The Definition of Points
in [Logic]
|
Prev: On Ultrafinitism
Next: Modal logic example
From: Tony Orlow on 27 Mar 2007 02:32 Lester Zick wrote: > 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. > Right, and then I suggested a clarification of it such that 0, 1 and oo can be related to one another, where 0 and oo are actually declared units Lil'Un and Big'Un. >>>> 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. > That would cause inconsistencies. :) >>> 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. > Which is so completely different from you, of course... >>> 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. > You might be surprised at how it relates to science. Where does mass come from, anyway? >>> 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. > Right, and they're not terribly significant. >>> 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. > It's the non-rigorous primitive thought patterns of fools like Leibniz and Newton that is so scoffed at today. >>> 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. > Those aren't geometrical expressions of addition, but iterative operations expressed linguistically. >> 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. > So, start with the straight line: R exists. >> 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. > Science doesn't prove anything true. It only eliminates false hypotheses. That's the way science works. If you want to declare ultimate truth with certainty, ask god for a revelation, but don't call it science. >>> 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. > Their size is finite for any finite number of subdivisions. > 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. > Sure, so n subdivisions produces 2^n subintervals. >> 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. > It's the same as Peano. Add 1 to a finite, and you get a finite, so adding 1 can never produce an infinite value, right? Wrong. Add 1 n times to 0 and you get n. If n is infinite, then n is infinite. If n is infinite, so is 2^n. If you actually perform an infinite number of subdivisions, then you get actually infinitesimal subintervals. >> 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. > Bijections have their applications. I just don't think bijection alone is all that significant for infinite sets. The actual mapping function describes the relationship between infinite sets. >>> 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. > You don't really seem interested in demonstrations of truth, are you? >>> 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. > Do you not assume anything? You sure do. You assume "not" is universally true. It's universally meaningless in isolation. not(x) simply means "complement of x" or "1-x". You assume something else to begin with, which is not demonstrably true. >>>> 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. How have you demonstrated any truth? > > 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. Been there and done that, but you seem to reject anything that is "establishment gospel", even if exhaustively analyzed for truth. > >> 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...) >> If you're interested in the breakdown of binary logical operators, lemme know... 01oo >>>> 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~~
From: Lester Zick on 27 Mar 2007 15:18 On Tue, 27 Mar 2007 01:32:07 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> 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. >> > >Right, and then I suggested a clarification of it such that 0, 1 and oo >can be related to one another, where 0 and oo are actually declared >units Lil'Un and Big'Un. Problem is that 0 and 1 are finites but so are lots of other numbers and your original contention was that 00*0=finites not 1 so that you still haven't clarified the process involved in 00*0 that makes it 1 and not some other finite or which makes the process reversible. >>>>> 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. >> > >That would cause inconsistencies. :) And 00*0=1 wouldn't cause inconsistencies? >>>> 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. >> > >Which is so completely different from you, of course... Difference is that I demonstrate the truth of what I'm talking about in mechanically reduced exhaustive terms whereas what you talk about is just speculative. >>>> 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. >> > >You might be surprised at how it relates to science. Where does mass >come from, anyway? Not from number rings and real number lines that's for sure. >>>> 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. >> > >Right, and they're not terribly significant. Which takes us right back to derivatives without identity ratios. >>>> 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. >> > >It's the non-rigorous primitive thought patterns of fools like Leibniz >and Newton that is so scoffed at today. May be. Arithmetic just is what it is, a derivative of geometric subdivision. >>>> 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. >> > >Those aren't geometrical expressions of addition, but iterative >operations expressed linguistically. Which means what exactly, that they aren't arithmetic axioms forming the foundation of modern math? The whole problem is that they don't produce straight lines