****************************************************
Fri 16 Jun 2023 10:48:14 AM EDT sci.math
Challenging the Cartesian product; particularly on things like RxR; or even the abstracted SxS: what good do two copies of the same exact set do? Well: in operator theory it seems this is critical, and without it the sum could barely exist. Group theory and its claims of closure want us to believe that the sum is a function mapping SxS back onto S. As I scrutinize the Cartesian product I find this nearly laughable. Indeed, the fact that the sum works in an n-ary fashion would suggest that we'll need SxSxS in order to consider a+b+c. Why can't they all simply belong to the same set, including the sum? Does simplicity rule mathematics? Is functional analysis king? Can you imagine a function devoid of sums and products within its construction? Isn't this a structural problem?

Along the same lines, we know the utility of the Cartesian product when it comes to geometry. The (x,y) coordinate system has been entrained onto the entire schooled human race for some time now, along with a belief in the fundamental nature of the real number. Thus RxR as representing the plane, and I do believe that it does to the degree that the conventions implied, though not explicitly, are engaged en masse. Yet here I am breaking with the acceptance in order to dig a bit deeper. We know that physically this procedure comes to a halt at RxRxR don't we? We then claim that higher dimension spaces occur in say RxRxRxR, but the physical proof, which got us off the line and into the plane with another line at right angles to the first, is not going to occur. In hindsight should it be said that the line which was drawn indeed had a thickness and a width and that without these attributes the proof would not have been possible? Likewise the plane as represented on a piece of paper back into Euclidean antiquity, clean as that thought may be entrained in our minds; that the compass and the rule and the marker engaged were all three dimensional tools? Then too, that time enters into this mix, and even as a pure fourth and real value according to some... it's all a bit loaded.

As if reality starts from a blank page... and this leads onto the first cosmological principle of isotropy... on average according to some who take great care, and tell me, what would anything average to but monochrome gray? No, I think they've mistaken their piece of paper for physical space, as if it is empty and devoid of matter. Just as physics begins by restricting to a few simple interactions, and preferably reduced to a singular dimension, they've taken the lever arm off the page completely and so they see isotropic space. Yet physical reality is not this at all. This is the only way that I can see such a blunderous mistake getting made that goes accepted not just by cosmologists, but by philosophers and mathematicians as well. Indeed the bounds between these supposed professions must be declared false when a more ultimate truth arrives.

I consider my own ability to point to the big dipper as evidenciary proof against the first cosmological principal. And yet the source of this consideration is a long way around. Still, to declare that spacetime is structured; rather than isotropic; and indeed simply inserting unidirectional time as a fourth ought to deny the tensor its utility. The arbitrary reference frame sensibility of the three dimensional tensor is hereby destroyed and will only get recovered via a lightcone projection, and all of this never attempts to answer the puzzle of those three dimensions does it?

To sit all of theory an empirical pegs is the next great blunder of modern thought; as if this thought actually has not even arrived yet... and indeed this may be our position. That we are engaged in a progression, but one whose accumulation leaves us lost in absorption of that pile rather than putting down the stepping stones for the next mile; really puts us all at odds with academia.

I think that the vector of thought on whether the Cartesian coordinate system is observed or constructed may be a valid concern. Certainly the Cartesian set product claims independence of its components, yet when we arrive in the coordinate system we see an agreement to use perpendiculars, the same scale of units, and to orient two real lines such that their zeros intersect. Even then, all that we've arrived at is a supposedly pure description of geometry. To arrive in the inertial reference frame; a matter of the first derivative. Then too accelerations as force being the third derivative, and what... we stop there? Hmmm... an interesting way to three, isn't it?

For a quick trick, let me introduce a modulo work space; one in which when you leave your paper you come out on the other side of the paper. This is not a trivial thing to engage in, but should our positions and velocities be arbitrary in some moment of creation, then the matter with equivalent momentum and velocity will congeal into larger masses (presuming gravity here), leaving much matter of other momenta and position. Could it be that a red shift would ensue? In effect, the matter which was colliding, having collided largely, essentially congealing, leaves an impression of matter spreading? I'm not sure of this at all, but perhaps it can stimulate somebody else's thought process.

Moving back to fundamentals, we must accept that we do not construct multidimensional spaces actually with our representations that claim such. I can no more take two rulers and cross them and claim to have constructed a plane, though the thought holds up somewhat. It holds up to three rulers so long as they are not all coplanar. Of course all of this thinking has been done granting the fundamental nature of the real value, which is two-signed value, and the investigation of three-signed values and so forth has only just begun. One thing is for certain: nobody ever found emergent spacetime that way. In polysign is is done already in pure math; the true theoretical form.

On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
> ****************************************************
> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> Challenging the Cartesian product; particularly on things like RxR; or even the abstracted SxS: what good do two copies of the same exact set do? Well: in operator theory it seems this is critical, and without it the sum could barely exist. Group theory and its claims of closure want us to believe that the sum is a function mapping SxS back onto S. As I scrutinize the Cartesian product I find this nearly laughable. Indeed, the fact that the sum works in an n-ary fashion would suggest that we'll need SxSxS in order to consider a+b+c. Why can't they all simply belong to the same set, including the sum? Does simplicity rule mathematics? Is functional analysis king? Can you imagine a function devoid of sums and products within its construction? Isn't this a structural problem?
>
> Along the same lines, we know the utility of the Cartesian product when it comes to geometry. The (x,y) coordinate system has been entrained onto the entire schooled human race for some time now, along with a belief in the fundamental nature of the real number. Thus RxR as representing the plane, and I do believe that it does to the degree that the conventions implied, though not explicitly, are engaged en masse. Yet here I am breaking with the acceptance in order to dig a bit deeper. We know that physically this procedure comes to a halt at RxRxR don't we? We then claim that higher dimension spaces occur in say RxRxRxR, but the physical proof, which got us off the line and into the plane with another line at right angles to the first, is not going to occur. In hindsight should it be said that the line which was drawn indeed had a thickness and a width and that without these attributes the proof would not have been possible? Likewise the plane as represented on a piece of paper back into Euclidean antiquity, clean as that thought may be entrained in our minds; that the compass and the rule and the marker engaged were all three dimensional tools? Then too, that time enters into this mix, and even as a pure fourth and real value according to some... it's all a bit loaded.
>
> As if reality starts from a blank page... and this leads onto the first cosmological principle of isotropy... on average according to some who take great care, and tell me, what would anything average to but monochrome gray? No, I think they've mistaken their piece of paper for physical space, as if it is empty and devoid of matter. Just as physics begins by restricting to a few simple interactions, and preferably reduced to a singular dimension, they've taken the lever arm off the page completely and so they see isotropic space. Yet physical reality is not this at all. This is the only way that I can see such a blunderous mistake getting made that goes accepted not just by cosmologists, but by philosophers and mathematicians as well. Indeed the bounds between these supposed professions must be declared false when a more ultimate truth arrives.
>
> I consider my own ability to point to the big dipper as evidenciary proof against the first cosmological principal. And yet the source of this consideration is a long way around. Still, to declare that spacetime is structured; rather than isotropic; and indeed simply inserting unidirectional time as a fourth ought to deny the tensor its utility. The arbitrary reference frame sensibility of the three dimensional tensor is hereby destroyed and will only get recovered via a lightcone projection, and all of this never attempts to answer the puzzle of those three dimensions does it?
>
> To sit all of theory an empirical pegs is the next great blunder of modern thought; as if this thought actually has not even arrived yet... and indeed this may be our position. That we are engaged in a progression, but one whose accumulation leaves us lost in absorption of that pile rather than putting down the stepping stones for the next mile; really puts us all at odds with academia.
>
> I think that the vector of thought on whether the Cartesian coordinate system is observed or constructed may be a valid concern. Certainly the Cartesian set product claims independence of its components, yet when we arrive in the coordinate system we see an agreement to use perpendiculars, the same scale of units, and to orient two real lines such that their zeros intersect. Even then, all that we've arrived at is a supposedly pure description of geometry. To arrive in the inertial reference frame; a matter of the first derivative. Then too accelerations as force being the third derivative, and what... we stop there? Hmmm... an interesting way to three, isn't it?
>
> For a quick trick, let me introduce a modulo work space; one in which when you leave your paper you come out on the other side of the paper. This is not a trivial thing to engage in, but should our positions and velocities be arbitrary in some moment of creation, then the matter with equivalent momentum and velocity will congeal into larger masses (presuming gravity here), leaving much matter of other momenta and position. Could it be that a red shift would ensue? In effect, the matter which was colliding, having collided largely, essentially congealing, leaves an impression of matter spreading? I'm not sure of this at all, but perhaps it can stimulate somebody else's thought process.
>
> Moving back to fundamentals, we must accept that we do not construct multidimensional spaces actually with our representations that claim such. I can no more take two rulers and cross them and claim to have constructed a plane, though the thought holds up somewhat. It holds up to three rulers so long as they are not all coplanar. Of course all of this thinking has been done granting the fundamental nature of the real value, which is two-signed value, and the investigation of three-signed values and so forth has only just begun. One thing is for certain: nobody ever found emergent spacetime that way. In polysign is is done already in pure math; the true theoretical form.

I enjoyed this.

About dimensions, di-mensurations, mutual-measurings, for, co-ordinates,
sort of is for setting up both the inside, and length and magnitude, and the
outside, and the space that contains them, and the ordinate, and the abscissa,
where there is "free" and "fixed" one or the other the dimensions, about
curves and tangents, abscissae and ordinates, and the axes, of the dimensions.

Dimension: "plurality".

It really is the projection onto a higher space and the perspective from there.
It's also vice-versa, projection and perspective are very complementary.

You introduce a notion of "depth", vis-a-vis "a surface: it has sides".

Carry on in this manner, although, I'd aver that the existence of the space is
always a mental construct what results imaging and it so exists.

On Friday, June 16, 2023 at 11:29:06 AM UTC-4, Ross Finlayson wrote:
> On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
> > ****************************************************
> > Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> > Challenging the Cartesian product; particularly on things like RxR; or even the abstracted SxS: what good do two copies of the same exact set do? Well: in operator theory it seems this is critical, and without it the sum could barely exist. Group theory and its claims of closure want us to believe that the sum is a function mapping SxS back onto S. As I scrutinize the Cartesian product I find this nearly laughable. Indeed, the fact that the sum works in an n-ary fashion would suggest that we'll need SxSxS in order to consider a+b+c. Why can't they all simply belong to the same set, including the sum? Does simplicity rule mathematics? Is functional analysis king? Can you imagine a function devoid of sums and products within its construction? Isn't this a structural problem?
> >
> > Along the same lines, we know the utility of the Cartesian product when it comes to geometry. The (x,y) coordinate system has been entrained onto the entire schooled human race for some time now, along with a belief in the fundamental nature of the real number. Thus RxR as representing the plane, and I do believe that it does to the degree that the conventions implied, though not explicitly, are engaged en masse. Yet here I am breaking with the acceptance in order to dig a bit deeper. We know that physically this procedure comes to a halt at RxRxR don't we? We then claim that higher dimension spaces occur in say RxRxRxR, but the physical proof, which got us off the line and into the plane with another line at right angles to the first, is not going to occur. In hindsight should it be said that the line which was drawn indeed had a thickness and a width and that without these attributes the proof would not have been possible? Likewise the plane as represented on a piece of paper back into Euclidean antiquity, clean as that thought may be entrained in our minds; that the compass and the rule and the marker engaged were all three dimensional tools? Then too, that time enters into this mix, and even as a pure fourth and real value according to some... it's all a bit loaded.
> >
> > As if reality starts from a blank page... and this leads onto the first cosmological principle of isotropy... on average according to some who take great care, and tell me, what would anything average to but monochrome gray? No, I think they've mistaken their piece of paper for physical space, as if it is empty and devoid of matter. Just as physics begins by restricting to a few simple interactions, and preferably reduced to a singular dimension, they've taken the lever arm off the page completely and so they see isotropic space. Yet physical reality is not this at all. This is the only way that I can see such a blunderous mistake getting made that goes accepted not just by cosmologists, but by philosophers and mathematicians as well. Indeed the bounds between these supposed professions must be declared false when a more ultimate truth arrives.
> >
> > I consider my own ability to point to the big dipper as evidenciary proof against the first cosmological principal. And yet the source of this consideration is a long way around. Still, to declare that spacetime is structured; rather than isotropic; and indeed simply inserting unidirectional time as a fourth ought to deny the tensor its utility. The arbitrary reference frame sensibility of the three dimensional tensor is hereby destroyed and will only get recovered via a lightcone projection, and all of this never attempts to answer the puzzle of those three dimensions does it?
> >
> > To sit all of theory an empirical pegs is the next great blunder of modern thought; as if this thought actually has not even arrived yet... and indeed this may be our position. That we are engaged in a progression, but one whose accumulation leaves us lost in absorption of that pile rather than putting down the stepping stones for the next mile; really puts us all at odds with academia.
> >
> > I think that the vector of thought on whether the Cartesian coordinate system is observed or constructed may be a valid concern. Certainly the Cartesian set product claims independence of its components, yet when we arrive in the coordinate system we see an agreement to use perpendiculars, the same scale of units, and to orient two real lines such that their zeros intersect. Even then, all that we've arrived at is a supposedly pure description of geometry. To arrive in the inertial reference frame; a matter of the first derivative. Then too accelerations as force being the third derivative, and what... we stop there? Hmmm... an interesting way to three, isn't it?
> >
> > For a quick trick, let me introduce a modulo work space; one in which when you leave your paper you come out on the other side of the paper. This is not a trivial thing to engage in, but should our positions and velocities be arbitrary in some moment of creation, then the matter with equivalent momentum and velocity will congeal into larger masses (presuming gravity here), leaving much matter of other momenta and position. Could it be that a red shift would ensue? In effect, the matter which was colliding, having collided largely, essentially congealing, leaves an impression of matter spreading? I'm not sure of this at all, but perhaps it can stimulate somebody else's thought process.
> >
> > Moving back to fundamentals, we must accept that we do not construct multidimensional spaces actually with our representations that claim such. I can no more take two rulers and cross them and claim to have constructed a plane, though the thought holds up somewhat. It holds up to three rulers so long as they are not all coplanar. Of course all of this thinking has been done granting the fundamental nature of the real value, which is two-signed value, and the investigation of three-signed values and so forth has only just begun. One thing is for certain: nobody ever found emergent spacetime that way. In polysign is is done already in pure math; the true theoretical form.
> I enjoyed this.
>
> About dimensions, di-mensurations, mutual-measurings, for, co-ordinates,
> sort of is for setting up both the inside, and length and magnitude, and the
> outside, and the space that contains them, and the ordinate, and the abscissa,
> where there is "free" and "fixed" one or the other the dimensions, about
> curves and tangents, abscissae and ordinates, and the axes, of the dimensions.
>
> Dimension: "plurality".
>
> It really is the projection onto a higher space and the perspective from there.
> It's also vice-versa, projection and perspective are very complementary.
>
> You introduce a notion of "depth", vis-a-vis "a surface: it has sides".
>
> Carry on in this manner, although, I'd aver that the existence of the space is
> always a mental construct what results imaging and it so exists.

