Isn't it clear that the integers
1, 2, 3, 4, ..., 98, 99, 100, 101, 102
and their smaller counterparts
0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
are merely differing by the usage of a little dot in the notation?

Doesn't this suggest that the reuse of the mechanism by which we develop large numbers to develop small numbers could simplify the development of numbers smaller than unity and the introduction of those interstices between the large integers as for instance
234.567 ?

The utter perfection of all number systems and the proof of such is compromised by the use of epsilon/delta as their witness position. This system is no different. The simplicity gained does seem grand; as in slash away virtually all of the mumblings and stumblings of the twentieth century. Return to numerical computations as school children can do. All is well, and operators needn't come into numbers any more. This is the travesty of modern mathematics: it is the failure to admit that things like the square root of two are an element and an operator. Let's face it: when a tree falls in the woods it does make a sound whether you were there to hear it or not. It may have fallen due to internal rot. It may have fallen from a beaver gnawing at it. It could have been a man with an axe that finished it off. The way that one goes about these things is to go and witness it for yourself.

The usage of operators as generational to an elemental number system is a conflict. We cannot mix operators and numbers at this early stage, for then we will not have any basis to work from. That the mathematician could so poison his own pot is readily established. Drinking down the broth does little to stop the problem. As elements are claimed to be sets so again the mathematician has snared himself in his own trap. As sets lack order there again how can any thinking mathematician claim to be able to do a set intersection cleanly? May the farce be with you.

tisdag 15 juni 2021 kl. 15:44:09 UTC+2 skrev timba...@gmail.com:
> Isn't it clear that the integers
> 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
> and their smaller counterparts
> 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
> are merely differing by the usage of a little dot in the notation?
>
> Doesn't this suggest that the reuse of the mechanism by which we develop large numbers to develop small numbers could simplify the development of numbers smaller than unity and the introduction of those interstices between the large integers as for instance
> 234.567 ?
>
> The utter perfection of all number systems and the proof of such is compromised by the use of epsilon/delta as their witness position. This system is no different. The simplicity gained does seem grand; as in slash away virtually all of the mumblings and stumblings of the twentieth century. Return to numerical computations as school children can do. All is well, and operators needn't come into numbers any more. This is the travesty of modern mathematics: it is the failure to admit that things like the square root of two are an element and an operator. Let's face it: when a tree falls in the woods it does make a sound whether you were there to hear it or not. It may have fallen due to internal rot. It may have fallen from a beaver gnawing at it. It could have been a man with an axe that finished it off. The way that one goes about these things is to go and witness it for yourself.
>
> The usage of operators as generational to an elemental number system is a conflict. We cannot mix operators and numbers at this early stage, for then we will not have any basis to work from. That the mathematician could so poison his own pot is readily established. Drinking down the broth does little to stop the problem. As elements are claimed to be sets so again the mathematician has snared himself in his own trap. As sets lack order there again how can any thinking mathematician claim to be able to do a set intersection cleanly? May the farce be with you.

you keep conflating notation with actual things

On Tuesday, 15 June 2021 at 09:44:09 UTC-4, timba...@gmail.com wrote:
> Isn't it clear that the integers
> 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
> and their smaller counterparts
> 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
> are merely differing by the usage of a little dot in the notation?

The usage gives different meaning. I don't understand your problem.

>
> Doesn't this suggest that the reuse of the mechanism by which we develop large numbers to develop small numbers could simplify the development of numbers smaller than unity and the introduction of those interstices between the large integers as for instance
> 234.567 ?

Radix systems were designed to eliminate having to learn the theory of fractions and arithmetic. Essentially, dumbing down the concepts so that stupid people can learn to use numbers and arithmetic admittedly not in a perfect way.

>
> The utter perfection of all number systems and the proof of such is compromised by the use of epsilon/delta as their witness position. This system is no different. The simplicity gained does seem grand; as in slash away virtually all of the mumblings and stumblings of the twentieth century. Return to numerical computations as school children can do. All is well, and operators needn't come into numbers any more. This is the travesty of modern mathematics: it is the failure to admit that things like the square root of two are an element and an operator. Let's face it: when a tree falls in the woods it does make a sound whether you were there to hear it or not. It may have fallen due to internal rot. It may have fallen from a beaver gnawing at it. It could have been a man with an axe that finished it off. The way that one goes about these things is to go and witness it for yourself.
>
> The usage of operators as generational to an elemental number system is a conflict. We cannot mix operators and numbers at this early stage, for then we will not have any basis to work from. That the mathematician could so poison his own pot is readily established. Drinking down the broth does little to stop the problem. As elements are claimed to be sets so again the mathematician has snared himself in his own trap. As sets lack order there again how can any thinking mathematician claim to be able to do a set intersection cleanly? May the farce be with you.

Well, the use of operators is justified in say sqrt(4) because it evaluates to the rational number 2. This is a far cry from writing sqrt(2) which is not a number of any kind. sqrt(2) is a NAME for the incommensurable magnitude that is realised IF AND ONLY IF, one attempts to measure a square's diagonal using its side.

> you keep conflating notation with actual things

Right zelos... your i is only a notation it is not real...

Mitchell Raemsch

onsdag 16 juni 2021 kl. 22:45:47 UTC+2 skrev mitchr...@gmail.com:
> > you keep conflating notation with actual things
> Right zelos... your i is only a notation it is not real...
>
> Mitchell Raemsch

i is a complex number and not in the real numbers.

but you know what else isn't real? Your infinitesimal :)

On Wednesday, June 16, 2021 at 11:24:55 AM UTC-4, Eram semper recta wrote:
> On Tuesday, 15 June 2021 at 09:44:09 UTC-4, timba...@gmail.com wrote:
> > Isn't it clear that the integers
> > 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
> > and their smaller counterparts
> > 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
> > are merely differing by the usage of a little dot in the notation?
> The usage gives different meaning. I don't understand your problem.

