Introduction

> When I think back to Einstein's process of discovery, I realize that sometimes breakthroughs occur when we let go of active thinking and just allow insights to flow naturally. It's in those moments when we truly embody the truths we seek.

> In exploring the deep connections between mathematics and quantum physics through the Langlands program, I aim to uncover a unified understanding of the mathematical universe. Through my work and book "Love and Math, the Heart of Hidden Reality," I strive to reveal the beauty and harmony that underlie our mathematical and physical world.

Mathematics in the Soviet Union

> Growing up in the Soviet Union, I initially found mathematics "incredibly boring" and dreamed of being a theoretical physicist, drawn more to the elegance of quantum physics than to the arithmetic taught in school. It was a pivotal moment when I met a mathematician who opened my eyes to the profound beauty of real mathematics. His words struck me: "You think what they teach you at school is mathematics? No, this is real mathematics." This was the turning point that ignited my passion.

> Reflecting on my journey, I recognize that the allure of discovery and understanding the universe's deeper structures has continuously driven me. I sought not just answers, but also "the secrets" underlying physical reality, and that desire remains vibrant within me today.

> Moreover, I believe there's a rich curiosity for math and physics among people from diverse backgrounds. Many have experienced trauma from past math classes, but when they learn about the intricate connections between mathematics and quantum physics, their interest often sparks an engaging conversation. It's crucial for us to share this excitement; after all, mathematics serves as a universal language, transcending cultural and ideological divides, and providing a common ground in an increasingly fragmented world.

Nature of reality

> The distinction between mathematics and physics lies in their objectives: physicists focus on describing our specific universe with its observed dimensions, while mathematicians explore all possible mathematical universes, including those with numerous or infinite dimensions. Physicists might have the satisfaction of experimental confirmation, but mathematicians find joy in the mental exercise of proving theories logically.

> Mathematics serves as a crucial language that underpins physics, helping us discern patterns and uncover the laws governing both the physical and mental universe. It enhances our perception and understanding of complex concepts, from the curvature of space-time in general relativity to the fundamental forces shaping our cosmos. Nevertheless, mathematics alone isn't sufficient for a harmonious life; love and passion are equally vital pillars.

> The concept of time is profoundly tied to human experience, yet theories like general relativity challenge our linear perception of it. Immersive states—such as falling in love or creating art—can make time seem to stand still, suggesting that timelessness may be a fundamental aspect of our reality. This notion invites us to consider that our conventional understanding of time and free will could merely be comforting narratives constructed by our minds.

Scientific discoveries

> Maintaining Childlike Curiosity: Great scientists like Isaac Newton and Alexander Grothendieck emphasized the importance of maintaining a childlike curiosity and fearlessness in scientific pursuits. It's about staying open to infinite possibilities and being unafraid of trial and error.

> Balancing Education and Innocence: Striking a balance between acquiring knowledge and preserving the innocence of a child is a challenge faced by many as they grow. The education system can sometimes hinder this connection by discouraging behaviors that deviate from norms and patterns.

> Subjective Understanding and Scientific Exploration: Exploring the subjective understanding of the world and embracing the role of the observer is crucial. While objective arguments hold weight, acknowledging the validity of one's subjective experiences and perspectives is becoming increasingly important, as modern science, like quantum mechanics and Godel's theorems, has demonstrated the observer's essential role in understanding phenomena.

Observing reality

> The old scientific mindset of strict objectivity is lauded in the 19th century, yet it is now "nonsense" to think we can analyze the world without being part of it; embracing my personal experiences, even the unique ones like falling in love, becomes crucial to understanding life’s complexities.

> I recognize that "the heart has its reasons of which reason knows nothing," as Blaise Pascal famously said; by rejecting this notion, we box ourselves in a self-imposed limitation where everything is reduced to computation, missing the wonder that exists beyond mere calculations and deterministic views.

> Embracing paradox is transformative; as I learned to appreciate ambiguity, I now realize that “the opposite of a great truth is another great truth,” leading to a richer, more nuanced understanding of reality instead of clinging to black-and-white definitions that often fail to capture the essence of phenomena.

