The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 28 Apr 2010 00:17 William Elliot wrote: > On Mon, 26 Apr 2010, Nam Nguyen wrote: >>>> >> The correct versions are: >> >> GC(x) <-> (x is even >=4) -> Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)] > > By that definition, GC(pi) is true. The correct definition is: > > GC(x) <-> > (x is even integer) & (x >= 4 -> Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)]) Thanks for the correction. In fact cGC(x) should be similarly corrected: cGC(x) <-> (x is even integer) & (x >= 4 -> ~Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)]) > >> (*)P <-> Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ (x < y))] >> >>>>> First observation: if we have any intuition about the naturals >>>>> then we'd also have the intuition that we can't know the >>>>> arithmetic truth or falsehood of (1). >> > Based upon that intuition, I'll concur that we don't know > the truth or falsehood of (1). "Don't know" is actually not a precise notion in reasoning. Today we might be in a "don't know" state of something that's still possible to know, but tomorrow we might know it. Otoh, "never know", "can't know", "impossible" is a precise notion: it means it's not possible to know within the reasoning framework. For example, it's impossible to know through syntactical proof the syntactical consistency of a consistent formal system. But that's your claim and not mine and so we could instead concentrate the arguments on what my observation would conclude: "we can't know the arithmetic truth or falsehood of (1)", which is what you mentioned below. > > Based upon that intuition, you claim we'll never know > the truth or falsehood of (1). > > That's an unsubstatiated claim. > Give some insights why I should accept it. OK. This won't be short and I'm still organizing my thoughts so I hope you don't mind giving me a bit more time on this. But I will substantiate the conclusion.
From: Tim Golden BandTech.com on 28 Apr 2010 09:53 On Apr 26, 9:37 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Apr 24, 1:12 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > > > Aatu Koskensilta wrote: > > > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > > >> First observation: if we have any intuition about the naturals > > >> then we'd also have the intuition that we can't know the arithmetic > > >> truth or falsehood of (1). > > > > This isn't an observation. It's a bald assertion that is, on the face of > > > it, vague, implausible, and entirely arbitrary. > > > For what it's worth, I wish Torkel Franzen were still with us so that > > arguments about foundational issues of reasoning don't deteriorate > > into Inquisition-like decrees, blasting the opponent' arguments without > > due analysis of what he has repeatedly said with some technical details, > > post after posts. > > > The kind of decrees you have done to my arguments - posts after posts. > > > I don't mind _if_ you point out precisely what's wrong with my arguments, > > with the basis of my arguments. Many times in many threads you've only > > made vague, subjective, decrees like what you've made above, without > > due respect to the points, counter points I've made or raised, and then > > kept silent. And then come back later to just make another similar "blasting" > > ones on the same subjects, without any analysis at all. How frustrating > > it is arguing with you! > > > As much as I respect you knowledge on mathematical formalism, I wish that > > when you don't substantially have anything to argue with me, you'd kind > > of keep in mind Wittgenstein's wisdom: > > > "Wovon man nicht sprechan kann, darüber muss man schweigen"! > > > *** > > In any rate, what are you clear, plausible, specific technical reasons > > for your belief that the knowledge of the naturals aren't of intuitive > > nature, or that you'd know the arithmetic truth or falsehood of (1)? > > > Iow, what are your _technical, non-bias, objective_ grounds for attacking > > my observation above? > > I'm not honestly able to follow your OP, but do take interest in the > idea. > I do not accept that the continuum is built out of a discrete basis, > as the traditional number theory develops it. Instead, granting a > continuum, it is much easier to yield the discrete. > On the observed continuum we see no standard unity value; it is > instead arbitrarily chosen. > > Near to the natural numbers are numbers with modulo behaviors, which > are in some ways simplifications of the natural number, where the > natural number arguably is within this class of the modulo family as > modulo infinity. So we could start out considering the mod-1, then the > mod-2, mod-3, ..., and wind up at the natural number eventually, its > predecessors being simpler and so more fundamental. Rather, the > components of the construction being more fundamental and yielding the > family which contains simpler predecessors. Even a modern computer has > an upper limit to how high it can count. Yes, its