As we beat on the standing system, and under the flogging we see damage done, what of the optional zero of the natural numbers?

With natural analysis as a radix-one affair, it is too primitive to distinguish one from zero. There is no such distinction.

I suspect there is a bit of work to be done here. That this issue interacts with notation and its choices is relevant.

As we begin a numerical representation mustn't we end the thing too? This would be the problem of the old interpretation:
1/3 = 0.333...
as no terminal punctuation has been given. The option to correct this via:
1/3 = 0.333...33
obviously presents itself now. That these issues as well are radix based issues; issues that perhaps should be brought down to radix-one operational ability: this is actually quite a tricky business.

As radix-one denies the operational effect of the decimal point, and yet as natural analysis arguably is radix-one, we are left stumbling over the meaning of the decimal point. It's ordinary independent interpretation, as I see it is a mark of unity placement. This is of course the post-rational interpretation. I think returning to the digital form with pertinent termination may modify the problem, as we strip away the technology we are left with the digits of the radix value as our raw basis. This principal returns all numbers to their natural form regardless of radix or radix point punctuation.

As we dig around down here, trying to slap the poor system sober again, we have as well to confront the fact that our numbers are historically generated most significant digit first; as we scrawl then down on paper. To what degree terminus and prelude must be swappable... or whether both must even exist... these can be cast as open problems in a new mathematics which is radix-concentric; radix-n; unless of course we must demolish unity as radix-one altogether. That would be a very peculiar choice.

On Tuesday, January 24, 2023 at 1:20:38 PM UTC-5, Timothy Golden wrote:
> As we beat on the standing system, and under the flogging we see damage done, what of the optional zero of the natural numbers?
>
> With natural analysis as a radix-one affair, it is too primitive to distinguish one from zero. There is no such distinction.
>
> I suspect there is a bit of work to be done here. That this issue interacts with notation and its choices is relevant.
>
> As we begin a numerical representation mustn't we end the thing too? This would be the problem of the old interpretation:
> 1/3 = 0.333...
> as no terminal punctuation has been given. The option to correct this via:
> 1/3 = 0.333...33
> obviously presents itself now. That these issues as well are radix based issues; issues that perhaps should be brought down to radix-one operational ability: this is actually quite a tricky business.
>
> As radix-one denies the operational effect of the decimal point, and yet as natural analysis arguably is radix-one, we are left stumbling over the meaning of the decimal point. It's ordinary independent interpretation, as I see it is a mark of unity placement. This is of course the post-rational interpretation. I think returning to the digital form with pertinent termination may modify the problem, as we strip away the technology we are left with the digits of the radix value as our raw basis. This principal returns all numbers to their natural form regardless of radix or radix point punctuation.
>
> As we dig around down here, trying to slap the poor system sober again, we have as well to confront the fact that our numbers are historically generated most significant digit first; as we scrawl then down on paper. To what degree terminus and prelude must be swappable... or whether both must even exist... these can be cast as open problems in a new mathematics which is radix-concentric; radix-n; unless of course we must demolish unity as radix-one altogether. That would be a very peculiar choice.

As we engage the numerical analysis through the new lens of interpretation the peculiarity of expressions like:
x = 333...33;
x * x = 111...10888...89;
x+1 = 333...34;
so that we have computable infinities via the inductive ellipsis that second line with the double ellipsis exposes additional structure to the numbers.. Under such digital analysis we witness that the two aleph form is distinct from the one aleph form. This would of course launch an entirely new attack back onto Cantor et al, but as well there is a possibility of generalization in this kernel of activity.

Under product the quantity of digits required to develop the result is typically the sum of the digits of the summands, and so in the case of the square the quantity of digits must double, and the very computation as well exposes this behavior. There is no escaping it. There are some b-adic freaks who seem to think that they have, but they are obviously mistaken. They fell for the two aleph equals one aleph goof of the Cantorian generation. I might have a misnomer there because I am not well versed in these parts, but conceptually I can vouch. Rather than focus on the blame game, let's move on to bigger and better things.

The usage of an aleph mark to fix up a possible ambiguity, such as:
10 x = 333...30;
x > 10 x? :
333...30 > 333...33?
and so we simply introduce the aleph mark '|', though its usage does seem to be optional in many circumstances; much like the decimal point; we will more formally have:
x = | 333...33
10 x = 3 | 333...33
and so you can see that on the supposition that the ellipses imply the complete expression engages one aleph of quantity we must be capable of making this reference in order to maintain coherency of the mathematics. Thencely there are two aleph marks in:
x * x = | 111...10 | 888...89
and this is of course consistent with digit counting under product. The results are readily seen by:
sage: x=33333333333333333
sage: x*x
1111111111111111088888888888888889
sage:
and little is bound to change as we approach infinity thanks to this simplest instance of induction.

