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- Herausgeber: Phoenix Publication Press
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The Bankruptcy of Mathematics
It is a book born from the fires of doubt, from the vast cracks of logic, and the fierce battle between certainty and uncertainty.
This work is a call to awakening for those who dare to rise from the heavy slumber of unquestioned beliefs and to look straight into the eyes of the false perfection we once accepted without question.
Have you ever wondered whether the glorious foundation of mathematics, that seemingly sacred and unshakable science, might itself stand precariously on a trembling edge?
This book invites you on a journey with no return, a voyage into the darkest corridors of logic, the collapse of propositions, and the crumbling pillars of certainty.
The Bankruptcy of Mathematics is not a book for mere entertainment, nor is it a tale for the comfort-seekers.
It is a siren of warning, a call to an unrelenting battle against blind faith in systems we once believed were eternally flawless.
In this work, the author, through simple yet powerful language, weaves an epic narrative of the struggle to uncover truth, a truth hidden behind the mask of formulas and proofs.
This book places a torch in your hand to enter the unknown lands where great mathematicians such as Gödel, Russell, and Cantor lost themselves in labyrinths of contradictions.
You will walk alongside the silent agonies of these thinkers, their hopes, their defeats, and the questions that forever changed the world.
The Bankruptcy of Mathematics is not just a book.
It is a flame that burns away the old certainties about mathematics and a spark that ignites your mind into deep and daring thought.
If deep within your soul, a smoldering fire still seeks the truth,
If you know that knowledge without questioning is captivity,
If you dare to cross beyond your safest beliefs and to fly toward the endless skies of doubt, then this book was written for you.
This book will be your weapon in the relentless battle between awareness and ignorance.
It will be your companion through endless nights as you pursue the distant light of understanding.
You need this journey.
The world we have built trembles at the border between doubt and certainty.
Be brave. Take the first step.
The Bankruptcy of Mathematics
A call for the resurrection of the mind.
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Veröffentlichungsjahr: 2025
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The Bankruptcy of Mathematics
The Hidden Truths; Never Told Us
Ajurdan Xax
Copyright © 2025
All rights reserved.
ISBN: 978-1-0695025-0-6
DEDICATED TO
Terence Tao
“The Mozart of Mathematics”
Table of Contents
Book Catalogue
Foreword
Chapter One
The Perfect World of Mathematics Began to Crack
From Mathematics as the Language of the Universe to Doubt at Its Core
An Earthquake Named Gödel
The End of Certainty, The Beginning of an Adventure
The Tale of a Cat Both Dead and Alive
Chapter Two
Foundations of Mathematics
Questions That Were Never Answered
The Opening Riddle: Can One Build a House on Quicksand?
Wounds Upon Wounds: From Number Theory to Abstract Algebra
Applied Mathematics; or the Illusion of Control?
The Connection Between Self-Awareness, Mathematical Logic, and the Philosophy of Mathematics
Chapter Three
Symbolic Logic
Does Man Still Understand What He Writes?
Logic Without Logic: The Tale of Paradoxes
Philosophical Pain: When Logic Forgets “Awareness”
Chapter Four
Russell’s Paradox
The First Wound on the Body of Mathematics
Bertrand Russell
“Mathematics used to seem to me a paradise of certainty. But it has turned out to be a field shaken to its foundations by paradoxes.”
The Story of the Barber, When Mathematics Could Not Fix Itself
Russell’s Paradox.
A Devastating Blow to Set Theory and the Foundations of Mathematics
How Does This Thought Experiment Undermine Traditional Mathematics?
The Fundamental Question
Possible Scenarios for Paradox Emergence in Algebra
A Famous Example of Logical Conflict in Naive Algebra
Chapter Five
The Rise of the Star Called Fuzzy Logic
Where Certainty Failed
Fuzzy Logic in Artificial Intelligence
Mathematics: A Set of Propositions or a Garden of Possibilities?
Gray Geometry
Is Fuzzy Logic a Defeat of Logic?
Chapter Six
The Uncertainty Principle
A Blow to Certainty with the Sword of Indetermancy
Where Physics Reaches a Borderline, Mathematics Falls Off the Edge
A Wormhole in the Heart of Geometry
Algebra and Group Theory in a Double Dead End
Chapter Seven
Gödel’s Blow
The End of the Perfect Dream of Mathematics
Hilbert’s Formalist Machine and Collapse from Within
Gödel’s Second Theorem: A System Cannot Guarantee Itself
A Concrete Example: Turing Machines and the Halting Problem
Gödel’s Shadow over Algebra
Chapter Eight
Axiomatic Mathematics
A Rebellion of Thought within the Prison of Principles
What Is an Axiom?
