Could dS be the polysign differential? Integral( dS over S ) = 0

[Timothy Golden]

What's interesting about this attempt on polysign calculus is that it starts with the lattice as the differential. It is obviously so, yet somehow the way that calculus came to be does not seem to naturally bleed over onto polysign. Nobody has done integrals d- yet. The one simple fact that we do have is that a step in every sign does bring us to naught again; the offset is naught under these conditions. The very stepping of a dS under this abstract form in effect covers the signon since it could have gone -+*# [P4], -*#+, ... in a list of length four factorial.

Possibly Ross's doubling puzzle does alight here. We have a natural over-coverage in polysign if we treat our next lattice position as being one of these fundamental units away: using dS you are right back where you started so don't do that. Using say -1 as your step you are going to develop a fresh signon that is overlapped with the first. By my own analysis done years ago the signon which packs is at a horse move (2,1,1,...,0) away from the original position. If this makes iterating awkward then what of it? Here an earlier version of me halts to ponder the situation. Of course before this realization we simply had no packing whatsoever. In the plane we have hexagonal packing. The steps -1,+1,*1 in P3 actually even form the little delta triangle of yore. Yet that one does not pack. Well, go up one more to the tetrahedron of P4 and try. https://en.wikipedia.org/wiki/Tetrahedron_packing
Yet in P4 we will develop the rhombic dodecahedron and it will pack perfectly. This is done again via the possible steps of {-,+,*,#}. These signa have hairs on them too. They are unidirectional in their nature. This is not an ordinary part of our geometry as we were taught. Yet now the vector rules; the wave, too. We need to get these waves going in polysign. Help, please.

[Kristjan Robam]

You like this ----------->

https://www.youtube.com/watch?v=BecnTRBAZl4

video ?

You know the singer ?

[Lettucio Van Picklish]

https://www.youtube.com/watch?v=sbk3QaIQtKM
https://archive.org/details/on_the_art_of_threeesomes
https://en.wikipedia.org/wiki/Symmetric_derivative
https://www.youtube.com/watch?v=EEOcT7jd-iI

;0

ask Ross, he is designated in the chair of Measure, time-scales and countermeasures

[Timothy Golden]

Working the plane (P3) we will have steps of the form [ | -, +, * ], where the '|' indicates that permutations of this sequence are in use. Each individual triangle traced will take their place in a hexagon built by six of those triangles. It is the hexagon which packs the plane. This same procedure takes place in volumetric space via tetrahedral steps yielding a rhombic dodecahedron; again a packing shape. Back down at P2 we see the lowly line segment, but this segment does show a bidirectional character. The move by ( | 2,0,1,1,1,...) would yield uniquely packed signons in Pn... so long as you don't step on them twice. This could be a rudiment of calculus, but I'd like to make it much farther.

The idea that we would stay computational rather than work out theoretical problems as is the ordinary course seems fine. Perhaps the fact that this work does not come along trivially is a mark in favor of polysign for if it were trivial then polysign itself would be a trivial modification. To me it is clear that polysign numbers are not trivial.

[Lettucio Van Picklish]

There is one possible style that follows.
https://en.wikipedia.org/wiki/Homogeneous_polynomial

Polynomial/functions like (superscript usage)

x²+y²+z²-xy-yz-zx = 1

x²+y²+z²+w²-(2/3)(xy+xz+xw+yz+yw+zw) = 1

x³+y³+z³+w³+ 2xyz + 2xyw + 2xzw + 2yzw – (x²y +x²z+x²w+xy²+y²z+y²w+xz²+yz²+z²w+xw²yw²zw²) = 1

In x²+y²+z²-xy-yz-zx = 1 you may set any variable to zero (not to one as is usual in dehomogeneizing)

x²+y²-xy = 1

and after you clear one of them to plot it in one p3 sector(The homogeneous perspective)


But before the above you may choose what is a function in the first place.

with signs as input

f(s1,s2,...,sn) = ...

with mags as input

f(x1,x2,...,xn) = ...

with (s)x as input

f(s1x1,s2x2,...,snxn) = ...

with p3 numbers as input

f( -x1+y1*z1 , -x2+y2*z2 , ... , -xn+yn*zn)

or some combination of the above. What is a function in Pn ?

When you decide what a function may be, then from that a trig, or a calculus can flow from that.

For example the "total space" of a p3 function is a 4d space ?

For example if you feed certain function with -2+3*4, then you feed it with -3+4*5, and
the output is different. Will you dismiss that function for not comply with the homogeneous standards.
Then, will this disobedient function necessarily corrupt the definition of some derivative(or
some other calculus artifact) ? Not necesarily. Things that do not work on the macro may work in the small.

" One reason Descartes' method fell from favor was the algebraic complexity it involved. On the other hand, this
method can be used to rigorously define the derivative for a wide class of functions using neither infinitesimal
nor limit techniques. It is also related to a completely general definition of differentiability given by
Carathéodory (Range 2011). " https://en.wikipedia.org/wiki/Method_of_normals

https://en.wikipedia.org/wiki/Equiangular_polygon

I guess you want to desginb something in the spitit of the sequential paths of signon.