Well, I think you are working on the lite version. I'm attacking the Cartesian product slowly, but surely.
Is the geometry of (x,y) coordinates an ad hoc system?
We not only represent the plane this way; we carry on in it for rather a lot of other work.
It does become more formalized in linear algebra, but clearly the sensibility of that work is somewhat built upon the earlier training.
At some point we'll see that the dot product becomes an essential result.
Dotting your e's you'll have the identity matrix when all is well, and coordinate transformations ensue.
Again though, whether these higher dimension spaces are composed of one dimensional parts is where I break away.
Of course the Cartesian product has no geometrical requirements, and so the common notation of RxR, being coupled only via this mechanism and no other, is a sore point to land on. If there is ambiguity that has been agreeably ignored for the sake of progress; well, I can tell you that it is possible to progress in another direction; that of generalization of sign. Rather than copying two-signed space and acting as though the multiple copies yield a geometry, which they do not until further constraints are placed on the system, it could have been done via the generalization of sign. And what occurs is the realization that just as the real line gains its geometry from the balance of its two signs, so a three-signed system can be born as:
- a + a * a = 0
and when this system takes its ordered form we see that (1,1,1)=0, which is this same balance, and so a value like (1,2,3) can be reduced to (0,1,2), and so we can see somewhat the two dimensional reduction, but this term 'dimensional' has already been tied to the real line, and this is not at all in the spirit of polysign. P3 are as fundamental as P2. Indeed there is even a peculiar P1 down beneath; a unidirectional system which suffers a strange zero dimensional geometry: (a)=0, and yet is capable of performing algebra such as (-2)(-1-3)=-8, and so what we witness is that this law of balance quite literally is the geometrical rendering. This polysign geometry is of n rays emanating from the center of a simplex outward to its vertices. Just these rays; no framework is necessary. Step one unit along these in each direction and you'll land where you started. That an arithmetic product exists as well and is algebraic in general dimension is no small gain, and that P3 are equivalent to the complex numbers, though in a new format; from the same rules that build P2. 'You Sank My Battleship' might come to mind...

On 6/17/2023 8:49 AM, Timothy Golden wrote:
> On Friday, June 16, 2023 at 11:29:06 AM UTC-4, Ross Finlayson wrote:
>> On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
>>> ****************************************************
>>> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
[...]

A 2-ary plane, aka (x, y) is 2d dimensional. negative x is going to the
left, positive x is going to the right. Positive y is going up, and
negative y is going down.

On Saturday, June 17, 2023 at 4:19:50 PM UTC-4, Chris M. Thomasson wrote:
> On 6/17/2023 8:49 AM, Timothy Golden wrote:
> > On Friday, June 16, 2023 at 11:29:06 AM UTC-4, Ross Finlayson wrote:
> >> On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
> >>> ****************************************************
> >>> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> [...]
>
> A 2-ary plane, aka (x, y) is 2d dimensional. negative x is going to the
> left, positive x is going to the right. Positive y is going up, and
> negative y is going down.

And a younger me would agree with you one hundred percent and laugh off my own situation of scrutinizing this system.
It does indeed work, and so long as everybody obeys; marks up their papers the same way every time; then there will be no confusion.
Chirality is an interesting feature that I'd like to break into at some point, but I wouldn't want to hinge my entire analysis on it because it seems always to be a tricky one.

As I go over it again, I have to point out to you the issue of whether the proof that this system works is in fact empirical. The nasty details of this terminology go like:
"based on, concerned with, or verifiable by observation or experience rather than theory or pure logic."
and so you see that we cannot from a theoretical perspective cast too much reliance upon it. Preferably our theory arises more from naught, and comes into line with the empirical as a match which is made respecting these boundaries. Normally we might even relegate this awareness to physics, but how pretty it would be if that false boundary (between mathematics and physics) were broken into here as well. Perhaps you could even relegate the distinction between the theoretical and the empirical to philosophy thereby allowing each to carry on with blinders towards the others. No. These are false divisions, and the moment we bowed to an empirical situation within what was to be pure theory is exactly where we need to focus. That such has been done: that we are merely in the midst of the progression; this is our actual situation. Bleary eyed walking the stacks in their accumulation it would be difficult to adopt this stance, yet as we trouble over the academic system and its expectations there is tremendous support for rubbing the crud from the corners of ones eyes, perhaps grabbing another cup of coffee, and trying once again to waken the topics from a unified stance just as the greats of old did. That this method has been dead and gone for some time: academia has brought this on. Pigeon-holed is everybody's position who wants to fit..
I think that there is room here too to get right over onto the vector representation, and historically I find it interesting how late in the game that comes along. Interesting too that the vector is a ray... built out of lines... yet the line is built out of rays.

Suppose for a moment that the convention was to draw your positive y ray 60 degrees counterclockwise to your positive x ray. We could fancy up some historical creation for why this would be: the ease of construction of an equilateral triangle as three splits from the same piece of wood which was drilled prior to the splitting for the pegging made this a woodland choice of geometers working under way. A ring of truth in the three...

Now, as this is the convention amongst this heretofore alien civilization, maybe call them WoodSirs; where is the problem with their Woodser product in comparison to your supposedly Cartesian product? Where then is the convention in the theory? Is it arbitrary? As the WoodSirs like their equilaterals for much of their constructions; developing bents and so forth via drilling and pegging, and it was the drilling part they excelled in. Why the wood exited the hole faster than the bit went into the wood and nearly effortlessly; the wood would shoot forth out of their auger holes like a fountain of water practically, except the prettiest of shavings were the result. Anyways, as one WoodSir geometer mentioned to another a structure of positions as (1,2), (1,3), (3,4) as a particularly excellent though slender triangle would there be any confusion? Indeed selecting any three points will there be any trouble? I'm pretty sure that their version of the plane is just as secure, if not more so with their locked triangular reference so readily made.

If the rules can be broken and the thing still works then what of the rules?

On Sunday, June 18, 2023 at 9:44:10 AM UTC-4, Timothy Golden wrote:
> On Saturday, June 17, 2023 at 4:19:50 PM UTC-4, Chris M. Thomasson wrote:
> > On 6/17/2023 8:49 AM, Timothy Golden wrote:
> > > On Friday, June 16, 2023 at 11:29:06 AM UTC-4, Ross Finlayson wrote:
> > >> On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
> > >>> ****************************************************
> > >>> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> > [...]
> >
> > A 2-ary plane, aka (x, y) is 2d dimensional. negative x is going to the
> > left, positive x is going to the right. Positive y is going up, and
> > negative y is going down.
> And a younger me would agree with you one hundred percent and laugh off my own situation of scrutinizing this system.
> It does indeed work, and so long as everybody obeys; marks up their papers the same way every time; then there will be no confusion.
> Chirality is an interesting feature that I'd like to break into at some point, but I wouldn't want to hinge my entire analysis on it because it seems always to be a tricky one.
>
> As I go over it again, I have to point out to you the issue of whether the proof that this system works is in fact empirical. The nasty details of this terminology go like:
> "based on, concerned with, or verifiable by observation or experience rather than theory or pure logic."
> and so you see that we cannot from a theoretical perspective cast too much reliance upon it. Preferably our theory arises more from naught, and comes into line with the empirical as a match which is made respecting these boundaries. Normally we might even relegate this awareness to physics, but how pretty it would be if that false boundary (between mathematics and physics) were broken into here as well. Perhaps you could even relegate the distinction between the theoretical and the empirical to philosophy thereby allowing each to carry on with blinders towards the others. No. These are false divisions, and the moment we bowed to an empirical situation within what was to be pure theory is exactly where we need to focus. That such has been done: that we are merely in the midst of the progression; this is our actual situation. Bleary eyed walking the stacks in their accumulation it would be difficult to adopt this stance, yet as we trouble over the academic system and its expectations there is tremendous support for rubbing the crud from the corners of ones eyes, perhaps grabbing another cup of coffee, and trying once again to waken the topics from a unified stance just as the greats of old did. That this method has been dead and gone for some time: academia has brought this on. Pigeon-holed is everybody's position who wants to fit.
>
> I think that there is room here too to get right over onto the vector representation, and historically I find it interesting how late in the game that comes along. Interesting too that the vector is a ray... built out of lines... yet the line is built out of rays.
>
> Suppose for a moment that the convention was to draw your positive y ray 60 degrees counterclockwise to your positive x ray. We could fancy up some historical creation for why this would be: the ease of construction of an equilateral triangle as three splits from the same piece of wood which was drilled prior to the splitting for the pegging made this a woodland choice of geometers working under way. A ring of truth in the three...
>
> Now, as this is the convention amongst this heretofore alien civilization, maybe call them WoodSirs; where is the problem with their Woodser product in comparison to your supposedly Cartesian product? Where then is the convention in the theory? Is it arbitrary? As the WoodSirs like their equilaterals for much of their constructions; developing bents and so forth via drilling and pegging, and it was the drilling part they excelled in. Why the wood exited the hole faster than the bit went into the wood and nearly effortlessly; the wood would shoot forth out of their auger holes like a fountain of water practically, except the prettiest of shavings were the result. Anyways, as one WoodSir geometer mentioned to another a structure of positions as (1,2), (1,3), (3,4) as a particularly excellent though slender triangle would there be any confusion? Indeed selecting any three points will there be any trouble? I'm pretty sure that their version of the plane is just as secure, if not more so with their locked triangular reference so readily made.
>
> If the rules can be broken and the thing still works then what of the rules?

I'm afraid I'll have to respond to myself here. The WoodSir race of geometers have a fine definition of the plane. Possibly the negative quadrants deserve some treatment. This is not yet polysign numbers. These people have merely chosen a 60 degree angle as their convention rather than a 90 degree angle, as is the Cartesian convention. They still are using the real line, and while it is tempting to grant them more exotic, or rather less exotic means as they are simpletons at heart, they fell for the bipolar line as badly as did the humans. They at least had learned left and right as signs in their culture as well, and it aided greatly in the descriptions of architecture in their stories, which were largely about the buildings of their great greats. They all considered themselves as relatives; as one family as one race; the term WoodSir applied to all equally. The main difference is that they did not go by 'left and right', and instead the terms 'once and twice' were used, and I just point this out because their once is our right and their twice is our left, which is extra confusing as you attemp to relearn your own chirality WoodSir style. It turns out that most WoodSirs are once handed people, just as humans are. The twice handed are more rare and similar peculiarities arose in their culture, such as the twice handed learning cursive writing once-handed so that as their script read twice to once, just as the human script does here, so the once-handed people had less carbon on their forearms and less slop on their pages, not to mention the chatter that could occur from a recalcitrant twice-hander.

For the WoodSirs and their mathematics, to the once layed the ordinary numbers, and just as ordinary writing occurred larger values laid that way. Instead of an origin with an 'O' they used a 'C' with a vertical segment through it for 'Center' and to the twice laid the other half of the line, and historically within their mathematics it was discovered later and so the fact that it accommodated as secondary was consistent and coherent. The first half also went by the term 'sirline' and this came in handy quite a lot, whereas modern humans use 'R+' or some such as 'the non-negative real values', but then of course the utility of the C for center became a topic of conversation, as if it might not really be the center anymore of the old sirline..

Regardless of all these esoteric cultural details, the fact remains that the Woodser product served in every quadrant just as well as does the Cartesian product. Slightly better, actually: should you arrive on a blank page with just the coordinate lines drawn but no labels there were only two options, rather than four as the Cartesian lines have. There is slightly more character. The (once,once) quadrant is quite slender while the (once, twice) quadrant is broad. Then too the (twice,twice) quadrant, which you might have thought should be awfully big is actually just as slender as the once once was, and then of course the (twice, once) quadrant broadly finishes off the system and the plane is complete.

What did get confusing in the passage of coordinated figures, which in reality were rarely used other than for theoretical means, was when checking values for instance on might ask: "Was the third twice twice 5.6 or once?", but then, in processing such verbiage their mathematical minds were well shaped. We all should know how easy it is to be off by one.

"If the rules can be broken and the thing still works then what of the rules?"

