John A. Shuster

• Bachelor of Arts
• Independent-Researcher at NewMexico+Minnesota in USA

A call for “a Math with a Heart” with some thoughts, theories, and findings for creating a ‘kNew eartH of Heart’ via the “eartHeart” and “MatheMagics”

My sincere Thanks to my long-time friend Eva Doucet for this excellent translation (into French) that she provided me in 1994. The original "The Story of Zero and One" was written in English by "One-for-Joy" (John A. Shuster), and can be found in the author's RG Research/Articles.

Theta Numbers (imaginary infinitesimals and infinites) and other aspects of ZerO are explored, revealing hidden potentials of an Evolving-ZerO, which contains them. Theta Numbers are the author's re-interpretation of Staley's Zero Numbers and Shah's Universal Numbers. They are an entire system of numbers "inside ZerO", which have algebraic quirks f...

This is a personal account by the author, giving background behind his reasons for ZeroStories, and for what he calls "MatheMagics -- a Mathematics with/of Heart". It first appeared in the 1990 zero-th edition of "The MatheMagical Powers of Zero (and Her Whole Family)". Later it was slightly edited to be the last chapter of "MatheMagics for our ear...

Real numbers are 1D numbers that might be thought of as reflecting our “Real” world. They do, but only to a point. Their extension, the 2D Complex numbers, do a better job in explaining reality existing in time. But they too lack explanatory power when it comes to handling intangible“real world” notions of: ideas, desires, dreams, beliefs, support...

Riemann extended the complexes by a 1:1 map between the C plane and a sphere, then treating its North Pole point as “infinity”, where "oo" additively absorbs any finite z: z+oo = oo = oo+z. Here, we re-postulate this as “z+Ф=Ф+z a unique z-individualized infinity”, where Ф is a basis vector representing “infinity”. This permits an isomorphic map of...

The nuReal Numbers (nuR) were developed in [1] by creating a real surrogate G = qp from non-real conjugates q and p. The result is a 3D vector space (over the Reals) and a commutative ring (with identity L=q+p-G). nuReal space is isomorphic to the cylindrical split-complexes. Although the “nuReals" contain 7 idempotents, they lack any i-like elem...

In [1] it is shown that a “slide-duality” exists within any field (or commutative ring). In [2], [3], The nuReal numbers (nuR) are defined as a 3D vector space and commutative ring (with x-identity). These numbers were developed by creating a surrogate G=qp (for “1”) from non-real idempotents (q, p), with L=q+p-G, (not G or “1”) serving as the mult...

It is shown that any field, <F; +, x; 0, 1> under the isomorphic map f* = f-1, creates a dual field <F=F-{1}; ++, xx;-1, 0>, where ++ and xx are dual binary ops to +, x, resp. This isomorphism establishes a duality via "unit-slide to the left". Under this dual multiplication, the multiplicative identity of xx is id(xx) = 0, and 1xx1 is not 1!

This is the first story in "The MatheMagical Powers of Zero" series by author "One-for-Joy" (John A. Shuster), who visited Indonesia in August, 1993. My joyful thanks to Dyah Palupi Probosari for her enthusiastic translation of this story into Indonesian, while she was a student at the U of Minnesota, USA. -- JAS // AUTHOR'S SPECIAL NOTE (2025-04-...

This Presentation document, “Revisiting ‘Testing Love For Experiment'…” (uploaded May 15-16, 2024) revisits the authors’ original work by reorganizing it to include some of the authors’ new ideas and their personal views. Their original document, "Testing Love for Experiment and a Possible Fractal DNA Antenna Mechanism of Observer Effect”, was publ...

This is the first story in "The MatheMagical Powers of Zero" series by author "One-for-Joy" (John A. Shuster). My especial Thanks to friend and co-researcher, Jens Koeplinger, for his careful translation into German. -- JAS

This PowerPoint file contains a revised version of the presentation done on April 26, 2024 at the section on "Quantum Mind" of The 30th Conference on the Science of Consciousness, in Tucson, Arizona, USA by Anatoly Goldstein, Ph.D. The authors formulate and suggest a specific way of testing of the following hypothesis, Ho: "An Observer’s love for h...

This is the first story in "The MatheMagical Powers of Zero" series by author "One-for-Joy" (John A. Shuster). My Thanks to Bil Linzie for his fine translation into Esperanto, which means "one who hopes". -- JAS

These stories reflect certain learnings. To the world, ‘zero’ is a symbol of “nothing physical”. But to me, ‘Zero’ is both an “Entry Point” and “Source of All”! Zero, the intersection of all dimensions, represents both a ‘finite point nothingness’, yet a ‘door to the infinite’. She (Zero) became my “door” to Inner-self and the greater understanding...

