What is Fundamental is the area of the imaginary unit by Jouko Harri Tiainen

FAQ -- read below before asking questions -- I tried to anticipate what would be commonly asked.

Errata

1. Such as (a+ib) times (c+id)=ac + ibc + ibs +iВІbd should be (a+ib) times (c+id) = ac + iad + ibc + iВІbd

2. For "That is same a and b but they behave so differently we measured. " it is "That is, the same a and b but they behave so differently when measured.

References (were left out entirely)

1. The quotes about the "Fundamental Theorem of Algebra" and the two distinguishable +i and -i of "the" imaginary unit are from wiki

2. The appendix - "watched pot" and "quantum immortality" set-ups are from Phillip Hoffmann's soon to be published book "A World of Possibility", buy it - easy to read and understand, basically a study about monads and qm

3. Quotes about monogamy from arXiv:1112.3967 and arXiv:1712.04608

4. Minkowski's Paper - Minkowsky, Hermann, German paper Raum und Zeit (1909), Jahresberichte der Deutschen Mathematiker-Vereinigung, 75-88. In the 1920 English translation...{b]We can clothe the essential nature of this postulate in the mystical, but mathematically significant formula 3x10^8(metre)=в€љ-1(second)... www.en.wikisource.org/wiki/Space_and_Time

Some more answers for some questions

I tried to put the essay into abstract form

Abstract

In dual Maths - areas are numbers not lengths as in current day mathematics, that is, much as in the same way in QM we associate "outcomes" with an "area" (the square area of the wave-function), we define "numbers" as "areas". This seemingly small change, gives us the idea that "the imaginary unit" can be thought for as an "area". Once we think of the imaginary unit as iВІ and as an area we can use the intrinsic characteristics of an "area" to define new "complex numbers". The role of the imaginary unit (which is a length) in our current mathematics, is to close the algebra on the geometry. In our current maths we just say (better declare) that the imaginary unit exists, and it solves this one equation xВІ + 1= 0 with a unique solution, with the symbol i, from which, we can define complex numbers as z=a+ib with zero=(0+i0). And using these complex numbers we can have complete knowledge of the Cartesian Number Plane defined as в„ќxв„ќ or в„ќВІ, using equations that satisfy the "Fundamental Theory of Algebra". Clearly using this method of determining knowledge for the real-numbers associated in a number field has been a complete and utter failure in physics. It is an established theoretical assumption that the Born rule (or areas are outcomes) comes last! It is an established theoretical assumption that there is "inherent uncertainty"of how a quantum pair p and q lengths, when viewed as areas i.e. the outcome О"pО"q, cannot be less than an "intrinsic constant or length" called the reduced Plank's constant (or a constant of nature) or the Uncertainty Principe. It is an established theoretical assumption that we need a wave-function consisting only of complex numbers (postulate of quantum mechanics), that is, outside "measurable outcomes" and by squaring the wave-function (or making an area, which is the Born rule) we only get "probabilities" within an "area". We cannot even work out, what happens, when we only have two outcomes in a box area: the only answer we get is that all outcomes are happening in the area, yet in QM, it is an established assumption, we can form operator vectors, like from the sides of an "imaginary unit that is an area", that can be used to show that within, that real-number box, there are two lengths when multiplied, using complex numbers, obtains 100% certainty, that the theoretical area, is always full, or the SchrГ¶edinger Cat Thought Experiment. We don't even understand why [a,b]=ih/2ПЂ in QM. In this essay by introducing the concept of numbers as areas we show that a dual mathematics can help clarify many of the paradoxes, oddities and, weirdnesses, in current scientific thinking.

From Phil

Hi Harri,

I know what you are getting at and really like the way you want to change math, but I want to challenge some of your claims. To some extent I'm playing devil's advocate here, but not entirely.