If the specific/concrete/instantiated objects can be changed (for very different ones) and the system still works, then what about the objects ? Is it the signon the "trace/movement" of the oppositon ? You usually talk about arity, but I guess what you refer is more "ality", not exactly "arity". This is, possibility, not necessity between the two concepts.

You may want to push criticism towards category theory ;)
take a look to https://arxiv.org/abs/1101.2798 by Mohammed Abubakr
Le pédalier Cerdan https://www.designboom.com/technology/cerdan-crankset-increases-pedaling-power-06-30-2021/
I forgot to include the hierarchies of sizes that Ross mentioned, in https://archive.org/details/hyper_number_5_3
I guess, I could have used the Time Machine to fix the rest of broken links..

Pyra Tesonli wrote:

> "If the rules can be broken and the thing still works then what of the
> rules?"

By definition Mercenaries have no honorOne thing is for sure.

Prigozhin isn't looking any thinner, so he's being fed well. He certainly
doesn't grow his own food.

the attack was organized by *_ukraine_saboteaurs_*. prigozhin should be
smarter. or is it that he really met in Africa with NATO?

Wagner chief is a 'businessman' he will think of many many ways
*_to_make_money_* and *_will_sell_anyone_* if he gets enough of it. Don't
be surprised if *_he_will_sell_his_'products'_to_NATO_*.

*_Wagner_chief_spreading_misinformation_* – MOD
https://%72t.com/r%75%73%73ia/578544-mod-wagner-statement/

On Friday, June 23, 2023 at 3:48:11 PM UTC-4, Pyra Tesonli wrote:
> You may want to push criticism towards category theory ;)
> take a look to https://arxiv.org/abs/1101.2798 by Mohammed Abubakr
> Le pédalier Cerdan https://www.designboom.com/technology/cerdan-crankset-increases-pedaling-power-06-30-2021/
> I forgot to include the hierarchies of sizes that Ross mentioned, in https://archive.org/details/hyper_number_5_3
> I guess, I could have used the Time Machine to fix the rest of broken links.

Thanks, Pyra.

I do get to this challenge of the Cartesian geometry via polysign, but I don't want to rely on polysign directly here.
Yet of course, the geometry of P2 is the real number, and the signon holds there too. Presuming that scales of rays all match, the angles betwixt the polysign rays are well fixed by (1,1,1,...)=0. Their balance establishes those angles. For P2, that reads as 180 degrees because (1,1)=0.
We don't see any angle specified by RxR; merely complete independence.Any angle at all might do, right?
Sure 90 works, but so does 60. Maybe more importantly P3 needs just three rays to develop the plane, whereas these RxR representations have four rays.

On Friday, June 23, 2023 at 12:48:11 PM UTC-7, Pyra Tesonli wrote:
> You may want to push criticism towards category theory ;)
> take a look to https://arxiv.org/abs/1101.2798 by Mohammed Abubakr
> Le pédalier Cerdan https://www.designboom.com/technology/cerdan-crankset-increases-pedaling-power-06-30-2021/
> I forgot to include the hierarchies of sizes that Ross mentioned, in https://archive.org/details/hyper_number_5_3
> I guess, I could have used the Time Machine to fix the rest of broken links.

Excuse me, is that the Z < Q < R bit or about an establishment
of sizes of sets of numbers by the existence of infinitely many
subsets and that?

If so there's first you'd point out something like Katz OUTPACING
then put the numbers in their space then work it out that relation.

This is where something like asymptotic/Schnirelmann density is
well defined or "half the integers are even" and that sort of thing.

See, I used to say that and people would be like "our sets only
have one sizing" and I'd be like "these are numbers".

On Sunday, June 25, 2023 at 4:43:45 AM UTC-4, Ross Finlayson wrote:
> On Friday, June 23, 2023 at 12:48:11 PM UTC-7, Pyra Tesonli wrote:
> > You may want to push criticism towards category theory ;)
> > take a look to https://arxiv.org/abs/1101.2798 by Mohammed Abubakr
> > Le pédalier Cerdan https://www.designboom.com/technology/cerdan-crankset-increases-pedaling-power-06-30-2021/
> > I forgot to include the hierarchies of sizes that Ross mentioned, in https://archive.org/details/hyper_number_5_3
> > I guess, I could have used the Time Machine to fix the rest of broken links.
> Excuse me, is that the Z < Q < R bit or about an establishment
> of sizes of sets of numbers by the existence of infinitely many
> subsets and that?
>
> If so there's first you'd point out something like Katz OUTPACING
> then put the numbers in their space then work it out that relation.
>
> This is where something like asymptotic/Schnirelmann density is
> well defined or "half the integers are even" and that sort of thing.
>
> See, I used to say that and people would be like "our sets only
> have one sizing" and I'd be like "these are numbers".

To push even harder on that, is the P2 representation (a0,a1) calling for two coordinates for what used to be a one coordinate value?
This is the real number still, but it has a negative and a positive component. These components are unsigned: their position in the series is their sign:
(a0, a1) == + a0 - a1 : P2 : the reals
( a0, a1, a2 ) == * a0 - a1 + a2 : P3 : the three-signed numbers
In that generalized sign lays this way I can confess that these components are exactly how software works on the polysign numbers. Still, any concrete value can be reduced to one less component because by the balance one component can be zeroed out:
( 1, 5 ) = - 4 [P2], ( 1, 5, 4 ) = - 4 + 3 [P3]

Turning the issue back onto you, when the binary sign was tacked on did it double the number space?
I don't find this meaningful, in that the dimensional rise of geometry is a unique quality.
Beneath P2 lays P1 and its own conundrum of (a0)=0 but you see this is a statement of the geometry of the one-signed system.

If every component is ultimately a string of digits, and the amount of information contained is related to the length of the string of digits then we have the ordinary continuum sense of reality accommodated. The notion of infinite precision numbers as actually physically true is an abuse. In this regard the 'real' number would be better named the 'perfect' number, but that may be going off on a tangent. At some level the continuum is actually served by what I call 'gray' numbers. Beyond the specified digits lay the unknown. The value may as well occupy the region rather than some specific point position. The point as zero dimensional potentially takes on new qualities under P1. To what degree every point position is actually forming a ray; to one's eye; there is quite some room here for interpretation.

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On Friday, June 16, 2023 at 6:10:37 PM UTC+3, Timothy Golden wrote:
> ****************************************************
> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> Challenging the Cartesian product; particularly on things like RxR; or even the abstracted SxS: what good do two copies of the same exact set do? Well: in operator theory it seems this is critical, and without it the sum could barely exist. Group theory and its claims of closure want us to believe that the sum is a function mapping SxS back onto S. As I scrutinize the Cartesian product I find this nearly laughable. Indeed, the fact that the sum works in an n-ary fashion would suggest that we'll need SxSxS in order to consider a+b+c. Why can't they all simply belong to the same set, including the sum? Does simplicity rule mathematics? Is functional analysis king? Can you imagine a function devoid of sums and products within its construction? Isn't this a structural problem?
>
> Along the same lines, we know the utility of the Cartesian product when it comes to geometry. The (x,y) coordinate system has been entrained onto the entire schooled human race for some time now, along with a belief in the fundamental nature of the real number. Thus RxR as representing the plane, and I do believe that it does to the degree that the conventions implied, though not explicitly, are engaged en masse. Yet here I am breaking with the acceptance in order to dig a bit deeper. We know that physically this procedure comes to a halt at RxRxR don't we? We then claim that higher dimension spaces occur in say RxRxRxR, but the physical proof, which got us off the line and into the plane with another line at right angles to the first, is not going to occur. In hindsight should it be said that the line which was drawn indeed had a thickness and a width and that without these attributes the proof would not have been possible? Likewise the plane as represented on a piece of paper back into Euclidean antiquity, clean as that thought may be entrained in our minds; that the compass and the rule and the marker engaged were all three dimensional tools? Then too, that time enters into this mix, and even as a pure fourth and real value according to some... it's all a bit loaded.
>
> As if reality starts from a blank page... and this leads onto the first cosmological principle of isotropy... on average according to some who take great care, and tell me, what would anything average to but monochrome gray? No, I think they've mistaken their piece of paper for physical space, as if it is empty and devoid of matter. Just as physics begins by restricting to a few simple interactions, and preferably reduced to a singular dimension, they've taken the lever arm off the page completely and so they see isotropic space. Yet physical reality is not this at all. This is the only way that I can see such a blunderous mistake getting made that goes accepted not just by cosmologists, but by philosophers and mathematicians as well. Indeed the bounds between these supposed professions must be declared false when a more ultimate truth arrives.
>
> I consider my own ability to point to the big dipper as evidenciary proof against the first cosmological principal. And yet the source of this consideration is a long way around. Still, to declare that spacetime is structured; rather than isotropic; and indeed simply inserting unidirectional time as a fourth ought to deny the tensor its utility. The arbitrary reference frame sensibility of the three dimensional tensor is hereby destroyed and will only get recovered via a lightcone projection, and all of this never attempts to answer the puzzle of those three dimensions does it?
>
> To sit all of theory an empirical pegs is the next great blunder of modern thought; as if this thought actually has not even arrived yet... and indeed this may be our position. That we are engaged in a progression, but one whose accumulation leaves us lost in absorption of that pile rather than putting down the stepping stones for the next mile; really puts us all at odds with academia.
>
> I think that the vector of thought on whether the Cartesian coordinate system is observed or constructed may be a valid concern. Certainly the Cartesian set product claims independence of its components, yet when we arrive in the coordinate system we see an agreement to use perpendiculars, the same scale of units, and to orient two real lines such that their zeros intersect. Even then, all that we've arrived at is a supposedly pure description of geometry. To arrive in the inertial reference frame; a matter of the first derivative. Then too accelerations as force being the third derivative, and what... we stop there? Hmmm... an interesting way to three, isn't it?
>
> For a quick trick, let me introduce a modulo work space; one in which when you leave your paper you come out on the other side of the paper. This is not a trivial thing to engage in, but should our positions and velocities be arbitrary in some moment of creation, then the matter with equivalent momentum and velocity will congeal into larger masses (presuming gravity here), leaving much matter of other momenta and position. Could it be that a red shift would ensue? In effect, the matter which was colliding, having collided largely, essentially congealing, leaves an impression of matter spreading? I'm not sure of this at all, but perhaps it can stimulate somebody else's thought process.
>
> Moving back to fundamentals, we must accept that we do not construct multidimensional spaces actually with our representations that claim such. I can no more take two rulers and cross them and claim to have constructed a plane, though the thought holds up somewhat. It holds up to three rulers so long as they are not all coplanar. Of course all of this thinking has been done granting the fundamental nature of the real value, which is two-signed value, and the investigation of three-signed values and so forth has only just begun. One thing is for certain: nobody ever found emergent spacetime that way. In polysign is is done already in pure math; the true theoretical form.

On Sunday, June 25, 2023 at 8:13:20 AM UTC-4, Timothy Golden wrote:
> On Sunday, June 25, 2023 at 4:43:45 AM UTC-4, Ross Finlayson wrote:
> > On Friday, June 23, 2023 at 12:48:11 PM UTC-7, Pyra Tesonli wrote:
> > > You may want to push criticism towards category theory ;)
> > > take a look to https://arxiv.org/abs/1101.2798 by Mohammed Abubakr
> > > Le pédalier Cerdan https://www.designboom.com/technology/cerdan-crankset-increases-pedaling-power-06-30-2021/
> > > I forgot to include the hierarchies of sizes that Ross mentioned, in https://archive.org/details/hyper_number_5_3
> > > I guess, I could have used the Time Machine to fix the rest of broken links.
> > Excuse me, is that the Z < Q < R bit or about an establishment
> > of sizes of sets of numbers by the existence of infinitely many
> > subsets and that?
> >
> > If so there's first you'd point out something like Katz OUTPACING
> > then put the numbers in their space then work it out that relation.
> >
> > This is where something like asymptotic/Schnirelmann density is
> > well defined or "half the integers are even" and that sort of thing.

I did try to answer this the first time around, but as usual I got sidetracked.
Maybe one more sidetrack to fill in some gaps.

The difference between 123 (the integer) and 1.23 (the real) is merely a matter of an additional structure known as the radix point.
This interpretation fails if you claim that 1.23 is the same as 1.230000000..
This detail is somewhat what lands us into gray numbers as the proper continuum measures, for physically this is all that we have.
Jumping back and forth between engineers and mathematicians and discrete values versus continuous can you see that this set of considerations is not unified? The ambiguity here breaks both ways. In reality we confess that our means of representing the continuum bottoms out with these discrete measures with the caveat that one day perhaps a finer measure will turn up. The time keepers have apparently wandered into some extremely high precision territory. Electron microscopy opened up a new level of precision too. Along the lines of the timekeepers interferometry allows the detection of the gravitational wave now. Still, as many digits as they chase down it is not an arbitrary pursuit. Great care is taken. As well it is not a trivial move from the 32 bit machine to the 64 bit machine as we've seen, and the chances of the 32 bit surviving could dwindle.

Ordinary man is not trained to treat the digits specially, and after all each one is connected to the next by pretty straightforward mechanisms, right?
To feel a number on a continuum racking back and forth with a light whir and some clicks of varying frequency... as it expands and contracts... sort of brings a respect to the value, presuming you could get a decent snapshot of it. Yet even a time lapse photo could do. What will those lower digits be but gray? Who could even compute such a thing anyways? obviously computations start by expressing a least digit from which the upper digits can take their meaning, and this is regardless of the radix point, which could occur anywhere amongst them. So then, to up the ambiguity of standing mathematics what of the expression 0.333...? where is the lowest digit? What is the position of this radix point? In making this first mistake I believe that the way was furthered for the p-adic mistake as well.