These stories reflect certain learnings. To the world, ‘zero’ is a symbol of “nothing physical”. But to me, ‘Zero’ is both an “Entry Point” and “Source of All”! Zero, the intersection of all dimensions, represents both a ‘finite point nothingness’, yet a ‘door to the infinite’. She (Zero) became my “door” to Inner-self and the greater understanding...

These stories reflect certain learnings. To the world, ‘zero’ is a symbol of “nothing physical”. But to me, ‘Zero’ is both an “Entry Point” and “Source of All”! Zero, the intersection of all dimensions, represents both a ‘finite point nothingness’, yet a ‘door to the infinite’. She (Zero) became my “door” to Inner-self and the greater understanding...

These stories reflect certain learnings. To the world, ‘zero’ is a symbol of “nothing physical”. But to me, ‘Zero’ is both an “Entry Point” and “Source of All”! Zero, the intersection of all dimensions, represents both a ‘finite point nothingness’, yet a ‘door to the infinite’. She (Zero) became my “door” to Inner-self and the greater understanding...

These stories reflect certain learnings. To the world, ‘zero’ is a symbol of “nothing physical”. But to me, ‘Zero’ is both an “Entry Point” and “Source of All”! Zero, the intersection of all dimensions, represents both a ‘finite point nothingness’, yet a ‘door to the infinite’. She (Zero) became my “door” to Inner-self and the greater understanding...

We consider finite fields of characteristic 2, GF(q) = < N^k, (+), (*) > over the set N^k with q=2^(2^k) elements, and construct a binary operation that distributes over multiplication. In loose analogy to complex number arithmetic, we call this operation exponentiation, with symbol (v). This operation is distinct from conventional use of integral...

The dual numbers are re-derived to form "Known-Unknown (KU) space", KU = {a+bu | a, b are real, u 2 = 0}, where u is considered a ".unit of the Unknown". A homomorphism (with kernal = u-axis) exists from KU space onto Real space, ensuring that the real (known) part in a product is exactly the product of (its factors') the real parts, while the prod...

These stories reflect certain learnings. To the world, ‘zero’ is a symbol of “nothing physical”. But to me, ‘Zero’ is both an “Entry Point” and “Source of All”! Zero, the intersection of all dimensions, represents both a ‘finite point nothingness’, yet a ‘door to the infinite’. She (Zero) became my “door” to Inner-self and the greater understanding...

These stories reflect certain learnings. To the world, ‘zero’ is a symbol of “nothing physical”. But to me, ‘Zero’ is both an “Entry Point” and “Source of All”! Zero, the intersection of all dimensions, represents both a ‘finite point nothingness’, yet a ‘door to the infinite’. She (Zero) became my “door” to Inner-self and the greater understanding...

These stories reflect certain learnings. To the world, ‘zero’ is a symbol of “nothing physical”. But to me, ‘Zero’ is both an “Entry Point” and “Source of All”! Zero, the intersection of all dimensions, represents both a ‘finite point nothingness’, yet a ‘door to the infinite’. She (Zero) became my “door” to Inner-self and the greater understandin...

Spherical Complex Numbers (C3) as defined in [1] are modified to ensure that the * map from {h(r, θ, Ø)} parameter space to C3* coordinate space {(x, y, z; θ)} is 1-to-1 and onto, making both spaces isomorphic. C3*\{0} is now a multiplicative group that C3\{0} was claimed to be, but its mult is not distributive over the addition, as in [2]. In corr...

In his 2008 book, Neomathematicon, Joe Staley lays an early foundation for what he calls "zero numbers". He begins with the assumption that 0x0 < 0, not the traditionally assumed 0x0 = 0. He then claims that "0" is an expanded-positive number while "-0" is a expanded-negative number, and that a "ϕ" separates these expanded number domains. (ϕ is the...

These stories reflect certain learnings. To the world, ‘zero’ is a symbol of “nothing physical”. But to me, ‘Zero’ is both an “Entry Point” and “Source of All”! Zero, the intersection of all dimensions, represents both a ‘finite point nothingness’, yet a ‘door to the infinite’. She (Zero) became my “door” to Inner-self and the greater understanding...

By defining a commutative binary-operation (bio) xx that distributes over multiplication x, an upward ladder of recursively defined distributive op pairs on C is revealed. Similarly, the ++ bio is defined for which addition + distributes over ++, revealing a downward ladder of recursively defined distributive bio pairs, creating an axis of discret...

Groups, "other" than the subgroups within the largest standard group <U; 1> of a commutative monoid M (having finite-length cycles) are considered. Each of these largest other-groups <T i ; t i > is isomorphic to U/K i , a quotient group of U, but each possesses a different identity element (t i ≠ 1) than U or its subgroups. Thus, the largest other...

Elements of non-associative nuR1 space are renamed (q = Q', p = D') and nuR's multiplication is redefined so that qp = G = pq, where G is either like "0" or like "1". Under these definitions, an associative nuRG space is developed from a "PQ space", where the new multiplication is associative, commutative, and distributive over the vector addition....

