The Definition of Points
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From: The Ghost In The Machine on 25 Mar 2007 13:06 In sci.logic, RLG <Junk(a)Goldolfo.com> wrote on Sun, 25 Mar 2007 00:52:06 -0800 <CMOdnaSEuvKrt5vbnZ2dnUVZ_s-rnZ2d(a)comcast.com>: > > "The Ghost In The Machine" <ewill(a)sirius.tg00suus7038.net> wrote in message > news:slchd4-8v1.ln1(a)sirius.tg00suus7038.net... >> >> >> Well, that's just it...there's no last digit. However, >> were there a last digit one might run into either >> >> ....999999999 >> ....999999999 >> >> which has no borrow, or >> >> ....999999990 >> ....999999999 >> >> which will need one. After all, we multiplied by 10... >> >> Of course that's why limits need to be used anyway; my >> strawman logic verges on the ridiculous. :-) > > Yes, on the standard real number line there are no infinitessimals. > Non-standard number lines have infinitessimals, like the surreal number > line, but they are not given decimal representations. I'm not up on surds, myself. Best I can do is dy/dx, which is not a number, but a concept relating to functions. ;-) > >> But a few paradoxes get messy without them: >> >> -1 = 1 + 2 + 4 + 8 + ... >> since if x = 1 + 2 + 4 + 8 + ... then 2x = x-1 >> ? = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ... >> could be 0, 1, 1/2. > > Yes, divergent series can be arranged to converge to any number one likes. > The series 1 - 1/2 + 1/3 - 1/4 + 1/5 + ... is conditionally convergent. Makes life even more interesting. :-) > > R > > -- #191, ewill3(a)earthlink.net Useless C++ Programming Idea #992381111: while(bit&BITMASK) ; -- Posted via a free Usenet account from http://www.teranews.com
From: doslong on 25 Mar 2007 20:45 On Mar 14, 1:52 am, Lester Zick <dontbot...(a)nowhere.net> wrote: > The Definition of Points > ~v~~ > > In the swansong of modern math lines are composed of points. But then > we must ask how points are defined? However I seem to recollect > intersections of lines determine points. But if so then we are left to > consider the rather peculiar proposition that lines are composed of > the intersection of lines. Now I don't claim the foregoing definitions > are circular. Only that the ratio of definitional logic to conclusions > is a transcendental somewhere in the neighborhood of 3.14159 . . . > > ~v~~ There is no point in the real world at all, so we cannot define it exactly. I mean , the concept of point is absolutely illusion of mankind at all.
From: Tony Orlow on 26 Mar 2007 12:47 Lester Zick wrote: > On Sat, 24 Mar 2007 08:00:17 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 22 Mar 2007 20:15:37 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> On Thu, 22 Mar 2007 17:15:35 -0500, Tony Orlow <tony(a)lightlink.com> >>>>> wrote: >>>>> >>>>>> Lester Zick wrote: >>>>>>> On Wed, 21 Mar 2007 22:24:22 -0500, Tony Orlow <tony(a)lightlink.com> >>>>>>> wrote: >>>>>>> >>>>>>>> Lester Zick wrote: >>>>>>>>> On Wed, 21 Mar 2007 07:40:46 -0400, Bob Kolker <nowhere(a)nowhere.com> >>>>>>>>> wrote: >>>>>>>>> >>>>>>>>>> Tony Orlow wrote: >>>>>>>>>>> There is no correlation between length and number of points, because >>>>>>>>>>> there is no workable infinite or infinitesimal units. Allow oo points >>>>>>>>>>> per unit length, oo^2 per square unit area, etc, in line with the >>>>>>>>>>> calculus. Nuthin' big. Jes' give points a size. :) >>>>>>>>>> Points (taken individually or in countable bunches) have measure zero. >>>>>>>>> They probably also have zero measure in uncountable bunches, Bob. At >>>>>>>>> least I never heard that division by zero was defined mathematically >>>>>>>>> even in modern math per say. >>>>>>>>> >>>>>>>>> ~v~~ >>>>>>>> Purrrrr....say! Division by zero is not undefinable. One just has to >>>>>>>> define zero as a unit, eh? >>>>>>> A unit of what, Tony? >>>>>>> >>>>>>>> Uncountable bunches certainly can attain nonzero measure. :) >>>>>>> Uncountable bunches of zeroes are still zero, Tony. >>>>>>> >>>>>>> ~v~~ >>>>>> Infinitesimal units can be added such that an infinite number of them >>>>>> attain finite sums. >>>>> And since when exactly, Tony, do infinitesimals equal zero pray tell? >>>>> >>>>> ~v~~ >>>> Only in the "standard" universe, Lester. >>> So 1-1="infinitesimal" Tony? Somehow I doubt that's exactly what >>> Newton and Leibniz had in mind with their calculus. >>> >>> ~v~~ >> 1=0.999...? > > Well, Tony, 0.999 . . . is only an approximation to 1 and not 1 per > se. However it's still the case that 0.999 . . . - 0.999 . . . = 0 and > not some infinitesimal