When you say "areas are numbers not lengths", that's confusing. Whoever said numbers are lengths in the first place? Lengths (i.e. distances) are one-dimensional and areas are two-dimensional, so this is a strange thing to say in more ways than one. "Areas (of squares) are numbers, not SQUARED lengths" presumably makes your point more accurately. Also, when you say the imaginary unit is an area, are you referring to i or iВІ? I assume you want to replace i with iВІ as the basic imaginary unit, but mathematicians will have a fit about this. How do you get iВІ without having i first, and if you do have i first, then surely IT, not iВІ, is the basic unit after all, and mathematicians will insist that i is a number because that's what they've called it all along. Finally, the idea that the Born Rule squaring of wave function amplitudes yields areas rather than probabilities is going to strike everyone as extremely weird. Defining operator vectors the sides of Pythagorean hypotenuse squares helps, especially since vectors are lines, but it's weird nevertheless. On the other hand, if your approach has enough explanatory virtue, it will win the day in the end. A big challenge, but you've never been afraid of challenges before! Bravo!

From Harri

Dear Phil

Yeah I have been reading and trying to understand the other top rating essays (shivers, they are really good) and the comments they receive are super-technical and "nit-pick" in the best possible way, that will be good for me.

In geometry we have the fact that lengths are one-d lines of length "a number", and that is what we use to do "geometry". Pythagoras's theorem is all about "ratio of natural-number lengths" to define "numbers" and clearly what the essay does is show that any pair of natural-numbered lengths (as in the sides of the triangle a and b) that form a ratio which defines a "number" can be part of the entanglement. Clearly if a,b are counting numbers as lengths then the hypotenuse box shows WHY (the area of the imaginary unit) and HOW we have "numbers" (the four sides of the area, two sides to get the area of the box and the other two sides to encode the properties of the objects called timeless or pure numbers) in the first place to actually count actual physical objects.

Yes you say it even better than me "Areas (of squares) are numbers, not SQUARED lengths" yes exactly, I'm going to use that thanks, mate.

Lengths (i.e. distances) are one-dimensional and areas are two-dimensional, so this is a strange thing to say in more ways than one, well how do we associate 3 a timeless number with three meters or three meters squared in the first place in our physics. This is the point of the essay, how do we attach a pure maths 3 with a physical "interval" of "three metres" in our physics or maths in the first place we don't know how to do this -- this essay addresses this "entanglement" with one pure ordinal state "the imaginary unit" with the impure n-tuplet objects (a and b) in a state labelled "cardinality measure" of "n definitional units".В  See my last essay in fxqi for more details

Yeah I can see about the confusion about that from our view point it is iВІ but yes it can be thought of as i but only if we "are looking from that area itself" which we aren't. Realise all the a and b we can observe and action are located on the outside of the common area, physical reality isn't pictured on that box, that diagram is what we cannot observe directly. So that is why I labelled it iВІ. I will think about this notation confusion. I suppose it is like this if all numbers are areas then the imaginary unit is an area, which we assume is a square, so for ease of notation we write that the "imaginary sides" of the imaginary area are the indistinguishable imaginary units hence the notation =iВІ for the area. Clearly it is called the imaginary unit, and think like this -- a "unit of what". If I ask you to draw 3 everybody would drawn a line "three length units long as compared to an invariant length" or just call an arbitrary length "as three units long". This essay does both ways at once -- the first way of doing the geometrical representation of a "number" where we are drawing the "constant of nature" as "the invariant length" of an active hypotenuse that has a unit of "c=i=h", the invariant is iВІ from our viewpoint. And the second way, we are within the actual area of the number called the imaginary unit formed from and the two "free sides" that encode the hypotenuse vector complex numbers, from this vintage point within it is =i that is, physical reality is not there, this is where the laws of nature are "located" in the hypotenuse box. One is the dual of the other and we "reside" on the outside of the box on those non-labelled hypotenuse. While within the hypotenuse box we have the pure states of "laws of nature", "constants of nature", etc. The "metaphysical" things (or states of the hypotenuse box) that are needed to describe a reality that doesn't look like "numbers" yet is "endued" within timeless numbers and laws. There are no other correlations allowed for a and b except timeless laws with physical natural constants.

