November 14, 2018 at 3:09 pm #296554
Wed 14 Nov 2018 08:39:42 AM EST tpg
Are dimensions real?
Yes, they are by definition tied to the real number within modern mathematical formalism. As such a one dimensional entity is bidirectional, and this poses a numerical error which is being passed down generation after generation and is in such a place that to challenge it is to challenge much of modern knowledge, for physics relies heavily upon the real number as fundamental; as does mathematics. Even simpler forms such as R+ are of course defined in terms of the real numbers. Mimicry rules us all, but variations (and especially variations which work) are worthy of investigation. This route outlined above is suggestive, and I don’t generally take it, but what appears as rhetoric will be substantiated below so that the definition of dimension deserves to be challenged. Particularly the ill treatment of time by moderns, whose behavior is unidirectional; whose formal treatment is bidirectional; so much so that great modern physicists puzzle over the action of their equations under time reversal; this absurdity must end.
If spacetime is to be accurately unified then the mathematical formalism which develops spacetime as fundamental must be discovered. The generalization of sign produces such a route, so that a unidirectional number known as the one-signed numbers exists alongside the ‘real’ two-signed numbers, followed by three-signed numbers. None of these is any more fundamental than the other, nor their subsequent siblings. They exist as a family
P1, P2, P3, P4, …
and lo and behold a behavioral arithmetic breakpoint does in fact exist after P3
P1, P2, P3 | P4, P5, P6, …
and this boundary thus provides the arithmetic support for spacetime with unidirectional time even while the entirety is general dimensional and subdimensional.
A prone mind should be asking at this point what that breakpoint is exactly, but to develop this requires learning polysign numbers
but can be briefed here in short form. The usual behavior of the real numbers
– 1.0 + 1.0 = 0 or – x + x = 0
does allow for generalized sign so that for a three-signed system
– 1.0 + 1.0 * 1.0 = 0
and strangely enough for the one-signed numbers
– 1.0 = 0 .
This peculiarity on P1 might seem cause for rejection, but let us please recall the geometrical meaning of the real value so familiar and sensible between its numeric form and its geometric form. Two rays opposed form the real line such that minus one plus one equals zero. Which is more fundamental: the ray or the line? Because the line is formed of two rays we should choose the ray as fundamental, and this is a necessary choice in the development of polysign numbers and their geometrical representation. In order to provide graphical support for P3 we already have enough to go on just from the simple generalization posed above. Three rays arranged in balanced opposition form in a plane. We can draw these rays and label them ‘-‘, ‘+’, and ‘*’. Travelling along the minus ray one unit, then along the + ray one unit, then along the star ray one unit brings one back to the origin. Thus we can argue the laws of balance posed above form the geometry or a rendering of the polysign numbers. It is apparent to the wary that the four-signed numbers will be three dimensional in nature, though this language of ‘dimension’ is being partially challenged at this point. With P1 we now have a zero dimensional unidirectional entity, whose correspondence to time still goes underappreciated; at least publically that is so. Is to publish to make public? Does an individual have such power? In a free modern society this ought to be true. Whether that is so or not is a valid concern as subauthoritatian power structures get exposed. Should I publish in a high priced journal that I have no access to? No, I should not. Certainly that is far less public than this method here.
The furtherance of polysign numbers shows that beyond the vector behaved simplex coordinate systems outlined above we have a sum and a product to cover, and until we study the product carefully the breakpoint at P4 cannot be exposed. The above construction is compact and a cause for further study, and if it is not readily understood then it should be gone over again. The simplicity thus far along with all of the consequences so far mentioned (except the breakpoint) develops multidimensional geometry from a singular equation
Sum over s ( s x ) = 0
where s is sign and x is magnitude. The geometry is inherent to this equation. No further constraints are required other than the instantiation as a family of number systems, but this comes naturally as one instantiates firstly the real (two-sign) number thencely the three-sign number, the four-sign number(whose rays emanate from the center of a tetrahedron outward to its vertices), and so forth. Oddly though there is enough already to puzzle over the little sister P1… or is she the elder?
Is multiplicity fundamental or ought it be derived via a generator? These are instances of paths to follow out which I have not exhausted though I certainly rely upon the multiplicity. Either way the real value, and so the language of dimension, no longer holds a special place, and this is the moral of the story.November 21, 2018 at 11:36 am #296612
This is not a physics forum. Not sure what you’re looking for.November 25, 2018 at 1:18 pm #296651
“The Center for Inquiry leads the charge on promoting reason, science, critical thinking, and humanist values. We work to defend the rights of atheists and non-believers around the world and advocate for public policy that’s rooted in evidence and objective truth.”
