Within the buildup of the reals we see a menagerie of numerical types.
Integers, rationals, and then irrationals.
If any were to be rejected it seems those irrationals were on the slate yet thanks to epsilon/delta they've made it in to seal the gaps.
Onward and upward we see even the reals subsumed as a subset of the complex numbers. There is an admitted dimensional rise to the plane and a two value format, yet back at the rational we have a two value format
a/b
and it is this dimensional fail first and foremost that could have caught some attention, yet historically this notion of dimension as we have it now did not exist.

"Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century.[9]"
- https://en.wikipedia.org/wiki/Decimal_separator
The decimal notation these days is considered to be rational. Yet when we witness a value such as
1.2345
and we remove the little dot we are looking at an integer. The meaning of the little dot; the decimal point; is in fact consistent with the preexistent radix ten notation. It is merely positing a unity position of an otherwise integer expression. The mechanics of computation with the decimal point has no need of the rational analysis that could be brought to bear. The fact that the symbolic notation is consistent with this analysis is obvious. If anything a new type is here tacked on to the integer type. It can be shown to be another integer, yet a much smaller integer most of the time. It can be an unsigned integer as well; unlike scientific notation.

In this day and age many have been exposed to structured programming of computers and data types. We perceive an informational conflict quite directly in terms of claiming a singular value to contain two values; obviously a single value does not contain two values. That division is the operation invoked here is not at all helpful either, for the product is the more fundamental over division. As we consider the introduction of operators to values it is clear that we cannot embed operators inside of values. This simplistic structural argument is offended by mathematicians going back to the very early stages of mathematics. Integer division does naturally yield a remainder as well; a second value; so no magic is taking place.

Well, this analysis is taken on an assumption of the real value as having fundamental status, which modern mathematicians go to great lengths to demonstrate. Now though we can at least get a date on the real value back to the mid 1600s. At that time the rational value was firmly established.

The structural argument deserves repetition: operators and values are distinct concepts and do not deserve to be mashed together; particularly not in the definition of a value. Yet mathematicians are practicing this mashup nearly universally.

The resolution as I see it is to allow in epsilon/delta analysis onto every number. The procedure of chasing down digits exposes quite a distinction between
4 versus 4.00000
which implies that the subsetting of number systems is not an accurate methodology. This then reflects on the continuum as a gray substrate of unknowns and unknowables down to infinite precision, which of course the mathematician has presumed of the paltry 4 instanced above. Clearly work on the continuum does suffer a precision issue. Ask any physicist and they will explain. Philosophically this argument exposes a false divide between the mathematician and the physicist. That division need not exist and in fact it does not exist. The mathematician's epsilon/delta thinking exposes this, and to reserve it only for certain values rather than apply it universally is a deep mistake. Chasing digits is what we do. Fortunately most times close enough is good enough.

On Sunday, October 10, 2021 at 8:27:48 AM UTC-7, timba...@gmail.com wrote:
> Within the buildup of the reals we see a menagerie of numerical types.
> Integers, rationals, and then irrationals.
> If any were to be rejected it seems those irrationals were on the slate yet thanks to epsilon/delta they've made it in to seal the gaps.
> Onward and upward we see even the reals subsumed as a subset of the complex numbers. There is an admitted dimensional rise to the plane and a two value format, yet back at the rational we have a two value format
> a/b
> and it is this dimensional fail first and foremost that could have caught some attention, yet historically this notion of dimension as we have it now did not exist.
>
> "Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century.[9]"
> - https://en.wikipedia.org/wiki/Decimal_separator
> The decimal notation these days is considered to be rational. Yet when we witness a value such as
> 1.2345
> and we remove the little dot we are looking at an integer. The meaning of the little dot; the decimal point; is in fact consistent with the preexistent radix ten notation. It is merely positing a unity position of an otherwise integer expression. The mechanics of computation with the decimal point has no need of the rational analysis that could be brought to bear. The fact that the symbolic notation is consistent with this analysis is obvious. If anything a new type is here tacked on to the integer type. It can be shown to be another integer, yet a much smaller integer most of the time. It can be an unsigned integer as well; unlike scientific notation.
>
> In this day and age many have been exposed to structured programming of computers and data types. We perceive an informational conflict quite directly in terms of claiming a singular value to contain two values; obviously a single value does not contain two values. That division is the operation invoked here is not at all helpful either, for the product is the more fundamental over division. As we consider the introduction of operators to values it is clear that we cannot embed operators inside of values. This simplistic structural argument is offended by mathematicians going back to the very early stages of mathematics. Integer division does naturally yield a remainder as well; a second value; so no magic is taking place.
>
> Well, this analysis is taken on an assumption of the real value as having fundamental status, which modern mathematicians go to great lengths to demonstrate. Now though we can at least get a date on the real value back to the mid 1600s. At that time the rational value was firmly established.
>
> The structural argument deserves repetition: operators and values are distinct concepts and do not deserve to be mashed together; particularly not in the definition of a value. Yet mathematicians are practicing this mashup nearly universally.
>
> The resolution as I see it is to allow in epsilon/delta analysis onto every number. The procedure of chasing down digits exposes quite a distinction between
> 4 versus 4.00000
> which implies that the subsetting of number systems is not an accurate methodology. This then reflects on the continuum as a gray substrate of unknowns and unknowables down to infinite precision, which of course the mathematician has presumed of the paltry 4 instanced above. Clearly work on the continuum does suffer a precision issue. Ask any physicist and they will explain. Philosophically this argument exposes a false divide between the mathematician and the physicist. That division need not exist and in fact it does not exist. The mathematician's epsilon/delta thinking exposes this, and to reserve it only for certain values rather than apply it universally is a deep mistake. Chasing digits is what we do. Fortunately most times close enough is good enough.