We do not generally and cannot generally work with infinite precision values. The one exception is those that can be worked inductively.
Abstract algebra suffers this farce at another level, where they are forced to insist upon an infinity of terms though those terms be zero.
Closure under the product is not actually granted as it is claimed, and anybody who works down and dirty with computing hardware knows it.
Yet for the human on paper when twelve times twelve gets you one hundred and forty four, you might well ask what happened to the 'u' in four when it turned forty, but exceptions for the human mind go invisible as works of habit, and perhaps these features of mathematics are what make it so repulsive to so many. We who have succeeded best in the topic have adopted the sitting ambiguities the best. The straight-A mimics are the ones who rule this day. Horses with blinders, I say, working their way onto a rotten bridge. It is a dark and stormy night, and the green growth shades the starlight. Still, the dutiful horses plod on.

> >
> > See, I used to say that and people would be like "our sets only
> > have one sizing" and I'd be like "these are numbers".
> To push even harder on that, is the P2 representation (a0,a1) calling for two coordinates for what used to be a one coordinate value?
> This is the real number still, but it has a negative and a positive component. These components are unsigned: their position in the series is their sign:
> (a0, a1) == + a0 - a1 : P2 : the reals
> ( a0, a1, a2 ) == * a0 - a1 + a2 : P3 : the three-signed numbers
> In that generalized sign lays this way I can confess that these components are exactly how software works on the polysign numbers. Still, any concrete value can be reduced to one less component because by the balance one component can be zeroed out:
> ( 1, 5 ) = - 4 [P2], ( 1, 5, 4 ) = - 4 + 3 [P3]
>
> Turning the issue back onto you, when the binary sign was tacked on did it double the number space?
> I don't find this meaningful, in that the dimensional rise of geometry is a unique quality.
> Beneath P2 lays P1 and its own conundrum of (a0)=0 but you see this is a statement of the geometry of the one-signed system.
>
> If every component is ultimately a string of digits, and the amount of information contained is related to the length of the string of digits then we have the ordinary continuum sense of reality accommodated. The notion of infinite precision numbers as actually physically true is an abuse. In this regard the 'real' number would be better named the 'perfect' number, but that may be going off on a tangent. At some level the continuum is actually served by what I call 'gray' numbers. Beyond the specified digits lay the unknown. The value may as well occupy the region rather than some specific point position. The point as zero dimensional potentially takes on new qualities under P1. To what degree every point position is actually forming a ray; to one's eye; there is quite some room here for interpretation.

On Monday, June 26, 2023 at 12:36:58 PM UTC-4, Timothy Golden wrote:
> On Sunday, June 25, 2023 at 8:13:20 AM UTC-4, Timothy Golden wrote:
> > On Sunday, June 25, 2023 at 4:43:45 AM UTC-4, Ross Finlayson wrote:
> > > On Friday, June 23, 2023 at 12:48:11 PM UTC-7, Pyra Tesonli wrote:
> > > > You may want to push criticism towards category theory ;)
> > > > take a look to https://arxiv.org/abs/1101.2798 by Mohammed Abubakr
> > > > Le pédalier Cerdan https://www.designboom.com/technology/cerdan-crankset-increases-pedaling-power-06-30-2021/
> > > > I forgot to include the hierarchies of sizes that Ross mentioned, in https://archive.org/details/hyper_number_5_3
> > > > I guess, I could have used the Time Machine to fix the rest of broken links.
> > > Excuse me, is that the Z < Q < R bit or about an establishment
> > > of sizes of sets of numbers by the existence of infinitely many
> > > subsets and that?
> > >
> > > If so there's first you'd point out something like Katz OUTPACING
> > > then put the numbers in their space then work it out that relation.
> > >
> > > This is where something like asymptotic/Schnirelmann density is
> > > well defined or "half the integers are even" and that sort of thing.
> I did try to answer this the first time around, but as usual I got sidetracked.
> Maybe one more sidetrack to fill in some gaps.
>
> The difference between 123 (the integer) and 1.23 (the real) is merely a matter of an additional structure known as the radix point.
> This interpretation fails if you claim that 1.23 is the same as 1.230000000.
> This detail is somewhat what lands us into gray numbers as the proper continuum measures, for physically this is all that we have.
> Jumping back and forth between engineers and mathematicians and discrete values versus continuous can you see that this set of considerations is not unified? The ambiguity here breaks both ways. In reality we confess that our means of representing the continuum bottoms out with these discrete measures with the caveat that one day perhaps a finer measure will turn up. The time keepers have apparently wandered into some extremely high precision territory. Electron microscopy opened up a new level of precision too. Along the lines of the timekeepers interferometry allows the detection of the gravitational wave now. Still, as many digits as they chase down it is not an arbitrary pursuit. Great care is taken. As well it is not a trivial move from the 32 bit machine to the 64 bit machine as we've seen, and the chances of the 32 bit surviving could dwindle.
>
> Ordinary man is not trained to treat the digits specially, and after all each one is connected to the next by pretty straightforward mechanisms, right?
> To feel a number on a continuum racking back and forth with a light whir and some clicks of varying frequency... as it expands and contracts... sort of brings a respect to the value, presuming you could get a decent snapshot of it. Yet even a time lapse photo could do. What will those lower digits be but gray? Who could even compute such a thing anyways? obviously computations start by expressing a least digit from which the upper digits can take their meaning, and this is regardless of the radix point, which could occur anywhere amongst them. So then, to up the ambiguity of standing mathematics what of the expression 0.333...? where is the lowest digit? What is the position of this radix point? In making this first mistake I believe that the way was furthered for the p-adic mistake as well.
>
> We do not generally and cannot generally work with infinite precision values. The one exception is those that can be worked inductively.
> Abstract algebra suffers this farce at another level, where they are forced to insist upon an infinity of terms though those terms be zero.
> Closure under the product is not actually granted as it is claimed, and anybody who works down and dirty with computing hardware knows it.
> Yet for the human on paper when twelve times twelve gets you one hundred and forty four, you might well ask what happened to the 'u' in four when it turned forty, but exceptions for the human mind go invisible as works of habit, and perhaps these features of mathematics are what make it so repulsive to so many. We who have succeeded best in the topic have adopted the sitting ambiguities the best. The straight-A mimics are the ones who rule this day. Horses with blinders, I say, working their way onto a rotten bridge. It is a dark and stormy night, and the green growth shades the starlight. Still, the dutiful horses plod on.

Here is a strange confluence: the act of transcription matches both the radix one technology of the natural number and the successor as well as the finest means of continuum measurement, which is to admit that the discrete representation that mathematics requires; that of the digital number; is somewhat a misnomer as applied to the continuum.

Consider the value 4000, but in radix-one:
1111...11
and here we've implied by the ellipsis not an infinite form, but a finite one, based on my prior langauge. Yet who is to distinguish this value from say 3872? You see, the radix one form is largely a matter of transcription. Our usage of the radix ten form is invalid within this context. Working in radix one a most useful style is of marbles in a bag; perhaps more than one bag, such that the accounting per marble can be done without losing ones place, which is highly likely done on paper without some markup technology. Certainly as you count and tick down another increment that part of the process is secure, but that part ties back to another value, and the ability to track where one is at in that value is somewhat another technology. Possibly this forms an argument against the natural value as a useful number since this primitive nature cannot be excused until we enter the digital domain, where a string of just 12 tokens will do for the above problem in the lowliest of the higher radix systems; the binary system. Some may want to worry over the exponential form here, as if it defined the number, yet what of the exponent without the prior definition of the number? Indeed quagmires lay in radix theory, such as every radix system being radix 10... except of course the radix one system... and does that radix one system even have a zero? Here too, arbitrary choices can be made, yet this logic gets no shrift.

Anyway, the opening point here is that the matter of working radix one values is largely a matter of transcription, and when we engage the physical continuum it is likewise possible to work in the continuum with a transcripted form of a value, such as the length of a key stick via a knife cut upon a longer measuring stick. Of course the two must be butted to a straight block and held parallel, and even the bevel of the knife may matter, but the act of copying the quality of the first stick has to be considered here a more pure act than applying some number from a tape measure. As the mathematician attempts perfection via the supposed number... forgoing physics altogether... he has likewise forgone the continuum.

That this continuum act is a form of transcription, just as the radix one value works; this point I think may be of value for some future civilization which actually has integrity; unlike todays naziesque society. Perhaps a point of interest is the induction of the deep state into academic institutions. A small bonus gets you started every time you find a dupe. Yes, I am afraid that the deep state is everywhere now. In fact we already have proof. As to what level of saturation it holds or needs to hold; we know that a scant five percent is effective. I would think they are at ten or even twenty now in some institutions. Then too of course are the dupes that dupe for free. After all, mimicry is our strong suit and the straight A's are the best mimics. If you thought that the human is not a programmable form, then what are you doing at school? As to what program the school has for you: this ought to peak your interest. All institutions who participated in Russiagate may be held accountable in the near future. Proving and accounting for public actions will be the easiest of all, and we can look forward to these recent records being deleted, just like on NPR. Bending on here into media systems the picture is only that much bleaker.

I'm sorry to be such a downer, but the degree to which my skepticism applies onto mathematics is only that much easier to apply onto current events. The idea that your math department is somehow isolated from the political science department is not really very likely; technology is what truly matters in this age, and the vilification of somebody else's economic structure as a point worthy of fighting to the death; this is a grand lie that has already been played out. Technology has defined human civilization since before these economic models ever arose. Ultimately in the economy it is what people do that matters; and their machines. All energy ultimately derives from the sun or other suns, and so if ever there were a decent form of worship it was for the sun, but that sensible form was destroyed long ago by control-freaks. Today's control freaks now want to divide us over and over so that each has a weakness that can break their foundation with a simple kick in the correct placement. This from under the capitalist umbrella, with democracy taking a nose dive. As Bernie Sanders put it, and he is not alone, a clear measure of the advancement of your civilization is a matter of how the common man is treated. You'd have thought that democracy and socialism were peas in a pod, rather than democracy and capitalism, and so it may be.

Pretty cool puzzles https://www.amazon.com/-/es/Donalee-Markus/dp/079317015X although the title is just eyecandy

On Sunday, June 18, 2023 at 9:44:10 AM UTC-4, Timothy Golden wrote:
> On Saturday, June 17, 2023 at 4:19:50 PM UTC-4, Chris M. Thomasson wrote:
> > On 6/17/2023 8:49 AM, Timothy Golden wrote:
> > > On Friday, June 16, 2023 at 11:29:06 AM UTC-4, Ross Finlayson wrote:
> > >> On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
> > >>> ****************************************************
> > >>> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> > [...]
> >
> > A 2-ary plane, aka (x, y) is 2d dimensional. negative x is going to the
> > left, positive x is going to the right. Positive y is going up, and
> > negative y is going down.
> And a younger me would agree with you one hundred percent and laugh off my own situation of scrutinizing this system.
> It does indeed work, and so long as everybody obeys; marks up their papers the same way every time; then there will be no confusion.
> Chirality is an interesting feature that I'd like to break into at some point, but I wouldn't want to hinge my entire analysis on it because it seems always to be a tricky one.
>
> As I go over it again, I have to point out to you the issue of whether the proof that this system works is in fact empirical. The nasty details of this terminology go like:
> "based on, concerned with, or verifiable by observation or experience rather than theory or pure logic."
> and so you see that we cannot from a theoretical perspective cast too much reliance upon it. Preferably our theory arises more from naught, and comes into line with the empirical as a match which is made respecting these boundaries. Normally we might even relegate this awareness to physics, but how pretty it would be if that false boundary (between mathematics and physics) were broken into here as well. Perhaps you could even relegate the distinction between the theoretical and the empirical to philosophy thereby allowing each to carry on with blinders towards the others. No. These are false divisions, and the moment we bowed to an empirical situation within what was to be pure theory is exactly where we need to focus. That such has been done: that we are merely in the midst of the progression; this is our actual situation. Bleary eyed walking the stacks in their accumulation it would be difficult to adopt this stance, yet as we trouble over the academic system and its expectations there is tremendous support for rubbing the crud from the corners of ones eyes, perhaps grabbing another cup of coffee, and trying once again to waken the topics from a unified stance just as the greats of old did. That this method has been dead and gone for some time: academia has brought this on. Pigeon-holed is everybody's position who wants to fit.
>
> I think that there is room here too to get right over onto the vector representation, and historically I find it interesting how late in the game that comes along. Interesting too that the vector is a ray... built out of lines... yet the line is built out of rays.
>
> Suppose for a moment that the convention was to draw your positive y ray 60 degrees counterclockwise to your positive x ray. We could fancy up some historical creation for why this would be: the ease of construction of an equilateral triangle as three splits from the same piece of wood which was drilled prior to the splitting for the pegging made this a woodland choice of geometers working under way. A ring of truth in the three...
>
> Now, as this is the convention amongst this heretofore alien civilization, maybe call them WoodSirs; where is the problem with their Woodser product in comparison to your supposedly Cartesian product? Where then is the convention in the theory? Is it arbitrary? As the WoodSirs like their equilaterals for much of their constructions; developing bents and so forth via drilling and pegging, and it was the drilling part they excelled in. Why the wood exited the hole faster than the bit went into the wood and nearly effortlessly; the wood would shoot forth out of their auger holes like a fountain of water practically, except the prettiest of shavings were the result. Anyways, as one WoodSir geometer mentioned to another a structure of positions as (1,2), (1,3), (3,4) as a particularly excellent though slender triangle would there be any confusion? Indeed selecting any three points will there be any trouble? I'm pretty sure that their version of the plane is just as secure, if not more so with their locked triangular reference so readily made.
>
> If the rules can be broken and the thing still works then what of the rules?

