The Virgin Birth of Points
in [Physics]
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From: G. Frege on 1 Dec 2007 17:47 On Sat, 01 Dec 2007 19:22:12 +0100, G. Frege <nomail(a)invalid> wrote: To avoid misunderstandings, I would like to ad a n additional clause... >> >> The collection of postulates do not have to be (jointly) true. They only >> have to be consistent. >> > On the other hand, _if_ they are consistent, then a model for that > theory [assuming we are talking about first-order theories] exists; > i.e. an interpretation that makes all axioms (and hence all theorems) > of this theory true. With other words, there might exist a "world" ~~~~~~~~~~~~~~ > (in the "modal" sense of the word) where all those axioms (and theorems) > actually are true. > One might think that this should be clear from the context. But Lester Zick proved me wrong! :-) F. -- E-mail: info<at>simple-line<dot>de
From: Lester Zick on 2 Dec 2007 10:29 On Sat, 01 Dec 2007 12:14:16 -0700, Quint Essential <QT(a)archangel.net> wrote: >On Sat, 01 Dec 2007 10:33:44 -0700, Lester Zick ><dontbother(a)nowhere.net> wrote: > >>On Fri, 30 Nov 2007 22:46:35 -0700, Quint Essential <QT(a)archangel.net> >>wrote: >> >>>On Fri, 30 Nov 2007 11:33:50 -0700, Lester Zick >>><dontbother(a)nowhere.net> wrote: >>> >>>>On Sun, 11 Nov 2007 14:40:29 -0700, Lester Zick >>>><dontbother(a)nowhere.net> wrote: >>>> >>>>> >>>>> The Virgin Birth of Points >>>>> ~v~~ >>>>> >>>>>The Jesuit heresy maintains points have zero length but are not of >>>>>zero length and if you don't believe that you haven't examined the >>>>>argument closely enough. >>>> >>>>The epistemological problem for modern math is where do an infinite >>>>number of points required to unionize points into lines come from? >>>> >>>>Only one solid is needed to produce one surface and one surface >>>>required to produce one line but an infinite number of points are >>>>required to produce one line. And the difficulty is that we can only >>>>produce finite numbers of points through tangency or intersection. So >>>>where are all the points supposed to come from? Imagination? Otherwise >>>>we can only be left with a finite number of straight line segments >>>>defined between points. >>>> >>>>~v~~ >>> >>>So let me give you a hypothetical. What's wrong with assuming an >>>infinite number of points from which we construct lines and so on? >> >>Well of course the objective of mathematics are demonstrations of >>truth and not merely hypothetical assumptions. And this is one of the >>main themes I've been trying to stress throughout these threads. >> >>However even hypothetically the problem is that lines have direction >>and points don't. Consequently any infinity of points that might be >>assumed couldn't also be assumed to lie on any line in any direction. >> >>In other words given some line, infinite subdivision is possible but >>those results would not be points; they would be line segments defined >>by points of intersection. And given any infinity of points all you >>could produce are various line segments not lying along any line >>unless the line itself is defined first regardless of the points. >> >>~v~~ > >Well then why not just hypothetically assume space is filled with >infinities of points. Since points would be everywhere wouldn't that >circumvent the problem of directionality? Doesn't really change the epistemological problem. It's a question of methodology. Finitists simply define line segments with points at the ends of segments. Infinitists also insist lines are composed of points but cannot show the infinities of points they claim lines are composed of. That's what I mean when I say "where are all the points supposed to come from?". It's one thing to claim such a thing but it's another to demonstrate the truth of the claim. Infinitists claim there a union of infinities of points which make up the line but when asked to show those infinities of points for any given line have nothing to offer. ~v~~