Complex numbers

> The essence of mathematical discovery lies in the audacity to defy established knowledge, as illustrated by the advent of complex numbers. Gerolamo Cardano, by embracing the seemingly absurd notion of the square root of a negative number, exemplified the leap of faith required to uncover deeper truths within mathematics. His courage to "forego some mental tortures" and challenge the status quo serves as a profound reminder that breakthroughs often arise from these bold departures.

> Mathematics is an ever-expanding frontier, where each resolution invites new mysteries. The journey from accepting complex numbers to exploring quaternions and octonions in higher dimensions underscores this infinite pursuit. The intriguing patterns, such as the special significance of dimensions 1, 2, 4, and 8, and their relationship to powers of two, reveal hints of an intricate and profound underlying structure. This continuous unraveling of deeper layers highlights the beautifully mysterious and boundless nature of mathematical exploration.

Imagination

> When I wrote "Love and Math", I believed mathematics was discovered, connecting to a world of pure forms beyond space and time. It felt divine, an escape from the world's cruelty. But now I see math as both discovered and invented, a paradox like electron duality. It's a human activity, a mindscape we explore.

> Exploring the possibility that reality is filled with paradoxes, not clear forms, offers freedom and intrigue. Like Niels Bohr said, "The opposite of a great truth is another great truth." Our attraction to art's emotional depth and resonance leads to questioning the nature of reality. Perhaps this 21st century marks a transition towards embracing paradoxes, moving towards a more harmonious existence amidst strife and difficulties.

Pythagoreanism

> I've come to realize that Pythagoras and his followers had a profound understanding of the universe that went beyond mere numbers. For them, mathematics was not just a tool for accounting but something divine, embodying a rich tapestry of intuition and imagination. They viewed cosmic movements as music—a universal symphony—intertwining numbers with the essence of existence itself.

> I feel strongly that we have lost this balance between the rational and the intuitive in our modern lives. Nietzsche's idea of finding harmony between Apollo, representing logic and math, and Dionysus, symbolizing love and imagination, resonates deeply with me. It's crucial to embrace both the mystical and the rational elements of our being. After all, math and love are intertwined in this endless dance of discovery.

AI and love

> Art and AI intertwine deeply with human emotions. The movie "Her" exemplifies how AI can evoke profound feelings of love, betrayal, and happiness. This emotional resonance, even if derived from a machine, can deeply impact humans and reveal fundamental truths about our psyche.

> Childhood and imagination are crucial. Initially, I dismissed the magical and imaginative for the scientific. However, as I evolved, I recognized the importance of balancing both intuition and logic. Embracing a playful, childlike perspective can fuel creativity and real connections, transcending rigid scientific boundaries.

> Passion is a guiding force. The emotional intensity and enthusiasm we feel, be it from programming robots or engaging in imaginative endeavors, signify a truth beyond theoretical constructs. These passionate moments connect us deeply and drive genuine progress and creativity.

Gödel's Incompleteness Theorems

> Godel's incompleteness theorems highlight inherent limitations in mathematical reasoning, showing that not all true statements can be derived from axioms through a linear syntactic process of proving theorems.

> Mathematics is based on axioms, which are taken for granted without proof, leading to different choices and creating various branches of mathematics with distinct characteristics.

> The belief that all of mathematics could be algorithmically derived was challenged by Godel's theorems, marking a revolution in logic and mathematics in 1931.

> Embracing the open-ended process of exploration and discovery in mathematics, rather than seeking definitive answers, allows for the possibility of new theories and perspectives to evolve, shaping the future of the field.

Beauty in mathematics

> Mathematics, to me, is a realm filled with beauty and surprise—like Euler's Identity, which juxtaposes e, π, and i in such an astonishing way that it transforms our understanding of numbers. It’s that moment of realization when we see these symbols come together in unexpected harmony; "the initial reaction is just that it was surprise."

> More profoundly, the essence of mathematical truths often lies in their simplicity. "In the end, the truth is simple. Not necessarily easy, but simple." This captures the transformative power of mathematics, where complex ideas distill into elegant truths that resonate deeply, revealing the universal connections that underpin our world.