or colinear straight line segments as claimed. >>> 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. >> > >So, start with the straight line: How? By assumption? As far as I know the only way to produce straight lines is through Newton's method of drawing tangents to curves. That means we start with curves and derivatives not straight lines.And that means we start with curved surfaces and intersections between them. >R exists. Nice but still an axiomatic assumption of truth. >>> 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. >> > >Science doesn't prove anything true. Sure it does. That's the purpose of science. Empiricism and modern math don't prove anything true. Mysticism in action. That's why modern mathematikers consider themselves neo platonists. They're just divines who intuit the truth and what's true and false and go on from there. > It only eliminates false >hypotheses. Well it would certainly do that if it quite knew what was true and false to begin with. Strictly speaking contemporary science eliminates hypotheses which contradict axiomatic assumptions of truth. But that doesn't mean it eliminates hypotheses which are false because it just doesn't know what is false in mechanically reduced exhaustive terms. > That's the way science works. If you want to declare >ultimate truth with certainty, ask god for a revelation, but don't call >it science. It's modern mathematikers and empirics who ask gods for revelations. I concentrate on demonstrating what's true and false in mechanically reduced exhaustive terms of finite tautological regression to self contradictory alternatives. Whole nuther kettle of fish. Your position on science and math seems to be that either we proceed according to naive and mechanically unreduced and inexhaustive assumptions of truth or we proceed by appeals to divine revelation. Six of one half dozen of the other. >>>> 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. >> > >Their size is finite for any finite number of subdivisions. And it continues to be finite for any infinite number of subdivisions as well.The finitude of subdivisions isn't related to their number but to the mechanical nature of bisective 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. >> > >Sure, so n subdivisions produces 2^n subintervals. Equal subdivisions. That's what gets us cardinal numbers. >>> 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. >> > >It's the same as Peano. Not it isn't, Tony. Cumulative addition doesn't produce straight lines or even colinear straight line segments. Some forty odd years ago at the Academy one of my engineering professors pointed out that just because there is a stasis across a boundary doesn't necessarily mean that there is no flow across the boundary only that the net flow back and forth is zero.I've always been impressed by the line of reasoning. In other words modern mathematikers just assume that because the Peano and suc( ) axioms produce successive straight line segments between numbers there is some kind of guarantee that the successive straight line segments will themselves line up colinearly on straight line segments and that we can thus just assume or infer the existence of straight line segments and straight lines from those axioms.Doesn't happen that way because even if we assume the existence of straight line segments between numbers that doesn't demand successive segments align in any particular direction colinearly along any common straight line segment. Same principle as above, different application. > Add 1 to a finite, and you get a finite, so >adding 1 can never produce an infinite value, right? Finite addition never produces infinites in magnitude any more than bisection produces infinitesimals in magnitude. It's the process which is infinite or infinitesimal and not the magnitude of results. Results of infinite addition or infinite bisection are always finite. > Wrong. Sure I'm wrong, Tony. Because you say so? > Add 1 n >times to 0 and you get n. If n is infinite, then n is infinite. This is reasoning per say instead of per se. > If n is >infinite, so is 2^n. If you actually perform an infinite number of >subdivisions, then you get actually infinitesimal subintervals. And if the process is infinitesimal subdivision every interval you get is infinitesimal per se because it's the result of a process of infinitesimal subdivision and not because its magnitude is infinitesimal as distinct from the process itself. Just ask yourself, Tony, at what magic point do intervals become infinitesimal instead of finite? Your answer should be magnitudes become infintesimal when subdivision becomes infinite. But the term "infinite" just means undefined and in point of fact doesn't become infinite until intervals become zero in magnitude. But that never happens. And until it does the magnitude of subdivisions remains finite. The fact that there is a limit to the process doesn't mean the process itself ever reaches that limit or ever can reach that limit. >>> 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. >> > >Bijections have their applications. I just don't think bijection alone >is all that significant for infinite sets. The actual mapping function >describes the relationship between infinite sets. Well if you mean "matching" don't say "bijection". I don't have any use for people whose only purpose in math is