Mathematics Is Built on Assumptions, Not Certainties
Where Paradoxes Outnumber Answers
Chapter Nine
The Philosophy of Mathematics – What Is Math? Invention or Discovery?
Imre Lakatos
A 3 A.M. Conversation with Myself
Chapter Ten
The Five Fault Lines , Undefined, Unacceptable, Paradox, Contradiction, Ambiguity
The Beginning of a Crisis
Unacceptable – When Reason Refuses
Ambiguity – On the Foggy Border of Meaning
Chapter Eleven
The Geometries Euclid Never Knew
The Bankruptcy of Mathematics in the Labyrinth of Curved Spaces
One Differential Equation, a Thousand Paths in a Single Space
Chapter Twelve
If the Caravan Leader of Geometry Had known About My Intuitive Theorem!
Relation of the Trough Theorem with Euclidean and Geodesic Geometry
The Footprints of Two Giants, Riemann and Lobachevsky
The True Culprit: Euclid’s Fifth Postulate
The Crisis of Group Theory in Different Spaces
Affine, Algebraic, Differential, and Topological Geometries Where Did the Story Begin?
An Example from Group Theory and Algebraic Geometry
Differential Geometry: Enchanting Curvatures and Catastrophic Errors
Topology: Infinite Freedom or Loss of Meaning?
A Hybrid Example from Geometry and Graph Theory
Chapter Thirteen
A Trace of Infinity in the Mirror of Finitude
The Endless Returns
Fractal: A shape that repeats itself
The Non-Integer Dimension
Entering the World of Irrational Dimensions!
Definition of Dimension in Fractal Geometry
Formula for Calculating Fractal Dimension (Dim):
Chapter Fourteen
Calculus
The Starting Point: Infatuation or Aversion?
Derivatives Without Meaning
Divergent Series: False Dreams of Infinity
From Linear Algebra to Number Theory: Doubting the Core Tool
At this point, Ramanujan arrives on the scene; The Magician of Numbers
The Casimir Effect: When Physics Takes -1/12 Seriously
The Bankruptcy of Integrals , A Trilogy
A Tale from Harmonic Analysis
Part Two of the Integral Bankruptcy Trilogy
A Paradox in Coordinate Transformation
Part Three of the Integral Bankruptcy Trilogy
The First Tremors: The Butterfly That Made a Storm
Partial Derivatives: The Exhausted Heroes
The Equation and Predictability
Chapter Fifteen
Counting Meaning, or the Meaning of Counting?
The Tale of the Man Who Counted Without Knowing Why
Title: Zero Factorial – When Nothing Becomes One
Chapter Sixteen
Abstract Algebra
Premature Abstraction
Two Groups with Identical Representations Are Not Necessarily Isomorphic
The 37% Rule
Intuition
Chapter Seventeen
The Achilles’ Heel of Mathematics
The Invisible Monster
How Infinity Devoured Mathematics
Geometry: Infinity in Space and Lines
Number Theory: Silent but Explosive Infinities
Infinity...
Infinity in Physics: The Forbidden That Slips Through
Infinity in Quantum Mechanics and Field Theory
Chapter Eighteen
When Continuity Gives Way to Discreteness
An Attack on Analysis, an Attack on Geometry, an Attack on the Principle of Continuity
Geometry: When Space Trembles
The Principle of Continuity: A Mathematical Fortress or a Mental Mirage?
Paradoxes: Quiet Pits in the Sidewalks of Mathematics
Discreteness: The Future of Math, or a Return to Reality?
Chapter Nineteen
Collapse or Renaissance?
Proposals for a More Human, Humble, and Polysemous Mathematics
Redefining Mathematics in a Quantum and Bio-Centric World
Polysemous Mathematics: A World Between Zero and One
Postscript
Book Catalogue
Title:The Bankruptcy of Mathematics
Author: Ajurdan Xax
Publisher:Phoenix Publication Press
ISBN: 978-1-0695025-0-6
Publication Date:May 2025
Number of Pages: 188
Subject:Mathematics – Philosophy – Logic
Foreword
A Personal Story of a Math Lover Whose Inner World Once Collapsed
Years ago, in my teenage days, mathematics was more than just formulas and problems. In those passionate high school years, math was like a secret language to me—a language through which the world whispered its truths. Every time I solved an equation, I felt as though I had uncovered a piece of the mystery of existence. It was as if God, before creation, had measured the universe with derivatives and integrals and then engineered it with geometry. But somewhere along the path, something cracked. Not a crack from confusion, but from heartbreak and longing.
When Math Was Still a Love
The first time I learned the concept of “limits,” I felt like I was standing at the edge of awareness. Infinity was no longer a number—it became a mental experience. Then I discovered derivatives. I understood how to extract instantaneous speed from average speed.
