there is no negative 1.

Mitchell Raemsch

On 8/25/2023 11:11 AM, mitchr...@gmail.com wrote:
> there is no negative 1.

[...]

Yawn...

On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail.com wrote:
> there is no negative 1.
>
> Mitchell Raemsch

Yeah there is.

Kevin S brought next idea :
> On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail.com wrote:
>> there is no negative 1.
>>
>> Mitchell Raemsch
>
> Yeah there is.

Indeed, since we have integers. One could claim that there is no minus
one though.

On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
> there is no negative 1.
>
> Mitchell Raemsch

*
How would you solve this equation?

x^2 + 1 = 0

Thanks,

On 8/25/2023 1:34 PM, FromTheRafters wrote:
> Kevin S brought next idea :
>> On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail.com
>> wrote:
>>> there is no negative 1.
>>> Mitchell Raemsch
>>
>> Yeah there is.
>
> Indeed, since we have integers. One could claim that there is no minus
> one though.

Not exactly sure what you mean. Zero minus one equals negative one.
Negative one minus negative one equals zero.

On Friday, August 25, 2023 at 1:34:56 PM UTC-7, FromTheRafters wrote:
> Kevin S brought next idea :
> > On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail..com wrote:
> >> there is no negative 1.
> >>
> >> Mitchell Raemsch
> >
> > Yeah there is.
> Indeed, since we have integers. One could claim that there is no minus
> one though.

There are no negatives there are positives with a minus sign attached
turning them into a subtraction in every equation. There is no magnitude below zero.

On 8/25/2023 6:55 PM, mitchr...@gmail.com wrote:
> On Friday, August 25, 2023 at 1:34:56 PM UTC-7, FromTheRafters wrote:
>> Kevin S brought next idea :
>>> On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail.com wrote:
>>>> there is no negative 1.
>>>>
>>>> Mitchell Raemsch
>>>
>>> Yeah there is.
>> Indeed, since we have integers. One could claim that there is no minus
>> one though.
>
> There are no negatives there are positives with a minus sign attached
> turning them into a subtraction in every equation. There is no magnitude below zero.

I think you have a minus sign attached to your brain, right?

On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
> > there is no negative 1.
> >
> > Mitchell Raemsch
> *
> How would you solve this equation?
>
> x^2 + 1 = 0
By your own terms negative one squared is one.
i is a formula that has no solution.
It shows why negative one does not exist.

> Thanks,

On 8/25/2023 6:59 PM, mitchr...@gmail.com wrote:
> On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
>> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
>>> there is no negative 1.
>>>
>>> Mitchell Raemsch
>> *
>> How would you solve this equation?
>>
>> x^2 + 1 = 0
> By your own terms negative one squared is one.
> i is a formula that has no solution.
> It shows why negative one does not exist.

You have never used complex numbers before, and you do not understand them.

on 8/25/2023, Chris M. Thomasson supposed :
> On 8/25/2023 1:34 PM, FromTheRafters wrote:
>> Kevin S brought next idea :
>>> On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail.com wrote:
>>>> there is no negative 1.
>>>> Mitchell Raemsch
>>>
>>> Yeah there is.
>>
>> Indeed, since we have integers. One could claim that there is no minus one
>> though.
>
> Not exactly sure what you mean.

One could claim...

The integers form a ring, instead of subtraction of naturals (a partial
function for them) we have addition of integers. The binary operator
'minus' (a-b) is essentially gone and we add a negative integer
(a+(-b)) instead.

> Zero minus one equals negative one.

Zero plus a negative one equals a negative one.

0 + (-1) = -1

> Negative one minus negative one equals zero.

Negative one plus a negated negative one equals zero.

-1 + (-(-1)) = 0

Sure, we still say minus for subtraction as if it exists but a ring has
only two binary operators addition and multiplication.

======================================================
Formally, a ring is an abelian group whose operation is called
addition, with a second binary operation called multiplication that is
associative, is distributive over the addition operation, and has a
multiplicative identity element.
======================================================

Sure, you can find many math sources which tell you just how to
subtract integers from one another, but some can and do claim that
there is no subtraction of integers due to the tenets of ring theory.

On Saturday, August 26, 2023 at 7:21:35 AM UTC-4, FromTheRafters wrote:
> on 8/25/2023, Chris M. Thomasson supposed :
> > On 8/25/2023 1:34 PM, FromTheRafters wrote:
> >> Kevin S brought next idea :
> >>> On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail.com wrote:
> >>>> there is no negative 1.
> >>>> Mitchell Raemsch
> >>>
> >>> Yeah there is.
> >>
> >> Indeed, since we have integers. One could claim that there is no minus one
> >> though.
> >
> > Not exactly sure what you mean.
> One could claim...
>
> The integers form a ring, instead of subtraction of naturals (a partial
> function for them) we have addition of integers. The binary operator
> 'minus' (a-b) is essentially gone and we add a negative integer
> (a+(-b)) instead.
> > Zero minus one equals negative one.
> Zero plus a negative one equals a negative one.
>
> 0 + (-1) = -1
> > Negative one minus negative one equals zero.
> Negative one plus a negated negative one equals zero.
>
> -1 + (-(-1)) = 0
>
> Sure, we still say minus for subtraction as if it exists but a ring has
> only two binary operators addition and multiplication.
>
> ======================================================
> Formally, a ring is an abelian group whose operation is called
> addition, with a second binary operation called multiplication that is
> associative, is distributive over the addition operation, and has a
> multiplicative identity element.
> ======================================================
>
> Sure, you can find many math sources which tell you just how to
> subtract integers from one another, but some can and do claim that
> there is no subtraction of integers due to the tenets of ring theory.