Do you work without 'discs', only with the sequential paths ?

Or maybe just work on the geometry and visuals of the concept ?

holly molly...

" Here is how you get started: Get a little diary and take it with you wherever
you go. Write down every time you're in a situation where you felt that if
something went wrong in that very moment, it would be socially awkward.
Most people wouldshy away from thosesituations and make sure those
things don't happen. Do the rule is quite simple: DO THAT AWKWARD THING!
if you're in a elevator and you need to fart and you're thinking "Oh,that
would be awkward". Fart. Let that fart go. LET IT GO! "

Extract of "the direct daygame bible"

https://wiki.tfes.org/Zeteticism

[Lettucio Van Picklish]

>> https://www.youtube.com/watch?v=EEOcT7jd-iI
(in min 3:20 ;0)

[Timothy Golden]

Well, actually the product is best studied on the disc in P3, or the sphere in P4, or on up in higher sign.
Of course this is because they are linear. z1 z2 as a general product does not preserve magnitude other than in P2 and P3. Still, to study the orientations and the variations in magnitude it makes sense that we use the unit shell to understand it. The product certainly is rotational, but as well the magnitudinal effects are present. Beyond this it is still true that to multiply by scalars those scalars can come out of the product:
( r1 z1)( r2 z2 ) = (r1 r2) (z1 z2)
where r are scalars and z are polysign values. In effect you might as well normalize the value rather than get carried away with nonunity values. Another claim for unity? Maybe. Whatever: it makes sense what you say in the terms of the product especially.

This said, paths work fine in Pn as a list of deltas. We can do that no problem. There is a peculiarity in polysign that nondecreasing functions become natural when we look at the path as signals in sign and as functions. When you increment the minus sign you've shifted one that way, but there is no negation other than to pulse the other signs each once. The idea that we would bother to reduce at this point is not realistic, nor pure, though it could be done. In effect each sign is pulsing on a staircase of variable width steps, for they are the signs of the path; each step in a sign is a step in that direction. In effect as counters they are racking up large values no matter what path you choose. At step 1000 they will total 1000, if you stepped one sign at a time, and started at zeros. If you are stepping as a vector of course all of this would be off, and as for curved systems that are following a parametric equation I would think that you have these vectors per dl length segment, or somehow or other. Yeah, whatever that term 'parametric' meant may need some work. The idea that we are going to bottom out in P2 is not realistic. We could bottom out in P1, however. Are P1 unsigned numbers? No: technically P1 are the one-signed numbers. They will always have their sole sign attached and via the law of balance these hold this P1 system as zero dimensional so long as we yield P2 as one dimensional. It is worse than this: literally -x=0 is the P1 balance. Large values are possible, but rendering these large values they plot to naught in their own geometry. Maybe better to just worry about the higher sign systems, but somehow P1 will come into it consistently.

>
> Or maybe just work on the geometry and visuals of the concept ?

I've done a fair amount of this. In that P3 are the complex plane there is this bleed into existing mathematics, and yet it's not as if I have complex analysis all redone in polysign. An awful lot is not done.
The Zeteticism paradigm sounds like physics to me. Somehow they got something wrong.

[Timothy Golden]

On Wednesday, September 14, 2022 at 1:05:38 PM UTC-4, Lettucio Van Picklish wrote:

> There is one possible style that follows.
> https://en.wikipedia.org/wiki/Homogeneous_polynomial
>
> Polynomial/functions like (superscript usage)
>
> x²+y²+z²-xy-yz-zx = 1
>
> x²+y²+z²+w²-(2/3)(xy+xz+xw+yz+yw+zw) = 1
>
> x³+y³+z³+w³+ 2xyz + 2xyw + 2xzw + 2yzw – (x²y +x²z+x²w+xy²+y²z+y²w+xz²+yz²+z²w+xw²yw²zw²) = 1
>
> In x²+y²+z²-xy-yz-zx = 1 you may set any variable to zero (not to one as is usual in dehomogeneizing)
>
> x²+y²-xy = 1
>
> and after you clear one of them to plot it in one p3 sector(The homogeneous perspective)
>
>
> But before the above you may choose what is a function in the first place.
>
> with signs as input
>
> f(s1,s2,...,sn) = ...

Yes: this is the simplest stepping arrangement. No, wait: is n=3 in P3 or is n a very large value?
Clearly s[n] will be in P3 sign space if we are working P3. So is
f( -,-,+,-,+,-,-,*)
what you had in mind? This is definitely a well constructed path. It lands at -5+2*1, or -4+1 in reduced form.

>
> with mags as input
>
> f(x1,x2,...,xn) = ...

