Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 6 Dec 2006 02:47 On 12/5/2006 7:06 PM, Tony Orlow wrote: > Eckard Blumschein wrote: >> On 12/1/2006 9:59 PM, Virgil wrote: >> >>> Depends on one's standard of "size". >>> >>> Two solids of the same surface area can have differing volumes because >>> different qualities of the sets of points that form them are being >>> measured. >> >> Both surface and volume are considered like continua in physics as long >> as the physical atoms do not matter. >> Sets of points (i.e. mathematical atoms) are arbitrarily attributed. >> There is no universal rule for how fine-grained the mesh has to be. >> Therefore one cannot ascribe more or less points to these quantities. >> >> Look at the subject: Galileo's paradox: The relations smaller, equally >> large, and larger are pointless in case of infinite quantities. >> >> > > Now, just a minute, Eckard. You're contradicting yourself, if these > objects are infinite sets of points. They have different measure. I did not consider measures. Let's get concrete. Are there more naturals than odd naturals? This question could easily be answered if there was a measure of size. Infinite means: There is no measure of overall-size. There are merely measures concerning single elements. > Therefore they are different "sized" sets in that respect. Where we can > establish an infinite unit of measure for the sets corresponding to a > finite unit of measure for the objects, then we can easily distinguish > the relative sizes of the sets, even if one cannot establish certain > facts pertaining to finite sets, such as divisibility by some finite > value. We can still say, for instance, if we have cubes A and B with the > edges of A half the length of those in B, that cube A contains 1/8 as > many points as cube B, that A's faces have 1/4 as many points as those > of B, and that the combined points within A's edges is 1/2 the points in > B's. The 8 remainign 0D points, of course, are the same in both figures. Galilei was more intelligent. >>> Sets can have the same cardinality but different 'subsettedness' because >>> different qualities are being measured. >> >> I do not know a German equivalent to subsettedness. Wiki did not know >> subsettedness or at least subsetted either. So I have to gues what you >> possibly meant. Proper subset means included but not all-including. >> While I consider cardinality a rather illusory notion, I distinguish >> between counted (alias finite), countable (alias potentially infinite) >> and uncountable (alias fictitious). Well, the counteds are a subset of >> the countables. So far, the mislead bulk of mathematicians regard the >> countables (i.e. genuine numbers) a subset of the uncountables, too. >> > They're all points on the infinite line, that's all. Perhaps you do not have a clear notion of what all means in case of something without any limit. You are ready to believe that all does not change its meaning if tranferred from finite case to the open-ended one. Likewise you can imagine god like a a very special human. I respect your belief. However, it is just a belief, no science.
From: cbrown on 6 Dec 2006 02:47 David Marcus wrote: > I suppose one solution would be to avoid using any concepts > that aren't already known to non-mathematicians. We wouldn't really get > very far, though. Or, the cranks could actually learn the concepts. But, > then they wouldn't be cranks. > That'd be the Catch 21.9999... Cheers - Chas
From: Eckard Blumschein on 6 Dec 2006 03:17 On 12/5/2006 7:20 PM, Tony Orlow wrote: > Eckard Blumschein wrote: >> On 11/24/2006 1:20 PM, Richard Tobin wrote: >>>> * If you have two sets of infinite size, is the union of these sets >>>> than larger than infinity? What would larger than infinity mean? >>> That this is not a stumbling block should be clear of you replace >>> "infinite" with "big". The union of two big sets can be bigger than >>> either of the original big sets. Like "big", "infinity" is not a >>> number; it's a description of certain numbers. >> >> According to Spinoza, infinity is something that cannot be enlarged >> (and also not exhausted). It is a quality, not a quantity. >> >> There is only one such ideal concept, not different levels of infinity. > > Isn't the purpose of math be to quantify? Well, the purpose to quantify is an important aspect of mathematics. > That description by Spinoza > doesn't lead to mathematics, does it? Spinoza lived from 1622 to 1677 in the Netherlands, which was less hurt from the 30 years lasting war and refuge to experts like Ren� Descartes. About at that time, Galilei (1564-1642), Kepler (1572-1630), Cavalieri (1598-1647), Fermat (1601-1665), Newton (1642-1727), and Leibniz (1646-1716) contributed to the real fundamentals of modern mathematics. The notion of infinity played an important role despite of worries uttered by Berkeley. > But, infinite is a term often > applied to quantities, whether of space, or time, or knowledge, or power... It belongs to the concept of continuity. > We have a unit interval, consisting of some infinity of distinct points ? > (more than any finite number). We have the unit square, consisting of an > equally infinite number of distinct parallel unit line segments. The > unit cube, again, has this infinite number of parallel unit squares > within it. Now, does it stand to reason that each added dimension to > this figure multiplies the number of points by this infinite number? oo * anything = oo. > When we divide the line by this number, we get one point, This is the trick which provides anything. Are you reaaly so naive? > the 0D square. > There's really no way you can convince me that the cube does not have > infinitely more points than the square, or the square than the segment, > or the segment than the point. Even Georg Cantor eventually accepted this while being hampered by the same kind of intuitive thinking like you. He wrote: Je le vois, mais je ne le crois pas. These are different levels of infinity. You did not even understand Cantor. How will you understand me? > As far as sequences go, we can also distinguish between different > infinities, certainly where one is a subset of the other, A subset inside the reals is comparable to a piece of sugar within tea. > So, Spinoza might be reconsidering a little, were he still around. No.
From: Bob Kolker on 6 Dec 2006 05:59 David Marcus wrote: > > > As Virgil pointed out, "no bijection to the naturals" is not a correct > definition of "uncountable". Eh? See: http://en.wikipedia.org/wiki/Uncountable In mathematics, an uncountable or nondenumerable set is a set which is not countable. Here, "countable" means countably infinite or finite, so by definition, all uncountable sets are infinite. Explicitly, a set X is uncountable if and only if there is an injection from the natural numbers N to X, but no injection from X to N. From the wiki article on countability. Do you see in error in the wikipedia entry on uncountability Bob Kolker
From: Bob Kolker on 6 Dec 2006 06:05
David Marcus wrote: > Bob Kolker wrote: > >>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. > > > The image of a space filling curve is a square, so has dimension 2. I sit corrected. The Peano curve has hausdorff dimension w. Bob Kolker |