It is somewhat interesting that the minus sign is capable of generating the plus sign, but not the other way around.
In this regard as abstract algebra seeks a minimal set of requirements, things will get inverted.
The fact is that an operator goes confused with a value when '+' means both summation and a sign value.
Likewise subtraction as non-fundamental, wrt aa, thence forms a primitive lockup.

As polysign numbers challenge all of this thinking directly, and recover general dimensional geometry directly from the generalization of sign, it turns out that the confusion over operator and value still remains, but the need for a zero sign is mnemonically required to dissipate an identity sign which keeps shifting. With signs accounted as the number of strokes to draw them we have -,+,*,#, and here I am out of standard symbols, but this will allow the development of P4 using ordinary algebraic arithmetic, and with the introduction of the zero sign '@' which is fairly reminiscent of a zero, then we arguably can do P5 as well, for the signs are modulo behaved. That you all operate with an assumption of P2... even into realms such as the construction of complex numbers, which are more coherently arrived at as P3, and on into aa, P2 are required to develop more complicated derivatives; whereas in polysign P2 and P3 are siblings; P4 is no less primitive than P2. P1 goes overlooked to the chagrine of physicists of the future who tortuously crawl through the accumulation in the stacks of another century; believing that they are going to uncover a kernel of truth.

As we ponder the discrete versus the continuous, is it instructive that the signs of the P2 (real) value are discrete while the magnitude is continuous?
Is it instructive that those signs are modulo two behaved under product?
Could it matter that they form a balanced geometry
- 1 + 1 = 0 ; - a + a = 0; - x + x = 0 [P2]
That the extension is already obvious?
- 1 + 1 * 1 = 0 [P3]
That their geometry is already built here in this simple balance is a purity which the Cartesian product lacks. Occam likes polysign.
That the product obeys the same rules here in P3 as was done in P2, just modulo three now instead of modulo two; and that these then are the complex numbers in a new format; this gain goes ignored by nearly every reader. Ahh, but this is only the beginning.

Mitch's argument that there is no negative one can remain within the concept of magnitude; within the concept of physical distance, even.
Really, humanity has tripped over the introduction of the negative value into the real value, and by tying its shoes together to each other too early it has forgone the richer options of polysign mathematics. Descartes himself never adopted the negative value nor the complex value and in some ways we could mark Mitch as a true Cartesian, whereas those who believe that Rene Descartes himself used RxR are completely misled.

As we witness propaganda in the media today in this country known as the U$ and watch it in a runaway state, to what degree could mathematics have undergone a similar trajectory? I assure you that through the lens of polysign numbers rather a lot falls away, for the complexity is introduced so early and in such a general form that we really are back in gradeschool arithmetic. Best of all these signs dance; they are active; they provide dynamics with surprises. They even provide support for emergent spacetime:
P1 P2 P3 | P4 P5 P6 ...
The breakpoint above is easily established and any with time to investigate who are of a clear mind will arrive with this observation. It takes many interpretations, the easiest of which is the product of two nonzero values yielding a zero value:
( - 1 + 1 )( + 1 # 1 ) = * 1 - 1 # 1 + 1 = 0 [P4]
and so we see that dimensional collapse is possible in the higher geometries. Yet their algebraic behaviors do hold up. Division is not so easy, but the product is easy-peasy. Just FOIL through and you'll have your answer. As for the geometry: the balance of the signs is found geometrically in the rays from the center of an n-verticed simplex outward to its vertices. Label each with a sign and you'll witness the balance is true. It's not that I wish this were true: in fact I'd rather find something non-Euclidean, but this ray based system is still yielding Euclidean space. Still, to realize that the ray is more fundamental than the line: here Mitch is correct.

On Friday, August 25, 2023 at 7:01:48 PM UTC-7, Chris M. Thomasson wrote:
> On 8/25/2023 6:59 PM, mitchr...@gmail.com wrote:
> > On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
> >> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
> >>> there is no negative 1.
> >>>
> >>> Mitchell Raemsch
> >> *
> >> How would you solve this equation?
> >>
> >> x^2 + 1 = 0
> > By your own terms negative one squared is one.
> > i is a formula that has no solution.
> > It shows why negative one does not exist.
> You have never used complex numbers before, and you do not understand them.

i is the no solution formula... what does it do?
complex numbers are named rightly...
They are imaginary.
Stop moving your i around it does not
change the imaginary...

On Saturday, August 26, 2023 at 9:13:26 PM UTC+2, mitchr...@gmail.com wrote:
> On Friday, August 25, 2023 at 7:01:48 PM UTC-7, Chris M. Thomasson wrote:
> > On 8/25/2023 6:59 PM, mitchr...@gmail.com wrote:
> > > On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
> > >> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
> > >>> there is no negative 1.
> > >>>
> > >>> Mitchell Raemsch
> > >> *
> > >> How would you solve this equation?
> > >>
> > >> x^2 + 1 = 0
> > > By your own terms negative one squared is one.
> > > i is a formula that has no solution.
> > > It shows why negative one does not exist.
> > You have never used complex numbers before, and you do not understand them.
> i is the no solution formula... what does it do?
> complex numbers are named rightly...
> They are imaginary.
> Stop moving your i around it does not
> change the imaginary...