terminological regression and the creation of buzzwords instead of mechanical reduction. First they say they can't use generic language because it isn't sufficiently precise then they turn right around and corrupt the usage of perfectly acceptable generic words such as "cardinality" on the same basis. Such "mathematikers" are just speaking in tongues. They don't understand what truth is so they just proclaim whatever they say is mathematical truth because the domain of their discussion is supposed to be 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. >> > >You don't really seem interested in demonstrations of truth, are you? What demonstrations of truth did you have in mind exactly, Tony? All I've seen so far are your ideas of truth per say and not per se.When I demonstrate truth the demonstration is per se by exhaustive mechanical reduction and not simply per say according to what seems plausible to me or anyone else just because I say so. What I don't seem interested in at the moment are more philosophical tracts when I've already shown the demonstration of universal truth by finite tautological reduction to self contradictory alternatives whereas all you've demonstrated is philosophical preferences for some variety of ideas apart form others. If I don't seem particularly interested in demonstrations of universal truth it's partly because you aren't doing any and I've already done the only ones which can matter. It's rather like the problem of 1+1=2 or the rac trisection of general angles. Once demonstrated in reduced mechanically exhaustive terms the problem if not its explication and implications loses interest. If you want to argue the problem itself go ahead. Just don't expect me to be interested in whether 1+1=2 or whether you can trisect general angles. What you're trying to do is argue the problem and not its resolution or my demonstration of the universal truth of the problem. And I just don't care what you think about the problem of universal truth when you refuse to discuss any demonstration of the problem of universal truth or my demonstration of universal truth in mechanically reduced exhaustive terms. What difference can your opinions on the subject possibly make? I argue A and you come right back and say A can't possibly be true because you like B. You can't even say whether A and B are really different. All you say is that you like binary logic and conjunctions. Well I like them too. I just say that they're specialized instances of universal truth demonstrated through finite tautological regression to self contradictory alternatives and so far you have yet to adduce any arguments to the contrary much less any demonstration to the contrary. So I suppose the short answer is no I don't really seem to care about problems I've already solved. >>>> 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. >> > >Do you not assume anything? You sure do. You assume "not" is universally >true. No I don't, Tony. I certainly do not assume "not" is universally true. I demonstrate "not" is universally true only to the extent "not not" is self contradictory and self contradiction is universally false. > It's universally meaningless in isolation. not(x) simply means >"complement of x" or "1-x". You assume something else to begin with, >which is not demonstrably true. No I don't, Tony.I demonstrate the universal truth of "not" per se in mechanically exhaustive terms through finite tautological reduction to self contradictory alternatives which I take to be false to the extent they're self contradictory. If you want to argue the demonstration per se that's one thing but if you simply want to revisit and rehash the problem per say without arguing the demonstration per se that's another because it's a problem per say I have no further interest in unless you can successfully argue against the demonstration per se. This is why science is so useful because you stop arguing isolated problems to argue demonstrations instead which subsume those isolated problems. There's simply no point to arguing such problems individually as to whether "not" is universally true of everything or whether there are such things as conjunctions not reducible to "not" in mechanically exhaustive terms unless the demonstration itself is defective and not true. And just claiming so per say won't cut it. >>>>> 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. > >How have you demonstrated any truth? See above. If you have per se arguments to show the demonstration itself is defective I'm all ears. But I've no interest in just per say assertions that the demonstration is defective without showing why and how. And naive assertions to the contrary of the demonstration are of no interest unless you can demonstrate how your assertions to the contrary invalidate the demonstration. In other words I'm just not interested in assertions to the effect that tautologies aren't what I take them to be since these kinds of terminological disputes don't address the mechanical characteristics involved and can always be resolved by the use of other terms. Nor am I interested in assertions such as the demonstration is incorrect because conjunctions are necessary to tautologies unless you can show how my demonstration that conjunctions are mechanically reducible to the application of successive combinations of "not" is defective. >> 