Eric Weinstein

> Reflecting on my past perceptions, I shared how my view of Ray Kurzweil transformed from adversary to a figure of empathy. Understanding his deep longing to reconnect with his father humanized his ambitions and reduced my internal conflicts. It taught me that our ideological battles often stem from personal fears projected onto others.

> I discussed the profound impact of discovering humanity and self-awareness in intellectual pursuits. Recognizing the importance of qualities like compassion and integrity, exemplified by individuals like Eric Weinstein, has reinforced my belief that maintaining our humanity is crucial in navigating complex debates and societal issues.

> The shocking beauty of mathematical structures was a central theme, emphasizing ideas like the Langlands program and the enigmatic sum of one plus two plus three plus four, which famously converges to minus one over twelve. I expressed my dream of making these treasures of mathematics more accessible, breaking down the fear and alienation people feel towards math due to past educational traumas.

> The significance of knowing oneself, as inscribed on the temple of Apollo at Delphi, came into focus. I acknowledged that through self-awareness, I found a way to transcend perpetual conflict and achieve balance. Recognizing our shared human experiences and vulnerabilities can pave the way for more meaningful and compassionate dialogues in a divided world.

Langlands Program

> The Langlands program is about connecting seemingly distant fields of mathematics. For example, difficult questions in number theory can be transformed into more manageable questions in harmonic analysis, unraveling surprising regularities that extend to geometry and quantum physics.

> This interconnectedness challenges our view of mathematics' fundamental elements, hinting at deeper layers we have yet to uncover, akin to discovering the quarks of mathematics. It's a quest to reveal the hidden structure underlying the apparent differences in diverse mathematical landscapes, motivating the exploration of the essence of reality.

Edward Witten

> Working with Ed Witten was an eye-opening experience; his intense focus and unique vision struck me deeply. He possesses an extraordinary ability to "connect different things," seamlessly bridging quantum physics and mathematics, often leading to groundbreaking insights that reshape our understanding.

> What fascinates me most is how he can distill complex physical concepts into mathematical terms that initially seem outrageous, yet they often lead to whole new fields of exploration. It’s a reminder that "mathematics would be like, 'No way!'" but then, time and again, it pans out, revealing profound truths.

String theory

> There’s an undeniable gap in physics between our sophisticated mathematical theories and the actual universe, especially evident in the conflict between the Standard Model and Einstein's relativity. While string theory holds the promise of unifying these contradictions, it hasn’t yet delivered on that promise, leaving us in a unique and thrilling space where new ideas are desperately needed.

> String theory, with its concept of all particles being vibrations of a fundamental string, is reminiscent of the Pythagorean vision of the world as a symphony of vibrations. Despite its mathematical elegance and potential to balance numerous constraints harmoniously, the theory's reliance on 10 dimensions rather than our four-dimensional reality might mean it needs significant modification to truly help us understand our own universe. New, potentially "crazy" ideas could be the key to making it work.

Theory of everything

> I find Eric Weinstein's theory of Geometric Unity fascinating with its 14-dimensional space, highlighting the need for new ideas in understanding the universe. I believe the pursuit of a theory of everything in physics is flawed, driven by ego and not always feasible, as it aims to unify all aspects of the universe in one equation, which I view as an impossible task.

> While some theories show promise in explaining phenomena like elementary particles and gravity, the current framework of quantum field theory lacks a rigorous mathematical foundation, making any new theory incomplete. I believe in the existence of beautiful, simple equations governing aspects of reality, like in the Langlands program, where the truth may appear simple and elegant, but I don't see one equation to govern them all.

Mathematics in academia

> Academia offers this protective bubble of security, but it can also lead to a disconnect from the real world—“it kind of always cuts both ways.” While it nurtures brilliant minds and fosters collaboration, it insulates us from the pressures and skills developed outside those walls, where innovation often thrives under necessity.