In the days of giant linear solvers it seems it's much the interval artihmetic,
what results formally that all sorts of numerical methods hold together.

Though that's freely enough error analysis, interval arithmetic or organization
often makes for in terms that the giant linear and vector is organizable, about
what is often or usually the floating point or fixed point or arithmetic units,
in terms of for example what could be in the past or future analog integrators,
what holds together for analytical log analysis, terms what compute and reduce.

Anyways after delta/epsilonics and numerical methods after for what we were
given approximations not dominated by their error terms, interval arithmetic
seems for what results in numbers going from "values" to "ranges", with respect
to that distinct or discrete values under indeterminacy, which of course _could_
be "determinate though indeterminate with respect to ours bounds and limits",
makes for usual estimators in orders of magnitude.

The infinite is relly great and I'd imagine you might look to particularly, only the
values between zero and one, first, in terms of how many there are and the distribution
of the distributions of all their sub-segments, as they are written as infinite
contiguous segments for example in binary .0000... to .1111.... (That for all
the representations in radix form, 1/2 is always in the middle.)

I.e., "is Cantor space transpose-able", or "the opposite diagonal is an argument",
makes for interesting questions about the Factorial/Exponential Identity what results
when half the sequences have equal 0/1 densities, that Cantor space is a space and
a probabilistic support, in quasi-invariant measure theory.

On 10/10/2021 11:27 AM, Timothy Golden wrote:

> Within the buildup of the reals we see a menagerie of
> numerical types. Integers, rationals, and then irrationals.

Add to your list
algebraics -- solutions to polynomials with integer coefficients
transcendentals -- the others

> If any were to be rejected it seems those irrationals were
> on the slate yet thanks to epsilon/delta they've made it in
> to seal the gaps.

The numbers which make it in are the numbers which serve a
certain purpose. With different purposes, you'll see different
types. Hence, the menagerie.

If one rejects some type, one should expect to fail at whatever
purpose that type was developed to serve. I'm not making a logical
argument here, only pointing out that there have been a lot of
very smart people who have addressed these questions in the past.
Consider the possibility that they knew what they were doing.

I am pushing "safety-pin points" as a description for
those that seal the gaps.

Consider a function f : [0,1] --> R
f(0) = 0, f(1) = 1, and continuous at every point in [0,1]

Our purpose in declaring f continuous, what we would like
to be true, is that f crosses every point between 0 and 1.

Suppose a safety-pin number is missing.
By that, I mean: suppose that [0,1] can be partitioned
into BEFORE and AFTER so that there is no point x between
BEFORE and AFTER,
no x which is after any other point in BEFORE and before any
other point in AFTER. It's that point-between I'm calling
a safety-pin point.

With a safety-pin point missing, it's not enough to
say that f is continuous. f could be defined, for example, as
f(x) =
{ 0 for x in BEFORE
{ 1 for x in AFTER

For each point x in BEFORE, there is a neighborhood around x
in which f is also 0.
For each point x in AFTER, there is a neighborhood around x
in which f is also 1.

So, this f would be continuous at every point, and still
skip over all the points between 0 and 1.

Without all the safety-pin points, "all the points"
are not enough points to serve our purpose, of having
continuous functions that are _what we mean_ by
"continuous function".