I thought I had reached the ultimate insight. When I encountered combinatorics, I realized even a simple count of seating arrangements around a round table was a universe full of subtleties. Graph theory showed me that sometimes the world is just dots and lines—and maybe the human heart is just a node in a much larger network. Up until this point, all was well. But the trouble began when mathematics not only asked questions—it began to question itself.
The Crisis Began When Math Asked: “Who Am I?”
My encounter with symbolic logic and the philosophy of mathematics was like stepping into a pitch-black night. Gödel taught me that “in any consistent mathematical system, there are statements that can neither be proven nor disproven.” I was stunned.
What did that even mean? I turned to one of the most distinguished professors of mathematics, but his explanation using the Goldbach conjecture only added to my confusion. This was around the 1980s. My mental turbulence had begun. The grand edifice of mathematics I had built in my mind was constructed on swampy ground. Set theory, long considered the foundation of mathematics, itself was riddled with paradoxes. Russell’s paradox—the famous set of all sets that do not contain themselves—shook me to the core.
I began to wonder: Are we building upon mere imagination and assumptions? Is math just a linguistic game? Are proofs more like poetry than truth? Does ∀x ∈ ℝ mean anything real—or is it, as Wittgenstein would say, just another language game?
When Geometry Shattered My Sense of Identity
I had grown up with Euclidean geometry. A straight line connecting two points is unique. That principle was the backbone of my belief in an orderly world.
Then came a book on non-Euclidean geometry. I discovered that in some worlds, from a single point outside a line, infinitely many parallel lines can pass. And these worlds weren’t just theories—they found real application in general relativity. As Rumi might say, they formed the geometry of the cosmos.
The world was no longer simple. And neither was I. Each new advanced chapter brought me both joy and confusion. Fractal geometry and multivariable calculus taught me how the world, in higher dimensions, is more complex—and more beautiful, much like what modern string theory suggests.
But was that beauty worth all this disorientation?
Abstract Algebra and Group Theory: Harmony or Imprisonment?
One summer, I learned group theory from a few pages of a high school math book. Group theory offered a strange kind of order. Sets that, under specific operations, adopted unique structures. Soon after, I discovered a group in which each element was its own inverse!
The deeper I dug, the harder it became to comprehend.
How could it be? How do you reconcile fire and water? How can Hegel’s thesis and antithesis be merged into one? On the advice of a mathematician friend, I corresponded with one of the leading experts on group theory. I described my work to him. The response was disheartening: “The groups you claim to have discovered are nothing but isomorphic to the well-known Klein four-group; Just approached through a different method.”
I was stunned. I decided to abandon mathematics forever. But gradually, stranger questions arose in my mind: Why are some groups abelian and others not? Why are some fields commutative and others not? And most importantly, do these structures reflect the real world? Did we discover them, or invent them?
On a Rainy Night, New Questions Emerged
One rainy evening, while reading a book by a philosopher of mathematics, new questions surfaced in my mind:
“If we created math, why does it work so well? And if we discovered it, why is it so full of contradictions? Does math represent truth—or is it merely a tool? Or perhaps just one of the many stories the human mind invents to give meaning to meaninglessness?”
“Students don’t hate mathematics. What they hate is being lost in a fog of meaningless symbols.”
Kent E. Everett
The Day I Fell Out of Love with Math
It began with something small. I was solving a combinatorics problem. My answer was wrong—again. Not because I didn’t know the method, but because the problem was inhumanly deceptive.
I felt like math was no longer speaking to me—it was toying with me. Right there, I put my pen down and said, “That’s enough. I’m tired.” I didn’t abandon math altogether, no. But the passion was gone. I could no longer blindly trust mathematics.
When I saw that even the most precise theories housed uncertainty, probability, paradox, and relativity, I asked myself: What’s the point of all this effort, only to be told, “You might never truly understand”?
Returning to Math; But This Time with Open Eyes and Caution
Years later, from a distance, I returned to mathematics. Not out of childish excitement—but with a deeper curiosity.
I came to understand that math is neither absolutely trustworthy nor absolutely misleading. It’s like a mirror: If you are ready, you will see yourself in it. If you’re confused, you’ll only see crooked lines. Now, through this book, I invite you to walk beside me.
A journey through the darkest corridors of logic, across many dead ends, and into doubts that might just lead to clarity.
Let us step together into a world of mathematics—not one that comforts us, but one that enlightens us. A world where sometimes you must accept not knowing—and yet still move forward.
“If you can’t solve a problem, solve a different one you can; In other words, If you can’t be with the one you love, then love the one you’re with.”
George Pólya, the great Hungarian mathematician
This book is designed so that its topics can be explored independently. Rather than reading it cover to cover, you may dip intothe sections most relevant to you at any given time.