There is not a problem. This is a simplification. Simplification is good.
In effect the rational numbers are not needed. This means that we do not need to rely upon a non-fundamental operator (division) in the construction of smaller numbers. We simply take the digital form seriously, and again under epsilon/delta we land in the same space as everyone else with their pages of verbosity claiming to prove the continuum. This same should actually occur should we for instance have asked Dedekind to work his cuts on an actual value such as the square root of two. All of the rhetoric collapses as we chase digits.

The idea that much of higher mathematics deserves such a collapse: here is a crux worthy of consideration. If mathematics is fundamental in nature then why should it accumulate such a conglomeration of terminology and separate branches that only assimilate via more accumulation? Compactness was sought after, yet the status quo system is divided into sectors that do not step upon each other. The inability of any to challenge what has been laid down on paper seems very well established and yet this is the conundrum of the human race in action at a position that claims immunity to that very situation. In fact no immunity can be granted: no free ticket away from human behavior is divined by declaring oneself a mathematician. In fact the burden which the mathematician carries ought to be quite heavy here under this awareness. Yet no such burden is under discussion even as en mass hypnosis is readily proven to exist.

There should be a bunch of readers here eventually who will declare that we are engaged in a progression. That a part of this progression is accumulative: this is obvious. Yet are we really to accept that the accumulation is going to grow in proportion to the quantity of PhDs in the pipeworks of academia? Is this authentic mathematics? This situation really does send us back to a time when individual works were lauded. Descartes for instance takes a very different position than todays redigested regurgitant. The level of mimicry which ensues in the current academic environment is narrow rather than wide. As one operates and attempts to fit into the pile of accumulation within a fully established branch with its own linguistic mimicry fully enforced by censors your only option is that same. Here at least we can attempt something quite different.

As to when exactly this over-abundance of verbosity began: hundreds of years ago certainly. On the other side of the coin to what degree should any concerned mathematician take the right to speak to his peers? Should the censors of the existing journals allow a fresh branch and so a fresh and lucrative publication? What if it was done as an act of kindness? These would be human concerns applied onto a problem of informational nature. More importantly if the progression of mathematics is getting stifled in an accumulation of assumptions that are preventing further freedoms and generalizations from taking place then certainly the snake is eating its own tail. The inability of the accumulated subject to receive criticism is due exactly to the assumption of immunity which all here will accept as a false assumption. Such false assumptions tend to generate rather a lot of bugs. That psychology, physics, and philosophy have to be the concern of the mathematician; this is conclusive. As to how much psychology has already been applied in this field... I suggest that the highly unmathematical has already taken place too many times. Here we see the usenet dodge over and over again, and in a medium that is right close to the printed form that the work has always been done in. Could it be that such dodges have gone on in the past?

It is as if we are working with an incomplete set of tools. Blind to the one that we grapple with and out of thin air we attempt a solution. If that solution works partially then it ought to be celebrated. Yet when that partial solution is imposed as an assumption for all to imprint upon under threat of failure and over the course of the formidable decades of ones life; back when your mind was fresh and vibrant; such a waste of humanity has taken place globally. Ultimately we each must demand the freedom to construct as an axiomatic stance that is so basic as to deserve a name as the 'axiom of choice'... and yet that language is taken already. Too much language and so much taken already badly named... and we are supposed to absorb this quagmire? No.

> >
> > Doesn't this suggest that the reuse of the mechanism by which we develop large numbers to develop small numbers could simplify the development of numbers smaller than unity and the introduction of those interstices between the large integers as for instance
> > 234.567 ?
> Radix systems were designed to eliminate having to learn the theory of fractions and arithmetic. Essentially, dumbing down the concepts so that stupid people can learn to use numbers and arithmetic admittedly not in a perfect way.
> >
> > The utter perfection of all number systems and the proof of such is compromised by the use of epsilon/delta as their witness position. This system is no different. The simplicity gained does seem grand; as in slash away virtually all of the mumblings and stumblings of the twentieth century. Return to numerical computations as school children can do. All is well, and operators needn't come into numbers any more. This is the travesty of modern mathematics: it is the failure to admit that things like the square root of two are an element and an operator. Let's face it: when a tree falls in the woods it does make a sound whether you were there to hear it or not. It may have fallen due to internal rot. It may have fallen from a beaver gnawing at it. It could have been a man with an axe that finished it off. The way that one goes about these things is to go and witness it for yourself.
> >
> > The usage of operators as generational to an elemental number system is a conflict. We cannot mix operators and numbers at this early stage, for then we will not have any basis to work from. That the mathematician could so poison his own pot is readily established. Drinking down the broth does little to stop the problem. As elements are claimed to be sets so again the mathematician has snared himself in his own trap. As sets lack order there again how can any thinking mathematician claim to be able to do a set intersection cleanly? May the farce be with you.
> Well, the use of operators is justified in say sqrt(4) because it evaluates to the rational number 2. This is a far cry from writing sqrt(2) which is not a number of any kind. sqrt(2) is a NAME for the incommensurable magnitude that is realised IF AND ONLY IF, one attempts to measure a square's diagonal using its side.

Timothy Golden expressed precisely :
> On Wednesday, June 16, 2021 at 11:24:55 AM UTC-4, Eram semper recta wrote:
>> On Tuesday, 15 June 2021 at 09:44:09 UTC-4, timba...@gmail.com wrote:
>>> Isn't it clear that the integers
>>> 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
>>> and their smaller counterparts
>>> 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
>>> are merely differing by the usage of a little dot in the notation?
>> The usage gives different meaning. I don't understand your problem.
>
> There is not a problem. This is a simplification. Simplification is good.
> In effect the rational numbers are not needed. This means that we do not need
> to rely upon a non-fundamental operator (division) in the construction of
> smaller numbers.

Oversimplification can be bad though.

In this case, rational fractions in convergent series are used to
classify such real numbers, algebraic and transcendental, as you
envision in their CDEs and CFEs. Seems a top down as opposed to bottom
up - deconstuctive rather than foundational - philosophy of what leads
to what. It is irrational to treat Q with such wanton disregard.