getting huge, but it > will never be infinite. > > These lower mod systems when coupled with continuous magnitude > generate raw support for spacetime (emergent spacetime) with > unidirectional time: > http://bandtechnology.com/PolySigned/index.html > > As far as I know you have the freedom to construct what you wish in > mathematics. If others don't like then so it is. Getting them to > understand what you are saying is more the point of the communication. > I think you should take the constructional freedom, and if somebody > isn't convinced of a step then it should shake out so that at least > both people have expressed themselves as cleanly as possible, and > perhaps so that others come to understand the two sides better. When I > think of my 'intuition of the naturals' I see two things: > 3 = 1 1 1 > and > 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 > will get so large that I won't be sure how large it is, but I feel > certain that it is large. > The next step in technology is actually the usage of modulo symbolism > within the number's representation, and so my argument on the modulo > family may bear heavily to some formal mathematician. We are taught > first to count and then to use the variable. Can it be any other way? > The variable whose qualities are unknown does not seem valid. The > variable whose qualities are limited does seem more valid. > > - Tim The usage of infinity in the lower mod systems is somewhat gone. For instance consider a mod 1 + 1 + 1 system. This is a mod three system, but since we are playing with number systems this language is less obscure, since every system is arguably a mod 10 system within its own native symbolism. Does the number three exist in a mod three system? Standard modulo math replaces the third element of a counting sequence with a zero 0, 1, 2, 0, 1, 2, ... but we are free to consider the symbology 1, 2, 3, 1, 2, 3, ... as pretty much the same meaning. I feel comfortable considering that even the standard left to right sequence with no end can be destroyed under the modulo form, so that the ellipsis suggesting continuation might better be replaced with a different symbol 0, 1, 2; or some such structural suggestion that this is a tied loop with an identifiable beginning element. In that unity is a better constructor than zero to get off the ground with then the choice to go with 1, 2, 3; makes a bit more sense, or even 1, 2, 0; This last feels the strongest to me, but I'm not arguing for any convention. The point to me is that the modulo forms are not built out of the natural numbers. They are simpler than the natural numbers and so deserve their own place. We should construct more complicated things from simpler things, and not the other way around. This is somewhat the structured thought from software engineering applied to mathematics. These modulo systems form a large family and with their sequencings coupled we see the chinese remainder theorem in a raw form from the get go. These numbers provide a dance akin to counting but with a much prettier coverage. The modulo one form or modulo ' yields the repetitious sequence 1, 1, 1, 1, ... or in the structural form 1; or 0; and is very close by to the grouping counting structure of 3 = 1 1 1 . If you see the mod-1 system as conflicted then that is good, for it presents a zero dimensional geometry when treated as sign. This is no different than the apparent conflict of the n-verticed simplex when n reaches unity, whose conflict is a remnant of cartesian real valued thinking. The need of radix usage (compound numbers like '231') within this consideration is not yet present. Simply consider the question: how high would you like to count to? Then assign an arbitrary sequence of unique symbols that exceeds or meets that count. This is like marbles in a bag method, but one step further along where the symbol's physical significance matches a quantity of marbles in a bag. This is a very primitive level. The polynomial of abstract algebra is nearby but relies upon trickery to get its yield. This may be an issue more of information theory than of mathematics per se, though the crossover must be undeniable. In that the mathematician may rely upon informational representations as grants then those qualities do deserve scrutiny, especially when their meanings become redundant. Statements like 'integers modulus infinity, integers modulus one' lose their meaning if we take the modulus forms as more primitive. From this perspective the natural numbers take the mod-infinity form, where every value has a unique symbollic representation. At some point a modern human language runs out of language to specify these large numbers and all that will spew out is a bunch of smaller numbers. One systematically structured form is the radix ten representation. Others exist and if we found that mother nature uses one then it probably isn't the one that we use. That would help explain the conundrums that humans have created. I believe that mother nature takes advantage of the progression; of the family of these mod-n systems. Their cyclic code needs to be taken to the continuum at which point the polysign number arrives. - Tim
