties? In this case, what kind of ordering - quasi, partial, simple, linear, etc. - is it? (But my original question was to know what kind of ordering TONY claims to exist, since in the past he's even claimed to prove the existence of a well ordering of the reals even without assuming an axiom such as choice or well ordering.) MoeBlee
From: Tony Orlow on 1 Dec 2006 14:38 Bob Kolker wrote: > Tony Orlow wrote: > >> Are x and y ordered? The Cartesian plane is ordered in two dimensions, >> not a linear order, but a 2D ordered plane with origin. > > > The plane is not a linearly ordered set of points. > > Bob Kolker That's what I just said.
From: MoeBlee on 1 Dec 2006 14:48
Tony Orlow wrote: > Bob Kolker wrote: > > Tony Orlow wrote: > > > >> Are x and y ordered? The Cartesian plane is ordered in two dimensions, > >> not a linear order, but a 2D ordered plane with origin. > > > > > > The plane is not a linearly ordered set of points. > > > > Bob Kolker > > That's what I just said. So what kind of ordering do you contend it is? MoeBlee |