On Thursday, June 17, 2021 at 10:25:32 AM UTC-4, FromTheRafters wrote:
> Timothy Golden expressed precisely :
> > On Wednesday, June 16, 2021 at 11:24:55 AM UTC-4, Eram semper recta wrote:
> >> On Tuesday, 15 June 2021 at 09:44:09 UTC-4, timba...@gmail.com wrote:
> >>> Isn't it clear that the integers
> >>> 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
> >>> and their smaller counterparts
> >>> 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
> >>> are merely differing by the usage of a little dot in the notation?
> >> The usage gives different meaning. I don't understand your problem.
> >
> > There is not a problem. This is a simplification. Simplification is good.
> > In effect the rational numbers are not needed. This means that we do not need
> > to rely upon a non-fundamental operator (division) in the construction of
> > smaller numbers.
> Oversimplification can be bad though.
>
> In this case, rational fractions in convergent series are used to
> classify such real numbers, algebraic and transcendental, as you
> envision in their CDEs and CFEs. Seems a top down as opposed to bottom
> up - deconstuctive rather than foundational - philosophy of what leads
> to what. It is irrational to treat Q with such wanton disregard.

Well thanks Rafters for the serious response. The continuum according to Dedekind and his cuts is guaranteed by epsilon/delta, which works perfectly fine as we chase those digits past the decimal point. It is wanton that modern mathematics arrives at the ring definition; claims the reals are ring behaved; all the while relying upon division within the formal construction. That operators and values are distinct concepts is not for the traditional mathematician is it? They much prefer a mix of the two so that the distinction between them is invalid. As to what this means for existing mathematical theory: I am fairly sure it secures bugs in the curriculum. A claim that the reals are ring behaved while they were constructed through the usage of division as an operator is a fine instance of such a bug. There are more, but this is a good start.

For myself as I regard polysign numbers through
s x
where s is discrete sign and x is continuous magnitude I am somewhat burdened to define that x portion, though I don't see it as controversial. Now though, as I dig through operator theory and attempt to disambiguate the accumulation it is obvious that the rationals are devolved. For instance to work in thirds and puzzle over the value one third we could simply check into the radix three number system and see that one third simply takes the representation 0.1 in radix-3 language. Meanwhile in radix ten notation there is no problem simply letting epsilon/delta in on the value 0.333. NOt sure how cde/cfe acronyms matter here. Equivalence can be had in a close enough is good enough way, and if you'd like more then take it. The buck stops there for everyone who believes in the continuum.

Timothy Golden brought next idea :
> On Thursday, June 17, 2021 at 10:25:32 AM UTC-4, FromTheRafters wrote:
>> Timothy Golden expressed precisely :
>>> On Wednesday, June 16, 2021 at 11:24:55 AM UTC-4, Eram semper recta wrote:
>>>> On Tuesday, 15 June 2021 at 09:44:09 UTC-4, timba...@gmail.com wrote:
>>>>> Isn't it clear that the integers
>>>>> 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
>>>>> and their smaller counterparts
>>>>> 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
>>>>> are merely differing by the usage of a little dot in the notation?
>>>> The usage gives different meaning. I don't understand your problem.
>>>
>>> There is not a problem. This is a simplification. Simplification is good.
>>> In effect the rational numbers are not needed. This means that we do not
>>> need to rely upon a non-fundamental operator (division) in the
>>> construction of smaller numbers.
>> Oversimplification can be bad though.
>>
>> In this case, rational fractions in convergent series are used to
>> classify such real numbers, algebraic and transcendental, as you
>> envision in their CDEs and CFEs. Seems a top down as opposed to bottom
>> up - deconstuctive rather than foundational - philosophy of what leads
>> to what. It is irrational to treat Q with such wanton disregard.
>
> Well thanks Rafters for the serious response. The continuum according to
> Dedekind and his cuts is guaranteed by epsilon/delta, which works perfectly
> fine as we chase those digits past the decimal point. It is wanton that
> modern mathematics arrives at the ring definition; claims the reals are ring
> behaved; all the while relying upon division within the formal construction.

From my POV the reals are a field, that is they are more than a ring.

Bottom up.

Naturals have closure for addition but not the inverse; you can't even
subtract a number from itself unless you include zero in the set
(necessary but not sufficient for an additve group structure). Even
then, it is still a partial function - we need closure for subtraction,
so we include negative numbers and call the set the set of integers. It
forms an abelian group a necessary sub-structure of a ring. You can
always multiply two integers and get a result in the integers, but
division is only a partial function. Not quite a multiplicative group.

Multiplication can be viewed as a systematic repeated addition, it has
closure, but the inverse does not (so it is a ring, not quite a field
yet) so we add inverse elements to obtain closure. Now we have a field
(more than a ring) and within it a ring (more than a group) and within
it an additive abelian group and a (not necessarily abelian)
multiplicative group.

Subtraction is just inverse addition and division is just inverse
multiplication. Once we have that, along with some other features for
groups and rings, we have a field. Then, within fields and within rings
and within groups we have sub-structures which are isomorphic to these.

The reals incorporate all of these structures through the embeddings,
the reals are "ring behaved" because that "behavior" is embedded from
the integers into the 'larger' structure of the field. Other elements
can have the same structure (isomorphisms) and can be treated the same
way and as such are "ring behaved" as you say.

> That operators and values are distinct concepts is not for the traditional
> mathematician is it? They much prefer a mix of the two so that the
> distinction between them is invalid.

Sometimes underlying isomorphic structures are abstracted. It is like
what I will call 'shuffling' for the time being. Three card monte,
walnut and pea shell game, nickels under styrofoam cups, tennis balls
with bells for cats or squeaky whistle ports for dog toys, running
chainsaws -- juggling reduced to merely taking the outside right hand
object and placing it in between the other two then the left then right
etcetera.

The shuffle.

ABC
ACB
CAB
CBA
BCA
BAC
ABC
....

We don't care about any of the bells or whistles inside the tennis
balls, or the objects being simply identical cups, unique nutshells, or
different weight chainsaws when we wish to address only the shuffling
aspect.