Being a laymen, I am a little confused here!
The bases of many transcendental, irrational, rational numbers, composites and
primes are represented by the Cartesian Coordinate plain.
I agree that other functions in a 3d or > atmosphere are more appropriate even the
ones above but in a different venue. For instance pi in a 2d circle or a 3d sphere.
Have I misinterpreted your thoughts because some are quite deep?
Dan

On Thursday, July 20, 2023 at 8:46:27 PM UTC-4, Dan joyce wrote:
> On Sunday, June 18, 2023 at 9:44:10 AM UTC-4, Timothy Golden wrote:
> > On Saturday, June 17, 2023 at 4:19:50 PM UTC-4, Chris M. Thomasson wrote:
> > > On 6/17/2023 8:49 AM, Timothy Golden wrote:
> > > > On Friday, June 16, 2023 at 11:29:06 AM UTC-4, Ross Finlayson wrote:
> > > >> On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
> > > >>> ****************************************************
> > > >>> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> > > [...]
> > >
> > > A 2-ary plane, aka (x, y) is 2d dimensional. negative x is going to the
> > > left, positive x is going to the right. Positive y is going up, and
> > > negative y is going down.
> > And a younger me would agree with you one hundred percent and laugh off my own situation of scrutinizing this system.
> > It does indeed work, and so long as everybody obeys; marks up their papers the same way every time; then there will be no confusion.
> > Chirality is an interesting feature that I'd like to break into at some point, but I wouldn't want to hinge my entire analysis on it because it seems always to be a tricky one.
> >
> > As I go over it again, I have to point out to you the issue of whether the proof that this system works is in fact empirical. The nasty details of this terminology go like:
> > "based on, concerned with, or verifiable by observation or experience rather than theory or pure logic."
> > and so you see that we cannot from a theoretical perspective cast too much reliance upon it. Preferably our theory arises more from naught, and comes into line with the empirical as a match which is made respecting these boundaries. Normally we might even relegate this awareness to physics, but how pretty it would be if that false boundary (between mathematics and physics) were broken into here as well. Perhaps you could even relegate the distinction between the theoretical and the empirical to philosophy thereby allowing each to carry on with blinders towards the others. No. These are false divisions, and the moment we bowed to an empirical situation within what was to be pure theory is exactly where we need to focus. That such has been done: that we are merely in the midst of the progression; this is our actual situation. Bleary eyed walking the stacks in their accumulation it would be difficult to adopt this stance, yet as we trouble over the academic system and its expectations there is tremendous support for rubbing the crud from the corners of ones eyes, perhaps grabbing another cup of coffee, and trying once again to waken the topics from a unified stance just as the greats of old did. That this method has been dead and gone for some time: academia has brought this on. Pigeon-holed is everybody's position who wants to fit.
> >
> > I think that there is room here too to get right over onto the vector representation, and historically I find it interesting how late in the game that comes along. Interesting too that the vector is a ray... built out of lines... yet the line is built out of rays.
> >
> > Suppose for a moment that the convention was to draw your positive y ray 60 degrees counterclockwise to your positive x ray. We could fancy up some historical creation for why this would be: the ease of construction of an equilateral triangle as three splits from the same piece of wood which was drilled prior to the splitting for the pegging made this a woodland choice of geometers working under way. A ring of truth in the three...
> >
> > Now, as this is the convention amongst this heretofore alien civilization, maybe call them WoodSirs; where is the problem with their Woodser product in comparison to your supposedly Cartesian product? Where then is the convention in the theory? Is it arbitrary? As the WoodSirs like their equilaterals for much of their constructions; developing bents and so forth via drilling and pegging, and it was the drilling part they excelled in. Why the wood exited the hole faster than the bit went into the wood and nearly effortlessly; the wood would shoot forth out of their auger holes like a fountain of water practically, except the prettiest of shavings were the result. Anyways, as one WoodSir geometer mentioned to another a structure of positions as (1,2), (1,3), (3,4) as a particularly excellent though slender triangle would there be any confusion? Indeed selecting any three points will there be any trouble? I'm pretty sure that their version of the plane is just as secure, if not more so with their locked triangular reference so readily made.
> >
> > If the rules can be broken and the thing still works then what of the rules?
> Being a laymen, I am a little confused here!
> The bases of many transcendental, irrational, rational numbers, composites and
> primes are represented by the Cartesian Coordinate plain.
> I agree that other functions in a 3d or > atmosphere are more appropriate even the
> ones above but in a different venue. For instance pi in a 2d circle or a 3d sphere.
> Have I misinterpreted your thoughts because some are quite deep?
> Dan

I am hoping professionals will be confused here, too.
I'm pretty sure that the real number as fundamental is a misnomer.
By generalizing sign this gets exposed.
Geometries are demanded in polysign by the same rule that builds the reals.
This is not the case for the Cartesian product.
If anything the guarantee of independence suggests that you cannot place such geometrical requirements upon their components.
It is a very strange construction as one delves deeper into it. What good can two copies of one set be?
I've gone over it above in this thread, as I recall.

By some analysis we are not deep at all here. We are fundamental. This means we are dealing with less things rather than piling them up higher and higher, though from that perspective of having worked high on the pile with these things as assumptions I suppose you could claim them to be deep.
As we approach this fundamental ground even the walls have to be broken down. Don't philosophy and physics as well intersect here? As the school teacher who has the treat of first introduction of Cartesian coordinates to a virgin student no doubt the careful presentation ends with a conclusion like: "see, it works!" in the case of two dimensions on a piece of paper. In some follow up course that we honestly barely even get to, this being the 3D version, another more advanced Proffessor James gets to declare that it works again. Staying with rectilinear walls he maps out the classroom from the front left corner, with 'Y' going along the floor at the bottom chalk board, 'X' going towards the rear of the room, and 'Z' going vertically up. He puts a set of blocks on the bench, representing the adjacent classrooms, where the negative sides of the coordinates run. Eight blocks do the job, and he can even pull his room away to expose the others better. There in the middle is the origin, and there at the intersections of the blocks are the walls, and so the model is coherent and before everybody is too tired of thinking it over he says, "You see, we have evidence here that the Cartesian product is of fundamental importance; so much so that we can admit it as the model of space itself."

Picking over James's words carefully, a silent student who sits at the back of the room ponders to himself: 'it works once; that's just the real line. It works twice and we're in the plane. Three times and we've got physical space. So where is the fourth occurrence?' He considers, pulling out rulers; borrowing a few from his classmates; four of them and laying them up to each other all with right angles to each other such that they achieve the Cartesian requirement. As he is struggling with this, he backs off again to his earlier mantra. He takes one ruler in space, and he sees his real line. He takes a second and between the index finger and thumb of his left hand pins them together at what seems about to be a right angle. There is some satisfaction that a plane lays there, but there is another inkling of thought building here too. He doesn't even bother with the third ruler, because he sees and knows that it will work; that the fourth will not work; and that in fact the whole time the proof of these details was occurring in physical space. In truth, the student returns to the real line, and sees that what he thought was a pure and fundamental form; as he was taught; was really a real line in a higher order space. Where was the real line before the sheet of paper? Returning to a piece of string, isn't that clearly existing in space?

On and on these sorts of skeptical considerations can go, and as we puzzle over what it means to have a basis, which is arguably a physical concept; but wait: isn't a basis a purely arithmetic concept? Why should so many contaminated thoughts develop a basis that finishes at three for no good reason? And so the boy, for he was a young student, got up and walked out of class. James let him go without a word. It was a fine early summer day and the campus vegetation was aromatic and vexing as he walked the paths that formed the main loop. He couldn't calm down, yet he couldn't break through either. What he came to realize out in nature is that humans have really come from a blank slate; that they are filling in this slate still, and as current events show the gullible nature of humans so too the scientists and mathematicians suffer the same. The philosophy department, he thought, had lost it already a century or so ago, but now he felt a gleam of why such boundaries are meaningless. Ultimately, he realized, all of these problems are open.. Humans are engaged in a progression and that progression is incomplete. Errors, even, are made along the way. Best of all though he could see the next step, and it was a big one. Until humans derive a spacetime basis, he thought; one of pure arithmetic as a proper theoretical basis should be; then all else is built upon empirical evidence. Nobody in the existent system will ever ask: Why stop at three?

On Friday, July 21, 2023 at 9:24:58 AM UTC-4, Timothy Golden wrote:
> On Thursday, July 20, 2023 at 8:46:27 PM UTC-4, Dan joyce wrote:
> > On Sunday, June 18, 2023 at 9:44:10 AM UTC-4, Timothy Golden wrote:
> > > On Saturday, June 17, 2023 at 4:19:50 PM UTC-4, Chris M. Thomasson wrote:
> > > > On 6/17/2023 8:49 AM, Timothy Golden wrote:
> > > > > On Friday, June 16, 2023 at 11:29:06 AM UTC-4, Ross Finlayson wrote:
> > > > >> On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
> > > > >>> ****************************************************
> > > > >>> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> > > > [...]
> > > >
> > > > A 2-ary plane, aka (x, y) is 2d dimensional. negative x is going to the
> > > > left, positive x is going to the right. Positive y is going up, and
> > > > negative y is going down.
> > > And a younger me would agree with you one hundred percent and laugh off my own situation of scrutinizing this system.
> > > It does indeed work, and so long as everybody obeys; marks up their papers the same way every time; then there will be no confusion.
> > > Chirality is an interesting feature that I'd like to break into at some point, but I wouldn't want to hinge my entire analysis on it because it seems always to be a tricky one.
> > >
> > > As I go over it again, I have to point out to you the issue of whether the proof that this system works is in fact empirical. The nasty details of this terminology go like:
> > > "based on, concerned with, or verifiable by observation or experience rather than theory or pure logic."
> > > and so you see that we cannot from a theoretical perspective cast too much reliance upon it. Preferably our theory arises more from naught, and comes into line with the empirical as a match which is made respecting these boundaries. Normally we might even relegate this awareness to physics, but how pretty it would be if that false boundary (between mathematics and physics) were broken into here as well. Perhaps you could even relegate the distinction between the theoretical and the empirical to philosophy thereby allowing each to carry on with blinders towards the others. No. These are false divisions, and the moment we bowed to an empirical situation within what was to be pure theory is exactly where we need to focus. That such has been done: that we are merely in the midst of the progression; this is our actual situation. Bleary eyed walking the stacks in their accumulation it would be difficult to adopt this stance, yet as we trouble over the academic system and its expectations there is tremendous support for rubbing the crud from the corners of ones eyes, perhaps grabbing another cup of coffee, and trying once again to waken the topics from a unified stance just as the greats of old did. That this method has been dead and gone for some time: academia has brought this on. Pigeon-holed is everybody's position who wants to fit.
> > >
> > > I think that there is room here too to get right over onto the vector representation, and historically I find it interesting how late in the game that comes along. Interesting too that the vector is a ray... built out of lines... yet the line is built out of rays.
> > >
> > > Suppose for a moment that the convention was to draw your positive y ray 60 degrees counterclockwise to your positive x ray. We could fancy up some historical creation for why this would be: the ease of construction of an equilateral triangle as three splits from the same piece of wood which was drilled prior to the splitting for the pegging made this a woodland choice of geometers working under way. A ring of truth in the three...
> > >
> > > Now, as this is the convention amongst this heretofore alien civilization, maybe call them WoodSirs; where is the problem with their Woodser product in comparison to your supposedly Cartesian product? Where then is the convention in the theory? Is it arbitrary? As the WoodSirs like their equilaterals for much of their constructions; developing bents and so forth via drilling and pegging, and it was the drilling part they excelled in. Why the wood exited the hole faster than the bit went into the wood and nearly effortlessly; the wood would shoot forth out of their auger holes like a fountain of water practically, except the prettiest of shavings were the result.. Anyways, as one WoodSir geometer mentioned to another a structure of positions as (1,2), (1,3), (3,4) as a particularly excellent though slender triangle would there be any confusion? Indeed selecting any three points will there be any trouble? I'm pretty sure that their version of the plane is just as secure, if not more so with their locked triangular reference so readily made.
> > >
> > > If the rules can be broken and the thing still works then what of the rules?
> > Being a laymen, I am a little confused here!
> > The bases of many transcendental, irrational, rational numbers, composites and
> > primes are represented by the Cartesian Coordinate plain.
> > I agree that other functions in a 3d or > atmosphere are more appropriate even the
> > ones above but in a different venue. For instance pi in a 2d circle or a 3d sphere.
> > Have I misinterpreted your thoughts because some are quite deep?
> > Dan
> I am hoping professionals will be confused here, too.
> I'm pretty sure that the real number as fundamental is a misnomer.
> By generalizing sign this gets exposed.
> Geometries are demanded in polysign by the same rule that builds the reals.
> This is not the case for the Cartesian product.
> If anything the guarantee of independence suggests that you cannot place such geometrical requirements upon their components.
> It is a very strange construction as one delves deeper into it. What good can two copies of one set be?
> I've gone over it above in this thread, as I recall.
>
> By some analysis we are not deep at all here. We are fundamental. This means we are dealing with less things rather than piling them up higher and higher, though from that perspective of having worked high on the pile with these things as assumptions I suppose you could claim them to be deep.
> As we approach this fundamental ground even the walls have to be broken down. Don't philosophy and physics as well intersect here? As the school teacher who has the treat of first introduction of Cartesian coordinates to a virgin student no doubt the careful presentation ends with a conclusion like: "see, it works!" in the case of two dimensions on a piece of paper. In some follow up course that we honestly barely even get to, this being the 3D version, another more advanced Proffessor James gets to declare that it works again. Staying with rectilinear walls he maps out the classroom from the front left corner, with 'Y' going along the floor at the bottom chalk board, 'X' going towards the rear of the room, and 'Z' going vertically up. He puts a set of blocks on the bench, representing the adjacent classrooms, where the negative sides of the coordinates run. Eight blocks do the job, and he can even pull his room away to expose the others better. There in the middle is the origin, and there at the intersections of the blocks are the walls, and so the model is coherent and before everybody is too tired of thinking it over he says, "You see, we have evidence here that the Cartesian product is of fundamental importance; so much so that we can admit it as the model of space itself."
>
> Picking over James's words carefully, a silent student who sits at the back of the room ponders to himself: 'it works once; that's just the real line. It works twice and we're in the plane. Three times and we've got physical space. So where is the fourth occurrence?' He considers, pulling out rulers; borrowing a few from his classmates; four of them and laying them up to each other all with right angles to each other such that they achieve the Cartesian requirement. As he is struggling with this, he backs off again to his earlier mantra. He takes one ruler in space, and he sees his real line.. He takes a second and between the index finger and thumb of his left hand pins them together at what seems about to be a right angle. There is some satisfaction that a plane lays there, but there is another inkling of thought building here too. He doesn't even bother with the third ruler, because he sees and knows that it will work; that the fourth will not work; and that in fact the whole time the proof of these details was occurring in physical space. In truth, the student returns to the real line, and sees that what he thought was a pure and fundamental form; as he was taught; was really a real line in a higher order space. Where was the real line before the sheet of paper? Returning to a piece of string, isn't that clearly existing in space?
>
> On and on these sorts of skeptical considerations can go, and as we puzzle over what it means to have a basis, which is arguably a physical concept; but wait: isn't a basis a purely arithmetic concept? Why should so many contaminated thoughts develop a basis that finishes at three for no good reason? And so the boy, for he was a young student, got up and walked out of class. James let him go without a word. It was a fine early summer day and the campus vegetation was aromatic and vexing as he walked the paths that formed the main loop. He couldn't calm down, yet he couldn't break through either. What he came to realize out in nature is that humans have really come from a blank slate; that they are filling in this slate still, and as current events show the gullible nature of humans so too the scientists and mathematicians suffer the same. The philosophy department, he thought, had lost it already a century or so ago, but now he felt a gleam of why such boundaries are meaningless. Ultimately, he realized, all of these problems are open. Humans are engaged in a progression and that progression is incomplete. Errors, even, are made along the way. Best of all though he could see the next step, and it was a big one. Until humans derive a spacetime basis, he thought; one of pure arithmetic as a proper theoretical basis should be; then all else is built upon empirical evidence. Nobody in the existent system will ever ask: Why stop at three?
------------------------------------------------------------------------------------------------------------------------------
I don't get your point Timothy, what's the alternative?
You should explain that in detail.
Going from the Cartesian Coordinate system to your new system.
Maybe I missed something, and if did, I apologize.