Yes: this will work as well for developing a path. Here we simply assume that n is the sign, so that the signs come through as MU^n:
f( * x1, - x2, + x3, * x4, - x5, + x6, ... sn xn)

>
> with (s)x as input
>
> f(s1x1,s2x2,...,snxn) = ...
>
> with p3 numbers as input
>
> f( -x1+y1*z1 , -x2+y2*z2 , ... , -xn+yn*zn)
>
> or some combination of the above. What is a function in Pn ?

Yes. All of these are good. And the question is good. In effect we are seeking f() which provide such data.
It seems that all that has been done really is calling a sequence a function.
The sequence is good and it naturally forms a path under all these formats in Pn. But we don't really have a function.

Clearly you are making a good attempt here. It is interesting, isn't it, that these options exist?
That we can march through space on MU^n is a very sweet option.
This would be making usage of the product at a fundamental level, yet it blurs what people used to think of as (x,y) and so the idea that
y = f(x)
won't show up. These are two P2 systems under polysign, where y are real and x are real. Well, possibly this is a start then.
I don't think the series from is necessarily the right starting position, but it will be relevant to some good work.
We can state
y = f( z )
Let f(z) = z^3 @ 2z^2 @ 5z @ 6
and this will work fine in Pn. Algebra works just fine on polysign numbers in any dimension. The nearby form:
f(z) = ( z @ 3 )( z @ 2)( z @ 1 )
but this does bring up the question of modifying the signs where these '@' symbols are sums. Well, arguably it is the constant terms which we wish to modify so that they become polysigned:
f(z) = ( z @ c1 )( z @ c2 )( z @ c3 )
and such a construction can certainly be computed on say P6 values, but to then construe those P6 values to say P4 values will be mangling them a bit. I guess the mangling part can be skipped without trouble. At some point you have to leave general sign behind for a computable solution.
Polynomials totally lend themselves to polysign because polysign are algebraically behaved:
( z @ c1 )( z @ c2 ) = zz @ zc1 @ zc2 @ c1c2 = zz @ z(c1 @ c2) @ c1c2
in any dimension.
If you threw me a list of z and a polynomial my computer would pump these out in no time in large Pn.
I see. I haven't even graphed polynomials yet. Of course the graphical representations will be challenging in terms of projection and data density.
Well, maybe this is an opportunity for a P3 plane to be injected into a P4 volume and just graph that.
This is how young polysign are, and that these seemingly simple things have not been done yet; well you can see how the complexity rises rapidly in general dimension. Data density as well. Keeping things sparse is about the only way to get a graphical interpretation. It makes sense that a hexagonal grid with say a hundred steps per leg would form a decently sparse coverage, then that plane swept through the source space as an animation. This being a P3 trace of a P4 polynomial. The connections at each of the hundred interstices are interconnected and the density of the hexagonal grid adjusted to the liking of the viewer.

I'm sorry to say that my own code will be using Cartesian coordinate as an interim to map that P3 plane onto P4 as I have no native projection algorithms yet. It was all done in Cartesian with conversions from polysign to cart and back.

>
> When you decide what a function may be, then from that a trig, or a calculus can flow from that.
>
> For example the "total space" of a p3 function is a 4d space ?

This is a new term to me. https://mathworld.wolfram.com/TotalSpace.html
Unfortunately Wolfram circularly defines base space and total space so that they each rely upon each other.
To actually have some data is helpful I would think. It's an interesting question whether a function f(z in P3) could spit out a P4 or a P5 result.
If f is composed of sums and products in P3, which seems traditional, then the result will be in P3. Somehow I guess you'd try to go back to one of those earlier sequence forms and get something convincing going. I don't think simply mangling signs is that convincing.
Possibly you'd have five P5 functions yielding magnitudes per sign?
I'm not feeling very strong about getting an interesting result here. It sounds like more mangling. Something two dimensional does not magically turn into something interesting in higher dimension. It's flat or its extruded through to the higher space. I don't even think the extruded option is on most people's radar. That's like dividing by zero and yielding everything. If for some reason your functions were convincing you might have something.

Here is a strange option: from P3 to P6 via the hexagonal form. A P3 hexagonal lattice goes through a polynomial and is morphed by the polynomial in strange ways. There are no hundred interconnects on the final interpretation; instead those morphed hexagonal points map into P6? Probably too complicated.

>
> For example if you feed certain function with -2+3*4, then you feed it with -3+4*5, and
> the output is different. Will you dismiss that function for not comply with the homogeneous standards.
> Then, will this disobedient function necessarily corrupt the definition of some derivative(or
> some other calculus artifact) ? Not necesarily. Things that do not work on the macro may work in the small.

Very good analysis. I suspect that it will work out without conflict. It almost seems as though the simplest of things like y=f(x) have been habituated into us in a way that is not analytical. Looking at the early works like Descartes as you mention does expose how complicated their analyses were.
Newton's too.