With all the safety-pin numbers, we have Dedekind completeness,
and what we usually call the real numbers.

söndag 10 oktober 2021 kl. 17:27:48 UTC+2 skrev timba...@gmail.com:
> Within the buildup of the reals we see a menagerie of numerical types.
> Integers, rationals, and then irrationals.
> If any were to be rejected it seems those irrationals were on the slate yet thanks to epsilon/delta they've made it in to seal the gaps.
> Onward and upward we see even the reals subsumed as a subset of the complex numbers. There is an admitted dimensional rise to the plane and a two value format, yet back at the rational we have a two value format
> a/b
> and it is this dimensional fail first and foremost that could have caught some attention, yet historically this notion of dimension as we have it now did not exist.
>
> "Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century.[9]"
> - https://en.wikipedia.org/wiki/Decimal_separator
> The decimal notation these days is considered to be rational. Yet when we witness a value such as
> 1.2345
> and we remove the little dot we are looking at an integer. The meaning of the little dot; the decimal point; is in fact consistent with the preexistent radix ten notation. It is merely positing a unity position of an otherwise integer expression. The mechanics of computation with the decimal point has no need of the rational analysis that could be brought to bear. The fact that the symbolic notation is consistent with this analysis is obvious. If anything a new type is here tacked on to the integer type. It can be shown to be another integer, yet a much smaller integer most of the time. It can be an unsigned integer as well; unlike scientific notation.
>
> In this day and age many have been exposed to structured programming of computers and data types. We perceive an informational conflict quite directly in terms of claiming a singular value to contain two values; obviously a single value does not contain two values. That division is the operation invoked here is not at all helpful either, for the product is the more fundamental over division. As we consider the introduction of operators to values it is clear that we cannot embed operators inside of values. This simplistic structural argument is offended by mathematicians going back to the very early stages of mathematics. Integer division does naturally yield a remainder as well; a second value; so no magic is taking place.
>
> Well, this analysis is taken on an assumption of the real value as having fundamental status, which modern mathematicians go to great lengths to demonstrate. Now though we can at least get a date on the real value back to the mid 1600s. At that time the rational value was firmly established.
>
> The structural argument deserves repetition: operators and values are distinct concepts and do not deserve to be mashed together; particularly not in the definition of a value. Yet mathematicians are practicing this mashup nearly universally.
>
> The resolution as I see it is to allow in epsilon/delta analysis onto every number. The procedure of chasing down digits exposes quite a distinction between
> 4 versus 4.00000
> which implies that the subsetting of number systems is not an accurate methodology. This then reflects on the continuum as a gray substrate of unknowns and unknowables down to infinite precision, which of course the mathematician has presumed of the paltry 4 instanced above. Clearly work on the continuum does suffer a precision issue. Ask any physicist and they will explain. Philosophically this argument exposes a false divide between the mathematician and the physicist. That division need not exist and in fact it does not exist. The mathematician's epsilon/delta thinking exposes this, and to reserve it only for certain values rather than apply it universally is a deep mistake. Chasing digits is what we do. Fortunately most times close enough is good enough.
are you on with your insanity yet again?

a/b is one value, you clearly do not what dimension means in vector fields.

On Sunday, October 10, 2021 at 10:16:57 AM UTC-7, Jim Burns wrote:
> On 10/10/2021 11:27 AM, Timothy Golden wrote:
>
> > Within the buildup of the reals we see a menagerie of
> > numerical types. Integers, rationals, and then irrationals.
> Add to your list
> algebraics -- solutions to polynomials with integer coefficients
> transcendentals -- the others
> > If any were to be rejected it seems those irrationals were
> > on the slate yet thanks to epsilon/delta they've made it in
> > to seal the gaps.
> The numbers which make it in are the numbers which serve a
> certain purpose. With different purposes, you'll see different
> types. Hence, the menagerie.
>
> If one rejects some type, one should expect to fail at whatever
> purpose that type was developed to serve. I'm not making a logical
> argument here, only pointing out that there have been a lot of
> very smart people who have addressed these questions in the past.
> Consider the possibility that they knew what they were doing.
>
> I am pushing "safety-pin points" as a description for
> those that seal the gaps.
>
> Consider a function f : [0,1] --> R
> f(0) = 0, f(1) = 1, and continuous at every point in [0,1]
>
> Our purpose in declaring f continuous, what we would like
> to be true, is that f crosses every point between 0 and 1.
>
> Suppose a safety-pin number is missing.
> By that, I mean: suppose that [0,1] can be partitioned
> into BEFORE and AFTER so that there is no point x between
> BEFORE and AFTER,
> no x which is after any other point in BEFORE and before any
> other point in AFTER. It's that point-between I'm calling
> a safety-pin point.
>
> With a safety-pin point missing, it's not enough to
> say that f is continuous. f could be defined, for example, as
> f(x) =
> { 0 for x in BEFORE
> { 1 for x in AFTER
>
> For each point x in BEFORE, there is a neighborhood around x
> in which f is also 0.
> For each point x in AFTER, there is a neighborhood around x
> in which f is also 1.
>
> So, this f would be continuous at every point, and still
> skip over all the points between 0 and 1.
>
> Without all the safety-pin points, "all the points"
> are not enough points to serve our purpose, of having
> continuous functions that are _what we mean_ by
> "continuous function".
>
> With all the safety-pin numbers, we have Dedekind completeness,
> and what we usually call the real numbers.

These "safety-pins" where "removable discontinuities" make for
the usual notion that "removable discontinuities" are what result
that no discontinuities means continuous. (Here a function.)