I'll move my i around to my heart's content. Without i there is no Laplace transform. Without Laplace transform solving differential equations with initial conditions is more hassle. Without i there is no FFT. Without FFT filtering signals takes quadratic time.

i can be imaginary for all I care. But there's no doubt its usefulness is real.

On 8/26/2023 4:21 AM, FromTheRafters wrote:
> on 8/25/2023, Chris M. Thomasson supposed :
>> On 8/25/2023 1:34 PM, FromTheRafters wrote:
>>> Kevin S brought next idea :
>>>> On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail.com
>>>> wrote:
>>>>> there is no negative 1.
>>>>> Mitchell Raemsch
>>>>
>>>> Yeah there is.
>>>
>>> Indeed, since we have integers. One could claim that there is no
>>> minus one though.
>>
>> Not exactly sure what you mean.
>
> One could claim...
>
> The integers form a ring, instead of subtraction of naturals (a partial
> function for them) we have addition of integers. The binary operator
> 'minus' (a-b) is essentially gone and we add a negative integer (a+(-b))
> instead.
>
>> Zero minus one equals negative one.
>
> Zero plus a negative one equals a negative one.
>
> 0 + (-1) = -1
>
>> Negative one minus negative one equals zero.
>
> Negative one plus a negated negative one equals zero.
>
> -1 + (-(-1)) = 0
>
> Sure, we still say minus for subtraction as if it exists but a ring has
> only two binary operators addition and multiplication.
>
> ======================================================
> Formally, a ring is an abelian group whose operation is called addition,
> with a second binary operation called multiplication that is
> associative, is distributive over the addition operation, and has a
> multiplicative identity element.
> ======================================================
>
> Sure, you can find many math sources which tell you just how to subtract
> integers from one another, but some can and do claim that there is no
> subtraction of integers due to the tenets of ring theory.

Fwiw, check this out; the times tables form mod rings and very
interesting formations:

https://youtu.be/qhbuKbxJsk8

Actually, they are connected to the Mandelbrot set...

On 8/26/2023 1:12 PM, Kevin S wrote:
> On Saturday, August 26, 2023 at 9:13:26 PM UTC+2, mitchr...@gmail.com wrote:
>> On Friday, August 25, 2023 at 7:01:48 PM UTC-7, Chris M. Thomasson wrote:
>>> On 8/25/2023 6:59 PM, mitchr...@gmail.com wrote:
>>>> On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
>>>>> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
>>>>>> there is no negative 1.
>>>>>>
>>>>>> Mitchell Raemsch
>>>>> *
>>>>> How would you solve this equation?
>>>>>
>>>>> x^2 + 1 = 0
>>>> By your own terms negative one squared is one.
>>>> i is a formula that has no solution.
>>>> It shows why negative one does not exist.
>>> You have never used complex numbers before, and you do not understand them.
>> i is the no solution formula... what does it do?
>> complex numbers are named rightly...
>> They are imaginary.
>> Stop moving your i around it does not
>> change the imaginary...
>
> I'll move my i around to my heart's content. Without i there is no Laplace transform. Without Laplace transform solving differential equations with initial conditions is more hassle. Without i there is no FFT. Without FFT filtering signals takes quadratic time.
>
> i can be imaginary for all I care. But there's no doubt its usefulness is real.

Agreed.