From: Lester Zick on 2 Dec 2007 10:42 On Sat, 1 Dec 2007 09:52:57 -0800 (PST), Marshall <marshall.spight(a)gmail.com> wrote: >On Dec 1, 9:36 am, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On Sat, 1 Dec 2007 08:54:26 -0800 (PST), Marshall >> >> >> >At least, if the scrivener is on usenet. >> >> One wouldn't necessarily expect universal agreement. One might however >> expect demonstrations of truth and not simply captious argumentation. > >Please remind me again what your definition of "demonstration >of truth" is. I am sure I read it some months ago but I don't >recall the particulars. Is it like a scientific experiment, an >observation of the natural world? Not at all. It's simply a mechanical insight that "not" must be true of everything because "not not" is self contradictory and thus false. The same is true whether we say "not" "contradiction" or "differences" because "not not" the "contradiction of contradiction" and "different from differences" are all self contradictory. I was recently able to demonstrate the truth of boolean conjunctions using compoundings of "not" alone but in ordinary mathematical contexts the concept of differences is probably more useful. >Do you have an example of a simple mathematical demonstration >of truth? I don't have an argument with most mathematical demonstrations in general. What I try to concentrate on instead are mathematical axioms which are assumed but are not be proven to be true. In mathematical contexts generally the flaw is that addition and summation are the mechanical basis of axioms whereas in true mechanics differences are the basis of mathematics. Conventional mathematics just tries to add up points or whatever one way or another starting with assumptions of truth for those points and the way they add or unionize or whatever whereas the correct approach is through differences. ~v~~
From: Lester Zick on 2 Dec 2007 11:06 On Sat, 01 Dec 2007 23:31:42 +0100, G. Frege <nomail(a)invalid> wrote: >On Sat, 01 Dec 2007 15:03:54 -0700, Lester Zick <dontbother(a)nowhere.net> >wrote: > >>>>>> >>>>>> The collection of postulates do not have to be (jointly) true. They only >>>>>> have to be consistent. >>>>>> >>>> On the other hand, _if_ they are consistent, then a model for that >>>> theory [assuming we are talking about first-order theories] exists; >>>> i.e. an interpretation that makes all axioms (and hence all theorems) >>>> > of this theory [of course, *sigh*] >>>> >>>> true. With other words, there might exist a "world" (in the "modal" >>>> sense of the word) where all those axioms (and theorems) actually >>>> are true. ^^^^^ >>> >>> So square circles are true after all? >>> >Note that geometrical OBJECTS can't be true or false. Sure they can if their definitions are true or false. All definitions are combinations of predicates and any combination of predicates can be true or false. > Only proposition >and/or statements (sentences) can be true or false. So I take your >question to mean > > So the claim that there are square circles is true after all? > >>> >>> Huh? Show me a _consistent_ theory of Euclidean Geometry where you can >>> prove the existence of square circles. :-) >>> >Which actually means: ...where a statement stating the existence of a >square circle can be proved. Well you're certainly welcome to rephrase what I said any way you feel like that makes you feel comfortable trying to explain what I said but that doesn't make it what I said. >> Oh, so you're only talking about Euclidean geometry? >> >Right. Since concerning Euclidean Geometry I know for sure that we can >disprove the existence of square circles. (So if Euclidean Geometry is >consistent, we _can't_ prove the existence of square circles in it. >Moreover then there is a interpretation of this theory which makes the >statement/theorem that there isn't a square circle true.) > >> >> Funny, from your words "all axioms (and theorems) >> > of the theory in question >> >> are actually true" >> >Are you on drugs, or what? I just assumed you were because you're babbling about things whose truth you can't demonstrate. ~v~~
From: Lester Zick on 2 Dec 2007 11:07
On Sat, 01 Dec 2007 23:42:12 +0100, G. Frege <nomail(a)invalid> wrote: >On Sat, 01 Dec 2007 23:31:42 +0100, G. Frege <nomail(a)invalid> wrote: > >>>>>> >>>>>> On the other hand, _if_ they are consistent, then a model for that >>>>>> theory [assuming we are talking about first-order theories] exists; >>>>>> i.e. an interpretation that makes all axioms (and hence all theorems) >>>>>> > of this theory >>>>>> >>>>>> true. With other words, there might exist a "world" (in the "modal" >>>>>> sense of the word) where all those axioms (and theorems) actually >>>>>> are true. >>>>>> >>>>> So square circles are true after all? >>>>> >>>> Huh? Show me a _consistent_ theory of Euclidean Geometry where you can >>>> prove the existence of square circles. :-) >>>> >>> Oh, so you're only talking about Euclidean geometry? >>> >Actually, _any_ theory would do where we can prove that the assumption >that an object is a square and also a circle leads to a contradiction. So there is no world etc. etc. where all axioms (and theorems) are actually true? ~v~~ |