magnitude. > > Now as to your underlying contention that "uncountable bunches of > zeroes can attain nonzero measure" by which I assume you mean finite > measure since zero itself is finite, division of finites by zero would > have to be defined for this to be possible.My reasoning is as follows. > > Let's assume we have some uncountable bunch of zeroes U and U*0 equals > some nonzero magnitude X. Thus U*0=X where I suppose X might equal a > finite cardinal like three.At least this is how I interpret the claim. > > So I suppose the next contention would concern whether the operation > is reversible and whether we could calculate U=3/0? If so 3/0 would > have to be defined and I don't see it is in any kind of formal math. > > I expect there are a couple different ways you might get around the > problem. You might try to define 3/0 in some way compatible with your > basic contention, for example claim that the operation involved is not > reversible. Or you might claim U*0 is an infinitesimal of some kind > and not finite at all. Or you might try to establish U as some kind of > weird uncountable not exactly related to counting. Or you might try to > suggest that "*" and "/" are not the common garden variety operations > we expect in ordinary math. However I think whichever route you take > you have to do considerably more than simply make a claim like the one > you did because all these considerations are inextricably linked. > > Nor is it readily apparent what the objective of all this might be. I > understand what you seem to be after with the +00-00 number ring and > all. But what's the point of all this if you have to corrupt the very > concepts you're trying to unite to do it? I mean we already have the > transfinite SOAP opera nonsense which has had to corrupt the meanings > of points and lines and points as constituents of lines, straight > lines, curves, irrationals, transcendentals, and so on to achieve its > aims of explaining geometry arithmetically.I just don't see the point. > > It looks to me as if mathematikers expect they can unite disparate > mathematical functions with a wink and a nod through some kind of > nomenclatural legerdemain such as simply calling lead gold and > pretending to have found the philosophers stone. > > The problem is then mathematikers have to invent all kinds of private > terminologies and regressional linguistic subterfuges, more commonly > known as dodges, to circumvent an obvious implication that they really > haven't done what they claimed. > > I've seen exactly analogous strategems in psychology and artificial > intelligence. Instead of actually solving problems in the disciplines > behaviorists and computer programmers quietly under the cover of > darkness exchange one set of problems which they can't explain for > another which they can or at least plausibly pretend they can. Then > they have to defend their evasions with the silliest terminological > regressions and excuses imaginable. > > The result is a vast network of language and terminology designed to > obfuscate the unitiated instead of answering questions. It's always > been the same with every faith based mystical epistemology. A priestly > caste emerges to guide believers in the faith with rote catechisms and > dogma whose actual rationales are safely cloaked and closeted within > ivory towers. Requiescant in pace. > > ~v~~ Well, all that commentary aside, you have a point. In order for a set of pure zeros to sum to anything nonzero, one would need to have a pure oo of them, but then the sum is unclear. When I say uncountably many zeros can have a finite sum, I really mean infinitesimals, which in the standard mathematical world are considered equal to zero. My proposal is to include infinitesimal and infinite units along with the finite unit, so as to relate these levels of scale in a unified theory. Maybe it doesn't appeal to you. Oh, well. After having read your Epistemology 401 essay, I still don't agree with your point. When you say (not A not C) you appear to be interpreting that as one would normally interpret not A OR not C, in which case the negation of that statement is indeed equivalent to A AND B. But, you have the OR implied to begin with. You also refer to NOT as a conjunction, but NOT is a one-place operator, whereas conjunctions relate two objects with each other. What is wrong with standard Boolean logic? Tony