So, that aleph mark will as disambiguate quite a lot of commonly seen expressions such as:

0.142857142857...
oops, and just here it did indeed turn into a decimal point! Do not worry: all is well, for the two as indistinguishable is felt by taking this seventh in its raw form. Computationally the value
|142857142857...142857
will maintain itself at aleph one, even though the length of the string of digits which is repeated is six. So we witness another disambiguation through the usage of the aleph mark, for a skeptic could interpret such a value as being six times longer than its simplest counterpart. Indeed the effects of confusion that run through the department of rational numbers, and it may be that such a department should be sequestered out of ordinary mathematics, and put in line for firing at first need of funds, for their own obscene ignorance of modulo principles leading into radix mathematics; their insistence upon working mixed radix forms and looking the other way every time a student puts them up to it; well; perhaps they should be superglued to the chalk board and tarred and feathered, or at least have a few cream pies thrown at them. Let the janitor scrape them off the chalk board so they can go home and wash up. One seventh, people, can simply be posited as:
0.1
in a radix seven system. What we witness in the complexity of the digits is a matter of inter-radix mechanics. The rational numbers then as counting numbers whose unital position has been arbitrarily raised up to say seven, are really doing just the same work as the decimal point; as the radix point; as the aleph point; in demarking a new form of unity. That we work in an agreed upon radix ten system without even discussing it is somewhat the cause of the problem. In doing number theory at the fundamental level the very usage of the glyphs even has to come under scrutiny, and so when the natural numbers are presented to you ala Peano as:
1, 2, 3, ...
you'll have to get the super-glue out; order the cream pies; block the doors; it may take some team-work to get this one done.

You see, those glyphs are a part of a radix ten counting system that is presumed, and it exists nowhere in Peano's axioms. Did you think that the value 321 is up there somewhere? How about infinity? This latter is the one that has peoples' attention it seems, or under tension, really, as a talking point that never ends. Onandon the threads of usenet's sci.math go; all the forums really; not that I spend much time on them, for I get censored off them so readily.

The disregard for the digits and yet the possibility that their structural options are actually wider than we allow... this is where I was trying to get to. We are close to establishing a value such as:
123.101.342
which does in fact carry dimensional correspondence, going back even to Descarte's time with his unfinished work in "rules for the direction of the mind".

As blocks go: each individual digit is a block of sorts. Now we see the infinite agglomeration of such blocks somehow sparks our interest; so why not the intermediates? Well: this is exactly what computing hardware looks like.. It is not as if we don't already have them; mathematicians are simply in denial of the facts. When the hardware multiplication yields one block instead of two there is trouble afoot. Rather: such a multiplier will have an exception. But for the fortunate values that do make it through, the rest will have to go through a bit more trouble. That this same can even be said of the sum; sure we can get into arbitrary sized spaces, sort of... the only ones who can escape these things are the pure modulo numbers. Our commitment to the digits is a trap of sorts leading onward to more digits. Chasing digits is what we do. But when we see these appendages to our numbers and realize that they do very little to the underlying digits; ahh, possibly I am about to attack sign from another perspective.