From: A on 28 Apr 2010 12:15 On Apr 28, 9:53 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > The usage of infinity in the lower mod systems is somewhat gone. For > instance consider a > mod 1 + 1 + 1 > system. This is a mod three system, but since we are playing with > number systems this language is less obscure, since every system is > arguably a mod 10 system within its own native symbolism. Does the > number three exist in a mod three system? Standard modulo math > replaces the third element of a counting sequence with a zero > 0, 1, 2, 0, 1, 2, ... > but we are free to consider the symbology > 1, 2, 3, 1, 2, 3, ... > as pretty much the same meaning. I feel comfortable considering that > even the standard left to right sequence with no end can be destroyed > under the modulo form, so that the ellipsis suggesting continuation > might better be replaced with a different symbol > 0, 1, 2; > or some such structural suggestion that this is a tied loop with an > identifiable beginning element. In that unity is a better constructor > than zero to get off the ground with then the choice to go with > 1, 2, 3; > makes a bit more sense, or even > 1, 2, 0; > This last feels the strongest to me, but I'm not arguing for any > convention. The point to me is that the modulo forms are not built out > of the natural numbers. They are simpler than the natural numbers and > so deserve their own place. We should construct more complicated The integers modulo n are obtained from the semiring of natural numbers by quotienting out the ideal generated by n. So the integers modulo n are indeed built out of the natural numbers. If you suggest that the natural numbers, including their addition and multiplication, can be constructed out of the integers modulo n for every n, please tell us how this works. I think it can be done but it is not as simple as you might at first guess. > things from simpler things, and not the other way around. This is > somewhat the structured thought from software engineering applied to > mathematics. These modulo systems form a large family and with their > sequencings coupled we see the chinese remainder theorem in a raw form > from the get go. These numbers provide a dance akin to counting but > with a much prettier coverage. > > The modulo one form or > modulo ' > yields the repetitious sequence > 1, 1, 1, 1, ... > or in the structural form > 1; > or > 0; > and is very close by to the grouping counting structure of > 3 = 1 1 1 . > If you see the mod-1 system as conflicted then that is good, for it > presents a zero dimensional geometry when treated as sign. This is no > different than the apparent conflict of the n-verticed simplex when n > reaches unity, whose conflict is a remnant of cartesian real valued > thinking. > > The need of radix usage (compound numbers like '231') within this > consideration is not yet present. Simply consider the question: how > high would you like to count to? Then assign an arbitrary sequence of > unique symbols that exceeds or meets that count. This is like marbles > in a bag method, but one step further along where the symbol's > physical significance matches a quantity of marbles in a bag. This is > a very primitive level. The polynomial of abstract algebra is nearby > but relies upon trickery to get its yield. > > This may be an issue more of information theory than of mathematics > per se, though the crossover must be undeniable. In that the > mathematician may rely upon informational representations as grants > then those qualities do deserve scrutiny, especially when their > meanings become redundant. Statements like > 'integers modulus infinity, integers modulus one' > lose their meaning if we take the modulus forms as more primitive. > From this perspective the natural numbers take the mod-infinity form, > where every value has a unique symbollic representation. At some point > a modern human language runs out of language to specify these large > numbers and all that will spew out is a bunch of smaller numbers. One > systematically structured form is the radix ten representation. Others > exist and if we found that mother nature uses one then it probably > isn't the one that we use. > That would help explain the conundrums that humans have created. I > believe that mother nature takes advantage of the progression; of the > family of these mod-n systems. Their cyclic code needs to be taken to > the continuum at which point the polysign number arrives. > > - Tim