From: Lester Zick on 26 Mar 2007 15:13 On Mon, 26 Mar 2007 11:47:06 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Well, all that commentary aside, you have a point. In order for a set of >pure zeros to sum to anything nonzero, one would need to have a pure oo >of them, but then the sum is unclear. No number of zeroes can produce a finite sum, Tony. But even so you need to determine what laws of arithmetic, calculus, and so on your ideas follow. If the product 00*0 is unclear you really need to make it clear whether finite, infinitesimal, or whatever. Then you need to decide whether the process is reversible and what functions processes obey. There has to be some structure to which all the elements and functions conform and are demonstrably reciprocally connected in mechanical terms. > When I say uncountably many zeros >can have a finite sum, I really mean infinitesimals, which in the >standard mathematical world are considered equal to zero. Well I can understand this technique. At least it's grounded in some kind of established mechanical approach and can take advantage of standard terminology and practices to some extent. But I think you're fudging when you try to equate zero and dr. These concepts are completely different because they originate in different functionality.Zero is the result of subtraction of identical numbers and dr the result of derivation in the calculus and it's the mxing and confusion in terminology which obscures what is actually going on with these kinds of terms and what they're supposed to mean. Furthermore I don't believe the two concepts are really confused in the standard mathematical world. I think there are some oddballs who would like to make believe they're functionally identical for purposes of their own but no one who really believes dr=zero. In fact just the other day I got a query from someone asking me for a college level textbook citation to support my contention that modern math considers lines to be made up of points. And the fact is all I had were various comments of Bob to support that idea. So it seems I may have been prejudging the issue. Unfortunately all any of us really have on the internet are these kinds of comments and prejudices to go by. And finally I suspect even if you go with a dr=zero approach you're going to find it difficult to implement because derivatives in general relate functions to one another and I don't see any significance of a function with respect to itself. But where I think you're light years ahead of standard mathematical usage lies in the subdivision of lines into infinitesimal fractions. At least there you're getting down to the basics of arithmetic and numbers instead of using some kind of fanciful addition because subdivision is at least consistent with geometry whereas addition is not because addition doesn't produce numbers on mutually colinear straight line segments. > My proposal is >to include infinitesimal and infinite units along with the finite unit, >so as to relate these levels of scale in a unified theory. Maybe it >doesn't appeal to you. Oh, well. Well as noted previously math has limited appeal. But that doesn't mean I don't have anything of significance to contribute. I've already pointed out there is no real number line which seems to completely befuddle and confound most common thinking on the subject. I think the most promising approach lies with a kind of infinitesimal subdivision and bisection of straight line segments for a definition of finite numbers because it's consistent with geometry. However this wouldn't be quite the same as the calculus even though we would be using the term "infinitesimal" to refer to results of it. I think if you take lines or boundaries in general to begin instead of points you'll be headed in the right direction. Points aren't anything at all without lines nor are lines without surfaces nor are surfaces without volumes. Then at least you could mechanically connect these figures via the calculus and subdivide straight line segments for the natural cardinal numbers. However what this means is that you don't start off counting with zero. The natural numbers begin with unity then two etc. And you'll also find that natural geometric figures bifurcate into curves and straight lines initially and various numeric concepts really start out as transcendentals and straight lines to begin with. >After having read your Epistemology 401 essay, I still don't agree with >your point. When you say (not A not C) you appear to be interpreting >that as one would normally interpret not A OR not C, in which case the >negation of that statement is indeed equivalent to A AND B. But, you >have the OR implied to begin with. Yes but an implied "or" is not the same as a stated OR. The fact is that there are no conjunctions at all "out there" nor do I assume any. The only things "out there" to begin with are different things and the only thing "in here" to begin with is "not" or tautological negation. What they are is the subject of science and science has to demonstrate what the relations are between them. That's what science does and the only way to do it is through tautological negation or contradiction and those processes compounded in terms of themselves. Now I agree I assume at least the two things A and C "out there". But I do not assume the OR or any necessary conjunctive relation between A and C because that's what is to be demonstrated and what I do demonstrate solely in the course of successive tautological negation. In the context of boolean and ordinary generic logic conjunctive relations as well as the significance of those conjunctive relations are only assumed between subjects "out there" whereas I demonstrate the actual mechanical significance of what boolean and generic logic only suppose is true. > You also refer to NOT as a >conjunction, but NOT is a one-place operator, whereas conjunctions >relate two objects with each other. Not necessarily, Tony. I can conjoin subjects through negation alone because the effective relation "out there" among subjects is the same as "or" "in here". In other words "A not C" means the same as what we consider "A or not C" to mean "in here" but without conjunctions. Thus tautological regressions don't have to consider conjunctions. For the same reason "A not A" is self contradictory just as is "not not". Thus the common assumption regarding "not" as a purely unary operator in the presence of other subjects is demonstrably incorrect because in mechanically reduced and exhautive terms there are no other conjunctive operators than "not". > What is wrong with standard Boolean >logic? What's wrong with standard boolean logic is that it treats ordinary conjunctions such as OR and AND as primitive functions whereas in point of fact as I demonstrate and prove they're not primitives at all but structural composites of tautological "not's". In other words the standard boolean logic is unreduced and undemonstrable except by typical empirical assumptions of truth. This doesn't mean boolean logic is wrong as far as it goes. It just means it relies on certain empirical assumptions of truth regarding conjunctions and the mechanical significance of conjunctions. The problem comes when we try to understand those conjunctions with undemonstrable mechanical significance. Then we just have to rely on whatever subjective interpretation the conjunctions seem to imply. Aristotle's law of non contradiction is a primary example. When we try to reduce it in boolean terms it becomes a hodgepodge of conjunctive nonsense because it relies on conventional inferences regarding the meaning of conjunctions instead of mechanically reduced meanings. Which is exactly why application of boolean conjunctive logic results in philosophy instead of science: it can't be reduced in mechanically exhaustive terms which result in unambiguous statements of truth. ~v~~
From: Lester Zick on 26 Mar 2007 15:21
On 25 Mar 2007 17:45:10 -0700, "doslong" <doslong(a)gmail.com> wrote: >On Mar 14, 1:52 am, Lester Zick <dontbot...(a)nowhere.net> wrote: >> The Definition of Points >> ~v~~ >> >> In the swansong of modern math lines are composed of points. But then >> we must ask how points are defined? However I seem to recollect >> intersections of lines determine points. But if so then we are left to >> consider the rather peculiar proposition that lines are composed of >> the intersection of lines. Now I don't claim the foregoing definitions >> are circular. Only that the ratio of definitional logic to conclusions >> is a transcendental somewhere in the neighborhood of 3.14159 . . . >> >> ~v~~ > >There is no point in the real world at all, so we cannot define it >exactly. >I mean , the concept of point is absolutely illusion of mankind at all. Yeah, yeah, yeah, reality is an illusion of mankind so what's the point of dealing with illusions. Science and mathematics are the point. They deal with our illusions of reality to determine which are illusions and which are facts, which are false illusions and which are true illusions. Just calling illusions of reality illusions can't tell us whether they are true or false illusions. Science and mathematics can. ~v~~ |