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. > >Been there and done that, but you seem to reject anything that is >"establishment gospel", even if exhaustively analyzed for truth. "Been there and done what" exactly, Tony? "Exhaustive analysis for truth" is not at all the same as "exhaustive demonstrations of truth". All "exhaustive analysis for truth" means or can mean is that you've looked the problem over and can find nothing amiss. It just doesn't matter whether the "you" is just yourself or a godzillion others when you don't have any demonstrable basis for truth to begin with. "You" can't very well analyze anything for truth when you don't know exactly what's true or may be true. All you can really do is guess.And that's exactly what seers, mystics, and empirics do because that's all they can do. >>> 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...) >>> > >If you're interested in the breakdown of binary logical operators, lemme >know... I have no special interest in things studied in college, Tony. They're well known and well demonstrated in their own terms and just not a substitute for truth in exhaustive mechanical terms. >01oo > >>>>> 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. ~v~~
From: Tony Orlow on 29 Mar 2007 10:37 Hi Lester - Glad you responded. I was afraid I put you off. This thread seems to have petered unlike previous ones I've participated in with you. I hope that's not entirely discouraging, as I think you have a "point" in saying points don't have meaning without lines, and that the subsequent definition of lines as such-and-such a set of points is somewhat circular. Personally, I think you need to come to grips with the universal circularity, including on the level of logic. Points and lines can be defined with respect to each other, and not be mutually contradictory. But, maybe I speak too soon, lemme see... Lester Zick wrote: > On Tue, 27 Mar 2007 01:32:07 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> 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. >>> >> Right, and then I suggested a clarification of it such that 0, 1 and oo >> can be related to one another, where 0 and oo are actually declared >> units Lil'Un and Big'Un. > > Problem is that 0 and 1 are finites but so are lots of other numbers > and your original contention was that 00*0=finites not 1 so that you > still haven't clarified the process involved in 00*0 that makes it 1 > and not some other finite or which makes the process reversible. > 0 is not really finite, but The Origin. It is not a finite distance from The Origin, because there is no distance between it and itself. 0 is less than any finite, or infinitesimal, distance. What I meant is I*i=1. Oh, that's much better. Infinity times iota equals 1. :) So, here's how it hangs. In any interval of the real line, [x, x+1) or (x,x+1], we have oo reals. Each real will then be assumed to occupy 1/oo of this line, and if the length of this line is oo, then there will be oo^2 reals on the line, instead of 2^aleph_0, as if aleph_0 means anything anyway. There's no smallest infinite any more than a smallest nonzero finite, or infinitesimal. It is a simple assumption that subtracting a positive number from any other decreases it. One thing you may notice is that somehow aleph_0-1=aleph_0. There is no smallest infinity, though that's what aleph_0's supposed to be. Aleph_0 is a phantom. Of course, as WM correctly insists, asserting that there are oo naturals starting with number 1 directly implies that there is a natural oo, since the nth is always equal to n, and saying there are n many is equivalent to saying there is an nth one in any sequential ordering, which is the last. In any case, it's quite reversible, and well-defined. 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. >>> >> That would cause inconsistencies. :) > > And 00*0=1 wouldn't cause inconsistencies? > I*i=1 doesn't. Well, as long as you know it's not the imaginary i.... >>>>> 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. >>> >> Which is so completely different from you, of course... > > Difference is that I demonstrate the truth of what I'm talking about > in mechanically reduced exhaustive terms whereas what you talk about > is just speculative. > You speculate that it's agreed that not is the universal truth. It's not. >>>>> 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. >>> >> You might be surprised at how it relates to science. Where does mass >> come from, anyway? > > Not from number rings and real number lines that's for sure. > Are you sure? What thoughts have you given to cyclical processes? >>>>> 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. >>> >> Right, and they're not terribly significant. > > Which takes us right back to derivatives without identity ratios. > Okay, make it interesting.... >>>>> 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. >>> >> It's the non-rigorous primitive thought patterns of fools like Leibniz >> and Newton that is so scoffed at today. > > May be. Arithmetic just is what it is, a derivative of geometric > subdivision. > I agree that roots of meaning lie in geometry, and of expression in language. >>>>> 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. >>> >> Those aren't geometrical expressions of addition, but iterative >> operations expressed linguistically. > > Which means what