On Saturday, August 26, 2023 at 7:46:40 AM UTC-7, Timothy Golden wrote:
> On Saturday, August 26, 2023 at 7:21:35 AM UTC-4, FromTheRafters wrote:
> > on 8/25/2023, Chris M. Thomasson supposed :
> > > On 8/25/2023 1:34 PM, FromTheRafters wrote:
> > >> Kevin S brought next idea :
> > >>> On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail.com wrote:
> > >>>> there is no negative 1.
> > >>>> Mitchell Raemsch
> > >>>
> > >>> Yeah there is.
> > >>
> > >> Indeed, since we have integers. One could claim that there is no minus one
> > >> though.
> > >
> > > Not exactly sure what you mean.
> > One could claim...
> >
> > The integers form a ring, instead of subtraction of naturals (a partial
> > function for them) we have addition of integers. The binary operator
> > 'minus' (a-b) is essentially gone and we add a negative integer
> > (a+(-b)) instead.
> > > Zero minus one equals negative one.
> > Zero plus a negative one equals a negative one.
> >
> > 0 + (-1) = -1
> > > Negative one minus negative one equals zero.
> > Negative one plus a negated negative one equals zero.
> >
> > -1 + (-(-1)) = 0
> >
> > Sure, we still say minus for subtraction as if it exists but a ring has
> > only two binary operators addition and multiplication.
> >
> > ======================================================
> > Formally, a ring is an abelian group whose operation is called
> > addition, with a second binary operation called multiplication that is
> > associative, is distributive over the addition operation, and has a
> > multiplicative identity element.
> > ======================================================
> >
> > Sure, you can find many math sources which tell you just how to
> > subtract integers from one another, but some can and do claim that
> > there is no subtraction of integers due to the tenets of ring theory.
> It is somewhat interesting that the minus sign is capable of generating the plus sign, but not the other way around.
> In this regard as abstract algebra seeks a minimal set of requirements, things will get inverted.
> The fact is that an operator goes confused with a value when '+' means both summation and a sign value.
> Likewise subtraction as non-fundamental, wrt aa, thence forms a primitive lockup.
>
> As polysign numbers challenge all of this thinking directly, and recover general dimensional geometry directly from the generalization of sign, it turns out that the confusion over operator and value still remains, but the need for a zero sign is mnemonically required to dissipate an identity sign which keeps shifting. With signs accounted as the number of strokes to draw them we have -,+,*,#, and here I am out of standard symbols, but this will allow the development of P4 using ordinary algebraic arithmetic, and with the introduction of the zero sign '@' which is fairly reminiscent of a zero, then we arguably can do P5 as well, for the signs are modulo behaved. That you all operate with an assumption of P2... even into realms such as the construction of complex numbers, which are more coherently arrived at as P3, and on into aa, P2 are required to develop more complicated derivatives; whereas in polysign P2 and P3 are siblings; P4 is no less primitive than P2. P1 goes overlooked to the chagrine of physicists of the future who tortuously crawl through the accumulation in the stacks of another century; believing that they are going to uncover a kernel of truth.
>
> As we ponder the discrete versus the continuous, is it instructive that the signs of the P2 (real) value are discrete while the magnitude is continuous?
> Is it instructive that those signs are modulo two behaved under product?
> Could it matter that they form a balanced geometry
> - 1 + 1 = 0 ; - a + a = 0; - x + x = 0 [P2]
> That the extension is already obvious?
> - 1 + 1 * 1 = 0 [P3]
> That their geometry is already built here in this simple balance is a purity which the Cartesian product lacks. Occam likes polysign.
> That the product obeys the same rules here in P3 as was done in P2, just modulo three now instead of modulo two; and that these then are the complex numbers in a new format; this gain goes ignored by nearly every reader. Ahh, but this is only the beginning.
>
> Mitch's argument that there is no negative one can remain within the concept of magnitude; within the concept of physical distance, even.
> Really, humanity has tripped over the introduction of the negative value into the real value, and by tying its shoes together to each other too early it has forgone the richer options of polysign mathematics. Descartes himself never adopted the negative value nor the complex value and in some ways we could mark Mitch as a true Cartesian, whereas those who believe that Rene Descartes himself used RxR are completely misled.
>
> As we witness propaganda in the media today in this country known as the U$ and watch it in a runaway state, to what degree could mathematics have undergone a similar trajectory? I assure you that through the lens of polysign numbers rather a lot falls away, for the complexity is introduced so early and in such a general form that we really are back in gradeschool arithmetic. Best of all these signs dance; they are active; they provide dynamics with surprises. They even provide support for emergent spacetime:
> P1 P2 P3 | P4 P5 P6 ...
> The breakpoint above is easily established and any with time to investigate who are of a clear mind will arrive with this observation. It takes many interpretations, the easiest of which is the product of two nonzero values yielding a zero value:
> ( - 1 + 1 )( + 1 # 1 ) = * 1 - 1 # 1 + 1 = 0 [P4]
> and so we see that dimensional collapse is possible in the higher geometries. Yet their algebraic behaviors do hold up. Division is not so easy, but the product is easy-peasy. Just FOIL through and you'll have your answer. As for the geometry: the balance of the signs is found geometrically in the rays from the center of an n-verticed simplex outward to its vertices. Label each with a sign and you'll witness the balance is true. It's not that I wish this were true: in fact I'd rather find something non-Euclidean, but this ray based system is still yielding Euclidean space. Still, to realize that the ray is more fundamental than the line: here Mitch is correct.

It's almost like there's infinity first and subtraction first that zero is just infinity minus infinity.

There are various deconstructive accounts to apply to the objects of what are called numbers.

For example, we all know the same integers, but there's a notion that they arise as from the
incommensurable magnitudes just as the incommensurable magnitudes get built of them.

Similarly, in deconstructive accounts, there's for addition and multiplication as inverse together
to division and subtraction, for example incr/add/mul and div/sub/decr, in integers, so that
besides where they happen to perfectly coincide in the usual ring then field for integers,
also where they just so differ that they hold each other up, where they meet in the middle.

For example, before getting to negative numbers or imaginary numbers, imagine the
first quadrant all positive that f(x) = x is the axis of the dimension, then what were x and y,
support either way the character of the length of the line, in the "identity dimension",
a sort of deconstructive account of magnitudes in a space, then with usual sorts of notions
"then the entire quadrant is scaled down to a square".

So, "inverse" and the existence of inverses, I would suggest to think not just how
they're defined but how they're implemented, in an infinite space of numbers,
where first it goes all the way out, before it returns at all.

But, it's a sort of higher-order approach in deconstructive elements so most people
are better off with extensions of finger-counting for practically their perspective on things.
That said though there's the entire real side or "completion of analysis" that make
for that usual finger-counting is never complete, and most people learn the word
"infinity" before they learn the word "googol", though "1 with a hundred zeros" is
pretty clear, if not necessarily resultingly tractable. Surely, most people just see that
as "closer to infinity" or "effectively infinite".