We all share a common giant lexicon that includes "removable discontinuities".

Getting into these "differentiable but not continuous" and vice-versa, is
mostly for not getting into those, about cadlag (continue a droite,
limite a gauche), kinks, the piece-wise, continuous functions that aren't
classical functions, the C^oo the usual milieu and so on.

The C^infty is about these days the most usual general realm of
continuous functions. But, there are many example functions not
C^infty that still have all their analytical character (g, courtesy geometry)
and under rotations and the meromorphic, for the inclusive class of
"continuous functions".

On Sunday, October 10, 2021 at 2:01:34 PM UTC-4, Ross A. Finlayson wrote:
> On Sunday, October 10, 2021 at 10:16:57 AM UTC-7, Jim Burns wrote:
> > On 10/10/2021 11:27 AM, Timothy Golden wrote:
> >
> > > Within the buildup of the reals we see a menagerie of
> > > numerical types. Integers, rationals, and then irrationals.
> > Add to your list
> > algebraics -- solutions to polynomials with integer coefficients
> > transcendentals -- the others
> > > If any were to be rejected it seems those irrationals were
> > > on the slate yet thanks to epsilon/delta they've made it in
> > > to seal the gaps.
> > The numbers which make it in are the numbers which serve a
> > certain purpose. With different purposes, you'll see different
> > types. Hence, the menagerie.
> >
> > If one rejects some type, one should expect to fail at whatever
> > purpose that type was developed to serve. I'm not making a logical
> > argument here, only pointing out that there have been a lot of
> > very smart people who have addressed these questions in the past.
> > Consider the possibility that they knew what they were doing.
> >
> > I am pushing "safety-pin points" as a description for
> > those that seal the gaps.
> >
> > Consider a function f : [0,1] --> R
> > f(0) = 0, f(1) = 1, and continuous at every point in [0,1]
> >
> > Our purpose in declaring f continuous, what we would like
> > to be true, is that f crosses every point between 0 and 1.
> >
> > Suppose a safety-pin number is missing.
> > By that, I mean: suppose that [0,1] can be partitioned
> > into BEFORE and AFTER so that there is no point x between
> > BEFORE and AFTER,
> > no x which is after any other point in BEFORE and before any
> > other point in AFTER. It's that point-between I'm calling
> > a safety-pin point.
> >
> > With a safety-pin point missing, it's not enough to
> > say that f is continuous. f could be defined, for example, as
> > f(x) =
> > { 0 for x in BEFORE
> > { 1 for x in AFTER
> >
> > For each point x in BEFORE, there is a neighborhood around x
> > in which f is also 0.
> > For each point x in AFTER, there is a neighborhood around x
> > in which f is also 1.
> >
> > So, this f would be continuous at every point, and still
> > skip over all the points between 0 and 1.
> >
> > Without all the safety-pin points, "all the points"
> > are not enough points to serve our purpose, of having
> > continuous functions that are _what we mean_ by
> > "continuous function".
> >
> > With all the safety-pin numbers, we have Dedekind completeness,
> > and what we usually call the real numbers.
> These "safety-pins" where "removable discontinuities" make for
> the usual notion that "removable discontinuities" are what result
> that no discontinuities means continuous. (Here a function.)
>
> We all share a common giant lexicon that includes "removable discontinuities".
>
>
> Getting into these "differentiable but not continuous" and vice-versa, is
> mostly for not getting into those, about cadlag (continue a droite,
> limite a gauche), kinks, the piece-wise, continuous functions that aren't
> classical functions, the C^oo the usual milieu and so on.
>
> The C^infty is about these days the most usual general realm of
> continuous functions. But, there are many example functions not
> C^infty that still have all their analytical character (g, courtesy geometry)
> and under rotations and the meromorphic, for the inclusive class of
> "continuous functions".

As well Jim's analysis is quite hollow in that the real value was to be considered a sort of complete type. Those many serious minds who had to struggle with this puzzle were already surfing the rational assumption as an old and resolved puzzle by their likewise highly intelligent and disciplined peers. No, I am sorry but by my analysis the structure of the problem is rather different. This is not a subtle philosophical detail. It upsets the system of thought at a different structural level. The upset is substantial and does not merely back us out of irrational values. Indeed it is the acceptance of those irrational values and the application of epsilon/delta to resolving them which allows epsilon/delta to arrive on the continuum. To now take on epsilon/delta for every value is meaningful. It does in fact yield these gray numbers. Poor Dedekind didn't want to get caught chasing digits yet that is all that epsilon/delta really does. If you don't like the square root of two out to eight digits take it out a few more places. If you don't like the number four out to eight digits take it a few more places.