exactly, that they aren't arithmetic axioms forming > the foundation of modern math? The whole problem is that they don't > produce straight lines or colinear straight line segments as claimed. > Uh, yeah, 'cause they're not expressed gemoetrically. >>>> 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. >>> >> So, start with the straight line: > > How? By assumption? As far as I know the only way to produce straight > lines is through Newton's method of drawing tangents to curves. That > means we start with curves and derivatives not straight lines.And that > means we start with curved surfaces and intersections between them. > Take long string and tie to two sticks, tight. >> R exists. > > Nice but still an axiomatic assumption of truth. > A declaration as foundation: "Assume A" >>>> 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. >>> >> Science doesn't prove anything true. > > Sure it does. That's the purpose of science. Empiricism and modern > math don't prove anything true. Mysticism in action. That's why modern > mathematikers consider themselves neo platonists. They're just divines > who intuit the truth and what's true and false and go on from there. > Um, same with us Scientifikers...sorry... >> It only eliminates false >> hypotheses. > > Well it would certainly do that if it quite knew what was true and > false to begin with. Strictly speaking contemporary science eliminates > hypotheses which contradict axiomatic assumptions of truth. But that > doesn't mean it eliminates hypotheses which are false because it just > doesn't know what is false in mechanically reduced exhaustive terms. > Again, define "mechanics"? And, this time, don't tell me it means that everything is derived from not, while you're proving that everything is derived from not, and complaining about circularity. :) >> That's the way science works. If you want to declare >> ultimate truth with certainty, ask god for a revelation, but don't call >> it science. > > It's modern mathematikers and empirics who ask gods for revelations. I > concentrate on demonstrating what's true and false in mechanically > reduced exhaustive terms of finite tautological regression to self > contradictory alternatives. Whole nuther kettle of fish. > Smells a little familiar... > Your position on science and math seems to be that either we proceed > according to naive and mechanically unreduced and inexhaustive > assumptions of truth or we proceed by appeals to divine revelation. > Six of one half dozen of the other. > Nah, I'll take two of the first, and five loaves...no need for the other. >>>>> 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. >>> >> Their size is finite for any finite number of subdivisions. > > And it continues to be finite for any infinite number of subdivisions > as well.The finitude of subdivisions isn't related to their number but > to the mechanical nature of bisective subdivision. > Only to a Zenoite. Once you have unmeasurable subintervals, you have bisected a finite segment an unmeasurable number of times. >>> 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. >>> >> Sure, so n subdivisions produces 2^n subintervals. > > Equal subdivisions. That's what gets us cardinal numbers. > Sure, n iterations of subdivision yield 2^n equal and generally mutually exclusive subintervals. >>>> 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. >>> >> It's the same as Peano. > > Not it isn't, Tony. Cumulative addition doesn't produce straight lines > or even colinear straight line segments. Some forty odd years ago at > the Academy one of my engineering professors pointed out that just > because there is a stasis across a boundary doesn't necessarily mean > that there is no flow across the boundary only that the net flow back > and forth is zero.I've always been impressed by the line of reasoning. The question is whether adding an infinite number of finite segments yields an infinite distance. > > In other words modern mathematikers just assume that because the Peano > and suc( ) axioms produce successive straight line segments between > numbers there is some kind of guarantee that the successive straight > line segments will themselves line up colinearly on straight line > segments and that we can thus just assume or infer the existence of > straight line segments and straight lines from those axioms.Doesn't > happen that way because even if we assume the existence of straight > line segments between numbers that doesn't demand successive segments > align in any particular direction colinearly along any common straight > line segment. Same principle as above, different application. > "Straight" doesn't even seem to mean anything in the context of Peano... >> Add 1 to a finite, and you get a finite, so >> adding 1 can never produce an infinite value, right? > > Finite addition never produces infinites in magnitude any more than > bisection produces infinitesimals in magnitude. It's the process which > is infinite or infinitesimal and not the magnitude of results. Results > of infinite addition or infinite bisection are always finite. > >> Wrong. > > Sure I'm wrong, Tony. Because you say so? > Because the results you toe up to only hold in the finite case. You can start with 0, or anything in the "finite" arena, the countable neighborhood around 0, and if you add some infinite value a finite number of