The Definition of Points
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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~~
From: Tony Orlow on 26 Mar 2007 17:19 Lester Zick wrote: > 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. > That's all pretty well figured out, as far as I'm concerned. If we declare some infinitesimal unit (I like to call it Lil'Un, but you can call it delta or epsilon or iota as you wish), then there is a unit infinity (Big'un's what I refers to it by, but some others might like to think of it as a form of c or some such), and the product of the two is one, the finite unit. There are further considerations regarding powers of these units, which produce countably or uncountably many relative levels of infinity, depending on whether one allows infinite powers. Where 0*oo=1, 0 and oo are reciprocals, obviously. >> 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. I think it's a mistake to confuse absolute 0 with an infinitesimal, and generally I am more careful than to call them "zeros", but the subject here is points, as wells as lines. Is a point really nothing? Can it not be assigned some infinitesimal measure, so as to allow things like uniform probability distributions over infinite sets, and cumulative distance from infinitudes of points? I think it's a useful concept, as long as it's quantified usefully. > > 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. Actually 0 is far more basic than that. It is the starting point, the "here and now", the "this" to your measurable "that". 0 is the Origin of your Coordinates, the Alpha to your Omega, the...oh, you get the point. Of all the axiomatic statements that may be of use, I can think of none more basic and important than, "0 exists". So, 0 isn't even infinitesimal. It's what infinitesimals are found to be infinitesimally different from, or infinitely close to, while still being nonzero. :) > > 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. Google point set topology". It's all about sets of points. And then, if you care to do historical research on my ideas, google "infinite induction and the limits of curves". Point set topology should be replaced with a multidimensional system of segment sequence topology, INMOSVHO. LMAOUICMP;) > > 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. Um, no, dr/dr=1. Looks like an identity function to me.... > > But where I think you're light years ahead of standard mathematical > usage lies in the subdivision of lines into infinitesimal fractions. Gee, thanks, but I actually think that has already been discovered, and needs to be dug up and reexamined. Maybe I'm light years behind, but then again, the most interesting processes are cyclical, no? > 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. > Addition is not consistent with the straightedge and compass? 1. Draw a line. 2. Pick a point, 0. 3. Pick a point, A. 4. Pick a point, B. 5. Place the point of the compass on 0, and either: a) open it so the pencil's on A, move the point to B, and draw the intersection to the line farthest from A, or b) open it so the pencil's on B, move the point to A, and draw the intersection to the line farthest from B. That point is A+B. Of course, bisection is simpler.... Trisection's impossible to do exactly.... >> 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. What DOES appeal to you, Lester? Truth? What IS truth to you, Lester? How do you measure it, if not within the base unit interval [0,1], as finite representation of the potentially infinite universe? Jes' wundrin'... > > 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 I see through your eye a little. Yes, picture the whole entire number line as a unit, and subdivide it into finite units. Then, do the same with each of those to get infinitesimals. Is that sort of the vision? The problem is, either the number line includes only finite values, in which case there are no definable endpoints, or, the number line includes infinite values, of which there also is no end, in an even more complicated way. My opinion at this point regarding the definition of "finite" is that the set theoretic definition is sufficient. A finite natural is essentially the size of a finite set, which is defined as a set which cannot be bijected with any subset. While bijections alone are not sufficient, in my opinion, to define infinitude, they are for defining finiteness. A finite real would lie between any two finite naturals. > > 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. I think I understand your gripe, but to say the more elemental things depend on the more complex and relatively infinite seems bass ackwards. No offense. Your intuition may be akin to mine that, physically, this is the way to universe progresses, from the roof down. That's not wrong, at least as far as we know. Science doesn't even dare guess what "preceded" the Big Bang. I've intimated a few of my thoughts on spatial generation, thought they're not directly germane to most of what we discuss 'round these parts. So, I don't talk 'bout it much. :) > > 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. > You start at zero before you begin counting. That's when you take your deep breath... >> 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. Lester, yes it is. Period. That is simply a matter of grammar. If you have a rule of interpretation that says, "if no conjunction, then we assume this", and then derive something from that, and then derive "this" from "that" ("this" being OR and "that" being AND), you are being more circular in logic than the many you accuse