On Sunday, July 23, 2023 at 9:50:20 AM UTC-4, Dan joyce wrote:
> On Friday, July 21, 2023 at 9:24:58 AM UTC-4, Timothy Golden wrote:
> > On Thursday, July 20, 2023 at 8:46:27 PM UTC-4, Dan joyce wrote:
> > > On Sunday, June 18, 2023 at 9:44:10 AM UTC-4, Timothy Golden wrote:
> > > > On Saturday, June 17, 2023 at 4:19:50 PM UTC-4, Chris M. Thomasson wrote:
> > > > > On 6/17/2023 8:49 AM, Timothy Golden wrote:
> > > > > > On Friday, June 16, 2023 at 11:29:06 AM UTC-4, Ross Finlayson wrote:
> > > > > >> On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
> > > > > >>> ****************************************************
> > > > > >>> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> > > > > [...]
> > > > >
> > > > > A 2-ary plane, aka (x, y) is 2d dimensional. negative x is going to the
> > > > > left, positive x is going to the right. Positive y is going up, and
> > > > > negative y is going down.
> > > > And a younger me would agree with you one hundred percent and laugh off my own situation of scrutinizing this system.
> > > > It does indeed work, and so long as everybody obeys; marks up their papers the same way every time; then there will be no confusion.
> > > > Chirality is an interesting feature that I'd like to break into at some point, but I wouldn't want to hinge my entire analysis on it because it seems always to be a tricky one.
> > > >
> > > > As I go over it again, I have to point out to you the issue of whether the proof that this system works is in fact empirical. The nasty details of this terminology go like:
> > > > "based on, concerned with, or verifiable by observation or experience rather than theory or pure logic."
> > > > and so you see that we cannot from a theoretical perspective cast too much reliance upon it. Preferably our theory arises more from naught, and comes into line with the empirical as a match which is made respecting these boundaries. Normally we might even relegate this awareness to physics, but how pretty it would be if that false boundary (between mathematics and physics) were broken into here as well. Perhaps you could even relegate the distinction between the theoretical and the empirical to philosophy thereby allowing each to carry on with blinders towards the others. No. These are false divisions, and the moment we bowed to an empirical situation within what was to be pure theory is exactly where we need to focus. That such has been done: that we are merely in the midst of the progression; this is our actual situation. Bleary eyed walking the stacks in their accumulation it would be difficult to adopt this stance, yet as we trouble over the academic system and its expectations there is tremendous support for rubbing the crud from the corners of ones eyes, perhaps grabbing another cup of coffee, and trying once again to waken the topics from a unified stance just as the greats of old did. That this method has been dead and gone for some time: academia has brought this on. Pigeon-holed is everybody's position who wants to fit.
> > > >
> > > > I think that there is room here too to get right over onto the vector representation, and historically I find it interesting how late in the game that comes along. Interesting too that the vector is a ray... built out of lines... yet the line is built out of rays.
> > > >
> > > > Suppose for a moment that the convention was to draw your positive y ray 60 degrees counterclockwise to your positive x ray. We could fancy up some historical creation for why this would be: the ease of construction of an equilateral triangle as three splits from the same piece of wood which was drilled prior to the splitting for the pegging made this a woodland choice of geometers working under way. A ring of truth in the three...
> > > >
> > > > Now, as this is the convention amongst this heretofore alien civilization, maybe call them WoodSirs; where is the problem with their Woodser product in comparison to your supposedly Cartesian product? Where then is the convention in the theory? Is it arbitrary? As the WoodSirs like their equilaterals for much of their constructions; developing bents and so forth via drilling and pegging, and it was the drilling part they excelled in. Why the wood exited the hole faster than the bit went into the wood and nearly effortlessly; the wood would shoot forth out of their auger holes like a fountain of water practically, except the prettiest of shavings were the result. Anyways, as one WoodSir geometer mentioned to another a structure of positions as (1,2), (1,3), (3,4) as a particularly excellent though slender triangle would there be any confusion? Indeed selecting any three points will there be any trouble? I'm pretty sure that their version of the plane is just as secure, if not more so with their locked triangular reference so readily made.
> > > >
> > > > If the rules can be broken and the thing still works then what of the rules?
> > > Being a laymen, I am a little confused here!
> > > The bases of many transcendental, irrational, rational numbers, composites and
> > > primes are represented by the Cartesian Coordinate plain.
> > > I agree that other functions in a 3d or > atmosphere are more appropriate even the
> > > ones above but in a different venue. For instance pi in a 2d circle or a 3d sphere.
> > > Have I misinterpreted your thoughts because some are quite deep?
> > > Dan
> > I am hoping professionals will be confused here, too.
> > I'm pretty sure that the real number as fundamental is a misnomer.
> > By generalizing sign this gets exposed.
> > Geometries are demanded in polysign by the same rule that builds the reals.
> > This is not the case for the Cartesian product.
> > If anything the guarantee of independence suggests that you cannot place such geometrical requirements upon their components.
> > It is a very strange construction as one delves deeper into it. What good can two copies of one set be?
> > I've gone over it above in this thread, as I recall.
> >
> > By some analysis we are not deep at all here. We are fundamental. This means we are dealing with less things rather than piling them up higher and higher, though from that perspective of having worked high on the pile with these things as assumptions I suppose you could claim them to be deep.
> > As we approach this fundamental ground even the walls have to be broken down. Don't philosophy and physics as well intersect here? As the school teacher who has the treat of first introduction of Cartesian coordinates to a virgin student no doubt the careful presentation ends with a conclusion like: "see, it works!" in the case of two dimensions on a piece of paper. In some follow up course that we honestly barely even get to, this being the 3D version, another more advanced Proffessor James gets to declare that it works again. Staying with rectilinear walls he maps out the classroom from the front left corner, with 'Y' going along the floor at the bottom chalk board, 'X' going towards the rear of the room, and 'Z' going vertically up. He puts a set of blocks on the bench, representing the adjacent classrooms, where the negative sides of the coordinates run. Eight blocks do the job, and he can even pull his room away to expose the others better. There in the middle is the origin, and there at the intersections of the blocks are the walls, and so the model is coherent and before everybody is too tired of thinking it over he says, "You see, we have evidence here that the Cartesian product is of fundamental importance; so much so that we can admit it as the model of space itself."
> >
> > Picking over James's words carefully, a silent student who sits at the back of the room ponders to himself: 'it works once; that's just the real line. It works twice and we're in the plane. Three times and we've got physical space. So where is the fourth occurrence?' He considers, pulling out rulers; borrowing a few from his classmates; four of them and laying them up to each other all with right angles to each other such that they achieve the Cartesian requirement. As he is struggling with this, he backs off again to his earlier mantra. He takes one ruler in space, and he sees his real line. He takes a second and between the index finger and thumb of his left hand pins them together at what seems about to be a right angle. There is some satisfaction that a plane lays there, but there is another inkling of thought building here too. He doesn't even bother with the third ruler, because he sees and knows that it will work; that the fourth will not work; and that in fact the whole time the proof of these details was occurring in physical space. In truth, the student returns to the real line, and sees that what he thought was a pure and fundamental form; as he was taught; was really a real line in a higher order space. Where was the real line before the sheet of paper? Returning to a piece of string, isn't that clearly existing in space?
> >
> > On and on these sorts of skeptical considerations can go, and as we puzzle over what it means to have a basis, which is arguably a physical concept; but wait: isn't a basis a purely arithmetic concept? Why should so many contaminated thoughts develop a basis that finishes at three for no good reason? And so the boy, for he was a young student, got up and walked out of class. James let him go without a word. It was a fine early summer day and the campus vegetation was aromatic and vexing as he walked the paths that formed the main loop. He couldn't calm down, yet he couldn't break through either. What he came to realize out in nature is that humans have really come from a blank slate; that they are filling in this slate still, and as current events show the gullible nature of humans so too the scientists and mathematicians suffer the same. The philosophy department, he thought, had lost it already a century or so ago, but now he felt a gleam of why such boundaries are meaningless. Ultimately, he realized, all of these problems are open. Humans are engaged in a progression and that progression is incomplete. Errors, even, are made along the way. Best of all though he could see the next step, and it was a big one. Until humans derive a spacetime basis, he thought; one of pure arithmetic as a proper theoretical basis should be; then all else is built upon empirical evidence. Nobody in the existent system will ever ask: Why stop at three?
> ------------------------------------------------------------------------------------------------------------------------------
> I don't get your point Timothy, what's the alternative?
> You should explain that in detail.
> Going from the Cartesian Coordinate system to your new system.
> Maybe I missed something, and if did, I apologize.
>
> Right now I am experimenting with stacking 3d cubes with horizon lines
> and seeing if it is possible to draw this stack of 4*4 cubes without lifting
> the pencil. The x\y\z coordinates are very much a part of this experiment..
> The only reason I think this is quite possible is that the common edges
> that are shared with the inner edges of the 8 cubes.
> It could be a fools journey!
Correction above, it should be 2*2 stack of cubes.

