Galileo's Paradox
in [Math]
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From: Tony Orlow on 8 Dec 2006 14:36 Virgil wrote: > In article <MPG.1fe02b24b2771b489899d1(a)news.rcn.com>, > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> Virgil wrote: >>> In article <45725b3c(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> I'm not talking more nonsense than you, if you're talking about this >>>> diagonal line through some "quandrant". What, in your least sloppy >>>> language, is the meaning of the anti-diagonal generated by the random >>>> list? >>> It has been explained often enough that even someone as dense as TO >>> could have understood it, if he were only willing to try. >> Do you really think so? I wonder. > > Well TO might only be trolling. Define that term, then explain why you think that's what I'm doing. You know, above all others, that it's not. :s Tony
From: Tony Orlow on 8 Dec 2006 14:42 Eckard Blumschein wrote: > 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? I'm sorry, Eckard. What do you mean by "more"? Is that a measure? 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. > Well, we COULD be that simplistic. I was shooting for something higher. I guess I missed. >> 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. > Oh. He disproved that? Did he have anything to say about that? Please give some reference to his opinion on that. Thanks. I'm sure Galileo Galilei had something to say, or you do. >>>> 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. I do. It means as far as anyone can logically see. 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. > It is science, and mathematics, to the extent that it can be formalized, formulated, expressed in language, and however many other ways you want to say that same thing. TONY
From: Tony Orlow on 8 Dec 2006 14:50 Eckard Blumschein wrote: > 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. One of the two? And the other, to express the quantity uniquely, in a given precise language? > >> 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. > Did that answer my question? I'm certainly impressed by the number of thinkers out of Hole Land, The Nether Lands. It's not surprising, for very many reasons. So, then, now, can we count ourselves out of the hole? >> 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. > > Only within the finite interval. >> We have a unit interval, consisting of some infinity of distinct points > > ? Uh, that was continuity..... > >> (more than any finite number). Finite measure=infinite count. 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. How Very Simplistic. Primitive, even. Ugh. I give you stick. I give rock. Bam Bam. > >> When we divide the line by this number, we get one point, > > This is the trick which provides anything. Are you reaaly so naive? > I am really so in touch with the root. >> 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? > > By hearing you express your thoughts in modern context. >> 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. > Yeah, well, the sugar is dissolved, but the leaves still float around. Read the fortune. It bodes....uh....inconclusive for your position. You have not bothered to make a decision yet. Eckard, some await you... TOny
From: Eckard Blumschein on 8 Dec 2006 14:52 On 12/8/2006 7:41 PM, Virgil wrote: > In article <4579A2D7.2030500(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> First of all I see mounting evidence for a lack of evidence which could >> support some bizarre fancies related to the elusory belief introduced by >> Dedekind, Cantor, et al. "there must be more reals than rationals >> because the latter are a subset of the former" > > That is an extremely convoluted way of saying nothing. If there were > evidence of anything wrong, present it. Otherwise quit carping. I am pointing my finger squarely on the root of all nonsense that has been confusing and hampering the fundamentals of mathematics for more than 100 years, and you are trying to belittle my attack? Take issue! >> I hide my daring smile and ask if |sign(0)|=1 might be correct, than >> people wonder why I have such a silly idea. > > As a "sign()" function is not standardized across mathematics, one can > chose to define one's own in any way one likes, but unless one has some > fairly good reasons for one's definition, it is not likely to gain much > acceptance. Slippery eel! The sign function is just an example.
From: Eckard Blumschein on 8 Dec 2006 14:55
On 12/8/2006 7:23 PM, Virgil wrote: > In article <457988FD.2060201(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > "Phucking up" is idiomatic English for making mistakes. Thank you. Meanwhile I already learned a lot from you. Exclusively hopefully exclusive words. |