Where in Jim's discussion or yours Ross are you really processing these claims? On a proper continuum when we ask you to pin 4 down it is not as easy as the mathematician assumes. The precision of your graphical art will come under scrutiny. Meanwhile the precision of your number 4 is terrible. What right do you have to claim that this number in your construction is infinitely perfect? Type theory suggests that you are using discrete values in a continuum. This is in direct conflict with the concept of a continuum. This is quite an inversion when we take your value 4 and take its precision literally as about as poor as it gets.

I do still think that the strongest argument is back on operators versus values. We are in an age of computing hardware that demands type safety. I merely am following that philosophy into mathematics and exposing conflicts. Still, this argument on operators versus values is deeper still than type conflicts. This level of confusion by mathematicians is a computational deficiency. It is like here in language we have verbs and nouns and mathematicians have blown their own snobenruvs and believe that they can still abide by nouns and verbs later in the game. I admit that it is a strange argument and it surprised me as I bumped into it yet it continues to gel. The logic is so simple and exposing the conflict is as simple.
If the construction of a number system requires an advanced operator it cannot possibly be a pure number system. This backs us all the way out to the integers. Lo and behold the interpretation of
4.00000001
as an integer is trivial. Yes there is a touch more structure present. That is all that is necessary. Epsilon/delta addresses all other concerns. This construction bypasses the rational and irrational numbers, both of which rely on operators in their construction. You int freaks should be quite pleased. Ordinals for Jim.

I did see it on irrational values ages ago. It is very easy to attack the square root of two as an operator and a value and so what right do people have to call it as pure value? This is a stage that academia is practiced at. To shove one falsehood at your students and have it accepted simply shove another bigger falsehood at them so that they forget the first falsehood. This incidentally is how magicians work too. It worked on me until recently.

Incidentally here as we discuss the term 'decimal' versus 'decimal point' there is linguistic confusion I think that I have participated in. The decimal number system can be simply our presumed mod-ten whole numbers if you wish as a first number system, whereas it is too easily confused with numbers that contain a 'fractional' part, or decimal point. Clearly it will not be proper under this interpretation to claim the terminology of 'real' value either since that obviously carries all the baggage. The essence of this interpretation is a rejection of the real value as it is taught, and yet the usage of the value nearly as in scientific notation is agreeable. However in scientific notation the ability to apply negative or positive direction to unity (a signed system) is not actually required to perform the duties we see here. As well to incorporate polysign into the numeric sign is generalized.

Clearly the structure of values such as
- 1.234 , + 34.56 , - 0.00001
is undergoing scrutiny here. Stripping away signs and decimal points we see natural values. One with some leading zeros, but large values can be expressed such as
+ 1000000000000.0
and so the coding of the decimal point as a positional thing from the right of the string of digits is direct and requires nearly no translation as a natural value. Likewise sign once generalized is yet another constrained natural value. To even require this language of 'unsigned' to describe sign is an obvious ambiguity and yet this is just how standard mathematics proceeds with numerous papers discussing R+ and zero or some such as if R were primitive. R is not primitive. R; the reals; are two-signed numbers. Their continuous nature as proven through epsilon/delta can hold generally across the whole lot, and this sort of universality; this sort of simplification; ought to be appealing, yet we know mathematicians pride themselves on the levels of complexity that they can achieve.

The full format of number is actually
s x e
where s is sign, x is a natural value, and e is the unit position(decimal point). Because though this interpretation is so ordinary the latter part can be abbreviated to
s x
where x is simply magnitude, which incorporates the decimal point into x. Certainly with the sign divorced from the x portion we will not be stating that the x is in R+ or some such contorted language. In some ways it has always been my burden to address that x portion, though the productive work was done in the s portion. It seems I have arrived in that interpretation here and it does include significant detail.

The idea that operators and values are distinct concepts is somehow not discussible by mathematicians. I have attempted to engage them here on this topic many times and been dodged regularly. Such a simple concept does go offended within the standard curriculum. Particularly the division operation, which is arguably not even fundamental since it is the reversal of the product operation which is more fundamental, is engaged in the rational value.

The bridge that is the operator as in
Value1 Operation1 Value2 = Value3
is not seriously adopted by the mathematician. It has to go avoided in abstract algebra with their polynomials that refuse to perform such evaluation.
This becomes an undiscussible topic. Believe me I have tried.

I see the sum as more fundamental than the product, while advanced mathematics attempts to blur the two ala group and ring theory. Meanwhile the language of a 'successor' hides addition. Here it seems they abide by a principal of avoiding operators in the definition of their number system, whereas within the rational value they don't mind mixing it up. There is such strong physical correspondence in the sum as fundamental; as in the consideration of the quantity of objects about you in sum or superposition in space. Of course the integral is a sum. Here too we land in criticism of the sum defined as a binary operator. Clearly the n-ary form of the sum is superior. This then as well gets us the singleton value as the first form
a a+b
a+b+c
a+b+c+d
....
The usage of a nonzero sign (sign two) as the sum as is done above is problematic notation other than in P2. Within polysign this is remedied with the implementation of a zero sign '@' which maintains its position as an identity sign; neutral in product. Here again with a nose for operator theory we have to ask whether convention and notation are ambiguous. It is not pretty rewriting convention, yet it does seem that this one works out, and again we have that physical concept of superposition as well.