[Timothy Golden]

On Wednesday, September 14, 2022 at 1:05:38 PM UTC-4, Lettucio Van Picklish wrote:

I think your questions are very good. I seem to be wandering into not quite calculus places, but just bumped into stream plots and polya plots on stackexchange: https://math.stackexchange.com/questions/607436/what-do-polynomials-look-like-in-the-complex-plane/1026657#1026657

Already at P3 the graphical problem ensues. Plotting z versus f(z) is a four dimensional plot. Of course this has nothing to do with computing f'(z) as a derivative, so it's off-topic, yet somehow very connected. If we can have a z can we simply have a dz in polysign? Then the title of this thread is off. I guess the idea that this dz is general dimensional out of the box, or signon, really, is part of the challenge. To what degree thinking in the terms of points or paths in general dimension holds up I think is trouble. These stream plots at the planar stage expose this. The path has become a sheet. I guess brane theory lays nearby. So is it bundles or is it branes that are the correct way? Branes I'd say. Of course I am feeling quite open about that especially since my mind refuses to wrap itself around the bundle terminology. Likely they come out to the same thing, and if that is the case then they are false branches. It's like one is impedance while the other is conductance. Throwing in these reciprocal views on an already challenging surface, to the human mind anyways, is not going to go well. We stuck with impedance as our primary language. That turns to complex numbers, you see, and we did poles and zeros, and I even took complex analysis, but I don't think we ever did try a stream plot.

I guess just by the algebraic behavior we can say that when f(z) = zz then f'(z) is 2z where z is in Pn.
lim as dz->0 (( f(z + dz) - f(z) ) / dz ) = ( zz + 2zdz +dzdz -zz ) / dz = 2z
though this usage of subtraction is in P2, and while the sensibility is good, technically in polysign this is bad language. Notationally a '~' can stand in as inversion. We've used a prime for inversion in the past but it won't do here with f' being a derivative. And of course the plus sign is implying summation and so with the zero sign '@' installed as the Neutral Unity (identity) sign the more general polysign version reads:
lim as dz->0 of (( f(z @ dz) ~ f(z) ) / dz ) = ( zz @ 2zdz @ dzdz ~ zz ) / dz = 2z
but really the math is all the same. There is a bit of a quagmire in that division is not so easily accomplished in polysign; neither is it so easily accomplished in complex analysis. Churchill and Brown have simplified things here too much, perhaps. And it is beyond me to visually see that this derivative will hold no matter which way we come at it. Then too, the old graphical interpretation of a slope is not on the radar. In that dz is a little triangle formed by dx, idy, and dx+idy, is not in their awareness. That's Cauchy-Riemann equations and they come at section 17 and I am before that section here working this example 1 of section 15, 5th edition Complex Variables And Applications.

[Timothy Golden]

I can't help but want to rhyme Picklish with tickelish; I mean what else is there?
Anyway I do feel thanful including the off-humor let it out type of thinking.
It is pretty good. Let loose. This is this place. This is the beauty of usenet, and yet so few are capable. Too shy? Too wrong? Afraid of getting picked on? Come On, Man!

The Bernie Bro's so quickly burnt out. Couldn't giver Bernie the benefit of the doubt. 2016 marked Russiagate on my calender how many months of that year? And when did it start? And we know nothing of FBI actions against any socialists, do we? Not to mention CIA activities overseas. No, C'mon, Maaaannn? Can't you see that old gray piece of deep state shit getting all into it? To sit as VP at the first Black President's desk as two in line, while your newfound friend jumps in two feet and two hands first. Youch, that must have hurt. That financial crisis: a thing of the distant past that nobody wants to revisit. Somethings out of whack and it just can't stop hiccuping, and it is getting real old now.

Well, hiccup here we can. Hiccup here we shall. Hiccup here most do. Repetition is our way, and my father was mistaken when he said that he would not chew his cabbage twice. He had no patience and passed that on to a son or two. What could we do? Imprint upon another? The imprint process; that pledge of allegiance said by every school student down through the ages back to when mimicry was, well, why not start them at age two?

There is a new class of human awaiting its place at the table. These are the receptive ones. They, unlike us, were given permission to use the internet for good. In their moment of curiosity an answer could almost always be found. Not so for you or me, and likely this is true, but there is a young who reads on and sees what I see. Rather than give him away I will grant him many times over into the future here and now as said by these words. It will be now as if I am speaking to him or her or even it, and I have willingly attempted to converse with bots here. It used to be there were some strange codes going across usenet. It's distributed media. It's a good thing. It's the pre-cloud. Make your deposit; make many if you must. Those who never deposit here are uncomfortable with this medium of who knows who will say what back to you, and much of the time nothing of interest occurs. Even this second order rendition; my got this is almost as old as bulletin boards. I missed those. Maybe they are about to come back.
Facebook is basically a bulletin board. It's just a question of complexity; storage space;

I guess looking at the polysign version of the derivative it does sort of expose how the usage of + and - in both values and as operators has locked up numerical systems by insisting that these remain coherent operators. In effect every system which obeys these is by definition not a polysign system, unless of course it generated an entirely new set of signs for its values. That these new signs would take the glyph of branches from a core point; the minus sign being a singlet escaping a core point. Sign two being simply one more escaping the core point. The angle really does not matter, but obviously could start out small rather than opposed since it needs to be elementally universal. To use an opposed is as well to confuse the second sign with the first sign, and so it could be that such things happen and others go overlooked. While this is a rhetorical statement, the truth in the graphics belays something there. That P1 has gone overlooked and yet that it sits at the base of these systems and not P2; that the first continuum somehow suffers that zero dimensional crash at -x=0. This ray gone astray is about to behave in another way, and as the world as off by one awaits us, in P2 we recover to the same old day.