of that evil sin. :) The truth is, some truth is circular. The fact is > that there are no conjunctions at all "out there" nor do I assume any. They are relations, expressed in human language as conjunctions, in various contexts. > 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. If you want to get down to basics, there are things, and things that things do. There are objects and their properties. There are sets of objects grouped by properties, and properties characterized by sets. Ask Leibniz. There are data points, and correlations. This is the root of your question, and the answer is "MU". The solution is Tao. > 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. > By assembling points into lines, and seeing where those lines intersect. > 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. No, you build it from AND, after having assumed it in the course of defining AND. Sorry, pal. > 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. > No, the functions can be categorized according to number of parameters and output conditions, as truth tables. Then the functions can be expressed algebraically, in which case an interesting question arises which might merit further investigation... >> 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". Ahem. So, now, you are admitting that there is an assumed "effective" OR in "not A not B"? That's good. 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". Why do you not consider "A not C" to be equivalent to "A AND not C"? You can build AND from OR as easily as OR from AND. The reason for explicitly stating OR or AND is to be entirely symbolically clear. > > 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". (sigh) Okay, here's a breakdown of binary operators. Each operator returns a value, false or true, which we shall encode, non-coincidentally, as 0 and 1. Each operator takes some natural number of truth values as parameters, and determines a set of outcomes for each combination, which is unique among operations of that number of parameters. We start with the zero-place operators, which take no parameters. There are two: 0 and 1, or true and false. Once we introduce some variable logical parameter x, we have f(x), and depending on whether x is 0 or 1, we can output 0 or 1. So, let's tabulate these two possible input situations, which number two: x f00 f01 f10 f11 0 0 0 1 1 1 0 1 0 1 Clearly, f00 and f11 are always false and true, respectively, and so are simply the first two unary functions we've already defined. Those are done. F01 is simply x, which we introduced as a starting concept as a parameter. That doesn't add anything new. F10 is the only actual unary operator, NOT. If x is 1, f10(x) is 0, and vice versa. NOT's the only unary operator. (I started to write you the whole breakdown of binary logical operators. later for that. Sorry...) > >> 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. Use truth tables, and the AND and OR aren't the elementary step... > > 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. Hmmm.... It seems to me you rest on assumptions as well. Such as, there is universal truth that can be ascertained directly. It can't. Grammar isa tool. The World is complex. Either sentences or vocabulary become complex, or we spend a lot of time talking. > > 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~~ Exhaustion would not necessarily seem to be the optimal goal. When it comes to ~(Pv~P), it's the same as P^~P, and boils down to the same questions of the uncertainty of assumed truths that your addressing. Explore values between 0 and 1, and then it will all become murkily clear. :) Tony 01oo
From: Lester Zick on 26 Mar 2007 22:36
On Mon, 26 Mar 2007 16:19:59 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> 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. >> > >That's all pretty well figured out, as far as I'm concerned. If we >declare some infinitesimal unit (I like to call it Lil'Un, but you can >call it delta or epsilon or iota as you wish), then there is a unit >infinity (Big'un's what I refers to it by, but some others might like to >think of it as a form of c or some such), and the product of the two is >one, the finite unit. There are further considerations regarding powers >of these units, which produce countably or uncountably many relative >levels of infinity, depending on whether one allows infinite powers. >Where 0*oo=1, 0 and oo are reciprocals, obviously. But, Tony, you just said 00*0 was 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. >> >> 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. > >I think it's a mistake to confuse absolute 0 with an infinitesimal, and >generally I am more careful than to call them "zeros", but the subject >here is points, as wells as lines. Is a point really nothing? Can it not >be assigned some infinitesimal measure, so as to allow things like >uniform probability distributions over infinite sets, and cumulative >distance from infinitudes of points? I think it's a useful concept, as >long as it's quantified usefully. Whatever. Is a point really nothing? Is a zero really nothing? Who cares. If you want to fudge things why not just say 1 is zero? Then we can all stop worrying about it one way or the other and go home. >> 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. > >Actually 0 is far more basic than that. It is the starting point, the >"here and now", the "this" to your measurable "that". 