Non-real D and Q are defined so that DQ = 1 = QD and real space {r = r.1} is extended to isomorphic copies: {rD} and {rQ}. All 3 subspaces are unified under a vector addition (+) to form a 3d vector space called nuReal space, nuR. Under a non-associative multiplication (x) which distributes over the vector addition (+), nuR is a commutative "non-as...

This algebraic system may be useful in modeling natural phenomena or behaviors which seemingly arise from or disappear to “nowhere” (e.g. the zerocenter ). In essence, A+zG is like a+bi (in C): A is the “real” or “visible” part, while zG is the “imaginary” or “invisible” part. Thus, PQG space may have applications which require such distinctions b...

The usual definition of the Rationals (Q) as { m/n: m, n in Z, n ≠ 0 } and its + and x ops is generalized on the set Q* := ZxZ with no restriction on "denominator", and no "reduction/equality" of m/n.. Such definition allows each extended rational to be expressed as "distinguishable or "distingt" elements"-each in an equivalency class in Q*, but "n...

Questioned are the properties: 1) 0.V = O, and 2) [radius, angle] = [0, ] = (0, 0) in the map of polar complex space onto its 2D Cartesian plane. Here, all supposed representations of zero, denoted as [0, ] = 0(cos , sin) = {O } = Φo, are projected onto (0, 0) = O, the directionless zero-vector. This mapping is often considered "equivalent" or "iso...

This is an update to an old topic: Message#727 (Nov/2009) and the paper "C with directed infinitesimals" (already in Files section of this site). Rather than post it, I've made it into this mini-paper.-John C+: C with directed-zeros I have always been a bit bothered by the vector space property: 0.X = O in C, for any X in C, the complex numbers. HC...

Our discussion postulates a societal ring S as a mathematical analog of ideas, actions, and behaviors in human society. The notion of a ring’s ‘Ideal’ is considered an algebraic analog containing support for a societal ‘ideal’. Such a math Ideal defines the ring’s landscape relative to that Ideal K, which forms Society’s quotient-ring S/K. Thus, a...

This is a draft of what was to be published as F.E.D.* Brief #8. It is another derivation of WQ, the Whole Qualifiers, but different in approach as was done by this author in F.E.D. Brief #5. The math of an “idea space” offers us a way to assign/interpret given ontologies and follow/predict their emergence via +, x interactions in Open Whole Quali...

Here, Whole Qualifier space, W Q := { q k | k in N } is shown to be a tropical algebra under a max addition (q m ++ q n := q max{m,n}) and the qualifier multiplication. Thus, W Q is now closed under both xx and ++ and Qualifier arithmetic is simplified.

A model extending and generalizing the Goertzel model [1] of "magician" agent-elements and an octonion algebra is offered. Proposed is a HeartMind system, consisting of a set of "felt-ideas" under various "defined (mathematical) and undefined (magical) processes. Such HeartMinds form a space of HeartMind Points, a collection of which forms a collec...

In this day of abundance of goods and services, we seem to be suffering from an abundance ("inundance") of numbers, numbers, numbers. The sheer use of whole and real numbers to count and produce this abundance is appalling: quantities of many outputs, quantities of many inputs, prices, market percentages, and on and on to the other numbers used: mi...

The purpose of this F.E.D.* "Brief #6" is to extend the N-Cum (CN) and N Q co-discoveries by invoking an "origin" element (C 0 = q 0) for their spaces, to obtain the W-Cum (CW) and W Q spaces. Surprisingly, this origin qualifier is like both 0 (under + + + +) and 1 (under × × × ×) in the Whole Numbers! The inclusion of C 0 then becomes the basis fo...

About E.D. Brief #7: This article is meant to extend the W-Cum (CW) and WQ co-discoveries made in Briefs #5 and #6 to co-discoveries of Z-Cum (CZ) and ZQ. By answering, "Is C 1 + + + + (C 1)-1 = = = = C 0 ?", it is possible to construct two versions of Z-Cum space, one for which q n + + + + q (-n) = = = = q 0 (the F.E.D. ++ formulation) based on th...

This mathematician/human has sought a way to give mathematical structure to ideas, concepts, beliefs, etc., under a “more humane way of thinking/acting” (a better “logic”). Soon after discovering “A Dialectical Theory of Everything,” I recognized its depth and beauty, its comprehensiveness and applications, and its potential to assist in Mankind’s...

In any scientific, or thought-based, theory of Reality, the “science” and/or the “mathematics” behind such a theory may often seem devoid of any “human feelings.” After all it is these physical and emotionalfeelings which a major theory is ultimately addressing / redressing as it attempts to ameliorate the human condition. The role of “feelings” as...