Does the claptrap on P1 even work?

f1 ~ f2 = f1 ; P1 ? This is by reverse induction at the moment I think.
f1 ~ f2 = f1 - f2 ; P2
f1 ~ f2 = f1 - f2 + f2 ; P3
f1 ~ f2 = f1 - f2 + f2 * f2 ; P4
~ f = sum( s != n ) s f ; Pn
A silent one in terms of this notation at P1. By f1~f2 I mean the difference as in the difference required to develop the derivative, which does not carry through into polysign trivially. Yes, we can write it as ~ meaning subtraction,yet it is inversion that we seek, and somehow the P1 instance is a disappearing instance.
f(z) = zz.
lim as dz->0 of (( f(z @ dz) ~ f(z) ) / dz ) = ( zz @ 2zdz @ dzdz ) / dz = zz/dz @ 2z = infinity I guess, in P1.

Alright. That's sort of weird, but it looks like taking things literally this will hold. This may be a moment of beauty as it is surprising, but I honestly don't see it yet. This literally just popped up here in the matter of coarse work.

Physics will be greatly relieved. It will be cause possibly of some P1 fundamentalism; just as P2 suffers now. Anyway it is unique that a singularity still means something in mathematics. That it could come seemingly from nowhere and yet in the midst of trying to do something serious that works generally, and so it should be that it does happen. At least I can hope that this is of use one day. Subtraction is not a fundamental operation. It is merely the reverse operator, and in that a forward operator does not necessarily afford reversibility, then any assumption of its existence is dubious. The same can be said of division. At a puritanical level a claim of a point particle as infinite in nature entirely maps to the wave theory of a perfect particle; that of a unit step impulse that is extremely tall. That time is its counterpart and these things will melange one day I can say with some confidence. Tomorrow, perhaps.

[Timothy Golden]

I'm onto a sidetrack here of implementing a derivative and seeing that P1 is a special case. That is not really a surprise. But the form of the surprise is new, so it has to be considered more carefully. Chewing the cabbage twice and thrice on this one will be necessary. Inversion is at the heart of this puzzle. Subtraction is the addition of the inverse. Subtraction is within the definition of the derivative. P1 is one-signed and so its inverse does not exist. It is a unidirectional system. It is as well geometrically a zero dimensional system, using this word dimension as it is done traditionally; tied to the real line as one dimensional. It is also true that time and P1 have correspondence, and that time is typically the independent variable that yields these sorts of analyses in the real world. So the coherence in the face of seeming decoherence is felt. The course onto geometry versus onto physics seems apt.

It would be strange, indeed, to wrap around onto this argument and land upon digital infinity and its aleph mark, yet I see it could be done this way. In effect, this interpretation would validate the claims of the infinite length decimal, though the junction of the aleph mark with the decimal point would be new. This is a route to the continuum via the infinite natural value. That the aleph mark is already necessary for coherent operation somewhat seals this deal. That P1 would come into play on its own and concurrent with that development is strange. Still, this is the operation of the human mind at some level. It's clickings and whirrings are largely hidden away. It is not a necessary step to congeal these two as a colloidal yet, but the option to do so can sort of relax things, maybe. I guess I'll have to post this continuum hypothesis over on another thread or something rather than force somebody into this corner, which isn't likely to happen.

[Timothy Golden]

Is subtraction a necessity? Are functions necessary? When it comes to the derivative both these things are included. Let's have a look at a discrete form for a moment. A P1 signal takes the following series:
2, 4, 7, 12, 14
Therefor the derivative takes the following series:
2, 3, 5, 2
In hindsight P1 does easily provide derivatives if a sequence can be found. Going back to f(x) = xx then can we have:
1, 4, 9, 15, 25
dx: 3, 5, 6, 10
and this derivative does approach 2x as for instance (25)(25) @ 2(25) @ 1 = (26)(26)
this is just the fact that (x@1)(x@1) = xx @ 2x @1
and of course that was a delta x of unity, so now getting finer on the scale:
f(2) = 4; f(2.01) = 4.0401
0.0401 / 0.01 = 4.01 =~= 2x

I think this shows that P1 is still behaved, but the derivative as f(x) summed with the inverse of f(x@dx) all over dx is not computable in P1 since the inverse requirement does not exist there.