If you want to fill out the multiplication table with familiar
'numerals' for a multiplicative group, you would need division -- you
have it from below (bottom up) whereas you expect it from above (top
down) it seems.

> As to what this means for existing
> mathematical theory: I am fairly sure it secures bugs in the curriculum. A
> claim that the reals are ring behaved while they were constructed through the
> usage of division as an operator is a fine instance of such a bug. There are
> more, but this is a good start.

To me it is not about having division but about having the inverse
elements needed no matter what symbol(s) eventually get used to
describe or handle them. Your polynomial ring is a good example, the
symbol zero is used in place of the actual zero polynomial and the
symbol one is used for the polynomial one because the abstraction
cleans things up on the chalkboard and the internals (what zero
actually means in expanded polynomial form) is like the bells in the
tennis ball - it has been shown to not matter when shuffling.

Okay, that crap above is not bullet proof, but the best I could do for
now. It is just my opinion, I am not a mathematician by any stretch of
the word.

On Thursday, June 17, 2021 at 6:55:37 PM UTC-4, FromTheRafters wrote:
> Timothy Golden brought next idea :
> > On Thursday, June 17, 2021 at 10:25:32 AM UTC-4, FromTheRafters wrote:
> >> Timothy Golden expressed precisely :
> >>> On Wednesday, June 16, 2021 at 11:24:55 AM UTC-4, Eram semper recta wrote:
> >>>> On Tuesday, 15 June 2021 at 09:44:09 UTC-4, timba...@gmail.com wrote:
> >>>>> Isn't it clear that the integers
> >>>>> 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
> >>>>> and their smaller counterparts
> >>>>> 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
> >>>>> are merely differing by the usage of a little dot in the notation?
> >>>> The usage gives different meaning. I don't understand your problem.
> >>>
> >>> There is not a problem. This is a simplification. Simplification is good.
> >>> In effect the rational numbers are not needed. This means that we do not
> >>> need to rely upon a non-fundamental operator (division) in the
> >>> construction of smaller numbers.
> >> Oversimplification can be bad though.
> >>
> >> In this case, rational fractions in convergent series are used to
> >> classify such real numbers, algebraic and transcendental, as you
> >> envision in their CDEs and CFEs. Seems a top down as opposed to bottom
> >> up - deconstuctive rather than foundational - philosophy of what leads
> >> to what. It is irrational to treat Q with such wanton disregard.
> >
> > Well thanks Rafters for the serious response. The continuum according to
> > Dedekind and his cuts is guaranteed by epsilon/delta, which works perfectly
> > fine as we chase those digits past the decimal point. It is wanton that
> > modern mathematics arrives at the ring definition; claims the reals are ring
> > behaved; all the while relying upon division within the formal construction.
> From my POV the reals are a field, that is they are more than a ring.
>
> Bottom up.
>
> Naturals have closure for addition but not the inverse; you can't even
> subtract a number from itself unless you include zero in the set
> (necessary but not sufficient for an additve group structure). Even
> then, it is still a partial function - we need closure for subtraction,
> so we include negative numbers and call the set the set of integers. It
> forms an abelian group a necessary sub-structure of a ring. You can
> always multiply two integers and get a result in the integers, but
> division is only a partial function. Not quite a multiplicative group.
>
> Multiplication can be viewed as a systematic repeated addition, it has
> closure, but the inverse does not (so it is a ring, not quite a field
> yet) so we add inverse elements to obtain closure. Now we have a field
> (more than a ring) and within it a ring (more than a group) and within
> it an additive abelian group and a (not necessarily abelian)
> multiplicative group.
>
> Subtraction is just inverse addition and division is just inverse
> multiplication. Once we have that, along with some other features for
> groups and rings, we have a field. Then, within fields and within rings
> and within groups we have sub-structures which are isomorphic to these.
>
> The reals incorporate all of these structures through the embeddings,
> the reals are "ring behaved" because that "behavior" is embedded from
> the integers into the 'larger' structure of the field. Other elements
> can have the same structure (isomorphisms) and can be treated the same
> way and as such are "ring behaved" as you say.
> > That operators and values are distinct concepts is not for the traditional
> > mathematician is it? They much prefer a mix of the two so that the
> > distinction between them is invalid.
> Sometimes underlying isomorphic structures are abstracted. It is like
> what I will call 'shuffling' for the time being. Three card monte,
> walnut and pea shell game, nickels under styrofoam cups, tennis balls
> with bells for cats or squeaky whistle ports for dog toys, running
> chainsaws -- juggling reduced to merely taking the outside right hand
> object and placing it in between the other two then the left then right
> etcetera.
>
> The shuffle.
>
> ABC
> ACB
> CAB
> CBA
> BCA
> BAC
> ABC
> ...
>
> We don't care about any of the bells or whistles inside the tennis
> balls, or the objects being simply identical cups, unique nutshells, or
> different weight chainsaws when we wish to address only the shuffling
> aspect.
>
> If you want to fill out the multiplication table with familiar
> 'numerals' for a multiplicative group, you would need division -- you
> have it from below (bottom up) whereas you expect it from above (top
> down) it seems.
> > As to what this means for existing
> > mathematical theory: I am fairly sure it secures bugs in the curriculum.. A
> > claim that the reals are ring behaved while they were constructed through the
> > usage of division as an operator is a fine instance of such a bug. There are
> > more, but this is a good start.
> To me it is not about having division but about having the inverse
> elements needed no matter what symbol(s) eventually get used to
> describe or handle them. Your polynomial ring is a good example, the
> symbol zero is used in place of the actual zero polynomial and the
> symbol one is used for the polynomial one because the abstraction
> cleans things up on the chalkboard and the internals (what zero
> actually means in expanded polynomial form) is like the bells in the
> tennis ball - it has been shown to not matter when shuffling.
>
> Okay, that crap above is not bullet proof, but the best I could do for
> now. It is just my opinion, I am not a mathematician by any stretch of
> the word.