times, or a finite value some infinite number of times, you're going to get an infinite product. If your set is one of cumulative sets of increments, like the naturals, then any infinite set is going to count its way up to infinite values. >> Add 1 n >> times to 0 and you get n. If n is infinite, then n is infinite. > > This is reasoning per say instead of per se. > Pro se, even. If the first natural is 1, then the nth is n, and if there are n of them, there's an nth, and it's a member of the set. Just ask Mueckenheim. >> If n is >> infinite, so is 2^n. If you actually perform an infinite number of >> subdivisions, then you get actually infinitesimal subintervals. > > And if the process is infinitesimal subdivision every interval you get > is infinitesimal per se because it's the result of a process of > infinitesimal subdivision and not because its magnitude is > infinitesimal as distinct from the process itself. It's because it's the result of an actually infinite sequence of finite subdivisions. One can also perform some infinite subdivision in some finite step or so, but that's a little too hocus-pocus to prove. In the meantime, we have at least potentially infinite sequences of subdivisions, increments, hyperdimensionalities, or whatever... > > Just ask yourself, Tony, at what magic point do intervals become > infinitesimal instead of finite? Your answer should be magnitudes > become infintesimal when subdivision becomes infinite. Yes. But the term > "infinite" just means undefined and in point of fact doesn't become > infinite until intervals become zero in magnitude. But that never > happens. But, but, but. No, "infinite" means "greater than any finite number" and infinitesimal means "less than any finite number", where "less" means "closer to 0" and "more" means "farther from 0". And until it does the magnitude of subdivisions remains > finite. The fact that there is a limit to the process doesn't mean the > process itself ever reaches that limit or ever can reach that limit. > There is a definable limit, but that limit is not reached. Because you cannot convey a point in every language with a finite alphabet in a finite string, this means that point doesn't exist? Then, no points exist, since there is always a number system which requires an infinite number of bits to specify that point. >>>> 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. >>> >> Bijections have their applications. I just don't think bijection alone >> is all that significant for infinite sets. The actual mapping function >> describes the relationship between infinite sets. > > Well if you mean "matching" don't say "bijection". I don't have any > use for people whose only purpose in math is terminological regression > and the creation of buzzwords instead of mechanical reduction. First > they say they can't use generic language because it isn't sufficiently > precise then they turn right around and corrupt the usage of perfectly > acceptable generic words such as "cardinality" on the same basis. Such > "mathematikers" are just speaking in tongues. They don't understand > what truth is so they just proclaim whatever they say is mathematical > truth because the domain of their discussion is supposed to be truth. > It's the proper and commonly used term. "Matching" is more colloquial and not well defined. A bijection between sets is a relation where each element of each set corresponds to a unique element of the other set. The mapping function is this relation, and a function from x to y can be inverted to form a function from y to x. Where the function is expressed as a formula, that formula characterizes the relative sizes of the sets. The only restriction with this approach is that the bijection must be in quantitative order for both sets. >>>>> 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. >>> >> You don't really seem interested in demonstrations of truth, are you? > > What demonstrations of truth did you have in mind exactly, Tony? All > I've seen so far are your ideas of truth per say and not per se.When I > demonstrate truth the demonstration is per se by exhaustive mechanical > reduction and not simply per say according to what seems plausible to > me or anyone else just because I say so. What I don't seem interested > in at the moment are more philosophical tracts when I've already shown > the demonstration of universal truth by finite tautological reduction > to self contradictory alternatives whereas all you've demonstrated is > philosophical preferences for some variety of ideas apart form others. > It would help if you could define a predicate, or mechanically demonstrate how "not" is universally true, instead of just axiomatically assuming it and subsequently deriving the fact through circular logic, without even defining what "truth" means to begin with. If you want to deal mathematically with logic, why don't you start by listing all the possible states a statement can have with respect to "truth"? Can it be true? Can it be false? Is there something in between? > If I don't seem particularly interested in demonstrations of universal > truth it's partly because you aren't doing any and I've already done > the only ones which can matter. It's rather like the problem of 1+1=2 > or the rac trisection of general angles. Once demonstrated in reduced > mechanically exhaustive terms the problem if