On Saturday, August 26, 2023 at 1:12:37 PM UTC-7, Kevin S wrote:
> On Saturday, August 26, 2023 at 9:13:26 PM UTC+2, mitchr...@gmail..com wrote:
> > On Friday, August 25, 2023 at 7:01:48 PM UTC-7, Chris M. Thomasson wrote:
> > > On 8/25/2023 6:59 PM, mitchr...@gmail.com wrote:
> > > > On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
> > > >> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
> > > >>> there is no negative 1.
> > > >>>
> > > >>> Mitchell Raemsch
> > > >> *
> > > >> How would you solve this equation?
> > > >>
> > > >> x^2 + 1 = 0
> > > > By your own terms negative one squared is one.
> > > > i is a formula that has no solution.
> > > > It shows why negative one does not exist.
> > > You have never used complex numbers before, and you do not understand them.
> > i is the no solution formula... what does it do?
> > complex numbers are named rightly...
> > They are imaginary.
> > Stop moving your i around it does not
> > change the imaginary...
> I'll move my i around to my heart's content.
It won't get you anywhere.
That no solution formula doesn't do anything in math
but behave like 1 in the complex plane.

Mitchell Raemsch

On Saturday, August 26, 2023 at 8:29:39 PM UTC-7, mitchr...@gmail.com wrote:
> On Saturday, August 26, 2023 at 1:12:37 PM UTC-7, Kevin S wrote:
> > On Saturday, August 26, 2023 at 9:13:26 PM UTC+2, mitchr...@gmail.com wrote:
> > > On Friday, August 25, 2023 at 7:01:48 PM UTC-7, Chris M. Thomasson wrote:
> > > > On 8/25/2023 6:59 PM, mitchr...@gmail.com wrote:
> > > > > On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
> > > > >> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
> > > > >>> there is no negative 1.
> > > > >>>
> > > > >>> Mitchell Raemsch
> > > > >> *
> > > > >> How would you solve this equation?
> > > > >>
> > > > >> x^2 + 1 = 0
> > > > > By your own terms negative one squared is one.
> > > > > i is a formula that has no solution.
> > > > > It shows why negative one does not exist.
> > > > You have never used complex numbers before, and you do not understand them.
> > > i is the no solution formula... what does it do?
> > > complex numbers are named rightly...
> > > They are imaginary.
> > > Stop moving your i around it does not
> > > change the imaginary...
> > I'll move my i around to my heart's content.
> It won't get you anywhere.
> That no solution formula doesn't do anything in math
> but behave like 1 in the complex plane.
>
> Mitchell Raemsch

Well, it's all about roots of unity and symmetries and reflection and that it models rotation
the roots and besides roots of numbers there are roots of equations, about "the fundamental
theorem of algebra", but then there's also "roots of zero" and the polar, about various diagrams
like the Argand and Wessell and for example a sort of complex-complex diagram out of the
identity dimension.

It's agreeable that Euler's formula isn't the end-all be-all and furthermore that it's sort of
contrived, about correspondences it makes in Argand diagrams, for things like screw theorems
and winding numbers and so on, then though lots of its utilities is for something like Elie Cartan's
and the usual reflections and rotations in the "hypercomplex", about inner and outer products
vis-a-vis usual products and particular vector products, raising and lowering, generalized
products and their inverses, in the various structures what usually enough result algebras,
but generalized go directly out of the algebras.

About roots and powers and things like incr/add, mul/pow, exp/tetr, about the log-linear,
I suppose a variety of functions are analytic in the complex, but, there are also various
functions with real character in the complex that aren't, and even that they're anti, analytic.

On 8/26/2023 8:48 PM, Ross Finlayson wrote:
> On Saturday, August 26, 2023 at 8:29:39 PM UTC-7, mitchr...@gmail.com wrote:
>> On Saturday, August 26, 2023 at 1:12:37 PM UTC-7, Kevin S wrote:
>>> On Saturday, August 26, 2023 at 9:13:26 PM UTC+2, mitchr...@gmail.com wrote:
>>>> On Friday, August 25, 2023 at 7:01:48 PM UTC-7, Chris M. Thomasson wrote:
>>>>> On 8/25/2023 6:59 PM, mitchr...@gmail.com wrote:
>>>>>> On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
>>>>>>> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
>>>>>>>> there is no negative 1.
>>>>>>>>
>>>>>>>> Mitchell Raemsch
>>>>>>> *
>>>>>>> How would you solve this equation?
>>>>>>>
>>>>>>> x^2 + 1 = 0
>>>>>> By your own terms negative one squared is one.
>>>>>> i is a formula that has no solution.
>>>>>> It shows why negative one does not exist.
>>>>> You have never used complex numbers before, and you do not understand them.
>>>> i is the no solution formula... what does it do?
>>>> complex numbers are named rightly...
>>>> They are imaginary.
>>>> Stop moving your i around it does not
>>>> change the imaginary...
>>> I'll move my i around to my heart's content.
>> It won't get you anywhere.
>> That no solution formula doesn't do anything in math
>> but behave like 1 in the complex plane.
>>
>> Mitchell Raemsch
>
> Well, it's all about roots of unity and symmetries and reflection and that it models rotation
[...]

Think of plotting pi/2 = 90 degrees at (0, 1), and plotting pi+pi/2 at
(0, -1) See, (0, 1) is i.