Let ~ mean inverse so that in Pn z ~ z is zero. Well, I will show that this is not possible for P1.
...
P5 : ~z = - z + z * z # z
P4: ~z = - z + z * z
P3: ~z = - z + z
P2: ~z = - z
P1: ~z =

It feels like being back at the first confusing interpretations or lack thereof of P1. Is this exceptional? No, but when it is called for in the derivative then we have a problem. Can math yield a blank? In P1 it can. P1 is zero dimensional. It has a modulo-one sign; just one sign. It is the identity sign. MU is NU in P1. That P1 has correspondence with time is of interest here. Many practical functions are f(t) type constructions. They don't bother with a derivative of time. It is the independent variable. Somehow we'd like to walk a consistent line that will as well be the cause of guidance here. I'm not claiming to be walking that line here.

[Ross A. Finlayson]

I think you only mean where differential operator theory applies for whatever reason.

Ah, then building in doubling, is an example of a means, that identifies
a system of equations, where formula in differential transforms and
particularly the functions in the systems of equations, why for example
a doubling space drops out to zero or a doubling space builds into measure.

Actions in doubling then points out the usual exponential in the differential,
about when people talk about "differential equations" like here you
want to make a notation holding according to "differential equations".

The packing and geometry, here it is lattice theory, though, the origin is
everywhere and it's the middle of a sphere. What that means is that there
are cases like 1- or 2-sided, for example 3, 4, 5, 6- sided in a plane,
and so on in higher dimensions, where the trunks and origins and branches
and so on, identify to that power or what affects dispersion or convergence.

Then, when I was writing about "Factorial/Exponential Identity, Infinity", in 2003,
what's key here is the example about the properties of the entire space of sequences,
with respect to otherwise there's no function over them, according to the finite,
or "Borel vs. Combinatorics". Then looking for formulas for the series terms and
the power terms, I was soon all over sums of powers in terms and writing Stirling
numbers, approximating factorial for large inputs with these simple terms I discovered,
and otherwise "Factorial/Exponential Identity, Infinity" and "Borel versus Combinatorics"
is itself a thing, here for what gets found in terms, in all sorts often numbers.

Then, back to differential equations and differential operator theory,
and of course for analysis, is what's amazing that besides e^x, that
also, this n/d d->oo, is its own antiderivative and "differential".

These are just some example parts what gets going about differential
systems here where in these dynamical systems , usually everything
is parameterized by t for time.

There' non-adiabatic and far field and long tail and so on, ....

(There's adiabatic and near field and central and so on, ....)

It's also a model like a windup of a chip sending a bit received in the chip, ....
This is usually called "systolic" what pumps, also parameterized by t.

[Timothy Golden]

On Friday, September 23, 2022 at 11:39:38 AM UTC-4, Ross A. Finlayson wrote:
> On Tuesday, September 6, 2022 at 10:54:12 AM UTC-7, timba...@gmail.com wrote:
> > What's interesting about this attempt on polysign calculus is that it starts with the lattice as the differential. It is obviously so, yet somehow the way that calculus came to be does not seem to naturally bleed over onto polysign. Nobody has done integrals d- yet. The one simple fact that we do have is that a step in every sign does bring us to naught again; the offset is naught under these conditions. The very stepping of a dS under this abstract form in effect covers the signon since it could have gone -+*# [P4], -*#+, ... in a list of length four factorial.
> >
> > Possibly Ross's doubling puzzle does alight here. We have a natural over-coverage in polysign if we treat our next lattice position as being one of these fundamental units away: using dS you are right back where you started so don't do that. Using say -1 as your step you are going to develop a fresh signon that is overlapped with the first. By my own analysis done years ago the signon which packs is at a horse move (2,1,1,...,0) away from the original position. If this makes iterating awkward then what of it? Here an earlier version of me halts to ponder the situation. Of course before this realization we simply had no packing whatsoever. In the plane we have hexagonal packing. The steps -1,+1,*1 in P3 actually even form the little delta triangle of yore. Yet that one does not pack. Well, go up one more to the tetrahedron of P4 and try. https://en.wikipedia.org/wiki/Tetrahedron_packing
> > Yet in P4 we will develop the rhombic dodecahedron and it will pack perfectly. This is done again via the possible steps of {-,+,*,#}. These signa have hairs on them too. They are unidirectional in their nature. This is not an ordinary part of our geometry as we were taught. Yet now the vector rules; the wave, too. We need to get these waves going in polysign. Help, please.
> I think you only mean where differential operator theory applies for whatever reason.
>
> Ah, then building in doubling, is an example of a means, that identifies
> a system of equations, where formula in differential transforms and
> particularly the functions in the systems of equations, why for example
> a doubling space drops out to zero or a doubling space builds into measure.
>
> Actions in doubling then points out the usual exponential in the differential,
> about when people talk about "differential equations" like here you
> want to make a notation holding according to "differential equations".
>
> The packing and geometry, here it is lattice theory, though, the origin is
> everywhere and it's the middle of a sphere. What that means is that there
> are cases like 1- or 2-sided, for example 3, 4, 5, 6- sided in a plane,
> and so on in higher dimensions, where the trunks and origins and branches
> and so on, identify to that power or what affects dispersion or convergence.