All of these details are only loosely woven, but they do fit together. There is only a loose need to confront the rational value, yet it has occurred through the lens of polysign number theory. This is because upon generalizing sign we are forced to bump into operator theory. Sum and product as fundamental rings true from polysign, yet their inversions are not so true. The field requirements of mathematics are wrong. They are not general. Abstract algebra seems to have the right beginning, yet they shove their foot in their mouths just as soon as they arrive in the polynomial form. Polysign clearly develops the form they were after naturally.

Our digital system of base ten ( base 10?) goes ignored and presumed.
Meanwhile every radix system is base 10.
This is how rapidly ambiguity can creep in and it certainly goes avoided.
To realize that sign is modulo behaved is not much of a step, yet humans have avoided that step until the advent of polysign.
Why is this? The extension to modulo three sign works perfectly. Modulo n sign works perfectly.
As I research the history of mathematics it's almost as if the two-signed reals were not taken seriously.
It's more like they crept their way in.

"The English mathematician, John Wallis (1616 - 1703) is credited with giving some meaning to negative numbers by inventing the number line, and in the early 18th century a controversy ensued between Leibniz, Johan Bernoulli, Euler and d'Alembert about whether log(−x) was the same as Log(x)." - https://nrich.maths.org/5961
"Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art (Jiu zhang suan-shu), which in its present form dates from the period of the Han Dynasty (202 BC – AD 220), but may well contain much older material." - https://en.wikipedia.org/wiki/Negative_number

Until you arrive with minus one times minus one is plus one the modulo behavior of sign is not established. These early forms are not that. Even the real value takes its prominence on claims of the continuum which need not really include the negative half. It's pretty clear that mathematics is abiding accumulation. The rational value as a subset of a continuum is a perfect instance. Rational values as taught in modernity form a re-radix system, and their more natural counterparts would be those originating radix systems; not in the base 1234567890 system. The ordinary decimal usage works but this does not mean that they are fundamental. Certainly they are computable and can abide by epsilon/delta.

On Sunday, October 10, 2021 at 11:27:48 AM UTC-4, Timothy Golden wrote:
> Within the buildup of the reals we see a menagerie of numerical types.

There must be more ground to cover here. Without going into various versions of rational analysis already I can see some trouble when somebody writes:
f(x) + f(x) = 2 f(x)
and now the precision of the value 2 as I claim it in the OP here is poor. Still as to whether this two is a continuum value? No: it is not a continuum two.
This notion that we are dealing in two types of values within the same notational system is problematic. The discrete and the continuous should not mix so readily. Yet the computation will work out so long as the 2 is interpreted as the discrete form which then extends to an infinite precision continuous form. Obviously I will not win any new notation from the masses here. Still, the physicist obviously does have to keep these types of values straight. That these two types of value coexist in our computations is an ambiguity.

The arguments that I am making are on the continuous form. The interpretation of those continuous values as gray rather than as perfect allows their integer representation such as
12.345
where the integer is
12345
and the little dot is indicating a unital position. Obviously the first unital position is at the tail of the value, so to call this a secondary unital position could possibly take hold if this interpretation is accepted as doing away with the rational analysis. Of course the little dot may as well be a whole value indicating which digit from the first unital position the secondary unital is. This is hardly any different than ordinary scientific notation except that large values need not be accounted for so that an unsigned value suffices.

More relevant than working out this notational system is the interpretation of epsilon/delta being operant on all continuum values. This leaves them as gray values though digits can be chased and this has been practiced on pi for instance out to extraordinary precision. This gray interpretation is an affront to the real value as it has been taught. Indeed it is not the real value, though the discrepancy is trite. Physical correspondence is coherency and the term 'real' suffers thanks to the naming of mathematicians. As stated above though ambiguity still remains. Still more ambiguity exists in the adoption of the rational and irrational as types within a type of number. I suppose a question of isotropy could be posed here. The resolution as I've secured it is to go gray on the continuum universally. This then returns all values to integers with just a bit more structure of the secondary unit.