Think like this nobody draws an area when you say draw 3 for example do they. Areas in our normal geometry are "Squared lengths" so an area of 9 has four sides that are 3 each, so clearly we can do "encoding" with the spare "two sides" for an "area" that isn't needed strictly to be tied to the 9, or these sides can do "hypotenuse-projections" or vectors of these copies of 3 as to get an "area" that can be used to ascribe (encode) properties. Basically we say that the other two sides encode properties from those operator vectors called 3 to the three objects we see around and about us. В 

In our geometry lengths are numbers, again nobody draws an area when asked to draw 3 everybody will draw a line. And the better point, is that nobody says that the hypotenuse of the Pythagoras triangle is the negative square root of two do they. And the best point, a complex number definition is z=a+ib with zero=(0+i0) it isn't choice because a-ib=z* is the complex conjugate of z by definition and the definition of zero is not defined as zero=(0-i0) there isn't choice in our maths about how we define complex numbers it must be "+i". In our maths recall the imaginary unit is the singular solution to the one and only equation xВІ + 1=0. There is by our own maths only one pure imaginary unit, that solves that one equation, what exactly is "i" in this complex number z=1+i9 or complex conjugate z*=1-i9 is it the same thing as the singular and solitary number that is the imaginary unit bare of all a and b trappings which is what solves the one equation xВІ + 1=0. Clearly not, so what "thing" we use in our complex numbers is a pale comparison to the actual imaginary unit as a singular solitary number which is an area in this essay. Clearly whatever thing we use in complex numbers labelled i is constricted by the very act of the way we write complex numbers. In our complex numbers there are two objects -i and +i that give the correct solution to xВІ + 1=0. Which now seems not to answer the question -В  what two same numbers when multiplied together gives us minus one, is it not assumed that there is only one number that solves the folk-version of the definition of the imaginary unit. And clearly we know that -ve times -ve gives +ve and that +ve times +ve gives +ve and -ve times +ve gives -ve and +ve times -ve equals -ve, clearly we know that Bell's theorem can be written as (a.b)=+/- which can be equated with the -i and +i obviously this is how we assume our maths shows its relationship to the singular and solidarity number called the imaginary unit. Well, when we use our complex numbers we have two different numbers that solve that question. While the definition of the imaginary unit states that there is only one i exists. So somehow we have got one unique i then in complex numbers we seem to have many i in the maths -- would you say that the one unique i that is defined as "i=the positive square root of minus one" that solves that one equation is the actual i we use in our complex numbers when we write z=a+b. Do we actually use the definitional object itself as the i in our complex numbers. Not a chance what thing we call i in complex numbers isn't the definitional object itself. But in dual maths idealisations are as much a part of the diagram as a and b, so we have a pure state or an active hypotenuse just for the singular i then we have encoding vectors to get "complex numbers" with hypotenuse vectors that are like "holographic" projections of the properties of the singular thing-in-itself so we can form the actual dual complex numbers. Dual maths is a totality theory. Idealisations are as much part of the whole as are a and b.

What dual maths is, in another way of putting it, is the physics of complex numbers and the existence of the imaginary unit. Monogamy explains why complex numbers: real parts & imaginary parts respectively act the way they do and why the "mixed" parts act the way they do. So duals maths is more like "we can explain mathematical objects behaviour using quantum concepts" and since complex numbers are the numbers that close algebra on the geometry, the complex numbers are the "objects" that contain the "necessary and necessary properties or information" that the timeless equations need within physical reality so they can describe actual a and b and the actions of a and b. Loosely we quantised our maths entirely we made the imaginary unit itself (or the definitional object) the actual "interval" or the actual invariant and not "the invariant length" of the wave-function or the "the invariant area" of the wave-function squared. The Born rule is ОЁ*ОЁ which is a-ib times a+ib equals aВІ + bВІ, and this area uses the hypotenuse vectors projections, not the singular imaginary unit which is in a pure state. The Born Rule gives us the area in the first place so we can have all "outcomes" in there. Why do we normalise to "1" or 100% since we know that the probability wave when made an "area" ОЁ*ОЁ orВ aВІ + bВІ is somewhere and some-when. Realise that within the box is the definitional objectВ ОЁ*ОЁ that our physical Born rule uses for all, any and, each & every "unit" intervalВ  when we do make a physical hypotenuse i.e. label the empty sides of the hypotenuse box. The definitional Born rule - rules. В 