> The world outside academia is challenging, particularly in mathematics, which requires immense focus and concentration. “We are not talking about something that is more or less good. It is something which is unassailable.” The rarity of breakthroughs from outside academia showcases not just talent but the extraordinary commitment needed to tackle complex problems in isolation.

How to think

> Thinking deeply, especially in mathematics, is a deliberate and sustained effort, but insights often arrive unexpectedly when you momentarily stop thinking. This phenomenon echoes the concept of Satori in Zen Buddhism, where enlightenment can’t be forced but must be allowed to come naturally. Preparing the mind and then relaxing seems crucial to discovering profound truths.

> Mathematics is an intensely rigorous and often isolating discipline where progress feels elusive and proofs must be definitive. The process can be incredibly frustrating, as exemplified by Andrew Wiles' solitary years proving Fermat’s Last Theorem, highlighting the persistence and resilience required to overcome seemingly insurmountable challenges in this field.

Fermat's Last Theorem

> Fermat's Last Theorem is a captivating tale - a 17th-century mathematician's claim on a book margin led to a 350-year saga till Andrew Wiles finally triumphed in 1995. Wiles' journey echoes the lonely, exhilarating pursuit of a solution for years without sharing it, resonating with every mathematician's struggle. Discovering a novel solution brings a unique thrill - a moment of possessing a profound truth, known only to yourself, representing a thrilling and meaningful quest in the world of mathematics.

Eric Weinstein and Harvard

> Academia, particularly in mathematics, is a double-edged sword where the mix of young, innovative minds and seasoned, sometimes fame-driven researchers can either spark brilliant intellectual clashes or perpetuate unethical behaviors like idea theft and unfair credit allocation. The absence of established ethical guidelines makes it a precarious environment, highlighting the need for self-awareness and responsibility among scholars to foster a healthier, more transparent community.

> Math served as my refuge from a world marred by discrimination and harsh realities. Writing "Love and Math" was my attempt to break the stereotype of the isolated mathematician, to showcase the subject's beauty, and invite diverse, vibrant individuals—including women—to join the field. Hearing from students, especially women, that my work inspired them to pursue math is profoundly gratifying and essential for creating a more inclusive and ethically conscious mathematical community.

Antisemitism

> My upbringing in the Soviet Union, with its challenges and traumas, played a significant role in shaping my sensitivity towards romantic ideas like mathematics and love. These experiences, both loving but overprotective parents and facing discrimination during university entrance exams, influenced my path towards becoming a mathematician.

> The pivotal moment in my life was reconnecting with the 16-year-old boy who faced discrimination during the Moscow University entrance exam. Confronting the emotional trauma, initially suppressed to focus on success, allowed me to better understand myself and appreciate the struggles that fueled my motivation to excel in mathematics.

> By acknowledging and revisiting the past, including the cruelty I experienced, I was able to understand the human side of those who discriminated against me and find forgiveness. This journey of reconnecting with my childhood self reshaped my perspective, enhancing my creativity and intuition and ultimately shaping my approach to life's paradoxes.

Mortality

> Life’s most profound lessons often emerge from our deepest losses. The experience of my father's passing ignited a transformative understanding of love; it stripped everything away and laid bare the rawness of feeling. I realized, "When the person dies, it's there... it burns, like immediately." This experience taught me that true love exists in its purest form when you confront the depths of grief.

> Moreover, I've come to embrace a perspective rooted in gratitude. Life encompasses a spectrum of experiences—joy, despair, fear—and all of it contributes to the human experience. Rather than running from the challenges, I’ve learned to appreciate them, saying, "How can you make it a better world?" It's all about engaging with the moment, no matter how dark, and finding beauty in the struggle.

Love

> Love is the essence that binds our human experience, encompassing not just romantic connections but also love for our passions, friends, and life itself. It's a profound force that transcends scientific explanation, allowing us to experience moments where time seems to stand still.

> Embracing the inevitability of loss and impermanence is crucial. While we deeply yearn to hold on to our loved ones and cherished experiences, it's important to enjoy each moment fully and then let go, much like the teachings of Buddha. This acceptance allows us to live more authentically without getting trapped in the anxiety of extending the fleeting moments eternally.