Galileo's Paradox
in [Math]
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From: Virgil on 4 Dec 2006 17:47 In article <45744411.2060408(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 10:50 PM, Virgil wrote: > > In article <45706F34.1070809(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> The word completeness is misleading. > > > > "Complete" for an ordered set has a precise mathematical definition. > > That mathematical meaning is the only relevant meaning in any > > mathematical discussion of ordered sets. Most words used in technical > > senses in mathematics mean something quite different from their common > > meanings, > > > Wiki: "In mathematics and related technical fields, > a mathematical object is complete if nothing needs to be added to it." That is not the same as a complete ordering, either partial or total. See: http://en.wikipedia.org/wiki/Completeness_%28order_theory%29 > What about the possibility to add numbers to "the" rational numbers, I > argue: > According to Archimedes, there is no limit for adding more precision. > However, infinite precision is as fictitious as is infinity. But no more so that 1 or 2. > Consequently, the system of genuine (rational) numbers exhibits open > ended acuity. It cannot at all be improved by adding genuine numbers > which are not yet part of it. maybe EB's irrational rationals cannot, but standard rationals can. > Continuum is a different quality. Tom some extent one may compare number > and continuum with stone and mortar in between. Physical analogies are all ultimately false. > > > >> > So of the sets mentioned above, the reals and only the reals are > >> > continuous in amy mathematically acceptable sense. > >> > >> I agree with the caveat that the meaning of the term set has been made > >> dubious. > > > > The meaning has been made precise by giving it a precise definition. > > A set in mathematical sense (including the infinite set) is something > which has precisely no valid definition. In set theories "set" is a primitive, just like "point' and "line" in Hilbert's completion of Euclidean geometry. > > > >> Our disagreement is based on different interpretation. > > > > In mathematics, the operant mathematical definitions determine the > > interpretations. To reject that is to reject mathematics entirely. > > Who does not obey rejects not just science entirely but even the holy > mathematics. Lucky Gauss, lucky Leibniz. They cannot be forced to > swallow stupid Virgilian definitions and pray to Georg the Lord. In order to discuss anything, there must be agreement on the meaning of the terms to be used in that discussion. Thousands of mathematicians have agreed on the meanings of certain mathematical terms, so that those who wish to discuss things mathematical cast themselves into outer darkness by insisting on using those terms to mean things at variance with their standard meanings. If EB wants to express different meanings he must come up with unused word or phrases to carry those meanings. Mathematics will not allow EB to rewrite mathematical dictionaries to suit his whims.
From: Virgil on 4 Dec 2006 17:52 In article <45744A4C.3020505(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 10:38 PM, Virgil wrote: > > In article <45706BFD.7090506(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 11/30/2006 10:02 PM, Virgil wrote: > >> > >> >> I consider Dedekind wrong, and he admitted to have no evidence in order > >> >> to justify his basic idea. > >> > > >> > The fact that Dedekind's definition of infiniteness of sets has been > >> > widely adopted indicates that many others have found it to be a useful > >> > definition. > >> > >> It was appealing even to Peirce. BTW, I referred to the lacking basis of > >> his cuts. What about the definition of an infinite set, I alredy > >> explained somewhere here why it tacitly implies an illusion. > > > > When you "explain" why 2 = 1, I am not persuaded. > > In this case you would probably guess that I misused division by zero. > No, I did not cheat you. You only tried to. > > > >> > >> And utility is the measure of the value of a definition. > >> > >> Utility for what? > > > > Mathematical definitions are abbreviations, they shorten things. If they > > are useful enough to be used often, they can save a great deal of time > > and space in mathematical writings. They are useful for that. > > So you meant: Usefulness is the measure ...? Of definitions, yes! Definitions which do not get used fade away. Those that get used, get spread around, and the more often used, the more spread around they get. > I am an electrical engineer Shocking!
From: Virgil on 4 Dec 2006 17:57 In article <45744E30.8090207(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 10:33 PM, Virgil wrote: > > In article <45706AD1.808(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > You claimed many "imperfectins" but did not justify those claims with > > anything mathematically valid. > > When I performed Fourier transform back and forth for a function > stepping at t=a, cf. > http://iesk.et.uni-magdeburg.de/~blumsche/M283.html > I correctly returned to the original function iff I ignored the > intermediate value at t=a and decided to extend integration from t<a > instead. That Fourier transforms do not do precisely as you wish they might, does not constitute an imperfection in the transforms so much as an imperfection of your understanding of what they can do. If you choose to hammer with a wrench, you may not always get the result a hammer would produce. > > >> >> I guess, point-set topology and measure > >> >> theory do not require the claim of set theory to rule all mathematics. > >> > > >> > They cannot exist without a foundation of set theory. > >> > >> In this case they could not exist. Set theory does not have a solid > >> basis. So I doubt. > > > > There are a lot of textbooks on point-set, and other, topologies and on > > measure theory. I have yet to see one of them that is not based on set > > theory. If EB claims these books do not exist, he is even more foolish > > than usual. > > I do not claim this. I just guess that not a single one really needs the > transfinite numbers and nonsense cardinalities like aleph_2. They certainly need the cardinality of the (real) continuum.
From: Virgil on 4 Dec 2006 18:03 In article <45745120.3020109(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 10:28 PM, Virgil wrote: > > In article <457069A0.7060208(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> Cantor did know that his fancy was rejected from all important figures > >> even those hundreds or even thousands of years ago. > > > > Is this supposed to mean something? > > Cantor was hospitalised in a mad house on a regular basis. He took > advantage of the possibility to read and comment on work by Aristotele, > Leibniz, etc. They were wrong altogether. He gave evidence for that just > by claiming to be more intelligent. His style and his promise: "The > essence of mathematics is just its freedom" in combination with > influencial friends made him very popular. He even founded the > mathematical society. During later depressive phases of his mind, he > withdrew from mathematics and dealt with the putative identity of > Shakespeare. In Newton's later years, he was quite odd, too, but that did not make his earlier work any less important. And how much of Cantor's depressions were brought on, or at least exacerbated, by Kronecker's relentless persecutions of him?
From: Virgil on 4 Dec 2006 18:09
In article <45745B16.40202(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> 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. Then length and area and volume comparisons of size must be fictional measurements. > > > > Sets can have the same cardinality but different 'subsettedness' because > > different qualities are being measured. > > I do not know a German equivalent to subsettedness. It is the quality of being a subset. It is one, but not the only, measure of set "size". > > TO>> You do? Do you mean that the addition of elements not already in a set > >> doesn't add to the size of the set in any sense? > > > > In the subsettedness sense yes, in the cardinality sense, not > > necessarily. In the sense of well-ordered subsettedness, not necessarily. > > > > TO seems to want all measures to give the same results, regardless of > > what is being measured. > > Standard mathematics may lack solid fundamentals. At least it is > understandable to me. However, I admit being not in position to likewise > easily understand what you mean with well-ordered subsettedness. The subset relation does not provide a well ordering of arbtrary sets. |