I think you are one of the most coherent regular posters here. Still you have dodged a crux of my argument and I suppose that is the strong part. Firstly though, on the field: if as you say division is defined via the product and subtraction via the sum, and I do believe that these reverse operators do deserve this interpretation, then they are not fundamental and do not deserve the treatment (nor the exceptions) that are required in the field definition. In effect the ring is sufficient, and yet as the ring definition describes operators it does nothing to tell us of their exact computations. This is something that school children learn and the 'high' mathematicians need not burden themselves with such mechanistic procedures, eh? Anyway, the problem of division and the enlargement of exceptions beyond zero into non-zero instances will be necessary to go general dimensional, whereas the product continues its good behaviors both algebraically and computationally.. This is fairly advanced mathematics outside of the ordinary communications here but well established within a branch known as associative algebra. I don't honestly understand how they can prove it but it does work. In effect all higher dimensional forms that carry good algebraic behavior will fit the progression
R, C, RxC, CxC, RxCxC, ...
with products simply defined thence if any one of these components is a zero the dimension will collapse, even though the value in its entirety is arguably non-zero. In effect the field requirements when established up in these high dimensional spaces will have to continue to grow exceptions simply due to zero valued sub-products. This is cause to reject the field requirements, beyond the earlier operator criticism.

The notion of values and operators being distinct things: this obvious truth continues to go betrayed by all here. Why is this discussion untouchable? While school children are graded on their very ability to firstly perform such computations and then for the priests to take a dodge on the whole matter: this is roughly a problem in a proportion that does paint the mathematician as a failed escapist; a purist run awry. Like Newton decoding his bible, or Godel starving himself to death; do such instances possibly infect the whole society en masse?

There are so many issues to bump into around here in this region of awareness. Dimensional analysis is one, and when the rational number as composed of two unique numbers and an operator (division) then are the real numbers two dimensional in fact? Is division thencely introduced as a fundamental operator? Isn't the product actually the fundamental operator? Shouldn't we admit that in the construction of numbers that we ought to admit that operators are to work upon numbers rather than be built into numbers? Is the blur that we witness absorbed into mathematics at such a base level going to be trouble? If we can do without this conflict then shouldn't we? Yes. And this returns us once again to the computations that school children know.

On Thursday, 17 June 2021 at 09:18:17 UTC-4, timba...@gmail.com wrote:
> On Wednesday, June 16, 2021 at 11:24:55 AM UTC-4, Eram semper recta wrote:
> > On Tuesday, 15 June 2021 at 09:44:09 UTC-4, timba...@gmail.com wrote:
> > > Isn't it clear that the integers
> > > 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
> > > and their smaller counterparts
> > > 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
> > > are merely differing by the usage of a little dot in the notation?
> > The usage gives different meaning. I don't understand your problem.
> There is not a problem. This is a simplification. Simplification is good.

I don't see any simplification. Decimal representation is for those who do not care to understand the concept of number.

> In effect the rational numbers are not needed.

WHAT?! Whatever you do in decimal relies on rational numbers, you imbecile! Without rational numbers, you don't have NUMBER. What part of this don't you get? LMAO.

> This means that we do not need to rely upon a non-fundamental operator (division) in the construction of smaller numbers.

There is no division operator, you moron. The vinculum is NOT a division operator. The obelus however, is a division operator and only applies when simplification is possible as in the case of p -:- q where p > q.

<drivel>

On Friday, June 18, 2021 at 11:18:09 AM UTC-4, Eram semper recta wrote:
> On Thursday, 17 June 2021 at 09:18:17 UTC-4, timba...@gmail.com wrote:
> > On Wednesday, June 16, 2021 at 11:24:55 AM UTC-4, Eram semper recta wrote:
> > > On Tuesday, 15 June 2021 at 09:44:09 UTC-4, timba...@gmail.com wrote:
> > > > Isn't it clear that the integers
> > > > 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
> > > > and their smaller counterparts
> > > > 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
> > > > are merely differing by the usage of a little dot in the notation?
> > > The usage gives different meaning. I don't understand your problem.
> > There is not a problem. This is a simplification. Simplification is good.
> I don't see any simplification. Decimal representation is for those who do not care to understand the concept of number.
> > In effect the rational numbers are not needed.
> WHAT?! Whatever you do in decimal relies on rational numbers, you imbecile! Without rational numbers, you don't have NUMBER. What part of this don't you get? LMAO.

No. I disagree. There are some details that were wiped over here, but the point to me is that the same mechanism which allows us to express large numbers such as
1234
which is one thousand two hundred and thirty four is in play as we work on extending this value on the right like:
1234.001
yet can't we see that already we have the freedom to move the little dot around?
123.4001
and that its position is a discrete quality within this symbolic notation? That what is left when we pull the decimal point is merely an integer that has no need of any rational construction? That the position of the decimal point can simply be another integer? That this is in fact how modern computers perform their computations?

Back in the traditional interpretation the purity of a value such as one third and the lack of correspondence in the decimal notation as say
0.333
is no different than the lack of purity that ensues when the mathematician considers the square root of two as a value in their set of real values. That the rational claims perfection when epsilon/delta is good enough for the other; no. Epsilon/delta is good enough for both. Simplicity is good. Mixing operators and numbers at the fundamental level of construction is bad. In fact it is a bug in all of higher mathematics. The inability of others here to confront this simple thought is indicative of the lameness. Numbers and operators are distinct concepts. They do not deserve to be blurred; particularly not in the stage of developing a number system. Operators work on numbers and yield other numbers. If they didn't then they wouldn't operate would they? Why can't anybody speak to this? This absurdity paints you as a flock and the wool is over the black sheep's eyes here as much as all the white sheeps'.

As I paint the simplicity of remaining in the integer form and topping it off with scientific notation it should be noted that the problems that ensue even under this simplification are considerable. Witness IEEE notation and the complexities that it undergoes as 128 bit floats and so forth have to become standards. As we analyze things like the Mandelbrot set and witness a product of say 10,000 terms in z and the loss of precision that takes place... are we looking at valid data or aren't we? On the edge it seems it could go either way, but just a bit, right?