On Sunday, January 29, 2023 at 8:40:02 AM UTC-8, timba...@gmail.com wrote:
> On Tuesday, January 24, 2023 at 1:20:38 PM UTC-5, Timothy Golden wrote:
> > As we beat on the standing system, and under the flogging we see damage done, what of the optional zero of the natural numbers?
> >
> > With natural analysis as a radix-one affair, it is too primitive to distinguish one from zero. There is no such distinction.
> >
> > I suspect there is a bit of work to be done here. That this issue interacts with notation and its choices is relevant.
> >
> > As we begin a numerical representation mustn't we end the thing too? This would be the problem of the old interpretation:
> > 1/3 = 0.333...
> > as no terminal punctuation has been given. The option to correct this via:
> > 1/3 = 0.333...33
> > obviously presents itself now. That these issues as well are radix based issues; issues that perhaps should be brought down to radix-one operational ability: this is actually quite a tricky business.
> >
> > As radix-one denies the operational effect of the decimal point, and yet as natural analysis arguably is radix-one, we are left stumbling over the meaning of the decimal point. It's ordinary independent interpretation, as I see it is a mark of unity placement. This is of course the post-rational interpretation. I think returning to the digital form with pertinent termination may modify the problem, as we strip away the technology we are left with the digits of the radix value as our raw basis. This principal returns all numbers to their natural form regardless of radix or radix point punctuation.
> >
> > As we dig around down here, trying to slap the poor system sober again, we have as well to confront the fact that our numbers are historically generated most significant digit first; as we scrawl then down on paper. To what degree terminus and prelude must be swappable... or whether both must even exist... these can be cast as open problems in a new mathematics which is radix-concentric; radix-n; unless of course we must demolish unity as radix-one altogether. That would be a very peculiar choice.
> As we engage the numerical analysis through the new lens of interpretation the peculiarity of expressions like:
> x = 333...33;
> x * x = 111...10888...89;
> x+1 = 333...34;
> so that we have computable infinities via the inductive ellipsis that second line with the double ellipsis exposes additional structure to the numbers. Under such digital analysis we witness that the two aleph form is distinct from the one aleph form. This would of course launch an entirely new attack back onto Cantor et al, but as well there is a possibility of generalization in this kernel of activity.
>
> Under product the quantity of digits required to develop the result is typically the sum of the digits of the summands, and so in the case of the square the quantity of digits must double, and the very computation as well exposes this behavior. There is no escaping it. There are some b-adic freaks who seem to think that they have, but they are obviously mistaken. They fell for the two aleph equals one aleph goof of the Cantorian generation. I might have a misnomer there because I am not well versed in these parts, but conceptually I can vouch. Rather than focus on the blame game, let's move on to bigger and better things.
>
> The usage of an aleph mark to fix up a possible ambiguity, such as:
> 10 x = 333...30;
> x > 10 x? :
> 333...30 > 333...33?
> and so we simply introduce the aleph mark '|', though its usage does seem to be optional in many circumstances; much like the decimal point; we will more formally have:
> x = | 333...33
> 10 x = 3 | 333...33
> and so you can see that on the supposition that the ellipses imply the complete expression engages one aleph of quantity we must be capable of making this reference in order to maintain coherency of the mathematics. Thencely there are two aleph marks in:
> x * x = | 111...10 | 888...89
> and this is of course consistent with digit counting under product. The results are readily seen by:
> sage: x=33333333333333333
> sage: x*x
> 1111111111111111088888888888888889
> sage:
> and little is bound to change as we approach infinity thanks to this simplest instance of induction.
>
> So, that aleph mark will as disambiguate quite a lot of commonly seen expressions such as:
>
> 0.142857142857...
> oops, and just here it did indeed turn into a decimal point! Do not worry: all is well, for the two as indistinguishable is felt by taking this seventh in its raw form. Computationally the value
> |142857142857...142857
> will maintain itself at aleph one, even though the length of the string of digits which is repeated is six. So we witness another disambiguation through the usage of the aleph mark, for a skeptic could interpret such a value as being six times longer than its simplest counterpart. Indeed the effects of confusion that run through the department of rational numbers, and it may be that such a department should be sequestered out of ordinary mathematics, and put in line for firing at first need of funds, for their own obscene ignorance of modulo principles leading into radix mathematics; their insistence upon working mixed radix forms and looking the other way every time a student puts them up to it; well; perhaps they should be superglued to the chalk board and tarred and feathered, or at least have a few cream pies thrown at them. Let the janitor scrape them off the chalk board so they can go home and wash up. One seventh, people, can simply be posited as:
> 0.1
> in a radix seven system. What we witness in the complexity of the digits is a matter of inter-radix mechanics. The rational numbers then as counting numbers whose unital position has been arbitrarily raised up to say seven, are really doing just the same work as the decimal point; as the radix point; as the aleph point; in demarking a new form of unity. That we work in an agreed upon radix ten system without even discussing it is somewhat the cause of the problem. In doing number theory at the fundamental level the very usage of the glyphs even has to come under scrutiny, and so when the natural numbers are presented to you ala Peano as:
> 1, 2, 3, ...
> you'll have to get the super-glue out; order the cream pies; block the doors; it may take some team-work to get this one done.
>
> You see, those glyphs are a part of a radix ten counting system that is presumed, and it exists nowhere in Peano's axioms. Did you think that the value 321 is up there somewhere? How about infinity? This latter is the one that has peoples' attention it seems, or under tension, really, as a talking point that never ends. Onandon the threads of usenet's sci.math go; all the forums really; not that I spend much time on them, for I get censored off them so readily.
>
> The disregard for the digits and yet the possibility that their structural options are actually wider than we allow... this is where I was trying to get to. We are close to establishing a value such as:
> 123.101.342
> which does in fact carry dimensional correspondence, going back even to Descarte's time with his unfinished work in "rules for the direction of the mind".
>
> As blocks go: each individual digit is a block of sorts. Now we see the infinite agglomeration of such blocks somehow sparks our interest; so why not the intermediates? Well: this is exactly what computing hardware looks like. It is not as if we don't already have them; mathematicians are simply in denial of the facts. When the hardware multiplication yields one block instead of two there is trouble afoot. Rather: such a multiplier will have an exception. But for the fortunate values that do make it through, the rest will have to go through a bit more trouble. That this same can even be said of the sum; sure we can get into arbitrary sized spaces, sort of... the only ones who can escape these things are the pure modulo numbers. Our commitment to the digits is a trap of sorts leading onward to more digits. Chasing digits is what we do. But when we see these appendages to our numbers and realize that they do very little to the underlying digits; ahh, possibly I am about to attack sign from another perspective.

Absolute zero math shows the no quantity to subtract from zero math limit.
Overshooting subtraction needs to take a correction in all math.