0 is the Origin of >your Coordinates, the Alpha to your Omega, the...oh, you get the point. >Of all the axiomatic statements that may be of use, I can think of none >more basic and important than, "0 exists". So, 0 isn't even >infinitesimal. It's what infinitesimals are found to be infinitesimally >different from, or infinitely close to, while still being nonzero. :) Okay, Tony. You've made it clear you don't care what anyone thinks as long as it suits your druthers and philosophical perspective on math. >> 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. > >Google point set topology". It's all about sets of points. And then, if >you care to do historical research on my ideas, google "infinite >induction and the limits of curves". Point set topology should be >replaced with a multidimensional system of segment sequence topology, >INMOSVHO. LMAOUICMP;) Tony, you don't care about my perspective and I don't care about yours. The only thing we have in common is that we disagree to a varying extent with establishment mathematikers. And that's not enough to build a serious extended conversation. You might just as well go back to your real number line, plus and minus infinity number ring, and infinitesimals without derivatives for all the good it's going to do me, science, or mathematics. >> 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. > >Um, no, dr/dr=1. Looks like an identity function to me.... So what. Most identity ratios are. >> But where I think you're light years ahead of standard mathematical >> usage lies in the subdivision of lines into infinitesimal fractions. > >Gee, thanks, but I actually think that has already been discovered, and >needs to be dug up and reexamined. Maybe I'm light years behind, but >then again, the most interesting processes are cyclical, no? I don't know as arithmetic ever had such a foundation. >> 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. >> > >Addition is not consistent with the straightedge and compass? >1. Draw a line. >2. Pick a point, 0. >3. Pick a point, A. >4. Pick a point, B. >5. Place the point of the compass on 0, and either: >a) open it so the pencil's on A, move the point to B, and draw the >intersection to the line farthest from A, or >b) open it so the pencil's on B, move the point to A, and draw the >intersection to the line farthest from B. > >That point is A+B. That isn't what the Peano and suc( ) axioms say is done. >Of course, bisection is simpler.... Obvously since it's the mechanical basis for fractional replication. Replicating line segments colinearly along a common straight line is a little harder unless you have the straight line to begin with. >Trisection's impossible to do exactly.... > >>> 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. > >What DOES appeal to you, Lester? Truth? What IS truth to you, Lester? >How do you measure it, if not within the base unit interval [0,1], as >finite representation of the potentially infinite universe? Jes' wundrin'... You know, Tony, you've read it over and over. So if you don't understand it by now there isn't much point to going over it. Assumptions of truth just don't cut it. Truth is what you're supposed to prove and not just what you can't imagine better alternatives to. >> 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 I see through your eye a little. Yes, picture the whole entire >number line as a unit, and subdivide it into finite units. Then, do the >same with each of those to get infinitesimals. Is that sort of the vision? Except the first time you subdivide in terms of an ongoing process of infinitesimal subdivision you already have infinitesimals because it's the idea of a process that makes them infinitesimals and not just the fact that they're small. Technically subdivision of this sort is never ending and we have no way to say that results are categorically different just because they're smaller or larger than other results. Their size is finite despite the process of infinitesimal subdivision. First I imagine you would bisect a line segment and get two line segments. Then you'd bisect both line segments to get four line segments. And so on. >The problem is, either the number line includes only finite values, in >which case there are no definable endpoints, or, the number line >includes infinite values, of which there also is no end, in an even more >complicated way. Tony, just ask yourself what the results of a process of infinitesimal subdivision amount to. Are the results of bisection finite? Yes. Does the process itself ever end? No. So any continuous bisection yields finite values only and all segments have defined endpoints. There is no infinity or infinitesimals in this sense apart from descriptions of the process itself. >My opinion at this point regarding the definition of "finite" is that >the set theoretic definition is sufficient. A finite natural is >essentially the size of a finite set, which is defined as a set which >cannot be bijected with any subset. While bijections alone are not >sufficient, in my opinion, to define infinitude, they are for defining >finiteness. A finite real would lie between any two finite naturals. I don't do bijections, Tony, any moe than I do axioms or assumptions of truth. >> 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. > >I think I understand your gripe, but to say the more elemental things >depend on the more complex and relatively infinite seems bass ackwards. >No offense. Your intuition may be akin to mine that, physically, this is >the way to universe progresses, from the roof down. That's not wrong, at >least as far as we know. Science doesn't even dare guess what "preceded" >the Big Bang. I've intimated a few of my thoughts on spatial generation, >thought they're not directly