On Friday, 18 June 2021 at 12:47:03 UTC-4, timba...@gmail.com wrote:
> On Friday, June 18, 2021 at 11:18:09 AM UTC-4, Eram semper recta wrote:
> > On Thursday, 17 June 2021 at 09:18:17 UTC-4, timba...@gmail.com wrote:
> > > On Wednesday, June 16, 2021 at 11:24:55 AM UTC-4, Eram semper recta wrote:
> > > > On Tuesday, 15 June 2021 at 09:44:09 UTC-4, timba...@gmail.com wrote:
> > > > > Isn't it clear that the integers
> > > > > 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
> > > > > and their smaller counterparts
> > > > > 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
> > > > > are merely differing by the usage of a little dot in the notation?
> > > > The usage gives different meaning. I don't understand your problem.
> > > There is not a problem. This is a simplification. Simplification is good.
> > I don't see any simplification. Decimal representation is for those who do not care to understand the concept of number.
> > > In effect the rational numbers are not needed.
> > WHAT?! Whatever you do in decimal relies on rational numbers, you imbecile! Without rational numbers, you don't have NUMBER. What part of this don't you get? LMAO.
> No. I disagree.

You can disagree all you like but it won't make your claims true.

> There are some details that were wiped over here,

You mean the hilarity of you claiming rational numbers are not needed? LMAO.. Yes, you swam over that one! Chuckle.

> but the point to me is that the same mechanism which allows us to express large numbers such as
> 1234
> which is one thousand two hundred and thirty four is in play as we work on extending this value on the right like:
> 1234.001
> yet can't we see that already we have the freedom to move the little dot around?
> 123.4001

Yes. But so what? "Moving the dot" around is the same as multiplying or dividing by an integral power of10.

Division for those who didn't learn about fractions:

123.4001 = 1234.001 -:- 10

Move dot left implies divide.
Move dot right implies multiply.

Am I missing something so utterly profound here that I can no longer be called a genius?! Ha, ha.

> and that its position is a discrete quality within this symbolic notation?

Not sure what you mean, but my genius mind tells me that you trying to say something about radix systems being positional. Well, yes they are. :) So what?

> That what is left when we pull the decimal point is merely an integer that has no need of any rational construction?

Gibberish. Sorry, I have to stop here, Timotheus. You need to learn how to express yourself properly but I suppose you first have to understand what you are talking about. Chuckle.

> That the position of the decimal point can simply be another integer? That this is in fact how modern computers perform their computations?
>
> Back in the traditional interpretation the purity of a value such as one third and the lack of correspondence in the decimal notation as say
> 0.333
> is no different than the lack of purity that ensues when the mathematician considers the square root of two as a value in their set of real values. That the rational claims perfection when epsilon/delta is good enough for the other; no. Epsilon/delta is good enough for both. Simplicity is good. Mixing operators and numbers at the fundamental level of construction is bad. In fact it is a bug in all of higher mathematics. The inability of others here to confront this simple thought is indicative of the lameness. Numbers and operators are distinct concepts. They do not deserve to be blurred; particularly not in the stage of developing a number system. Operators work on numbers and yield other numbers. If they didn't then they wouldn't operate would they? Why can't anybody speak to this? This absurdity paints you as a flock and the wool is over the black sheep's eyes here as much as all the white sheeps'.
>
> As I paint the simplicity of remaining in the integer form and topping it off with scientific notation it should be noted that the problems that ensue even under this simplification are considerable. Witness IEEE notation and the complexities that it undergoes as 128 bit floats and so forth have to become standards. As we analyze things like the Mandelbrot set and witness a product of say 10,000 terms in z and the loss of precision that takes place... are we looking at valid data or aren't we? On the edge it seems it could go either way, but just a bit, right?

Recalling from memory, as far as 2003 someone make the suggestion of the abstract polynomial.
Well, in the begining of the theory of congruences was a bit different, I guess Tim is more critical on
the later turn of abstract algebra (X), one may google galois imaginaire(or maybe, galois imaginaries).

Back to topic.

"Knuth has pointed out that truncation and rounding are the same operation in balanced ternary—they produce exactly
the same result (a property shared with other balanced numeral systems). The number 1/2 is not exceptional;
it has two equally valid representations, and two equally valid truncations: 0.1 (round to 0, and truncate to 0)
and 1.T (round to 1, and truncate to 1). With an odd radix, double rounding is also equivalent to directly
rounding to the final precision, unlike with an even radix." ( https://en.wikipedia.org/wiki/Balanced_ternary )
also https://simple.wikipedia.org/wiki/Affine_arithmetic , https://en.wikipedia.org/wiki/Saturation_arithmetic
or alike, propagation of errors or optimization maybe, why systems with redundant representations, mutibase, etc.

Your past mention to iconic arithmetic makes me remember the place-value vs sign-value debate(sign-value notation, Napier)
Or perhaps for what is intended the arithmetic, for humans or machines.

Your own view seems a bit Type theory.

You can imagine, for a moment, how would be, for example, if you translate to an epoch, like in old Egypt,
how is their arithmetic and computation,how are their "fractions", how is the accessible machinery to do
computations, or how is the sophisticated machinery to do computations (in that period), and the connection
with that machinery and the view of arithmetic of that time.

On Friday, June 18, 2021 at 6:44:35 PM UTC-4, michael Rodriguez wrote:
> Recalling from memory, as far as 2003 someone make the suggestion of the abstract polynomial.
> Well, in the begining of the theory of congruences was a bit different, I guess Tim is more critical on
> the later turn of abstract algebra (X), one may google galois imaginaire(or maybe, galois imaginaries).
>
> Back to topic.
>
> "Knuth has pointed out that truncation and rounding are the same operation in balanced ternary—they produce exactly
> the same result (a property shared with other balanced numeral systems). The number 1/2 is not exceptional;
> it has two equally valid representations, and two equally valid truncations: 0.1 (round to 0, and truncate to 0)
> and 1.T (round to 1, and truncate to 1). With an odd radix, double rounding is also equivalent to directly
> rounding to the final precision, unlike with an even radix." ( https://en..wikipedia.org/wiki/Balanced_ternary )
> also https://simple.wikipedia.org/wiki/Affine_arithmetic , https://en.wikipedia.org/wiki/Saturation_arithmetic
> or alike, propagation of errors or optimization maybe, why systems with redundant representations, mutibase, etc.