On 8/26/2023 8:58 PM, Chris M. Thomasson wrote:
> On 8/26/2023 8:48 PM, Ross Finlayson wrote:
>> On Saturday, August 26, 2023 at 8:29:39 PM UTC-7, mitchr...@gmail.com
>> wrote:
>>> On Saturday, August 26, 2023 at 1:12:37 PM UTC-7, Kevin S wrote:
>>>> On Saturday, August 26, 2023 at 9:13:26 PM UTC+2,
>>>> mitchr...@gmail.com wrote:
>>>>> On Friday, August 25, 2023 at 7:01:48 PM UTC-7, Chris M. Thomasson
>>>>> wrote:
>>>>>> On 8/25/2023 6:59 PM, mitchr...@gmail.com wrote:
>>>>>>> On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
>>>>>>>> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
>>>>>>>>> there is no negative 1.
>>>>>>>>>
>>>>>>>>> Mitchell Raemsch
>>>>>>>> *
>>>>>>>> How would you solve this equation?
>>>>>>>>
>>>>>>>> x^2 + 1 = 0
>>>>>>> By your own terms negative one squared is one.
>>>>>>> i is a formula that has no solution.
>>>>>>> It shows why negative one does not exist.
>>>>>> You have never used complex numbers before, and you do not
>>>>>> understand them.
>>>>> i is the no solution formula... what does it do?
>>>>> complex numbers are named rightly...
>>>>> They are imaginary.
>>>>> Stop moving your i around it does not
>>>>> change the imaginary...
>>>> I'll move my i around to my heart's content.
>>> It won't get you anywhere.
>>> That no solution formula doesn't do anything in math
>>> but behave like 1 in the complex plane.
>>>
>>> Mitchell Raemsch
>>
>> Well, it's all about roots of unity and symmetries and reflection and
>> that it models rotation
> [...]
>
> Think of plotting pi/2 = 90 degrees at (0, 1), and plotting pi+pi/2 at
> (0, -1) See, (0, 1) is i.
>

(0, 1) + pi = (0, -1)

Think polar...

On Saturday, August 26, 2023 at 8:59:52 PM UTC-7, Chris M. Thomasson wrote:
> On 8/26/2023 8:58 PM, Chris M. Thomasson wrote:
> > On 8/26/2023 8:48 PM, Ross Finlayson wrote:
> >> On Saturday, August 26, 2023 at 8:29:39 PM UTC-7, mitchr...@gmail.com
> >> wrote:
> >>> On Saturday, August 26, 2023 at 1:12:37 PM UTC-7, Kevin S wrote:
> >>>> On Saturday, August 26, 2023 at 9:13:26 PM UTC+2,
> >>>> mitchr...@gmail.com wrote:
> >>>>> On Friday, August 25, 2023 at 7:01:48 PM UTC-7, Chris M. Thomasson
> >>>>> wrote:
> >>>>>> On 8/25/2023 6:59 PM, mitchr...@gmail.com wrote:
> >>>>>>> On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
> >>>>>>>> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
> >>>>>>>>> there is no negative 1.
> >>>>>>>>>
> >>>>>>>>> Mitchell Raemsch
> >>>>>>>> *
> >>>>>>>> How would you solve this equation?
> >>>>>>>>
> >>>>>>>> x^2 + 1 = 0
> >>>>>>> By your own terms negative one squared is one.
> >>>>>>> i is a formula that has no solution.
> >>>>>>> It shows why negative one does not exist.
> >>>>>> You have never used complex numbers before, and you do not
> >>>>>> understand them.
> >>>>> i is the no solution formula... what does it do?
> >>>>> complex numbers are named rightly...
> >>>>> They are imaginary.
> >>>>> Stop moving your i around it does not
> >>>>> change the imaginary...
> >>>> I'll move my i around to my heart's content.
> >>> It won't get you anywhere.
> >>> That no solution formula doesn't do anything in math
> >>> but behave like 1 in the complex plane.
> >>>
> >>> Mitchell Raemsch
> >>
> >> Well, it's all about roots of unity and symmetries and reflection and
> >> that it models rotation
> > [...]
> >
> > Think of plotting pi/2 = 90 degrees at (0, 1), and plotting pi+pi/2 at
> > (0, -1) See, (0, 1) is i.
> >
> (0, 1) + pi = (0, -1)
>
> Think polar...

If you're going to overload addition in arithmetic of various types
that's fine, basically reflecting pi as some "constant of rotation".
(As a value of a type of some "constant of rotation".)

If you're going to overload arithmetic for between two types otherwise
unrelated by the symbols, the goal of "arithmetization" then is sort of
"whatever are the relations, they're implemented by the operations of
arithmetic", so it's a sort of, "reverse arithmetization", but it's operator
overloading, that the semantics what you define are whatever are as so,
where the syntax of the language includes operator overloads.

Then, maybe the operation has a natural semantic that makes for
"multiplication is repeated addition" or "subtraction is inverse addition"
or whatever other manner makes well-defined behavior insofar as
you care the behavior is well-defined.

Polar then I think of polar and first of all I miss my winters where the
temperature does not go above freezing for at least three months in
a row, and I'm thinking about doing something about that, but about
polar there are two sorts notions, sort of like there are two sorts notions
of roots and roots, though they're related. There's "the asymptote" or
there's "north-south". So anyways the polar we all know is radial,
it goes around in two dimensions, at all, vis-a-vis the Cartesian, which
is sums of the objects already in the space, two dimensions.