> Integers, rationals, and then irrationals.
> If any were to be rejected it seems those irrationals were on the slate yet thanks to epsilon/delta they've made it in to seal the gaps.
> Onward and upward we see even the reals subsumed as a subset of the complex numbers. There is an admitted dimensional rise to the plane and a two value format, yet back at the rational we have a two value format
> a/b
> and it is this dimensional fail first and foremost that could have caught some attention, yet historically this notion of dimension as we have it now did not exist.
>
> "Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century.[9]"
> - https://en.wikipedia.org/wiki/Decimal_separator
> The decimal notation these days is considered to be rational. Yet when we witness a value such as
> 1.2345
> and we remove the little dot we are looking at an integer. The meaning of the little dot; the decimal point; is in fact consistent with the preexistent radix ten notation. It is merely positing a unity position of an otherwise integer expression. The mechanics of computation with the decimal point has no need of the rational analysis that could be brought to bear. The fact that the symbolic notation is consistent with this analysis is obvious. If anything a new type is here tacked on to the integer type. It can be shown to be another integer, yet a much smaller integer most of the time. It can be an unsigned integer as well; unlike scientific notation.
>
> In this day and age many have been exposed to structured programming of computers and data types. We perceive an informational conflict quite directly in terms of claiming a singular value to contain two values; obviously a single value does not contain two values. That division is the operation invoked here is not at all helpful either, for the product is the more fundamental over division. As we consider the introduction of operators to values it is clear that we cannot embed operators inside of values. This simplistic structural argument is offended by mathematicians going back to the very early stages of mathematics. Integer division does naturally yield a remainder as well; a second value; so no magic is taking place.
>
> Well, this analysis is taken on an assumption of the real value as having fundamental status, which modern mathematicians go to great lengths to demonstrate. Now though we can at least get a date on the real value back to the mid 1600s. At that time the rational value was firmly established.
>
> The structural argument deserves repetition: operators and values are distinct concepts and do not deserve to be mashed together; particularly not in the definition of a value. Yet mathematicians are practicing this mashup nearly universally.
>
> The resolution as I see it is to allow in epsilon/delta analysis onto every number. The procedure of chasing down digits exposes quite a distinction between
> 4 versus 4.00000
> which implies that the subsetting of number systems is not an accurate methodology. This then reflects on the continuum as a gray substrate of unknowns and unknowables down to infinite precision, which of course the mathematician has presumed of the paltry 4 instanced above. Clearly work on the continuum does suffer a precision issue. Ask any physicist and they will explain. Philosophically this argument exposes a false divide between the mathematician and the physicist. That division need not exist and in fact it does not exist. The mathematician's epsilon/delta thinking exposes this, and to reserve it only for certain values rather than apply it universally is a deep mistake. Chasing digits is what we do. Fortunately most times close enough is good enough.

On Sunday, October 10, 2021 at 11:27:48 AM UTC-4, Timothy Golden wrote:
> Within the buildup of the reals we see a menagerie of numerical types.
> Integers, rationals, and then irrationals.
> If any were to be rejected it seems those irrationals were on the slate yet thanks to epsilon/delta they've made it in to seal the gaps.
> Onward and upward we see even the reals subsumed as a subset of the complex numbers. There is an admitted dimensional rise to the plane and a two value format, yet back at the rational we have a two value format

> a/b

By what right is closure delayed for this a/b? Then it is allowed thereafter?
Is this somebodies idea of a clean theory?
Everybody's idea of clean theory?
Not me.
No thanks.

Squeak We Must!
> and it is this dimensional fail first and foremost that could have caught some attention, yet historically this notion of dimension as we have it now did not exist.
>
> "Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century.[9]"
> - https://en.wikipedia.org/wiki/Decimal_separator
> The decimal notation these days is considered to be rational. Yet when we witness a value such as
> 1.2345
> and we remove the little dot we are looking at an integer. The meaning of the little dot; the decimal point; is in fact consistent with the preexistent radix ten notation. It is merely positing a unity position of an otherwise integer expression. The mechanics of computation with the decimal point has no need of the rational analysis that could be brought to bear. The fact that the symbolic notation is consistent with this analysis is obvious. If anything a new type is here tacked on to the integer type. It can be shown to be another integer, yet a much smaller integer most of the time. It can be an unsigned integer as well; unlike scientific notation.
>
> In this day and age many have been exposed to structured programming of computers and data types. We perceive an informational conflict quite directly in terms of claiming a singular value to contain two values; obviously a single value does not contain two values. That division is the operation invoked here is not at all helpful either, for the product is the more fundamental over division. As we consider the introduction of operators to values it is clear that we cannot embed operators inside of values. This simplistic structural argument is offended by mathematicians going back to the very early stages of mathematics. Integer division does naturally yield a remainder as well; a second value; so no magic is taking place.
>
> Well, this analysis is taken on an assumption of the real value as having fundamental status, which modern mathematicians go to great lengths to demonstrate. Now though we can at least get a date on the real value back to the mid 1600s. At that time the rational value was firmly established.
>
> The structural argument deserves repetition: operators and values are distinct concepts and do not deserve to be mashed together; particularly not in the definition of a value. Yet mathematicians are practicing this mashup nearly universally.
>
> The resolution as I see it is to allow in epsilon/delta analysis onto every number. The procedure of chasing down digits exposes quite a distinction between
> 4 versus 4.00000
> which implies that the subsetting of number systems is not an accurate methodology. This then reflects on the continuum as a gray substrate of unknowns and unknowables down to infinite precision, which of course the mathematician has presumed of the paltry 4 instanced above. Clearly work on the continuum does suffer a precision issue. Ask any physicist and they will explain. Philosophically this argument exposes a false divide between the mathematician and the physicist. That division need not exist and in fact it does not exist. The mathematician's epsilon/delta thinking exposes this, and to reserve it only for certain values rather than apply it universally is a deep mistake. Chasing digits is what we do. Fortunately most times close enough is good enough.