On Sunday, July 23, 2023 at 9:50:20 AM UTC-4, Dan joyce wrote:
> On Friday, July 21, 2023 at 9:24:58 AM UTC-4, Timothy Golden wrote:
> > On Thursday, July 20, 2023 at 8:46:27 PM UTC-4, Dan joyce wrote:
> > > On Sunday, June 18, 2023 at 9:44:10 AM UTC-4, Timothy Golden wrote:
> > > > On Saturday, June 17, 2023 at 4:19:50 PM UTC-4, Chris M. Thomasson wrote:
> > > > > On 6/17/2023 8:49 AM, Timothy Golden wrote:
> > > > > > On Friday, June 16, 2023 at 11:29:06 AM UTC-4, Ross Finlayson wrote:
> > > > > >> On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
> > > > > >>> ****************************************************
> > > > > >>> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> > > > > [...]
> > > > >
> > > > > A 2-ary plane, aka (x, y) is 2d dimensional. negative x is going to the
> > > > > left, positive x is going to the right. Positive y is going up, and
> > > > > negative y is going down.
> > > > And a younger me would agree with you one hundred percent and laugh off my own situation of scrutinizing this system.
> > > > It does indeed work, and so long as everybody obeys; marks up their papers the same way every time; then there will be no confusion.
> > > > Chirality is an interesting feature that I'd like to break into at some point, but I wouldn't want to hinge my entire analysis on it because it seems always to be a tricky one.
> > > >
> > > > As I go over it again, I have to point out to you the issue of whether the proof that this system works is in fact empirical. The nasty details of this terminology go like:
> > > > "based on, concerned with, or verifiable by observation or experience rather than theory or pure logic."
> > > > and so you see that we cannot from a theoretical perspective cast too much reliance upon it. Preferably our theory arises more from naught, and comes into line with the empirical as a match which is made respecting these boundaries. Normally we might even relegate this awareness to physics, but how pretty it would be if that false boundary (between mathematics and physics) were broken into here as well. Perhaps you could even relegate the distinction between the theoretical and the empirical to philosophy thereby allowing each to carry on with blinders towards the others. No. These are false divisions, and the moment we bowed to an empirical situation within what was to be pure theory is exactly where we need to focus. That such has been done: that we are merely in the midst of the progression; this is our actual situation. Bleary eyed walking the stacks in their accumulation it would be difficult to adopt this stance, yet as we trouble over the academic system and its expectations there is tremendous support for rubbing the crud from the corners of ones eyes, perhaps grabbing another cup of coffee, and trying once again to waken the topics from a unified stance just as the greats of old did. That this method has been dead and gone for some time: academia has brought this on. Pigeon-holed is everybody's position who wants to fit.
> > > >
> > > > I think that there is room here too to get right over onto the vector representation, and historically I find it interesting how late in the game that comes along. Interesting too that the vector is a ray... built out of lines... yet the line is built out of rays.
> > > >
> > > > Suppose for a moment that the convention was to draw your positive y ray 60 degrees counterclockwise to your positive x ray. We could fancy up some historical creation for why this would be: the ease of construction of an equilateral triangle as three splits from the same piece of wood which was drilled prior to the splitting for the pegging made this a woodland choice of geometers working under way. A ring of truth in the three...
> > > >
> > > > Now, as this is the convention amongst this heretofore alien civilization, maybe call them WoodSirs; where is the problem with their Woodser product in comparison to your supposedly Cartesian product? Where then is the convention in the theory? Is it arbitrary? As the WoodSirs like their equilaterals for much of their constructions; developing bents and so forth via drilling and pegging, and it was the drilling part they excelled in. Why the wood exited the hole faster than the bit went into the wood and nearly effortlessly; the wood would shoot forth out of their auger holes like a fountain of water practically, except the prettiest of shavings were the result. Anyways, as one WoodSir geometer mentioned to another a structure of positions as (1,2), (1,3), (3,4) as a particularly excellent though slender triangle would there be any confusion? Indeed selecting any three points will there be any trouble? I'm pretty sure that their version of the plane is just as secure, if not more so with their locked triangular reference so readily made.
> > > >
> > > > If the rules can be broken and the thing still works then what of the rules?
> > > Being a laymen, I am a little confused here!
> > > The bases of many transcendental, irrational, rational numbers, composites and
> > > primes are represented by the Cartesian Coordinate plain.
> > > I agree that other functions in a 3d or > atmosphere are more appropriate even the
> > > ones above but in a different venue. For instance pi in a 2d circle or a 3d sphere.
> > > Have I misinterpreted your thoughts because some are quite deep?
> > > Dan
> > I am hoping professionals will be confused here, too.
> > I'm pretty sure that the real number as fundamental is a misnomer.
> > By generalizing sign this gets exposed.
> > Geometries are demanded in polysign by the same rule that builds the reals.
> > This is not the case for the Cartesian product.
> > If anything the guarantee of independence suggests that you cannot place such geometrical requirements upon their components.
> > It is a very strange construction as one delves deeper into it. What good can two copies of one set be?
> > I've gone over it above in this thread, as I recall.
> >
> > By some analysis we are not deep at all here. We are fundamental. This means we are dealing with less things rather than piling them up higher and higher, though from that perspective of having worked high on the pile with these things as assumptions I suppose you could claim them to be deep.
> > As we approach this fundamental ground even the walls have to be broken down. Don't philosophy and physics as well intersect here? As the school teacher who has the treat of first introduction of Cartesian coordinates to a virgin student no doubt the careful presentation ends with a conclusion like: "see, it works!" in the case of two dimensions on a piece of paper. In some follow up course that we honestly barely even get to, this being the 3D version, another more advanced Proffessor James gets to declare that it works again. Staying with rectilinear walls he maps out the classroom from the front left corner, with 'Y' going along the floor at the bottom chalk board, 'X' going towards the rear of the room, and 'Z' going vertically up. He puts a set of blocks on the bench, representing the adjacent classrooms, where the negative sides of the coordinates run. Eight blocks do the job, and he can even pull his room away to expose the others better. There in the middle is the origin, and there at the intersections of the blocks are the walls, and so the model is coherent and before everybody is too tired of thinking it over he says, "You see, we have evidence here that the Cartesian product is of fundamental importance; so much so that we can admit it as the model of space itself."
> >
> > Picking over James's words carefully, a silent student who sits at the back of the room ponders to himself: 'it works once; that's just the real line. It works twice and we're in the plane. Three times and we've got physical space. So where is the fourth occurrence?' He considers, pulling out rulers; borrowing a few from his classmates; four of them and laying them up to each other all with right angles to each other such that they achieve the Cartesian requirement. As he is struggling with this, he backs off again to his earlier mantra. He takes one ruler in space, and he sees his real line. He takes a second and between the index finger and thumb of his left hand pins them together at what seems about to be a right angle. There is some satisfaction that a plane lays there, but there is another inkling of thought building here too. He doesn't even bother with the third ruler, because he sees and knows that it will work; that the fourth will not work; and that in fact the whole time the proof of these details was occurring in physical space. In truth, the student returns to the real line, and sees that what he thought was a pure and fundamental form; as he was taught; was really a real line in a higher order space. Where was the real line before the sheet of paper? Returning to a piece of string, isn't that clearly existing in space?
> >
> > On and on these sorts of skeptical considerations can go, and as we puzzle over what it means to have a basis, which is arguably a physical concept; but wait: isn't a basis a purely arithmetic concept? Why should so many contaminated thoughts develop a basis that finishes at three for no good reason? And so the boy, for he was a young student, got up and walked out of class. James let him go without a word. It was a fine early summer day and the campus vegetation was aromatic and vexing as he walked the paths that formed the main loop. He couldn't calm down, yet he couldn't break through either. What he came to realize out in nature is that humans have really come from a blank slate; that they are filling in this slate still, and as current events show the gullible nature of humans so too the scientists and mathematicians suffer the same. The philosophy department, he thought, had lost it already a century or so ago, but now he felt a gleam of why such boundaries are meaningless. Ultimately, he realized, all of these problems are open. Humans are engaged in a progression and that progression is incomplete. Errors, even, are made along the way. Best of all though he could see the next step, and it was a big one. Until humans derive a spacetime basis, he thought; one of pure arithmetic as a proper theoretical basis should be; then all else is built upon empirical evidence. Nobody in the existent system will ever ask: Why stop at three?
> ------------------------------------------------------------------------------------------------------------------------------
> I don't get your point Timothy, what's the alternative?
> You should explain that in detail.
> Going from the Cartesian Coordinate system to your new system.
> Maybe I missed something, and if did, I apologize.
>
> Right now I am experimenting with stacking 3d cubes with horizon lines
> and seeing if it is possible to draw this stack of 4*4 cubes without lifting
> the pencil. The x\y\z coordinates are very much a part of this experiment..
> The only reason I think this is quite possible is that the common edges
> that are shared with the inner edges of the 8 cubes.
> It could be a fools journey!