I find this paragraph above impressive. It's general dimensional.
That origin; I see you have brought it alive.
That does seem wise. That null and zero and balance are all tightly related;
To deny that everything is ultimately in balance would be, well, to build off of that balanced form, and in doing so arbitrary origins will have been devised and possibly global origins, cosmic origins; galactic origins are getting obscenely interesting. I wonder when they will see a star get eaten at SagitariusA*? Apparently there's mostly just orbital dynamics going on and pretty spacious too.

>
> Then, when I was writing about "Factorial/Exponential Identity, Infinity", in 2003,
> what's key here is the example about the properties of the entire space of sequences,
> with respect to otherwise there's no function over them, according to the finite,
> or "Borel vs. Combinatorics". Then looking for formulas for the series terms and
> the power terms, I was soon all over sums of powers in terms and writing Stirling
> numbers, approximating factorial for large inputs with these simple terms I discovered,
> and otherwise "Factorial/Exponential Identity, Infinity" and "Borel versus Combinatorics"
> is itself a thing, here for what gets found in terms, in all sorts often numbers.
>
> Then, back to differential equations and differential operator theory,
> and of course for analysis, is what's amazing that besides e^x, that
> also, this n/d d->oo, is its own antiderivative and "differential".
>
> These are just some example parts what gets going about differential
> systems here where in these dynamical systems , usually everything
> is parameterized by t for time.
>
> There' non-adiabatic and far field and long tail and so on, ....

Jeeze the sound of these words is even good.
Near field sets up puzzles too.

>
> (There's adiabatic and near field and central and so on, ....)
>
> It's also a model like a windup of a chip sending a bit received in the chip, ....
> This is usually called "systolic" what pumps, also parameterized by t.

Now hang on, you telling me somebody's using 'systolic' in mathematics?
Sounds like the works of Russians! Must be wrong! Take that back, sir!
https://en.wikipedia.org/wiki/Systolic_geometry
Absolutely amazing.
I would like to cover the ground here definitely.
I am trying to wrap my head around the Shottky problem, but the language is intense. I don't know it all. I'm sure I read about the Jacobian in the past but it does not really mean much.

I'll try again soon.

[Ross A. Finlayson]

Yes, studying perspective and geometry is critical for, the critical.


Reading is fundamental, perspective is critical.


It's sure that usual people's numbers are integer-part and non-integer part,
lattice, and vector space, where the ordered fiield makes the modular in fractions,
while, the lattice makes the modular in increments.

Reducing definition of the non-integer to [0,1] can help a lot then for
a usual [0, infinity).

Defining "least upper bound" or simply getting completeness from "[0,1] iota values",
is that basically "iota" is introduced as a formal term, ..., with the same usual meaning
it has as "infinitesimal" and "arbitrarily small" while "non-zero".

The polydimensional is that the zero-dimensional, for example, and one-dimensional,
happens to also sit in all the dimensions it is in, that basically is a lattice of dimensions,
all the way to infinite dimensions, but more than less max-ing out analytical character
in three dimensions, the "point" is in all the dimensions at once, but, must be defined
by its sides or else for example it's a line or higher dimensional surface. The idea is
that this is built directly from a continuum, it's a synthetic geometry in the sense of
that from the principles of the "points, in a spiral-space-filling-curve define space,
in lower dimensions and higher dimensions".

This isn't "different" from "geometry is points, lines, ..., Euclidean with constructions
after classical constructions", it's "before".

Nobody told me there is one this thing but I found it here.

Really though a strong mathematical platonism for mathematical philosophy,
technical philosophy, "axiomless", it's that the "axiomless" properties of geometry,
work out about great, all free in terms.

This is no retreat from rigor, indeed the opposite, and very strong in foundations,
among all _zero_ or _one_ candidates for mathematical foundations.





That sounds great.

[Timothy Golden]

The idea of an axiomless system does seem interesting, but ultimately we are constructing these things. Yes, you could claim that we are discovering them; the good ones anyways. Constructing from thin air in mathematics, somewhat. Still, upon granting something carefully you'll have an axiom, whether you call it that or not.

I am having a problem at https://en.wikipedia.org/wiki/Jacobian_variety#Construction_for_complex_curves.
The integral of omega over gamma could be / should be zero.
To what degree then this is a discussion of zeros masquerading as things...
For instance, The integral of B dot dS over a closed surface is zero. Yes, this is magnetics, but the effect is mathematical.
In effect, going around a closed curve brings you back to your starting position again. Claiming that there is something there should be investigated on a simple example.

Of course I am open to misinterpreting things. This is pretty mature work. I must be oversimplifying.

[Ross A. Finlayson]

Any equation or formula is its own little theory, ....

In a universe of theories the empty one.

Really, philosophers keep arriving at axiomless natural deduction,
and so on, _long after_ already knowing mathematical platonism
and abstraction, that platonism is strong.