not its explication and > implications loses interest. If you want to argue the problem itself > go ahead. Just don't expect me to be interested in whether 1+1=2 or > whether you can trisect general angles. You assume OR in defining AND, and then derive OR from AND, all the while claiming all you've done is NOT. > > What you're trying to do is argue the problem and not its resolution > or my demonstration of the universal truth of the problem. And I just > don't care what you think about the problem of universal truth when > you refuse to discuss any demonstration of the problem of universal > truth or my demonstration of universal truth in mechanically reduced > exhaustive terms. > > What difference can your opinions on the subject possibly make? I > argue A and you come right back and say A can't possibly be true > because you like B. You can't even say whether A and B are really > different. All you say is that you like binary logic and conjunctions. > Well I like them too. I just say that they're specialized instances of > universal truth demonstrated through finite tautological regression to > self contradictory alternatives and so far you have yet to adduce any > arguments to the contrary much less any demonstration to the contrary. > > So I suppose the short answer is no I don't really seem to care about > problems I've already solved. > Alright. Take your basic assumptions and build something useful, or derive a new, or old, result from them. >>>>> 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. >>> >> Do you not assume anything? You sure do. You assume "not" is universally >> true. > > No I don't, Tony. I certainly do not assume "not" is universally true. > I demonstrate "not" is universally true only to the extent "not not" > is self contradictory and self contradiction is universally false. > So you assume "not not" is self contradictory, even though that sentence no verb, so it not statement. "not not" is generally taken like "--", as the negation of negation, and therefore taken as positive. So, that assumption doesn't ring true. That's the root issue with this. >> It's universally meaningless in isolation. not(x) simply means >> "complement of x" or "1-x". You assume something else to begin with, >> which is not demonstrably true. > > No I don't, Tony.I demonstrate the universal truth of "not" per se in > mechanically exhaustive terms through finite tautological reduction to > self contradictory alternatives which I take to be false to the extent > they're self contradictory. If you want to argue the demonstration per > se that's one thing but if you simply want to revisit and rehash the > problem per say without arguing the demonstration per se that's > another because it's a problem per say I have no further interest in > unless you can successfully argue against the demonstration per se. > not(not("not not")) "not not" is not self-contradictory-and-therefore-false. > This is why science is so useful because you stop arguing isolated > problems to argue demonstrations instead which subsume those isolated > problems. There's simply no point to arguing such problems > individually as to whether "not" is universally true of everything or > whether there are such things as conjunctions not reducible to "not" > in mechanically exhaustive terms unless the demonstration itself is > defective and not true. And just claiming so per say won't cut it. > Your "not a not b" has an assumed OR in it. >>>>>> 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. >> How have you demonstrated any truth? > > See above. If you have per se arguments to show the demonstration > itself is defective I'm all ears. But I've no interest in just per say > assertions that the demonstration is defective without showing why and > how. And naive assertions to the contrary of the demonstration are of > no interest unless you can demonstrate how your assertions to the > contrary invalidate the demonstration. In other words I'm just not > interested in assertions to the effect that tautologies aren't what I > take them to be since these kinds of terminological disputes don't > address the mechanical characteristics involved and can always be > resolved by the use of other terms. Nor am I interested in assertions > such as the demonstration is incorrect because conjunctions are > necessary to tautologies unless you can show how my demonstration that > conjunctions are mechanically reducible to the application of > successive combinations of "not" is defective. > See above. >>> 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. >> Been there and done that, but you seem to reject anything that is >> "establishment gospel", even if exhaustively analyzed for truth. > > "Been there and done what" exactly, Tony? "Exhaustive analysis for > truth" is not at all the same as "exhaustive demonstrations of truth". > All "exhaustive analysis for truth" means or can mean is that you've > looked the problem over and can find nothing amiss. It just doesn't > matter whether the "you" is just yourself or a godzillion others when > you don't have any demonstrable basis for truth to begin with. "You" > can't very well analyze anything for truth when you don't know exactly > what's true or may be true. All you can really do is guess.And that's > exactly what seers, mystics, and empirics do because that's all they > can do. > It helps if you at least define your terms. >>>> 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...) >>>> >> If you're interested in the breakdown of binary logical operators, lemme >> know... > > I have no special interest in things studied in college, Tony. They're > well known and well demonstrated in their own terms and just not a > substitute for truth in exhaustive mechanical terms. > They didn't teach me this analysis of the logical operators in college, just the mechanics. This particular analysis is independent and fairly recent. >> 01oo >> >>>>>> 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. > > ~v~~ 01oo