On Thursday, December 16, 2021 at 9:34:25 AM UTC-5, Timothy Golden wrote:
> On Sunday, October 10, 2021 at 11:27:48 AM UTC-4, Timothy Golden wrote:
> > Within the buildup of the reals we see a menagerie of numerical types.
> > Integers, rationals, and then irrationals.
> > If any were to be rejected it seems those irrationals were on the slate yet thanks to epsilon/delta they've made it in to seal the gaps.
> > Onward and upward we see even the reals subsumed as a subset of the complex numbers. There is an admitted dimensional rise to the plane and a two value format, yet back at the rational we have a two value format
>
> > a/b
> By what right is closure delayed for this a/b? Then it is allowed thereafter?
> Is this somebodies idea of a clean theory?
> Everybody's idea of clean theory?
> Not me.
> No thanks.
>
> Squeak We Must!

Now I see this on wikipedia though I don't see it in the link:
"In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals (for more, see Construction of the real numbers https://en.wikipedia.org/wiki/Construction_of_the_real_numbers)."
- https://en.wikipedia.org/wiki/Rational_number

I think I am still different than the last option. I don't believe we need to insist upon infinite decimals. Finite decimals will do.
Nonlinear systems are no problem are they? I guess if you cannot resolve a curved value well or you insist on some obnoxious step function... No. These are completely computable. As to physics and claiming perfect prediction of physical events: that is not what this math is about. This math is about a basis on which to work those events. In that mathematics ought to form the basis of physics then all is well in academia. In that we still struggle badly; well we creep along and occasionally get catapulted too.

It would almost seem to a newcomer that some C coder arrived here naively insisting on compiler level integrity in mathematics. Indeed; this is the case.

> > and it is this dimensional fail first and foremost that could have caught some attention, yet historically this notion of dimension as we have it now did not exist.
> >
> > "Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century.[9]"
> > - https://en.wikipedia.org/wiki/Decimal_separator
> > The decimal notation these days is considered to be rational. Yet when we witness a value such as
> > 1.2345
> > and we remove the little dot we are looking at an integer. The meaning of the little dot; the decimal point; is in fact consistent with the preexistent radix ten notation. It is merely positing a unity position of an otherwise integer expression. The mechanics of computation with the decimal point has no need of the rational analysis that could be brought to bear. The fact that the symbolic notation is consistent with this analysis is obvious.. If anything a new type is here tacked on to the integer type. It can be shown to be another integer, yet a much smaller integer most of the time. It can be an unsigned integer as well; unlike scientific notation.
> >
> > In this day and age many have been exposed to structured programming of computers and data types. We perceive an informational conflict quite directly in terms of claiming a singular value to contain two values; obviously a single value does not contain two values. That division is the operation invoked here is not at all helpful either, for the product is the more fundamental over division. As we consider the introduction of operators to values it is clear that we cannot embed operators inside of values. This simplistic structural argument is offended by mathematicians going back to the very early stages of mathematics. Integer division does naturally yield a remainder as well; a second value; so no magic is taking place.
> >
> > Well, this analysis is taken on an assumption of the real value as having fundamental status, which modern mathematicians go to great lengths to demonstrate. Now though we can at least get a date on the real value back to the mid 1600s. At that time the rational value was firmly established.
> >
> > The structural argument deserves repetition: operators and values are distinct concepts and do not deserve to be mashed together; particularly not in the definition of a value. Yet mathematicians are practicing this mashup nearly universally.
> >
> > The resolution as I see it is to allow in epsilon/delta analysis onto every number. The procedure of chasing down digits exposes quite a distinction between
> > 4 versus 4.00000
> > which implies that the subsetting of number systems is not an accurate methodology. This then reflects on the continuum as a gray substrate of unknowns and unknowables down to infinite precision, which of course the mathematician has presumed of the paltry 4 instanced above. Clearly work on the continuum does suffer a precision issue. Ask any physicist and they will explain. Philosophically this argument exposes a false divide between the mathematician and the physicist. That division need not exist and in fact it does not exist. The mathematician's epsilon/delta thinking exposes this, and to reserve it only for certain values rather than apply it universally is a deep mistake. Chasing digits is what we do. Fortunately most times close enough is good enough.