Yeah surely I know that pi/2 radians rotations, involving rotation matrices,
work out more exactly contriving the pi/2, pi, 3pi/2, ..., rotation matrices
or quaternions, vis-a-vis plugging in the transcendental number as a floating
point, getting a floating point, and never getting exactly 90 or pi/2 radians.

Then this is usually called orientations often for example with the "F", i.e.,
that the letter "F" can be shown in each of the eight orientations and it's
unambiguous the 0, pi/2, pi, 3pi/2, and whether it's reversed, orientations..

Then, back to arithmetizations and operator overloading, there's lots to be said
for implementing arithmetizations because then the objects or attributes
can have values in integers and run on the integer unit, for example the vector unit.
The most usual types that have those already are the numbers already,
then for example for indeterminate forms, say, i.e. reserving the semantics of
cases in the semantics of indeterminate forms or exceptions, about the use
of performing usual functions on the integer units while falling out for cases.

Or, that's a use case, say if the only machine provided was an integer unit..

On 8/26/2023 9:25 PM, Ross Finlayson wrote:
> On Saturday, August 26, 2023 at 8:59:52 PM UTC-7, Chris M. Thomasson wrote:
>> On 8/26/2023 8:58 PM, Chris M. Thomasson wrote:
>>> On 8/26/2023 8:48 PM, Ross Finlayson wrote:
>>>> On Saturday, August 26, 2023 at 8:29:39 PM UTC-7, mitchr...@gmail.com
>>>> wrote:
>>>>> On Saturday, August 26, 2023 at 1:12:37 PM UTC-7, Kevin S wrote:
>>>>>> On Saturday, August 26, 2023 at 9:13:26 PM UTC+2,
>>>>>> mitchr...@gmail.com wrote:
>>>>>>> On Friday, August 25, 2023 at 7:01:48 PM UTC-7, Chris M. Thomasson
>>>>>>> wrote:
>>>>>>>> On 8/25/2023 6:59 PM, mitchr...@gmail.com wrote:
>>>>>>>>> On Friday, August 25, 2023 at 4:53:09 PM UTC-7, Earle wrote:
>>>>>>>>>> On Fri Aug 25 11:11:06 2023 "mitchr...@gmail.com" wrote:
>>>>>>>>>>> there is no negative 1.
>>>>>>>>>>>
>>>>>>>>>>> Mitchell Raemsch
>>>>>>>>>> *
>>>>>>>>>> How would you solve this equation?
>>>>>>>>>>
>>>>>>>>>> x^2 + 1 = 0
>>>>>>>>> By your own terms negative one squared is one.
>>>>>>>>> i is a formula that has no solution.
>>>>>>>>> It shows why negative one does not exist.
>>>>>>>> You have never used complex numbers before, and you do not
>>>>>>>> understand them.
>>>>>>> i is the no solution formula... what does it do?
>>>>>>> complex numbers are named rightly...
>>>>>>> They are imaginary.
>>>>>>> Stop moving your i around it does not
>>>>>>> change the imaginary...
>>>>>> I'll move my i around to my heart's content.
>>>>> It won't get you anywhere.
>>>>> That no solution formula doesn't do anything in math
>>>>> but behave like 1 in the complex plane.
>>>>>
>>>>> Mitchell Raemsch
>>>>
>>>> Well, it's all about roots of unity and symmetries and reflection and
>>>> that it models rotation
>>> [...]
>>>
>>> Think of plotting pi/2 = 90 degrees at (0, 1), and plotting pi+pi/2 at
>>> (0, -1) See, (0, 1) is i.
>>>
>> (0, 1) + pi = (0, -1)
>>
>> Think polar...
>
> If you're going to overload addition in arithmetic of various types
> that's fine, basically reflecting pi as some "constant of rotation".
> (As a value of a type of some "constant of rotation".)
>
> If you're going to overload arithmetic for between two types otherwise
> unrelated by the symbols, the goal of "arithmetization" then is sort of
> "whatever are the relations, they're implemented by the operations of
> arithmetic", so it's a sort of, "reverse arithmetization", but it's operator
> overloading, that the semantics what you define are whatever are as so,
> where the syntax of the language includes operator overloads.
>
> Then, maybe the operation has a natural semantic that makes for
> "multiplication is repeated addition" or "subtraction is inverse addition"
> or whatever other manner makes well-defined behavior insofar as
> you care the behavior is well-defined.
>
> Polar then I think of polar and first of all I miss my winters where the
> temperature does not go above freezing for at least three months in
> a row, and I'm thinking about doing something about that, but about
> polar there are two sorts notions, sort of like there are two sorts notions
> of roots and roots, though they're related. There's "the asymptote" or
> there's "north-south". So anyways the polar we all know is radial,
> it goes around in two dimensions, at all, vis-a-vis the Cartesian, which
> is sums of the objects already in the space, two dimensions.
>
> Yeah surely I know that pi/2 radians rotations, involving rotation matrices,
> work out more exactly contriving the pi/2, pi, 3pi/2, ..., rotation matrices
> or quaternions, vis-a-vis plugging in the transcendental number as a floating
> point, getting a floating point, and never getting exactly 90 or pi/2 radians.
>
> Then this is usually called orientations often for example with the "F", i.e.,
> that the letter "F" can be shown in each of the eight orientations and it's
> unambiguous the 0, pi/2, pi, 3pi/2, and whether it's reversed, orientations.
>
> Then, back to arithmetizations and operator overloading, there's lots to be said
> for implementing arithmetizations because then the objects or attributes
> can have values in integers and run on the integer unit, for example the vector unit.
> The most usual types that have those already are the numbers already,
> then for example for indeterminate forms, say, i.e. reserving the semantics of
> cases in the semantics of indeterminate forms or exceptions, about the use
> of performing usual functions on the integer units while falling out for cases.
>
> Or, that's a use case, say if the only machine provided was an integer unit.