From: Tim Golden BandTech.com on 29 Apr 2010 07:37 On Apr 28, 12:15 pm, A <anonymous.rubbert...(a)yahoo.com> wrote: > On Apr 28, 9:53 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > wrote: > > > > > The usage of infinity in the lower mod systems is somewhat gone. For > > instance consider a > > mod 1 + 1 + 1 > > system. This is a mod three system, but since we are playing with > > number systems this language is less obscure, since every system is > > arguably a mod 10 system within its own native symbolism. Does the > > number three exist in a mod three system? Standard modulo math > > replaces the third element of a counting sequence with a zero > > 0, 1, 2, 0, 1, 2, ... > > but we are free to consider the symbology > > 1, 2, 3, 1, 2, 3, ... > > as pretty much the same meaning. I feel comfortable considering that > > even the standard left to right sequence with no end can be destroyed > > under the modulo form, so that the ellipsis suggesting continuation > > might better be replaced with a different symbol > > 0, 1, 2; > > or some such structural suggestion that this is a tied loop with an > > identifiable beginning element. In that unity is a better constructor > > than zero to get off the ground with then the choice to go with > > 1, 2, 3; > > makes a bit more sense, or even > > 1, 2, 0; > > This last feels the strongest to me, but I'm not arguing for any > > convention. The point to me is that the modulo forms are not built out > > of the natural numbers. They are simpler than the natural numbers and > > so deserve their own place. We should construct more complicated > > The integers modulo n are obtained from the semiring of natural > numbers by quotienting out the ideal generated by n. So the integers > modulo n are indeed built out of the natural numbers. This is a fine instance of using a very complicated construction to develop a form that is simpler than what you started with. Modulo three numbers are much simpler than the natural numbers. Why should you require so much top heavy math to construct them? Is that really the way that you think of such a limited construction? I'm sorry, but the mechanism of the modulo math is so simple and primitive that it does not deserve this treatment. If a gradeschool child can understand them should it take college mathematics to describe them? > > If you suggest that the natural numbers, including their addition and > multiplication, can be constructed out of the integers modulo n for > every n, please tell us how this works. I think it can be done but it > is not as simple as you might at first guess. Well, we see that the product should not wrap, so the concept of a square has to be accomodated somehow without using the natural numbers if we are to construct them. In other words the mod-m space will handle arithmetic multiply up to n without wrapping, where n is less than the sqrt(m). It's probably just better to consider the progression mod-1, mod-2, mod-3, mod-4, mod-5, ... The progression reaches toward mod-infinity where we can see that any finite value will not have looped. So long as we use variables with finite values and operators that yield finite values out of finite values then we ought to be OK. Then we just look at n+1 and allow for the generalization on every n. Besides the argument of simplicity (above), I believe that the language of abstract algebra is flawed. The X of the polynomial which allows for the ring quotient manipulation is farcical. Still, it does formally introduce the operators, which is worth discussing. Back in real analysis there was a development of numbers too, right? It did not bother to worry much about the operators in such a formal way. Was the real analysis method wrong? This is becoming a dissection of the structure of mathematics, and I will argue that simpler things should not necessitate the construction of more complicated things to recover those simpler things. Either of us might be guilty of making use of the natural number prior to its construction unless we get very formal. I see your request as valid but don't see it as a requirement to make my statement. Concrete values of large numbers will require usage of modulo principles to express them, and so the variable form will never be instantiable using modern representation, if the modulo form is truly built out of the natural number. To count to three we have no need of the value four. - Tim > > > things from simpler things, and not the other way around. This is > > somewhat the structured thought from software engineering applied to > > mathematics. These modulo systems form a large family and with their > > sequencings coupled we see the chinese remainder theorem in a raw form > > from the get go. These numbers provide a dance akin to counting but > > with a much prettier coverage. > > > The modulo one form or > > modulo ' > > yields the repetitious sequence > > 1, 1, 1, 1, ... > > or in the structural form > > 1; > > or > > 0; > > and is very close by to the grouping counting structure of > > 3 = 1 1 1 . > > If you see the mod-1 system as conflicted then that is good, for it > > presents a zero dimensional geometry when treated as sign. This is no > > different than the apparent conflict of the n-verticed simplex when n > > reaches unity, whose conflict is a remnant of cartesian real valued > > thinking. > > > The need of radix usage (compound numbers like '231') within this > > consideration is not yet present. Simply consider the question: how > > high would you like to count to? Then assign an arbitrary sequence of > > unique symbols that exceeds or meets that count. This is like marbles > > in a bag method, but one step further along where the symbol's > > physical significance matches a quantity of marbles in a bag. This is > > a very primitive level. The polynomial of abstract algebra is nearby > > but relies upon trickery to get its yield. > > > This may be an issue more of information theory than of mathematics > > per se, though the crossover must be undeniable. In that the > > mathematician may rely upon informational representations as grants > > then those qualities do deserve scrutiny, especially when their > > meanings become redundant. Statements like > > 'integers modulus infinity, integers modulus one' > > lose their meaning if we take the modulus forms as more primitive. > > From this perspective the natural numbers take the mod-infinity form, > > where every value has a unique symbollic representation. At some point > > a modern human language runs out of language to specify these large > > numbers and all that will spew out is a bunch of smaller numbers. One > > systematically structured form is the radix ten representation. Others > > exist and if we found that mother nature uses one then it probably > > isn't the one that we use. > > That would help explain the conundrums that humans have created. I > > believe that mother nature takes advantage of the progression; of the > > family of these mod-n systems. Their cyclic code needs to be taken to > > the continuum at which point the polysign number arrives. > > > - Tim
From: A on 29 Apr 2010 11:14
On Apr 29, 7:37 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Apr 28, 12:15 pm, A <anonymous.rubbert...(a)yahoo.com> wrote: > > > > > On Apr 28, 9:53 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > > wrote: > > > > The usage of infinity in the lower mod systems is somewhat gone. For > > > instance consider a > > > mod 1 + 1 + 1 > > > system. This is a mod three system, but since we are playing with > > > number systems this language is less obscure, since every system is > > > arguably a mod 10 system within its own native symbolism. Does the > > > number three exist in a mod three system? Standard modulo math > > > replaces the third element of a counting sequence with a zero > > > 0, 1, 2, 0, 1, 2, ... > > > but we are free to consider the symbology > > > 1, 2, 3, 1, 2, 3, ... > > > as pretty much the same meaning. I feel comfortable considering that > > > even the standard left to right sequence with no end can be destroyed > > > under the modulo form, so that the ellipsis suggesting continuation > > > might better be replaced with a different symbol > > > 0, 1, 2; > > > or some such structural suggestion that this is a tied loop with an > > > identifiable beginning element. In that unity is a better constructor > > > than zero to get off the ground with then the choice to go with > > > 1, 2, 3; > > > makes a bit more sense, or even > > > 1, 2, 0; > > > This last feels the strongest to me, but I'm not arguing for any > > > convention. The point to me is that the modulo forms are not built out > > > of the natural numbers. They are simpler than the natural numbers and > > > so deserve their own place. We should construct more complicated > > > The integers modulo n are obtained from the semiring of natural > > numbers by quotienting out the ideal generated by n. So the integers > > modulo n are indeed built out of the natural numbers. > > This is a fine instance of using a very complicated construction to > develop a form that is simpler than what you started with. Modulo > three numbers are much simpler than the natural numbers. Why should > you require so much top heavy math to construct them? Is that really > the way that you think of such a limited construction? I'm sorry, but > the mechanism of the modulo math is so simple and primitive that it > does not deserve this treatment. If a gradeschool child can understand > them should it take college mathematics to describe them? > > > > > If you suggest that the natural numbers, including their addition and > > multiplication, can be constructed out of the integers modulo n for > > every n, please tell us how this works. I think it can be done but it > > is not as simple as you might at first guess. > > Well, we see that the product should not wrap, so the concept of a > square has to be accomodated somehow without using the natural numbers > if we are to construct them. In other words the mod-m space will > handle arithmetic multiply up to n without wrapping, where n is less > than the sqrt(m). It's probably just better to consider the > progression > mod-1, mod-2, mod-3, mod-4, mod-5, ... > The progression reaches toward mod-infinity where we can see that any > finite value will not