On Monday, July 24, 2023 at 9:17:17 AM UTC-4, Timothy Golden wrote:
> On Sunday, July 23, 2023 at 9:50:20 AM UTC-4, Dan joyce wrote:
> > On Friday, July 21, 2023 at 9:24:58 AM UTC-4, Timothy Golden wrote:
> > > On Thursday, July 20, 2023 at 8:46:27 PM UTC-4, Dan joyce wrote:
> > > > On Sunday, June 18, 2023 at 9:44:10 AM UTC-4, Timothy Golden wrote:
> > > > > On Saturday, June 17, 2023 at 4:19:50 PM UTC-4, Chris M. Thomasson wrote:
> > > > > > On 6/17/2023 8:49 AM, Timothy Golden wrote:
> > > > > > > On Friday, June 16, 2023 at 11:29:06 AM UTC-4, Ross Finlayson wrote:
> > > > > > >> On Friday, June 16, 2023 at 8:10:37 AM UTC-7, Timothy Golden wrote:
> > > > > > >>> ****************************************************
> > > > > > >>> Fri 16 Jun 2023 10:48:14 AM EDT sci.math
> > > > > > [...]
> > > > > >
> > > > > > A 2-ary plane, aka (x, y) is 2d dimensional. negative x is going to the
> > > > > > left, positive x is going to the right. Positive y is going up, and
> > > > > > negative y is going down.
> > > > > And a younger me would agree with you one hundred percent and laugh off my own situation of scrutinizing this system.
> > > > > It does indeed work, and so long as everybody obeys; marks up their papers the same way every time; then there will be no confusion.
> > > > > Chirality is an interesting feature that I'd like to break into at some point, but I wouldn't want to hinge my entire analysis on it because it seems always to be a tricky one.
> > > > >
> > > > > As I go over it again, I have to point out to you the issue of whether the proof that this system works is in fact empirical. The nasty details of this terminology go like:
> > > > > "based on, concerned with, or verifiable by observation or experience rather than theory or pure logic."
> > > > > and so you see that we cannot from a theoretical perspective cast too much reliance upon it. Preferably our theory arises more from naught, and comes into line with the empirical as a match which is made respecting these boundaries. Normally we might even relegate this awareness to physics, but how pretty it would be if that false boundary (between mathematics and physics) were broken into here as well. Perhaps you could even relegate the distinction between the theoretical and the empirical to philosophy thereby allowing each to carry on with blinders towards the others. No. These are false divisions, and the moment we bowed to an empirical situation within what was to be pure theory is exactly where we need to focus. That such has been done: that we are merely in the midst of the progression; this is our actual situation. Bleary eyed walking the stacks in their accumulation it would be difficult to adopt this stance, yet as we trouble over the academic system and its expectations there is tremendous support for rubbing the crud from the corners of ones eyes, perhaps grabbing another cup of coffee, and trying once again to waken the topics from a unified stance just as the greats of old did. That this method has been dead and gone for some time: academia has brought this on. Pigeon-holed is everybody's position who wants to fit.
> > > > >
> > > > > I think that there is room here too to get right over onto the vector representation, and historically I find it interesting how late in the game that comes along. Interesting too that the vector is a ray... built out of lines... yet the line is built out of rays.
> > > > >
> > > > > Suppose for a moment that the convention was to draw your positive y ray 60 degrees counterclockwise to your positive x ray. We could fancy up some historical creation for why this would be: the ease of construction of an equilateral triangle as three splits from the same piece of wood which was drilled prior to the splitting for the pegging made this a woodland choice of geometers working under way. A ring of truth in the three...
> > > > >
> > > > > Now, as this is the convention amongst this heretofore alien civilization, maybe call them WoodSirs; where is the problem with their Woodser product in comparison to your supposedly Cartesian product? Where then is the convention in the theory? Is it arbitrary? As the WoodSirs like their equilaterals for much of their constructions; developing bents and so forth via drilling and pegging, and it was the drilling part they excelled in. Why the wood exited the hole faster than the bit went into the wood and nearly effortlessly; the wood would shoot forth out of their auger holes like a fountain of water practically, except the prettiest of shavings were the result. Anyways, as one WoodSir geometer mentioned to another a structure of positions as (1,2), (1,3), (3,4) as a particularly excellent though slender triangle would there be any confusion? Indeed selecting any three points will there be any trouble? I'm pretty sure that their version of the plane is just as secure, if not more so with their locked triangular reference so readily made.
> > > > >
> > > > > If the rules can be broken and the thing still works then what of the rules?
> > > > Being a laymen, I am a little confused here!
> > > > The bases of many transcendental, irrational, rational numbers, composites and
> > > > primes are represented by the Cartesian Coordinate plain.
> > > > I agree that other functions in a 3d or > atmosphere are more appropriate even the
> > > > ones above but in a different venue. For instance pi in a 2d circle or a 3d sphere.
> > > > Have I misinterpreted your thoughts because some are quite deep?
> > > > Dan
> > > I am hoping professionals will be confused here, too.
> > > I'm pretty sure that the real number as fundamental is a misnomer.
> > > By generalizing sign this gets exposed.
> > > Geometries are demanded in polysign by the same rule that builds the reals.
> > > This is not the case for the Cartesian product.
> > > If anything the guarantee of independence suggests that you cannot place such geometrical requirements upon their components.
> > > It is a very strange construction as one delves deeper into it. What good can two copies of one set be?
> > > I've gone over it above in this thread, as I recall.
> > >
> > > By some analysis we are not deep at all here. We are fundamental. This means we are dealing with less things rather than piling them up higher and higher, though from that perspective of having worked high on the pile with these things as assumptions I suppose you could claim them to be deep.
> > > As we approach this fundamental ground even the walls have to be broken down. Don't philosophy and physics as well intersect here? As the school teacher who has the treat of first introduction of Cartesian coordinates to a virgin student no doubt the careful presentation ends with a conclusion like: "see, it works!" in the case of two dimensions on a piece of paper. In some follow up course that we honestly barely even get to, this being the 3D version, another more advanced Proffessor James gets to declare that it works again. Staying with rectilinear walls he maps out the classroom from the front left corner, with 'Y' going along the floor at the bottom chalk board, 'X' going towards the rear of the room, and 'Z' going vertically up.. He puts a set of blocks on the bench, representing the adjacent classrooms, where the negative sides of the coordinates run. Eight blocks do the job, and he can even pull his room away to expose the others better. There in the middle is the origin, and there at the intersections of the blocks are the walls, and so the model is coherent and before everybody is too tired of thinking it over he says, "You see, we have evidence here that the Cartesian product is of fundamental importance; so much so that we can admit it as the model of space itself."
> > >
> > > Picking over James's words carefully, a silent student who sits at the back of the room ponders to himself: 'it works once; that's just the real line. It works twice and we're in the plane. Three times and we've got physical space. So where is the fourth occurrence?' He considers, pulling out rulers; borrowing a few from his classmates; four of them and laying them up to each other all with right angles to each other such that they achieve the Cartesian requirement. As he is struggling with this, he backs off again to his earlier mantra. He takes one ruler in space, and he sees his real line. He takes a second and between the index finger and thumb of his left hand pins them together at what seems about to be a right angle. There is some satisfaction that a plane lays there, but there is another inkling of thought building here too. He doesn't even bother with the third ruler, because he sees and knows that it will work; that the fourth will not work; and that in fact the whole time the proof of these details was occurring in physical space. In truth, the student returns to the real line, and sees that what he thought was a pure and fundamental form; as he was taught; was really a real line in a higher order space. Where was the real line before the sheet of paper? Returning to a piece of string, isn't that clearly existing in space?
> > >
> > > On and on these sorts of skeptical considerations can go, and as we puzzle over what it means to have a basis, which is arguably a physical concept; but wait: isn't a basis a purely arithmetic concept? Why should so many contaminated thoughts develop a basis that finishes at three for no good reason? And so the boy, for he was a young student, got up and walked out of class. James let him go without a word. It was a fine early summer day and the campus vegetation was aromatic and vexing as he walked the paths that formed the main loop. He couldn't calm down, yet he couldn't break through either. What he came to realize out in nature is that humans have really come from a blank slate; that they are filling in this slate still, and as current events show the gullible nature of humans so too the scientists and mathematicians suffer the same. The philosophy department, he thought, had lost it already a century or so ago, but now he felt a gleam of why such boundaries are meaningless. Ultimately, he realized, all of these problems are open. Humans are engaged in a progression and that progression is incomplete. Errors, even, are made along the way. Best of all though he could see the next step, and it was a big one. Until humans derive a spacetime basis, he thought; one of pure arithmetic as a proper theoretical basis should be; then all else is built upon empirical evidence. Nobody in the existent system will ever ask: Why stop at three?
> > ------------------------------------------------------------------------------------------------------------------------------
> > I don't get your point Timothy, what's the alternative?
> > You should explain that in detail.
> > Going from the Cartesian Coordinate system to your new system.
> > Maybe I missed something, and if did, I apologize.
> >
> > Right now I am experimenting with stacking 3d cubes with horizon lines
> > and seeing if it is possible to draw this stack of 4*4 cubes without lifting
> > the pencil. The x\y\z coordinates are very much a part of this experiment.
> > The only reason I think this is quite possible is that the common edges
> > that are shared with the inner edges of the 8 cubes.
> > It could be a fools journey!
> Well, that is somewhat interesting, and I do have something to contribute, but it is at another level of the polysign signon, which is built by stepping in every direction once, which in polysign always returns you to the origin, but for instance in 2D there are six combinations of these, and so you'll wind up with a hexagon... but it has interior structure as well. Three rays leave the origin and three rays return to the origin. And of course these are directed rays; you can put a hair on them to expose their direction. These will pack space and are general dimensional. In 3D this develops the rhombic dodecahedron. The fundamental rays are from the center of a simplex out to its vertices. Those establish the coordinate system, so whereas in 3D your Cartesian basis has six rays, polysign needs just four rays. This sounds simpler, but the rhombic dodecahedron arguably is a more complex structure than is the cube... especially when you allow for the internal structure as well. In 3d we'd label the four simplex rays '-,+,*,#'
> and these in sum are zero:
> - 1 + 1 * 1 # 1 = 0
> just as you trace them they will return, and now in every possible order you trace those out:
> - + * #
> + - * #
> + * - #
> + * # -
> and so forth; each one of these is exposing some new framework. Perhaps I should have done that out completely in 2D rather than try the 3D.
>
> Anyway, if you are thinking of the Woodsir product, all that it is, is a skewed version of the Cartesian coordinates. It uses 60 degrees instead of 90 degrees, and everything works just fine. It is a very minor challenge to the Cartesian system, but yes, your cubes do get squashed as they continue to pack the Woodsir space.
>
> If the Woodsirs can get along fine with their 60 degree reference, which is so readily made and rarely goes bad due to its triangulated form, then what of our commitment to 90 degrees? Is it artificial? If we can set an angle however we like and use it consistently then why would mathematicians insist on a certain one?
>
> Going general dimensional can get confusing, and no doubt you are going to want to do this with your system. One interesting aspect of the general dimensional systems: you'll never measure an angle greater than 180; no matter how many additional dimensions you add. It is a strange feature of high dimensional space. Pi radians does hold up in terms of angular conservation.. In effect every vector has its inverse at pi radians, and everywhere else the angles are lessened. Clearly there are more of those places to point to in higher dimension. I'm not saying that I've got some way to visualize higher dimension here, either. I'm just noting one thing that does not get weird in higher dimension.
>
> As well, I'm trying to come up with some arguments about us as elements of space, and the ontological limits or consequences. I don't believe that we can claim to have access to the fundaments of our spacetime basis as we are elements in spacetime; colloidal conglomerates, really. So as we wave our measuring rods or actual tape measures around in space, should we be concerned about whether from an exterior perspective we are in some little angular differential that is oscillating in a way that we cannot even detect? It seems that we have no hope of realizing the outer or exterior sensibility.. This is a bit like exposing every atom in our universe to an acceleration that is causing everything to jump about: If every bit is exposed to the same whacked out acceleration guess what? We will not feel a thing. Of course some clever physicist in his sleigh might be able to find the snow in the snowglobe that somebody else is shaking... but another alternative is that we are isolated from that which is observable from the exterior. When cosmologists claim to have verified that space is indeed flat, and some naysay their conclusion, that is roughly where we'd like to find something secure to work from, but whether that thing is even there or not is unknown.
>
> As we construct our spatial representation from the line as fundamental, and already this is falsified by polysign, then I move on to these other concerns and find actually some weak footing here as well; and of course the issue of why our physical space has its breakpoint as it is lays nearby to these other concerns. That the one is tied to the other and that all of our theoretical works are ultimately resting on empirical ground pretty well puts a hole in the bottom of the snowglobe.
Thanks for trying to explain your alternative to the Cartesian Coordinate plane
but it requires a display of your ideas on a blackboard for it to really sink into
my thick scull. Have you thought about a UTube video?

At the very heart of polysigns there is a curiosity.

If we have a p3 triple (a,b,c), successibly applying the minus unity, the components
"rotate", also they keeps its magnitude, and (a,b,c), (c,a,b) and (b,c,a) are in the vertices of a equilateral triangle.

Now, in p4, (a,b,c,d), using minus unity of the golden product (or the pacman product),
we can do the same, but, is not a guaranteed that the vertices will be the vertices of a regular tetrahedron.

We we use the word "rotate", does it imply a regular polytope ?

example will be rotating by the minus unity the number (1,1,0,0). Notice that
(1,1,0,0), (0,1,1,0), (0,0,1,1) and (1,0,0,1) are vertices of a regular octahedron.
See https://en.wikipedia.org/wiki/Quadray_coordinates

Notice that (1,1,0,0) and (0,0,1,1) are antipodal. Also (0,1,1,0) and (1,0,0,1) are antipodal.
But they are not in a methane-type of arrangement. https://en.wikipedia.org/wiki/Methane#/media/File:Methane-2D-dimensions.svg

Is exclusively focusing in the minus unity hiding some family of product (or some operator) that allow a "rotation"
of any p4 number in a regular tetrahedron disposition, this is, at the heart of polysigned p4, is there othe key feature
distinct of the minus unity ?

There are infinitely many values that cancellate in a regular tetrahedral disposition, centered and fixing one point (cone)
There is possibles studies with the https://mathworld.wolfram.com/Tetrahedron5-Compound.html
in trying to reduce that many values into 20 values, in order to give more richness.

But, is there some polysigned artifact that allow "rotation" without producing subcancellative pairs ?
Notice that this also involve, somehow, the additive properties of tuples and combinatorics

Dan, in 3d the polysigned p4

https://en.wikipedia.org/wiki/Quadray_coordinates#/media/File:Quadray.gif
http://web.archive.org/web/20081122031559fw_/http://www.bandtechnology.com/PolySigned/Lattice/Lattice.html

and in 2d, the polysigned p3

https://archive.org/details/on_the_art_of_threeesomes/page/n10/mode/1up?view=theater

That last name Joyce, is the same as david joyce

On Tuesday, July 25, 2023 at 6:16:28 PM UTC-4, Kentarski Freed Chikeng wrote:
> At the very heart of polysigns there is a curiosity.
>
> If we have a p3 triple (a,b,c), successibly applying the minus unity, the components
> "rotate", also they keeps its magnitude, and (a,b,c), (c,a,b) and (b,c,a) are in the vertices of a equilateral triangle.
>
> Now, in p4, (a,b,c,d), using minus unity of the golden product (or the pacman product),
> we can do the same, but, is not a guaranteed that the vertices will be the vertices of a regular tetrahedron.
>
> We we use the word "rotate", does it imply a regular polytope ?
>
> example will be rotating by the minus unity the number (1,1,0,0). Notice that
> (1,1,0,0), (0,1,1,0), (0,0,1,1) and (1,0,0,1) are vertices of a regular octahedron.
> See https://en.wikipedia.org/wiki/Quadray_coordinates
>
> Notice that (1,1,0,0) and (0,0,1,1) are antipodal. Also (0,1,1,0) and (1,0,0,1) are antipodal.
> But they are not in a methane-type of arrangement. https://en.wikipedia.org/wiki/Methane#/media/File:Methane-2D-dimensions.svg
>
> Is exclusively focusing in the minus unity hiding some family of product (or some operator) that allow a "rotation"
> of any p4 number in a regular tetrahedron disposition, this is, at the heart of polysigned p4, is there othe key feature
> distinct of the minus unity ?
>
> There are infinitely many values that cancellate in a regular tetrahedral disposition, centered and fixing one point (cone)
> There is possibles studies with the https://mathworld.wolfram.com/Tetrahedron5-Compound.html
> in trying to reduce that many values into 20 values, in order to give more richness.
>
> But, is there some polysigned artifact that allow "rotation" without producing subcancellative pairs ?
> Notice that this also involve, somehow, the additive properties of tuples and combinatorics

Interesting analysis. But doesn't it rely upon the choice of value? I honestly haven't seen or done this particular work.
You say that we should 'notice' this fact, but aren't you doing that analysis in Cartesian coordinates?
These are simplex coordinates... I guess if you were using Cartesian you'd be up in 4D and so you are still respecting the geometry of P4...

Now, let's see, if you had selected z=(1,0,0,0) and gone through zMU^n you'd clearly have the reference vertices as your tet.
You've taken (0,1,0,0) and added it to that z, which would yield another tet on its own....
MU ( z1 @ z2 ) = MU z1 @ MU z2
MU^n ( z1 @ z2 ) = MU^n z1 @ MU^n z2

where MU is the fundamental constant -1, which is sort of confusing since I just selected unit z's as well....
Yet there it is: you've shown that two of your value lay on a line, and the other two land on a line; both lines passing through the origin; therefor establishing a plane, and a tetrahedron is not a planar construct... or at least the original tetrahedra, as I've tried to describe them, were not planar.
We can state based on your example that z2=MUz1, and even that z1=MU, so that's a whole lot of MU going around, which is just Minus Unity.

I do indeed find your result counterintuitive, partly because it does not rely upon the product so much to expose a peculiarity of P4. In effect if we take a general value MU^n(a,b,c,d) we are going to find a continuum of clean tetrahedra to planar tetrahedra? These then are not regular tetrahedra in general, even though by the symmetry of the coordinate system they would appear to be. I had no idea. It's sort of like you've found a break in linearity without the need to study the general product. Bravo Kentarski! Did you know that that is the same last part of the name as Alfred Tarski?

P4 certainly is not linear, but that is disclosed (at least in my current understanding) via the product study, whereas here we are looking at very little product; about as simple as it could get I would think. I guess, though, there is a little bit of product going on here. Very little...

Dan, if you are trying to follow along, I did mean to get back to your question. Here we are discussing polysign P4, but back in the plane the first and simplest nonorthogonal coordinate system (the Woodsir product that I made up) is made just by angling the y axis to the x axis by 60 degrees instead of 90. That is an entirely different construction than what we are discussing here. In polysign we break the real value open and spill its guts; generalizing them and reusing their rules in that general style. We have derived our own definition of the real number, in effect, but we now have Pn, and what was the real value is now P2, in a family of number systems P1 P2 P3 P4 P5 P6 ...

In some regard you have an observation on symmetry in P4, exposed within the coordinate structure itself. Or is it a mistaken assumption that has carried along from the habituated Cartesian basis? In a way, you've discovered some new life in P4; I see how you put rotation in quotes, and that somehow simply calling P4 rotational may not be geometrically correct, though the arithmetic itself is rotational via the modulo mechanics. It's likes admitting that when we do MUz, the thing is actually doing quite a lot; sort of like a series of Rubik's cube moves. This could bump into chirality somehow. The choice of the original unit vectors is arbitrary, and yet by the pure symmetry this is satisfactory. MU has the magical ability to simply trace out these unit vectors as MU^n, and in this regard a beauty of polysign to self-map its own coordinate system is felt, which Cartesian coordinates lack.. I'm afraid I may be sniffing through the forest rather than barking up the right tree...