Galileo's Paradox
in [Math]
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From: Virgil on 10 Dec 2006 19:48 In article <457c1eb6(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Bob Kolker wrote: > > Virgil wrote: > > > >> In article <457b8a2f(a)news2.lightlink.com>, > >> Tony Orlow <tony(a)lightlink.com> wrote: > >> > >> > >>> Language is a set of strings from an alphabet. > >> > >> > >> Like Chinese and Japanese? > >> > >> Languages are acoustic at first, and are still learnt that way. > >> Ideogramatics comes later. > > > > All languages (used by humans in general discourse) are acoustic because > > they are primarily spoken and heard. Writing of any kind is a very > > recent development (in the last 6000 years or so). You are refering to > > the writing systems. It turns out that Japanese has two phonetic > > alphabets (Katakana and Hirugana) but the Chinese style ideographs > > (Kanjii) are preferred for traditional reasons. If the Japanese wished > > to, they could use a totally phonetic graphology. They just do not want to. > > > > The Chinese have used ideograms for much longer, but there is a phonetic > > system for representing the Mandarin pronounciation. The problem with > > Chinese is there are so many dialects which are mutually incompatable > > the only common written language is ideographic. The Japanese are under > > no such constraint. > > > > The alphabet derived from the Phonecian graphology was originally > > ideographic. The letters were pictures (or highly stylized > > representations) of things that eventually took on conventional sound > > values. > > > > Bob Kolker > > > > You guys both need to take your meds, and take note of a few facts. > > 1) While a symbol is generally considered to be a visual pattern, it is > most generally any uniquely distinguishable sensory input. It could be > an auditory sound, a latin letter, arabic numberal, hieroglyph, or asian > pictogram. It can be a series of beeps and clicks, like Morse code. > > 2) While Western languages have a small alphabet, and words with an > averga of a large number of characters, one can have a large alphabet, > the size of the entire vocabulary, where each word is a single > character. So, Asian pictographic languages is not excluded by that > definition. > > 3) I have specifically been talking about digital number systems, which > are a special case of languages, complete, and with a unique > quantitative interpretation for any finite string. It doesn't help to > wander off into sociological and psychological realms when discussing > digital number system. Digital number representations are not languages, as they consist of only proper nouns, with no verbs, adjectives adverbs pronouns, etc.
From: Virgil on 10 Dec 2006 19:52 In article <457c1f0c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <457b8a2f(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Language is a set of strings from an alphabet. > > > > Like Chinese and Japanese? > > > > Languages are acoustic at first, and are still learnt that way. > > Ideogramatics comes later. > > An acoustic sound IS a symbol, which we string together in time, as > opposed to doing it on paper. A word is a symbol, only such combinations of sounds which are recognizable as words are such verbal symbols, most combinations of sounds are not words. and it is one recognizable combination of sounds to one word.
From: Virgil on 10 Dec 2006 19:59 In article <457c1fa0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <457b8ccf(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <457b5606(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> There is nothing wrong with saying E and N have the same > >>>>>>>> cardinality. > >>>>>>>> It's a fact. Six Letters is essentially suggesting IFR, my Inverse > >>>>>>>> Function Rule, which indeed can parametrically compare sets mapped > >>>>>>>> onto > >>>>>>>> the real line, using real valued functions, as a generalization to > >>>>>>>> set > >>>>>>>> density. It works for finite and infinite sets. So, what is wrong > >>>>>>>> with > >>>>>>>> trying to form a more cohesive theory of infinite set size, which > >>>>>>>> distinguishes set sizes that cardinality cannot? There certainly > >>>>>>>> seems > >>>>>>>> to be intuitive impetus for such a theory. > >>>>>>> Fine. Please state your rule. Let's take a look. > >>>>>> I've been over this a lot, but hey, what's one more time. Practice > >>>>>> makes > >>>>>> perfect. > >>>>>> > >>>>>> We start with the notion of infinite-case induction, such that a > >>>>>> equation proven true for all n greater than some finite k holds also > >>>>>> for > >>>>>> any positive infinite n. > >>>>> What is a "positive infinite n"? > >>>>> > >>>> A value greater than any finite value. > >>> But if "n", is, as usual, reserved for indicating natural numbers, those > >>> which are members of every inductive set, there is no such thing. > >>> > >> What, now mathematics has declared what the single letter n means? It's > >> a variable. > > > > Every variable has a domain of definition. The variable 'n' is quite > > commonly required to have the set of finite naturals as its domain. > > > > I merely remarked that when that is the case, there are no values for n > > other than finite ones. > > > >>>> Why do you have a problem with the mere suggestion of an infinite value? > >>> It is arithmetical operations with infinite values absent any > >>> definition of what those operations mean or what properties they have, > >>> to which we make legitimate objections. > >>> > >> Not really. > > > > Really. > > > > No. Reread the following: > > >> If the expressions used can themselves be ordered using > >> infinite-case induction, then we can say that one is greater than the > >> other, even if we may not be able to add or multiply them. Of course, > >> most such arithmetic expressions can be very easily added or multiplied > >> with most others. Can you think of two expressions on n which cannot be > >> added or multiplied? > > > > I can think of legitimate operations for integer operations that cannot > > be performed for infinites, such as omega - 1. > > Omega is illegitimate schlock. Read Robinson and see what happens when > omega-1<omega. I have read Robinson. On what page of what book does he refer to omega - 1 in comparison to omega? I do not find any such reference. > > >>>> Surely you must have guessed enough what I meant to follow the > >>>> paragraph? (sigh) > >>> Guessing is not a reliable way of finding things out. > >> Of course, reading is a little more reliable, but when one gets stuck on > >> such words as "positive" and "infinite", maybe one needs to do a little > >> guessing. > > > > If TO's descriptive powers are so insubstantial as to leave us > > perpetually guessing what he means, he will never be able to convey > > anything of mathematical interest or significance. > > If you don't know what "positive" or "infinite" mean, that's not my fault. My uncertainty about what TO means by a term is directly TO's fault. I know what mathematics usually means by many terms, but as TO frequently does not mean the same thing by any mathematical term as mathematics means, I cannot say that I know what TO means by them.
From: Virgil on 10 Dec 2006 20:03 In article <457c48ef(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Odd powers of i equal i Or -i. For integers n, i^(4*n + 1) = 1 but i^(4*n - 1) = -1 Try learning some arithmetic before pontificating.
From: Virgil on 10 Dec 2006 20:08
In article <457c658c$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> I don't need to construct these numbers. Consider them > >>>> axiomatically declared. > >>> Then list the axioms for them. > >> (sigh) > >> > >> infinite(x) <-> A yeR x>y > > > > Is that your only axiom? If so, then state your first theorem about > > them and give the proof. > > > > That's the only one necessary for what defining a positive infinite > n. It is a definition, not an axiom. And as it does not imply existence of anything, one has no more that other axioms will supply. > A whole array of theorems pop forth from infinite-case induction > and IFR, such as that the size of the even naturals is half that of > the naturals. Not unless you declare them as axioms, as neither holds without being assumed. |