I suppose some critical thinking will do.November 26, 2018 at 11:01 pm #296657
You ask “Are dimensions real?” length, width, depth, time.
Seems as real as real can get.
Then comes the math, which springs from the human mindscape. There is much more possible within the imagination and with math, then there is within the confines of the natural laws and order.
So, I’m no mathematician, and I tried but I get lost in all those words. Can you explain to a non-mathematician what it means?December 6, 2018 at 7:38 pm #296788
Very nice piece ( Missing Key). I agree with your level of awareness and often ponder whether humans have overgranted their own abilities. We are far more great mimics than we are great analysts and thinkers. For instance we are all taught newton’s law of gravitation so that it is familiar, but if our minds were stronger we would derive it for ourselves. This applies ad nauseum into a vast accumulation of human ‘knowledge’ that at this point is weighing us down rather heavily. The pile is deep, and academicians are expected to add to this pile. Working up high in the pile is safe, but what if there is an error beneath? The pile ought to collapse according to scientific thinking, but let’s not forget that we are humans practicing science.
We are all taught the real value; it is a number that can be either positive or negative, and it takes on a satisfactory graphical representation in the line which is by definition the ‘dimension’. When we talk of two-dimensional space the formality of the plane is notated as RxR, though it is also true that Euclidean geometry did not require the cartesian product.
With regard to reality and physical space we find that RxRxR is sufficient. We need no more data to address the positions of space within a box or a room, and we can thence stack boxes or rooms to extend this to arbitrary distances. As you say: length,width, and depth… though these are typically unsigned figures whereas vector space wants the option of positive or negative values Otherwise there is a corner in the system, and we observe no such corner. (x), (x,y), and (x,y,z) are all in common usage as one-dimensional, two-dimensional, and three-dimensional space where those letters imply real values.
Time is peculiar though isn’t it? You list it as a dimension, and it has become commonly known; credited to Einstein though you could go back to Kant if you wanted to. The unidirectional nature of time is not accomodated by the real line, which is inherently bidirectional. Furthermore tridirectional coordinates have been overlooked too. They lead to a native description of the complex plane from the same laws that develop the real value. These laws take a harsh stance on the unidirectional geometry: they require it to be zero dimensional, and yet it does arithmetic of the unidirectional form. Time likewise shows no such freedoms as space does, and it is these freedoms which allow us to claim 3D space. Thus the claim of time as zero dimensional is realistic. Take an object such as a mug and move it about a table top. Two degrees of freedom are observed. Were the mug not free to travel in one of them then we would claim the mug to exist in a 1D space. Regretably this is a large object that already consumes 3D space so perhaps I should have picked a smaller object. Loosely though these are the means at our disposal. Now I would ask for your interpretation of moving the mug ahead in time by one second, and likewise back in time one second. We will arrive in numerous contradictory circumstances, and to the best of my knowledge neither freedom exists. Therefor we find support for the zero dimensional nature of time even without the polysign analysis. The unidirectional nature of time carries direct correspondence with the one-signed numbers; both arithmetically and geometrically.
I know the about polysign is foreign, but I’ve kept it to a minimum for you. The math is actually very simple. The main problem is that we have been trained on the real number as fundamental for roughly four hundred years. Both mathematicians and physicists feel this way. According to polysign the real number is not fundamental. It is a member in a family of number systems
P1, P2, P3, P4, …
which are one-signed, two-signed, three-signed, and so forth. Of course the real values are P2. I hope I have peaked your interest in P1, and possibly in P3 which turn out to be the complex numbers in a new format. But you see P3 reside as siblings with P2 and are not dependent upon P2 for their constuction. Likewise P1 and really Pn. These systems are built from sign and magnitude; two very simple components which when married together yield this family. It gets a bit more complicated from there, but not really that much more. Why does (–) = + in P2? I don’t really answer why, but the generalization to Pn is that signs in product yield their modulo-n sum. In P2 that is modulo two. In P3 that is modulo three, and so forth. Beyond this we see the familiar
– 1 + 1 = 0
clearly generalizes so that in P3
– 1 + 1 * 1 = 0
and strangely in P1
– 1 = 0
which is the zero dimensional nature. These balanced systems turn out to have inherent geometry based on the simplex. In other words these statements above already imply their geometry. Remember the real line? P3 has three rays instead of two and it is planar.