When this student first began studying F.E.D.'s N Q qualifiers and their additions and multiplications, he soon asked himself the typical early-questions, as you also may have asked-questions such as: 1) Why is N Q := := := := {q 1 , q 2 , q 3 , … } in sequential correspondence with N := := := := {1, 2, 3, …}? 2) Why does q n + + + + q n = = = = q...

Dialectics is used to resolve various problems, given thesis and counter-thesis, a new thesis or synthesis results that surpasses or includes both others. qNumbers are a tool to explain and clarify this process. In our world where ‘making a difference’ may seem to be increasingly difficult, this ‘qBook’ shows how, using “Heartful Dialog”, we each/a...

Society (persons and cultural creations) is mapped onto its “math-essences”, and societal interactions are defined as binary operations + and x, whose closure on a base set P forms a finite commutative ring. Identity elements 0 nd 1 are viewed as reflections of Society’s “group identity”. The closure set, along with a proposed topology of finite “o...

Investigated are number systems based on the Cornu spiral, with corresponding numbers Λ 1 := {λ t } defined by a parametric function that defines the spiral: λ(t) = λ t. The spiral has the property that its length from the origin (t + u) can be treated as the product as the numbers of length t (λ t) and length u (λ u): λ t+u = λ t λ u. The < Λ 1 ,...

Investigated is a number system which contains a non-real, idempotent m, a "dual of 1": m n = m. Termed "M space" = <M x , *; " M + , +>, where M contains both a vector space M + , and a multiplicative space M x with a projection ": M x  M +. M consists of three key, disjoint component regions: M 8 ("figure 8") or "zero-modulus set, which separat...

Investigated are hypernumber systems which result as non-complex solution to the general parametric equation: z 2 = az + b, where a, b in {0, 1,-1}. Specifically discussed are complex i, counter-complex ϵ, Muses' w, p/q, m, Ω, as well as elliptical complements: ±u, and hyperbolic complements ±Φ 1 and ±Φ 0 to the real golden-means. Attention is also...

Using an "ortho-plus" binary operation ⦹, any diagonalized quadratic form over the reals can be transformed into all equivalent linear forms over a minimal space, C ax = R U iR. This space, when "folded" into {±x} and {±iy} set-pairs, is isomorphic to the real field. As this update draft makes clear, the circular and hyperbolic "sums", when lineari...

This book began with the desire to share some observations and analogies to human life with friends. In particular, to me, the Riemann surface for log(z), suggests many possible behaviors or ‘selves’ of each of us humans. Also, to me, Riemann’s mapping of the complex number plane onto the Riemann sphere (less its upper pole) suggests a possible...

In our world where ‘making a difference’ may seem to be increasingly difficult, this ‘qBook’ shows how, using “Heartful Dialog”, we each/all can create a quality thinking/acting which actually gives us ‘real solutions’ (‘resolutions’) rather than the ‘non-solutions’ offered by our current ‘all or nothing’ thinking.

While the Reals provide for closure of any Cauchy sequence, and for exact solution for sqrt(2), seldom is such precision needed outside mathematics. Here, we discuss that the precision provided by the Rationals is "Enough". Btw: See Comment. The "Rational Complexes" is probably a better term since it is the C basis {1, i} over Q: C^Q := {x(1)+y(i)...

The complex numbers, C, is used in quantum mechanics, but perhaps we need to go beyond C to represent quantum systems. A number system based on lattice is proposed.

Entry to FQXi essay contest (2012). Abstract: Fundamental questions in physics can be asked anytime, anywhere. Often they arise at the interface of physics, mathematics, and philosophy – where scrapping conversation turns into testable hypothesis. This essay explores the idea that the primitive act of counting "1, 2, 3 ..." makes an implicit assump...

Dual numbers, split-quaternions, split-octonions, and other number systems with nilpotent spaces have received sporadic yet persistent interest, beginning from their roots in the 19th century, to more recent attention in connection with supersymmetry in physics. In this paper, a number system in the 2D plane is investigated, where the squares of it...

Investigated is a number system in which the square of a basis number: (w)2, and the square of its additive inverse: (−w)2, are not equal. Termed W space, a vector space over the reals, this number system will be introduced by restating defining relations for complex space C, then changing a defining conjugacy relation from conj(z) + z = 0 in the c...

These stories reflect certain learnings. To the world, ‘zero’ is a symbol of “nothing physical”. But to me, ‘Zero’ is both an “Entry Point” and “Source of All”! Zero, the intersection of all dimensions, represents both a ‘finite point nothingness’, yet a ‘door to the infinite’. She (Zero) became my “door” to Inner-self and the greater understanding...

A non-standard research approach directed at the general nature of the principle of Multiplication, in mathematics and in all aspects of human life. Multiplication is seen is the energy from which addition is made, or is merely a reflection, and not the other way around, i.e. multiplication as a compacted form of addition. To promote a more convers...