Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 30 Nov 2006 10:09 On 11/30/2006 6:25 AM, Virgil wrote: > In article <456e475e(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >> Given a >> set density, value range determines count. > > Compare the "set densities" of the set of naturals, the set of > rationals, the set of algebraics, the set of transcendentals, the set of > constructibles, and the set of reals. Either discrete or continuous. Nothing in between.
From: Bob Kolker on 30 Nov 2006 11:13 Eckard Blumschein wrote: > > The argument Cantors transfinite numbers are somthing positive something > progressive is old and has proven wrong. Not even aleph_2 has found an > application. So what? The criterion for goodness in pure mathematics is consistency, not usability. After that aesthetic issues dominate. Are the systems interesting. Do they have a kind of beauty? etc. etc. Bob Kolker
From: Bob Kolker on 30 Nov 2006 11:15 Eckard Blumschein wrote: > > > Either discrete or continuous. Nothing in between. You obviously have no knowledge of fractal dimension or Hausdorf dimension. For example the Peano space filling curves have a dimension between 1 and 2. Bob Kolker
From: Lester Zick on 30 Nov 2006 12:19 On Thu, 30 Nov 2006 11:15:09 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Eckard Blumschein wrote: >> >> >> Either discrete or continuous. Nothing in between. > >You obviously have no knowledge of fractal dimension or Hausdorf >dimension. For example the Peano space filling curves have a dimension >between 1 and 2. Don't forget about fractional dimensions and fractional cardinality. ~v~~
From: Lester Zick on 30 Nov 2006 12:20
On Thu, 30 Nov 2006 10:09:10 +0100, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >On 11/29/2006 7:19 PM, Lester Zick wrote: > > > I mean if you ask "how much gas" and get the answer "two >> gallons" you've certainly measured the gas. Or if you ask "how much >> space" and get the answer "two inches" you've certainly measured the >> space. > >Common sense terminology tends to be nearly as logical as the >mathematical one but more subtle. >The question "how much gas" refers to the insight that the amount of gas >is not directly countable. The amount of gas or even space is as directly countable as anything else. All you have to do is superimpose a metric.The fact that certain things come neatly prepackaged suggests a metric nothing more. > Nowadays, almost any continuous measurement >yields in the end a numerical result because processing and transmission >of digital signals is superior to processing and transmission of >analogous signals. That kind of efficiency has nothing to do with the issue of a metric and its application. If you want to count you need some metric to do it. It's that simple and the metric is what establishes cardinality in mathematics. ~v~~ |