have looped. So long as we use variables with > finite values and operators that yield finite values out of finite > values then we ought to be OK. Then we just look at n+1 and allow for > the generalization on every n. > I do not see anywhere here where you have described the multiplication on the integers modulo n without referring to the multiplication on the natural numbers. I don't think this is anywhere near as simple as you seem to think. > Besides the argument of simplicity (above), I believe that the > language of abstract algebra is flawed. The X of the polynomial which > allows for the ring quotient manipulation is farcical. Every undergraduate mathematics major takes an abstract algebra course in which they prove basic facts about R[X] and quotients of R[X]. If you think it's "farcical," that's too bad--it's a very useful thing to be able to do. > Still, it does formally introduce the operators, which is worth > discussing. Back in real analysis there was a development of numbers > too, right? It did not bother to worry much about the operators in > such a formal way. Was the real analysis method wrong? > > This is becoming a dissection of the structure of mathematics, and I > will argue that simpler things should not necessitate the construction > of more complicated things to recover those simpler things. Either of Let's just bring it back to this point: you contend that the integers modulo n are "simpler" than the natural numbers. I think this isn't true, if you want to have the addition and the multiplication on the integers modulo n. How do you describe the multiplication on the integers modulo n without referring to the multiplication on the natural numbers? > us might be guilty of making use of the natural number prior to its > construction unless we get very formal. I see your request as valid > but don't see it as a requirement to make my statement. Concrete > values of large numbers will require usage of modulo principles to > express them, and so the variable form will never be instantiable > using modern representation, if the modulo form is truly built out of > the natural number. To count to three we have no need of the value > four. > > - Tim > > > > > > things from simpler things, and not the other way around. This is > > > somewhat the structured thought from software engineering applied to > > > mathematics. These modulo systems form a large family and with their > > > sequencings coupled we see the chinese remainder theorem in a raw form > > > from the get go. These numbers provide a dance akin to counting but > > > with a much prettier coverage. > > > > The modulo one form or > > > modulo ' > > > yields the repetitious sequence > > > 1, 1, 1, 1, ... > > > or in the structural form > > > 1; > > > or > > > 0; > > > and is very close by to the grouping counting structure of > > > 3 = 1 1 1 . > > > If you see the mod-1 system as conflicted then that is good, for it > > > presents a zero dimensional geometry when treated as sign. This is no > > > different than the apparent conflict of the n-verticed simplex when n > > > reaches unity, whose conflict is a remnant of cartesian real valued > > > thinking. > > > > The need of radix usage (compound numbers like '231') within this > > > consideration is not yet present. Simply consider the question: how > > > high would you like to count to? Then assign an arbitrary sequence of > > > unique symbols that exceeds or meets that count. This is like marbles > > > in a bag method, but one step further along where the symbol's > > > physical significance matches a quantity of marbles in a bag. This is > > > a very primitive level. The polynomial of abstract algebra is nearby > > > but relies upon trickery to get its yield. > > > > This may be an issue more of information theory than of mathematics > > > per se, though the crossover must be undeniable. In that the > > > mathematician may rely upon informational representations as grants > > > then those qualities do deserve scrutiny, especially when their > > > meanings become redundant. Statements like > > > 'integers modulus infinity, integers modulus one' > > > lose their meaning if we take the modulus forms as more primitive. > > > From this perspective the natural numbers take the mod-infinity form, > > > where every value has a unique symbollic representation. At some point > > > a modern human language runs out of language to specify these large > > > numbers and all that will spew out is a bunch of smaller numbers. One > > > systematically structured form is the radix ten representation. Others > > > exist and if we found that mother nature uses one then it probably > > > isn't the one that we use. > > > That would help explain the conundrums that humans have created. I > > > believe that mother nature takes advantage of the progression; of the > > > family of these mod-n systems. Their cyclic code needs to be taken to > > > the continuum at which point the polysign number arrives. > > > > - Tim > > |