From: Lester Zick on 29 Mar 2007 14:34 On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Hi Lester - > >Glad you responded. I was afraid I put you off. This thread seems to >have petered unlike previous ones I've participated in with you. I hope >that's not entirely discouraging, as I think you have a "point" in >saying points don't have meaning without lines, and that the subsequent >definition of lines as such-and-such a set of points is somewhat >circular. Personally, I think you need to come to grips with the >universal circularity, including on the level of logic. Points and lines >can be defined with respect to each other, and not be mutually >contradictory. But, maybe I speak too soon, lemme see... Hey, Tony - Yeah I guess I'm a glutton for punishment with these turkeys. The trick is to get finite regressions instead of circular definitions. We just can't say something like lines are the set of all points on lines because that's logically ambiguous and doesn't define anything. I don't mind if we don't know exactly what points are in exhaustive terms just that we can't use them to define what defines them in the intersection of lines and in the first place. The problem isn't mathematical it's logical. In mathematics we try to ascertain truth in exhaustively demonstrable terms. That's what distinguishes mathematics from physics and mathematicians from mathematikers and empirics. (By the way, Tony, I'm chopping up these replies for easier access and better responsiveness.) ~v~~
From: Lester Zick on 29 Mar 2007 14:58
On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >> Problem is that 0 and 1 are finites but so are lots of other numbers >> and your original contention was that 00*0=finites not 1 so that you >> still haven't clarified the process involved in 00*0 that makes it 1 >> and not some other finite or which makes the process reversible. >> > >0 is not really finite, but The Origin. It is not a finite distance from >The Origin, because there is no distance between it and itself. 0 is >less than any finite, or infinitesimal, distance. What I meant is I*i=1. >Oh, that's much better. Infinity times iota equals 1. :) > >So, here's how it hangs. In any interval of the real line, [x, x+1) or >(x,x+1], we have oo reals. Each real will then be assumed to occupy 1/oo >of this line, and if the length of this line is oo, then there will be >oo^2 reals on the line, instead of 2^aleph_0, as if aleph_0 means >anything anyway. There's no smallest infinite any more than a smallest >nonzero finite, or infinitesimal. > >It is a simple assumption that subtracting a positive number from any >other decreases it. One thing you may notice is that somehow >aleph_0-1=aleph_0. There is no smallest infinity, though that's what >aleph_0's supposed to be. Aleph_0 is a phantom. > >Of course, as WM correctly insists, asserting that there are oo naturals >starting with number 1 directly implies that there is a natural oo, >since the nth is always equal to n, and saying there are n many is >equivalent to saying there is an nth one in any sequential ordering, >which is the last. > >In any case, it's quite reversible, and well-defined. Sure, Tony. But only because you say it is and not because you show how any of the mechanics associated with subtraction, addition, and multiplication, and division are the same as those in ordinary finite mathematics. In other words it's a lot more than just saying 00*0=1 and presumably that 00=1/0 and 0=1/00. You're mixing up finites and things you call infinites without defining them in terms which are mechanically reciprocally exhaustive and true of each other. In this regard you can't just say 00*0=1 without showing the extent to which infinites like 00 and 0 can participate in ordinary finite arithmetic operations with 1 and other finites and do so unambiguously. There is a reason division of finites like n by zero are not defined. It's because any n*0=0 so that finite division by zero is ambiguous. In other words any n*0=0 so we can't just reverse the operation concluding n/0= any specific value. Infinites mean in-finite or not defined with respect to magnitude. And the only way we can address relations between zeroes and in-finites is through L'Hospital's rule where derivatives are not zero or in-finite. And all I see you doing is sketching a series of rules you imagine are obeyed by some of the things you talk about without however integrating them mechanically with others of the things you and others talk about. It really doesn't matter whether you put them within the interval 0-1 instead of at the end of the number line if there are conflicting mechanical properties preventing them from lying together on any straight line segment. ~v~~ |