(0, 1) in polar form is radius one with an angle of pi/2.

So, go to polar form, gain the angle, go to rectangular form, plot the
sucker on your plane. No problem.

(0, 1) is a point on the unit circle, i, this is how to plot the
imaginary unit. 0 + pi/2 radians wrt unit circle plots point (0, 1) in
rectangular form.

FromTheRafters <FTR@nomail.afraid.org> writes:
> on 8/25/2023, Chris M. Thomasson supposed :
>> On 8/25/2023 1:34 PM, FromTheRafters wrote:
>>> Kevin S brought next idea :
>>>> On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail.com wrote:
>>>>> there is no negative 1.
>>>>> Mitchell Raemsch
>>>>
>>>> Yeah there is.
>>>
>>> Indeed, since we have integers. One could claim that there is no
>>> minus one though.
>>
>> Not exactly sure what you mean.
>
> One could claim...
>
> The integers form a ring, instead of subtraction of naturals (a
> partial function for them) we have addition of integers. The binary
> operator 'minus' (a-b) is essentially gone and we add a negative
> integer (a+(-b)) instead.

That X can be defined in terms of Y, where Y is known to exist, does not
mean that X does not exist. One might even say it's pretty good proof
that X does exist.

Phil
--
We are no longer hunters and nomads. No longer awed and frightened, as we have
gained some understanding of the world in which we live. As such, we can cast
aside childish remnants from the dawn of our civilization.
-- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/

On Sunday, August 27, 2023 at 6:07:42 AM UTC-4, Phil Carmody wrote:
> FromTheRafters <F...@nomail.afraid.org> writes:
> > on 8/25/2023, Chris M. Thomasson supposed :
> >> On 8/25/2023 1:34 PM, FromTheRafters wrote:
> >>> Kevin S brought next idea :
> >>>> On Friday, August 25, 2023 at 8:11:12 PM UTC+2, mitchr...@gmail.com wrote:
> >>>>> there is no negative 1.
> >>>>> Mitchell Raemsch
> >>>>
> >>>> Yeah there is.
> >>>
> >>> Indeed, since we have integers. One could claim that there is no
> >>> minus one though.
> >>
> >> Not exactly sure what you mean.
> >
> > One could claim...
> >
> > The integers form a ring, instead of subtraction of naturals (a
> > partial function for them) we have addition of integers. The binary
> > operator 'minus' (a-b) is essentially gone and we add a negative
> > integer (a+(-b)) instead.
> That X can be defined in terms of Y, where Y is known to exist, does not
> mean that X does not exist. One might even say it's pretty good proof
> that X does exist.
>
> Phil
> --
> We are no longer hunters and nomads. No longer awed and frightened, as we have
> gained some understanding of the world in which we live. As such, we can cast
> aside childish remnants from the dawn of our civilization.
> -- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/

What it means to a mathematician 'to exist' is quite a controversial problem, especially when physical correspondence is cast aside as irrelevant, and what would be next to cast aside but philosophy? Certainly the quantum physicists have done exactly this, and as if these broad divides were even accurate in the third place, and which deserves the first place: each will claim it for their own, right?

If, on the other hand, we dismiss these status quo assumptions, and witness these divides as false, then we ought to reflect a bit deeper upon the negative integer. Of course sensibility is lost as regards sheep in a field, whereby no physical correspondence can be laid by -5 sheep. A discussion of inverse sheep I suppose could ensue, and in some cases this argument does take physical relevance going over to antimatter, landing in an open puzzle as to how such an imbalance has occurred as we seem to observe. The hope that we will one day unwind this problem and land in fertile mathematics which provides physical correspondence is obviously where many would like to land.

A broader treatment of sign will disentangle the confusion between sign and inversion; the conflict that exists as an operator versus a value in the discrete symbol '-' is an ambiguity within our notation. If anyone ought to abhor such ambiguity would it be the mathematician? The philosopher? The physicist? As to whose burden this one is; that is pretty easy to pinpoint. This is mathematics. As to who ought to be concerned about it: we all should..

Operator theory itself seems to contain additional quagmire. The idea that an operator will map SxS onto S is a functional blunder. To some all is a function apparently, and yet will you even be able to use addition within your function? Where is the need for this Cartesian product? To sum three elements we'll be needing SxSxS? I'm sorry but one S will do very nicely, thank you. As if closure didn't even mean closure in the first place, certainly it doesn't mean it in the second place, and by the time you've gone three-dimensional on the thing to call this closure any more is just an abuse of the human mind. That we all could buy into such clap-trap at the base of mathematics is a sad citing of modernity.