I was just entertaining the notion of ternary P3 as a simplification last couple of days actually. Not terribly convinced though. Still, for the sake of sharing if we have already stated that the sign is mod-3 why shouldn't the entire value be mod-3? This is sort of heading toward the discrete end of things whereas
s x
was originally thought of as discrete sign and continuous magnitude joined together. That we happen to use radix-10 notation for the x portion is happenstance... right? Here I could way-lay the conversation in the emergent conflicts of the usual radix interpretation, such as building large numbers from large numbers such as
123456789012
which is too large. I'll leave that cryptic here for the fun of it. The mechanism works fine, but it is a mechanical feature of the numbers which is extensive as opposed to the usual algebraic interpretation in powers of ten, which of course carries its own notational conundrum as every radix system is radix-10 in its own terms: this language is essentially meaningless. The number of conflicts here is rather much, and yet, like the conflicts of the polynomial in AA, things work out even while the theory has wronged itself. Polysign does disambiguate some of the system, and it does disambiguate some of their system.
That they would choose to insist upon an infinite dimensional basis to construct a two dimensional instance, for instance; is not necessary.

>
> Your past mention to iconic arithmetic makes me remember the place-value vs sign-value debate(sign-value notation, Napier)

Pretty neat find. Never seen it, or forgotten. That is some old math too: https://en.wikipedia.org/wiki/Rabdology
Very interesting. 1617 is getting really far back. Near the formalization of the real number.

> Or perhaps for what is intended the arithmetic, for humans or machines.
>
> Your own view seems a bit Type theory.
>
> You can imagine, for a moment, how would be, for example, if you translate to an epoch, like in old Egypt,
> how is their arithmetic and computation,how are their "fractions", how is the accessible machinery to do
> computations, or how is the sophisticated machinery to do computations (in that period), and the connection
> with that machinery and the view of arithmetic of that time.

Certainly, but the divorce of the higher mathematician from this level has some consequence too. Work in raw values obviously is in the instantiated form. Work that fails to instantiate is in danger of collapsing upon instantiation. I do believe that I see this occurring in modernity, and really, back in time, too. I understand that it sounds immature to think so primitively, yet without a primitive level to land on second order workings are dubious. Third order workings are complete trash, but for any ability to harken back to something instantiable.

On Friday, June 18, 2021 at 1:52:30 PM UTC-4, Eram semper recta wrote:
> On Friday, 18 June 2021 at 12:47:03 UTC-4, timba...@gmail.com wrote:
> > On Friday, June 18, 2021 at 11:18:09 AM UTC-4, Eram semper recta wrote:
> > > On Thursday, 17 June 2021 at 09:18:17 UTC-4, timba...@gmail.com wrote:
> > > > On Wednesday, June 16, 2021 at 11:24:55 AM UTC-4, Eram semper recta wrote:
> > > > > On Tuesday, 15 June 2021 at 09:44:09 UTC-4, timba...@gmail.com wrote:
> > > > > > Isn't it clear that the integers
> > > > > > 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
> > > > > > and their smaller counterparts
> > > > > > 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
> > > > > > are merely differing by the usage of a little dot in the notation?
> > > > > The usage gives different meaning. I don't understand your problem.
> > > > There is not a problem. This is a simplification. Simplification is good.
> > > I don't see any simplification. Decimal representation is for those who do not care to understand the concept of number.
> > > > In effect the rational numbers are not needed.
> > > WHAT?! Whatever you do in decimal relies on rational numbers, you imbecile! Without rational numbers, you don't have NUMBER. What part of this don't you get? LMAO.
> > No. I disagree.
> You can disagree all you like but it won't make your claims true.
> > There are some details that were wiped over here,
> You mean the hilarity of you claiming rational numbers are not needed? LMAO. Yes, you swam over that one! Chuckle.
> > but the point to me is that the same mechanism which allows us to express large numbers such as
> > 1234
> > which is one thousand two hundred and thirty four is in play as we work on extending this value on the right like:
> > 1234.001
> > yet can't we see that already we have the freedom to move the little dot around?
> > 123.4001
> Yes. But so what? "Moving the dot" around is the same as multiplying or dividing by an integral power of10.

I guess I will try to entertain the idea that your own idiocy is authentic. I rarely use such terse language, but as a reflectance it is meek still. After moving the little dot around enough do you not see that
1234001
is merely an integer and that
1234
is another integer that has the same digits at the head of the value? The little dot is quite literally just a little dot. It need not take the interpretation that you call for. That the little dot mimics exactly the interactions of radix digits under their modulo behavior: here is the crux. That the digital interpretation requires this mechanistic approach: I find this to be provable, though there are some interesting options under the hood.

For now, so as not to divert the conversation to badly, though of course such tangents ought to be savored if they can be presented convincingly, and I believe that they can, yet they will demand that we arrive at a mechanistic system rather than that algebraic system of thought through which you cannot see my argument. Any body of non-mathematical stature should be able to witness my argument as truthful. Having pulled the dot from the notation we can now reintroduce it onto the integer with another integer value:
1234001 e - 3
From a polysign persepective this could be opening quite a can of worms, yet for now to keep things simple and to recover polysign numbers in their ordinary form let's just admit that this digital representation is sufficient to form epsilon/delta convergence though it is requiring now two values. Still these values are of distinct type. Under multiplication in fact the second value will be added while the first will be multiplied. This happens to be exactly how we perform signed products:
( s1 x1 )( s2 x2 ) = ( s1 @ s2 )( x1 x2 )
where '@' implies the zero-signed sum(mod-3 in P3), which you could replace with '+' if you wish to remain in the cave man club(we need the zero sign instead of the mnemonic '+' as two for a fixed identity symbol).
To what degree is the introduction of general sign an introduction of another integer into the numerical format? As a numerical format aren't we now up to:
s x e
as a general form, where s is sign, x is an integer(which I would normally claim is a continuous magnitude), and e is this new integer which causes the interpretation of continuous x to have occurred previously. Well, to keep with that notation I should not be reusing x in that middle position. So let's just leave sxe as a mistake to be corrected to some transitional form
s n e
where n is an unsigned unexponented integer; a whole number that is arguably not so wholy when it has to be augmented with two others; one of which is normally two-signed (e) which is problematic to my own interpretation's purity. This has to be a transitional form because it is lacking the purity that we want. Possibly because we are in pure mathematics working this we can restrict e to simply being jumps from the right to the left, and certainly this is where we get that continuum sense, whereas the other way the values are scaling into the upward sizing, which the raw n can do anyways.