I could go on and on and I’ve already gone too far. Thanks though for your interest. The full and careful explanation is at http://bandtechnology.com/PolySignedDecember 9, 2018 at 10:10 pm #296817
Looked at a couple of the other topic – very humbling. I’m pretty good at simple math, but that stuff makes my head spin.
Okay, I’ll take it back to a level I can follow. You start this thread with the question: “Are Dimensions Real?”
Can you define what you mean by “real”?December 9, 2018 at 10:12 pm #296818
Or “un-real” for that matter.
Oh and thanks for taking the time to read that Missing Key essay.December 11, 2018 at 10:36 am #296836
What is real… I guess for a lot of philosophers here that becomes an abstract problem, but within mathematics and physics the ‘real’ number is taken to be fundamental. Even discussion of unsigned numbers (magnitudes) within mathematics discuss R+ and 0 as their working set. Modern computer programming takes us into structured thought at a compiler level of integrity. As such we build more complicated things from simpler things. Anyone witnessing a few concrete instances of real values such as
-1.23, +0.001, -0.056, +345.23
can see that there are two substructures present within the real value, and so its treatment as fundamental is invalid. Those substructures are sign and magnitude. The format
where s is discrete sign and x is continuous magnitude has gone overlooked by mathematicians and physicists. Most will carry on insistently that binary sign is fundamental. Geometrical representation via cartesian coordinates supports their belief to the point that we in modernity associate physical space as ‘3D’ space, but this value three is the result of the usage of the real number as a single dimension. We have lost our ability to discriminate between the cartesian form as a representation versus an actual construction. We do not actually witness the ability to physically construct multidimensional spaces from lines. The tie is cleanly broken when sign is generalized, for an alternative means of begetting multidimensional geometry arises; one which relies upon the simplex coordinate system rather than the orthogonal cartsian one.
The lead into polysign is as simple as introducing a third sign ‘*’. What does one do with it? Well, clearly we have the old two-signed instance
– a + a = 0
– a + a * a = 0
is the natural form of three-signed system. Already geometry in the plane is implied. Three rays are sufficient to address the plane, whereas the cartesian system requires four. Upon developing the arithmetic product which follows even more naturally then the complex numbers alight though they are in a new format P3; they are built from the same laws that build R or P2. This singular gain is sufficient to demand attention. But there is quite a lot more that promises a new understanding of physics. I do not have that formalism built out but enough signals are present.
The terminology of ‘dimension’ implies the usage of the real number as fundamental when we discuss 2D and 3D space, yet 3D space is arguable an a priori when we are actually speaking of physical space. Here the philosopher can alight a bit, for polysign is breaking open this system. It arguably does recover the real number as P2, but finds no need to build higher spaces from P2. They are instead siblings, and the family of polysign numbers
P1, P2, P3, P4, P5, P6, …
demonstrate a general dimensional system which includes a unidirection time like constituent in P1. A breakpoint is observed at P4 allowing for emergent spacetime from an arithmetic basis. So where is the physics? I do hope to live long enough to find someone who will make this contribution. It likely involves a rework of electromagnetics, whose properties will be more clearly and purely laid down in terms of structured spacetime. Possibly even requiring additional intangible dimensions. Possibly it is a mistake to regard spacetime still as the fundamental substrate. Already time is proven zero dimensional three different ways; is mistreated in the tensor format formally under relativity theory; so why shouldn’t we take even more freedom… if it begets a theory more naturally… well you say the string theorists have gone to 10D silly man. Well, I say, the polysign progression lands one in such a place:
P1 : 0D ; 0D total ; 1R total
P2 : 1D ; 1D total ; 3R total
P3 : 2D ; 3D total ; 6R total
P4 : 3D ; 6D total ; 10R total
P5 : 4D ; 10D total ; 15R total
and so here on CFI you get the first usage of the nR format. I’m not sure that it is solid yet, but there is little argument to make with it. That humans are off by one in several ways over; that I find completely believable. The R is of course for Rays; the real line being composed of two such rays cannot be regarded as fundamental. Possibly my language has been off a bit. Perhaps we should be using the terminology ‘sign product’ or some such as a replacement to ‘cartesian product’, but the fallout of the sign product is more broad. It is a full algebra, geometry, and number system which carries physical correspondence i.e. emergent spacetime support.December 11, 2018 at 1:41 pm #296840
I’m sure CC will want to keep this going, but philosophers don’t have a problem with the mathematical definition of real numbers, and mathematicians don’t have a problem with sign and dimensions. I kinda quite reading after that.
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