Introduction

This is the first of three planned vol­umes in Arith­moí – A Study of Co­he­sive Math­e­mat­ics in Physics and Meta­physics. The orig­i­nal in­ten­tion of these pro­ceed­ings was to be writ­ten in the form of a few aca­d­e­m­ic papers, but I quick­ly re­al­ized there was much more to convey and de­cid­ed it was better suited to the book format for a more in-depth dis­course. The writ­ings are suited to the reader with a rea­son­able grasp of un­der­grad­u­ate math­e­mat­ics and or physics, and a back­ground in linear al­ge­bra, group theory, and topol­o­gy will be help­ful, but ef­forts are made through­out the book to ex­plain con­cepts with thor­ough­ness. This first volume is a math­e­mat­i­cal mono­graph that is in­tend­ed to be a syn­the­sis of many fields in math­e­mat­ics that brings a co­he­sive ap­proach to our study into the nature of being. What I have found, and others will echo this, is that a syn­the­sis of the math­e­mat­i­cal sub­jects is sorely miss­ing at the aca­d­e­m­ic grad­u­ate level. I hope that stu­dents and life-long learn­ers will see math­e­mat­ics in a new light and in­spire those who wish to un­der­take the jour­ney of self-ac­tu­al­iza­tion.

His­tor­i­cal­ly in both math­e­mat­ics and physics, there has been a ten­den­cy for an­thro­po­mor­phic biases re­lat­ed to how things are in­ter­pret­ed. Eu­clid­ean geom­e­try is a good case in point, where sci­ence and math­e­mat­ics were biased to how humans per­ceived space as flat; it was not until the early 19th cen­tu­ry that the work of Carl Friedrich Gauss and others re­vealed ex­ist­ing robust and non-Eu­clid­ean forms of geom­e­try. Sim­i­lar­ly, human his­to­ry has biased real num­bers over the com­plex, and de­spite the imag­i­nary number $i$ being note­worthi­ly present in the fun­da­men­tal equa­tions of quan­tum physics, the ten­den­cy to make all things real is om­nipresent.

Physics has been fo­cused on find­ing a uni­fy­ing theory of every­thing but con­tin­ues to run astray by the ob­sta­cle of making quan­tum field theory and gen­er­al rel­a­tiv­i­ty mean­ing­ful­ly co­ex­ist, and this dis­con­nect is not from the lack of con­cert­ed ef­forts. It begs the ques­tion of whether the uni­fi­ca­tion sought is at all pos­si­ble from an em­piri­cist ap­proach based on sen­so­ry ob­ser­va­tions. The last sev­er­al decades have seen an omi­nous re­treat of the phi­los­o­phy of math­e­mat­ics and physics that his­tor­i­cal­ly sought to find reason in the progress of the sci­ences. For your author, this re­treat has been at the ex­pense of mean­ing­ful for­ward progress, and to find ample foot­ing, means coming to terms with the re­al­iza­tion that sen­so­ry-based progress is quick­ly reach­ing its dead-end. Math­e­mat­i­cal rea­son­ing and its philo­soph­i­cal un­der­pin­nings might be just what is re­quired to get back on the path and ven­ture fur­ther onward. The goal is to find that which lies beyond what the most pow­er­ful and sen­si­tive sci­ence in­stru­ments can detect, while hold­ing to a high stan­dard of rea­soned and log­i­cal con­tem­pla­tion. This co­he­sive ap­proach must also embody the foun­da­tion to ex­plain the phys­i­cal man­i­fes­ta­tions that sur­round our sen­so­ri­um.

It is ev­i­dent that the mind is painstak­ing­ly miss­ing in much of modern-day em­piri­cism. The notion that some­thing is not real unless we can detect it by ex­per­i­ment is flawed, as it ig­nores the whole­ness of the math­e­mat­i­cal and meta­phys­i­cal frame­work, which is the essence of what we will study in Arith­moí. The mind is much more pow­er­ful than any giant par­ti­cle col­lid­er that physi­cists might build be­cause it tran­scends the phys­i­cal states of time and space that these em­piri­cist ap­pa­ra­tus­es are con­fined to.

Making all at­tempts to cast aside math­e­mat­i­cal bias is cen­tral to the ap­proach of this un­der­tak­ing, where em­brac­ing com­plex num­bers and non-Eu­clid­ean geome­tries are the tools uti­lized. We ul­ti­mate­ly find this to be a valu­able ap­proach to find­ing in­ter­con­nect­ed Lie group struc­tures that are both ex­tend­ed and nonex­tend­ed, de­riv­ing much of the Stan­dard Model matrix groups along the way and yield­ing a math­e­mat­i­cal rep­re­sen­ta­tion of a co­he­sive sin­gu­lar­i­ty that cannot be probed as modern physics ap­proach­es break down.

Arith­moí is a modern-day Pythagore­an ap­proach to study­ing what we can and cannot see around us and within us. We ul­ti­mate­ly find that the mind begets body, and the phys­i­cal uni­verse does not exist with­out the holo­graph­ic ex­ten­sion of the sin­gu­lar­i­ty. We arrive at these con­clu­sions with math­e­mat­i­cal and log­i­cal rigor, which is, for those who seek the im­pli­cate order, a much-needed de­par­ture from re­ly­ing on re­li­gions plagued with con­tra­dic­tion and false­hoods.

Apolo­gies to any math­e­mati­cian, physi­cist, or meta­physi­cian who have not been cred­it­ed for their work but should have been. I have made every effort to ensure that my col­leagues have not al­ready in­tro­duced new con­cepts pre­sent­ed in the book.

The choice was made to self-pub­lish this work to main­tain con­trol of the con­tent, to add new dis­course with ease, and to dis­trib­ute it to the reader out­side of the for-profit model.

The abil­i­ty to learn and gain knowl­edge is the best gift that has been be­stowed upon us. May you see math­e­mat­ics as the light!