sne is still not a good enough format. The issuance of Pn as general cannot work will with n moving into the magnitude. Possibly
s m e
since 'm' as magnitude does have correspondence. Dimensional analysis says that we are engaging a three dimensional system here, though the s portion is quite limited normally. I don't really like it. I'm used to looking at the
continuous / discrete
paradigm and seeing that sx maps to it. Here we are landing in
discrete/discrete/discrete
though the qualities of each of these discretes is unique. I guess from this position which is the position of the creator of a numerical system to now state that epsilon/delta is the source of continuum feels like putting ones foot in one's mouth, which is at least better than the rectum's situation here. Still the context of a snake eating its own tail... the notion that the continuum can authentically be built from the discrete... no: I prefer sx where x is continuous by its own nature. Yet we have no representative as such other than true graphical measures. Is this then the human cause? That when we work material in our hands we deal in the continuum truly? That all our work on paper is sacrificed: yes, possibly, and the notion that physics is not separable from mathematics; that all of this is a human lie perpetrated by escapist perfectionists who have thrown away their actual dealing with the continuum. Have they likewise thrown away their qualities as tool-makers? Then have they thrown away their humanity? When did the post-human arrive? At least a couple hundred years ago it seems.

>
> Division for those who didn't learn about fractions:
>
> 123.4001 = 1234.001 -:- 10
>
> Move dot left implies divide.
> Move dot right implies multiply.
>
> Am I missing something so utterly profound here that I can no longer be called a genius?! Ha, ha.
> > and that its position is a discrete quality within this symbolic notation?
> Not sure what you mean, but my genius mind tells me that you trying to say something about radix systems being positional. Well, yes they are. :) So what?
> > That what is left when we pull the decimal point is merely an integer that has no need of any rational construction?
> Gibberish. Sorry, I have to stop here, Timotheus. You need to learn how to express yourself properly but I suppose you first have to understand what you are talking about. Chuckle.
> > That the position of the decimal point can simply be another integer? That this is in fact how modern computers perform their computations?
> >
> > Back in the traditional interpretation the purity of a value such as one third and the lack of correspondence in the decimal notation as say
> > 0.333
> > is no different than the lack of purity that ensues when the mathematician considers the square root of two as a value in their set of real values.. That the rational claims perfection when epsilon/delta is good enough for the other; no. Epsilon/delta is good enough for both. Simplicity is good. Mixing operators and numbers at the fundamental level of construction is bad. In fact it is a bug in all of higher mathematics. The inability of others here to confront this simple thought is indicative of the lameness. Numbers and operators are distinct concepts. They do not deserve to be blurred; particularly not in the stage of developing a number system. Operators work on numbers and yield other numbers. If they didn't then they wouldn't operate would they? Why can't anybody speak to this? This absurdity paints you as a flock and the wool is over the black sheep's eyes here as much as all the white sheeps'.
> >
> > As I paint the simplicity of remaining in the integer form and topping it off with scientific notation it should be noted that the problems that ensue even under this simplification are considerable. Witness IEEE notation and the complexities that it undergoes as 128 bit floats and so forth have to become standards. As we analyze things like the Mandelbrot set and witness a product of say 10,000 terms in z and the loss of precision that takes place... are we looking at valid data or aren't we? On the edge it seems it could go either way, but just a bit, right?

On Tuesday, June 15, 2021 at 9:44:09 AM UTC-4, Timothy Golden wrote:
> Isn't it clear that the integers
> 1, 2, 3, 4, ..., 98, 99, 100, 101, 102
> and their smaller counterparts
> 0.01, 0.02, 0.03, 0.04, ..., 0.98, 0.99, 1.00, 1.01, 1.02
> are merely differing by the usage of a little dot in the notation?
>
> Doesn't this suggest that the reuse of the mechanism by which we develop large numbers to develop small numbers could simplify the development of numbers smaller than unity and the introduction of those interstices between the large integers as for instance
> 234.567 ?
>
> The utter perfection of all number systems and the proof of such is compromised by the use of epsilon/delta as their witness position. This system is no different. The simplicity gained does seem grand; as in slash away virtually all of the mumblings and stumblings of the twentieth century. Return to numerical computations as school children can do. All is well, and operators needn't come into numbers any more. This is the travesty of modern mathematics: it is the failure to admit that things like the square root of two are an element and an operator. Let's face it: when a tree falls in the woods it does make a sound whether you were there to hear it or not. It may have fallen due to internal rot. It may have fallen from a beaver gnawing at it. It could have been a man with an axe that finished it off. The way that one goes about these things is to go and witness it for yourself.
>
> The usage of operators as generational to an elemental number system is a conflict. We cannot mix operators and numbers at this early stage, for then we will not have any basis to work from. That the mathematician could so poison his own pot is readily established. Drinking down the broth does little to stop the problem. As elements are claimed to be sets so again the mathematician has snared himself in his own trap. As sets lack order there again how can any thinking mathematician claim to be able to do a set intersection cleanly? May the farce be with you.

"But I see nothing meritorious — and this was just as far from Dirichlet's thought — in actually performing this wearisome circumlocution and insisting on the use and recognition of no other than rational numbers."

- Richard Dedekind