germane to most of what we discuss 'round >these parts. So, I don't talk 'bout it much. :) Well when you can intimate demonstrations of truth you'll have something to discuss. >> 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. >> > >You start at zero before you begin counting. That's when you take your >deep breath... Yeah, I've already explained my approach to counting. Show me the subdivision process that results in zero or even additive process that results in zero without assuming zero to begin with. Not happening. >>> 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. > >Lester, yes it is. Period. That is simply a matter of grammar. If you >have a rule of interpretation that says, "if no conjunction, then we >assume this", and then derive something from that, and then derive >"this" from "that" ("this" being OR and "that" being AND), you are being >more circular in logic than the many you accuse of that evil sin. :) Tony, we might as well stop right here. I've read over your comments and the one thing I see missing is any demonstration of truth for what you opine. I've demonstrated what I say to be true. Demonstrations of truth are what science and mathematics is all about. You talk about it as if your opinions were anything more than establishment gospel. And it may be establishment gospel but that doesn't make it true gospel. Just because it's honky dorry and plausible to you doesn't make it so. And without some mechanically exhautive reduction in mutually self consistent terms for all the things you describe and talk about there can be no truth to what you say. Just make a list of every function you describe and show me, yourself, and everyone else how they're demonstrably true of one another and demonstrably true in general. Otherwise you might just as well be blowing smoke and pipe dreams. >The truth is, some truth is circular. > >The fact is >> that there are no conjunctions at all "out there" nor do I assume any. > >They are relations, expressed in human language as conjunctions, in >various contexts. > >> 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. > >If you want to get down to basics, there are things, and things that >things do. There are objects and their properties. There are sets of >objects grouped by properties, and properties characterized by sets. Ask >Leibniz. There are data points, and correlations. This is the root of >your question, and the answer is "MU". The solution is Tao. > >> 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. >> > >By assembling points into lines, and seeing where those lines intersect. > >> 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. > >No, you build it from AND, after having assumed it in the course of >defining AND. Sorry, pal. > >> 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. >> > >No, the functions can be categorized according to number of parameters >and output conditions, as truth tables. Then the functions can be >expressed algebraically, in which case an interesting question arises >which might merit further investigation... > >>> 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". > >Ahem. So, now, you are admitting that there is an assumed "effective" OR >in "not A not B"? That's good. > >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". > >Why do you not consider "A not C" to be equivalent to "A AND not C"? You >can build AND from OR as easily as OR from AND. The reason for >explicitly stating OR or AND is to be entirely symbolically clear. > >> >> 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". > >(sigh) > >Okay, here's a breakdown of binary operators. Each operator returns a >value, false or true, which we shall encode, non-coincidentally, as 0 >and 1. Each operator takes some natural number of truth values as >parameters, and determines a set of outcomes for each combination, which >is unique among operations of that number of parameters. We start with >the zero-place operators, which take no parameters. There are two: 0 and >1, or true and false. > >Once we introduce some variable logical parameter x, we have f(x), and >depending on whether x is 0 or 1, we can output 0 or 1. So, let's >tabulate these two possible input situations, which number two: > >x f00 f01 f10 f11 >0 0 0 1 1 >1 0 1 0 1 > >Clearly, f00 and f11 are always false and true, respectively, and so are >simply the first two unary functions we've already defined. Those are done. > >F01 is simply x, which we introduced as a starting concept as a >parameter. That doesn't add anything new. > >F10 is the only actual unary operator, NOT. If x is 1, f10(x) is 0, and >vice versa. NOT's the only unary operator. >(I started to write you the whole breakdown of binary logical operators. >later for that. Sorry...) > >> >>> 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. > >Use truth tables, and the AND and OR aren't the elementary step... > >> >> 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. > >Hmmm.... > > >It seems to me you rest on assumptions as well. Such as, there is >universal truth that can be ascertained directly. It can't. Grammar isa >tool. The World is complex. Either sentences or vocabulary become >complex, or we spend a lot of time talking. > >> >> 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~~ > >Exhaustion would not necessarily seem to be the optimal goal. > >When it comes to ~(Pv~P), it's the same as P^~P, and boils down to the >same questions of the uncertainty of assumed truths that your >addressing. Explore values between 0 and 1, and then it will all become >murkily clear. > >:) Tony > >01oo ~v~~ |