The major point is that unlike our modern day equations, the equations in the hypotenuses are fixed in that box in that order and position, we cannot move things about like we do in "normal equations" used in maths and physics, these are invariant in all respects to "our physical reality" these are the invariants that are the units that the singular definitional imaginary unitВ  projects timeless by hypotenuse projection (i.e. perfect holographic) vectors, that is, there is no loss of fidelity, the projection is perfect in the sense that we used "complex numbers that close equations for the equations deductive in the mathematical sense" for what we call "the current equations of physics" like E=mcВІ and "the current counting maths equations" like one plus one gives two while counting an apple a and a banana b" in the physical sense. Since the hypotenuse vectors project the "property of the definitional number as the speed of light as i=c" we can actually have a real-life speed limit that is a physical behaviour defining perfectly (via equations) any and, each& every interactions of the duality of light. In our equations we use the actual physical manifestations of the hypotenuse box (i.e. on the outside of the hypotenuse box, that is, what we call measurable physical reality or more clearly outcome space) not the definitional objects themselves in a pure state which are in the hypotenuse box. Yes, the definitional objects aren't where we are but are as much part of the whole as we are, and where we are is the dual of the area of the imaginary unit. We ain't where omni-everywhere and omni-every-when objects in a pure state are. We are, where we can measure using "invariant units" of the singular definitional objects, that is, on the outside the hypotenuse box. That is where our equations reside that can be used with real-life light and the quanta of action.

"On the other hand, if your approach has enough explanatory virtue, it will win the day in the end. A big challenge, but you've never been afraid of challenges before! Bravo!

Hope that answers your concerns -- and it also shows the power of the explanations which picture

1. how the hypotenuse box handles the metaphysical and the physical within one geometry, derived from,

2.the actual definitional objects, which project, via:

3.two hypotenuse vectors, the necessary and sufficient properties to account for all aspects of a totality theory.В 

There are no separate parts or objects in a pure quantum system. All is drawn as one whole.

I need some support I am troubled this is sounding like it might have some merit to consider . And not just because it is a good simple idea. Can I use this email exchange as my first post in the competition since the questions are well covered by the reply. It might help quell fears that it is just a trick of the notation or something like smoke and mirrors.

Yours Harri

From Phil

Dear Harri

Yes, of course you may use it, Harri, no problem.

On another note, how do you extend and generalize the new math to volumes, circles, spheres, etc.? The area of a circle also involves a squared length, of course, so that fits nicely, but are volumes numbers, too? Presumably yes. And as always, I'm most interested in relating it all to the incommensurability of the discrete and continuous. I assume the key has to be i, or better, the two kinds of complex numbers that involve i. Since i=c and i=h, are these equations encoding the "fudge factor" that ties together the discrete and continuous, or is it the quantum commutator [ih]...or something else again?

Forgive these random and spontaneous questions. I'm obviously still thinking through things...

Yours Phil

To Harri

From Phil

On another note, how do you extend and generalise the new math to volumes, circles, spheres, etc.? The area of a circle also involves a squared length, of course, so that fits nicely, but are volumes numbers, too? Presumably yes.

Well I think this is outside the present scope of the level of discussion. But I have to clarify, there are no dimensions in the hypotenuse box so when you mention that a square is two-dimensional and well as a circle, and that volumes, could be considered as the "template" for the definitional imaginary unit or a number - these "dimensions" are "definitional units" as well, how do you think we get dimensionless numbers like i to have properties like metres or Joules via "definitional units" that act are like mini-me "constants of nature" for the outside of the box.

In our maths we have the indistinguishable definitional i then we can get the

The full version of the respond is here -- it seems to have cut of the last sectionAttachment #1: RESPONSE_Final_POST.pdf

To Peter Jackson your trick is here -- thanks for the problem and solution.

Sorry couldn't attach file

again

Red/Green Sock

Sorry I forgot to login hence the "Anonymous" in the above post -- Harri

a 1000 facts in isolation

1000 fact in isolation _ hard to know what to sayAttachment #1: Attachment_Edwin.pdf