So, these days it was after "square of opposition", then,
we point at Corcoran, then besides that also it's for all
the physics and so on.

For lots of philosophers some "technical axiomless natural deduction"
is the _only_ way to have a mathematics.

Then after that axiomatics are pretty simple, of course.

It's like the prinicple of relativity: no privileged axioms, while
at the same time, each moment is its own frame, relativity of theory.


So, ton's of philosophers have already adopted some
"strong mathematical platonism" after some "axiomless natural deduction".

Then it's just a general theory with that primary objects are primitive mathematically.

(Or elementary, primitive or elementary.)

Sure, axiomless natural deduction can start "nothing isn't a theory is a theory".

All the other theories are in it, ....

It's the same time a universal theory, it's dual. In its parts or elements, ....



Dually-Self-Infraconsistency

[Timothy Golden]

Well, as this applies to number you could ponder a leather bag of pebbles with the drawstring tied shut, sealed with wax.
The primitive axiomless person wonders whether the quantity of pebbles might change in the bag.
A bag that is empty; a bag that is full; a bag somewhere in between...
This concept of stability underlies most of our work, and if this is upset then little could be done.
As experiments proceed holding more and more things stable becomes relevant; This becomes physics.
As the mind realizes that the pebbles in the bag are in fact conserved, then the concept of number does resolve to something known as a constant, yet that some form of memory or other quality does exist with than number form: here we are actually lacking in our language. The mathematician cares little whether there are m marbles in the bag and whether m is fixed or variable is just a matter of appending this language. That these appendages could be in the number itself rather than in the mind: our hardware representations do allow for this, whereas prior to marks on paper did suffice. I think the paper language is more fraudulent and that a lot can be slipped under the rug without even knowing it. Particularly in the climate of the moderner; when you roam the math stacks and witness the accumulation; even and especially here on wolfram or the wikipedia; how much of that language becomes circular in nature and fails to bottom out to anything of substance? Then too, where ambiguities can be exposed atop this circular nature then the failings of that branch are ready for exposure. Yet will it happen? Or does academia allow for this since the fingerpointing could ensue back onto your own branch of specialty? As I have plowed down onto the rational number and find it to be an ambiguity in the terms of structured thought then I have come back to the paper version. Yet the type of structured thinking that I apply is due to our modern hardware.

Who on Earth would deny that you can divide a pie into six equal pieces? That each piece then is one sixth of the whole? Here then, like marbles in a bag, we can perform the physical experiment and discover problems with the procedure, even idealized out to sectors on a piece of paper, even worked out on a computer, for that sixty degree angle will not ever resolve perfectly. If you think sixths are too easy to get perfect maybe try for sevenths. Lucky seven; another wrap toward unity hidden in the glyph. For perfect sixths or sevenths in your number system you want to go after modulo-six or modulo-seven, and in hindsight these are the correct forms that will yield perfect sixths or sevenths. That you work in a modulo-ten number system does interfere with this thinking and that the assumption goes unstated is great cause for confusion. Indeed that all these systems are modulo-10 is an inconvenient fact in the base of mathematics. That every power representation of number which claims to dissect number will require the usage of 10 actually explains some of the glyphs that are in use; the correspondence of the eight with the infinity symbol is unmistakable.
That the modulo quality of what we call 'number' is present in our systems of mathematics exposes a substructure that is not necessarily fully leveraged yet, as evidenced by polysign numbers, whose modulo sign behaviors fall out directly and ought to have been recovered by every great who pondered them. The fact that they went undiscovered (and still unappreciated) is quite a large statement on human behavior. That we operate via habituation and are highly programmable are features we don't care to concede. In this day and age discussions on consciousness as untouchable are underway. Free will as limited is too easily established. To come down to ground level is somewhat the course that has to be recommended. That the world is in a grand reset; a time when the confrontation of lies and misinformation at all levels demands that we seek the truth. That we challenge everything that we operate on as assumptions; particularly when one country wreaks havoc on so many others, causing whoever is left to be swept under its wing. It gives me the creeps to be under that wing now. I'm sorry to devolve into the politics, but as to whose wing a pure mathematician is under: this is where I was going. The wings are many aren't they? How strong are those wings is the question. Can we develop a stronger wing is the question. To return to the ground and wipe the board clean and start again from scratch is the ideal position. This is the position that many greats did write from. Descartes for instance. Kant for instance. So many more but I think some of the modern mathematics will fail to return to ground level successfully, or rather fail to be recovered from ground level successfully. Possibly I have difficulties with their language, but possibly their language is flawed. I'm willing to go either way here, but I will carry on with the falsifications where I can see them. I will carry on attempting to remedy those as well. Shouldn't we all take this position rather than worshiping our predecessors?

Geeze I've had a spider outbreak here. I've got little baby spiders racing around my computer screen like little pixels on legs. Gremlins. Fortunately our pixels are stable enough that they don't go running off, eh?

[Eram semper recta]

It's garbage. Don't waste your time.