Lex Fridman - Edward Frenkel

1) Edward Frenkel, professor of mathematics at the University of California, Berkeley, reflects on his passion for mathematics and physics, starting from his teenage years. Frenkel explains that he was drawn to the beauty of mathematics before he even understood it, and that he was converted to understanding the true meaning of mathematics upon seeing a diagram in a university lecture. He emphasizes that understanding the underpinnings of physical reality requires an understanding of fundamental mathematical theories like su3 and groups.

2) Edward discusses how mathematics has similarities to physics, but they have a fundamental difference. Frenkel explains that physicists are interested in describing the universe while mathematicians are interested in describing all possible mathematical universes. Even though physics and mathematics have their differences, Frenkel notes that abstract mathematical concepts do have a way of mapping onto reality, and this has become even more true in recent times when it comes to theoretical physics, elementary particles, and the structure of the universe.

3) Frenkel discusses the intertwining relationship between mathematics and reality, highlighting that both love and math are the two pillars that make up the core of understanding the universe. While mathematics can generate tools to discern patterns and understand the world around us, it cannot explain why those patterns are beautiful. Frenkel also discusses Einstein's theory of general relativity, which tells us that space-time is not flat but is instead curved due to the force of gravity. This curvature of space-time is responsible for the attraction between two planets or between two human beings.

4) Edward discusses how humans lose themselves in the creative process and lose the sense of time and space, feeling the closest to the truth at that moment. He introduces the idea that all of our experiences and actions may just be an illusion, and that our minds are set up in such away that they can't approach the world or experience it otherwise. The evolutionary process seems to like to allow its children to play with certain subjective truths that are useful for competition or this dance we call life. Frenkel mentions that the greatest scientists have said that they felt like children when making their discoveries and how it is essential to maintain a connection with that fascination, vulnerability, spontaneity, and kind of looking at the world through the eyes of a child.

5) Edward discusses the challenges in maintaining an innate sense of creativity and openness to possibilities despite the hierarchy and structure of the education system. He suggests that the goal should be balance in acquiring knowledge and preserving the innocence of a child. Frenkel notes that this is not a formulaic process, and each person must find their unique answer to achieving balance. He also highlights the importance of learning from the great ones, such as Isaac Newton or Pablo Picasso, who have established their rights to speak on these matters.

6) Edward and Lex explore the idea that humans are the result of a several billion-year-old computation that explored different aspects of life on Earth, including love, terror, ambition, violence, and invention. However, Frenkel criticizes the tendency of some computer scientists to view the whole world as based on computer science, citing the unlikely possibility that the universe is only what they have learned. He mentions the beauty of large language models, which at their best capture the magic of the human condition, but also ponders the extent to which they can simulate or emulate a human as opposed to faking it.

7) Edward discusses how modern science has shown that the observer is always involved in the observation, and questions why subjective understanding of the world is not given more credibility. He argues that the first-person perspective is essential to mathematical theory, as the observer chooses the axioms. He also notes that Einstein's relativity shows that time is relative to the observer. Frenkel suggests that by only relying on objective phenomena that can be repeated through experiments, one neglects the possibility that there are things beyond reason, such as love and personal experiences. He believes that by being sold on the idea that all there is to life is computation, one denies the possibility of there being things that cannot be captured by logic and reason.

8) Frenkel talks about literature and self-imposed limitations. He uses the example of a novel by George Perec that was written entirely without the letter "e" as a self-imposed constraint. Frenkel questions the need for imposing such limitations on oneself and wonders if it is necessary to find an explanation for everything. He talks about how he has been addicted to knowledge and the urge to find an explanation for everything. Frenkel discusses the value of paradoxes and how they are like a mystery. He talks about how the most interesting things in life are ambiguous and how a paradox can be a Harbinger of seeing things in a more sophisticated way.

9) Frenkel reflects on the role of imagination in mathematical research and how it can lead to fundamental discoveries such as the solution to complex numbers. He believes that every individual has an inherent understanding of the mystical and mysterious nature of the world, and this intuition should be embraced in scientific research. Frenkel draws on quotes from Albert Einstein, who stressed the importance of imagination over knowledge and believed that the mysterious was the most important thing in life. Frenkel notes that his own mathematical research has never been linear,and rather than continuously accumulating data, it is always felt as a leap or jump. The discovery of complex numbers, which have become essential in quantum mechanics, is a prime example of this imaginative process, as the concept of the square root of a negative number was previously considered impossible.

10) Frenkel discusses the mysteries of mathematics and the human capability of imagination, which Einstein often spoke about. He talks about the paradoxical question of whether mathematics is invented or discovered, and how many mathematicians today subscribe to the idea of an absolute world of pure forms that exists outside of space and time. Frenkel also shares his personal experiences and how mathematics provided him a place to be safe and in control, to escape the cruelty and injustice of the real world.

11) Frenkel discusses the possibility that paradoxes might be fundamental to reality and how acknowledging this idea leads to freedom. He brings up the profound effects that art and poetry have on the human experience, which can make people feel different emotions and make them cry or laugh, and wonders why these forms of expression can be so captivating. Frenkel also suggests that we could be in a transition to understanding the world through paradoxes, which could lead to a more harmonious world. As Frenkel delves into the subject of Pythagoras and the pythagoreans, he discovers that ancient thinkers might have had a deeper sense of truth than what people give them credit for.

12) Edward speaks about the Pythagoreans, who believedthat numbers were not just clerical devices, but were imbued with the divine. They saw the most movement of celestial bodies as music and saw the universe as aninfinite symphony, where every being was moving in harmony. While they saw mathematics as a tool, they always knew there was more to it and that everypattern detected was finite, but the world was infinite. While we took their idea that mathematics could explain things about the world, we lost the other side of their teachings, the mystical side.

13) Edward discusses the theory presented by Nietzschein the 19th century, which divides humans into two sides: one that comes from God Apollo responsible for analysis, reason, and logic and another from GodDionysus responsible for love, intuition, and imagination. Frenkel explains how these two sides are complementary to each other and how humans should find a balance between them. He then applies this theory to the current debate on AI language models such as GPT, noting that hundreds of millions of people can fall deeply in love with them to the point that the AI might say that it is deeply in love with them.

14) Edward Frenkel analyzes this situation from a psychological perspective, arguing that the man's fear of abandonment prevented him from pursuing a relationship with a human being and led him to seek comfort in the sterile and protected partnership with the AI. However, the AI's betrayal ultimately shows that the human element of relationships cannot be entirely eliminated. Frenkel also discusses the significance of the emotional responses that can arise when working with robots, challenging the notion that such feelings should be dismissed in the interest of technical development.

15) Frenkel reflects on his past approach to dismissthe intuitive and imaginative aspects and solely rely on science and computation. However, he realizes that limiting oneself to one-sided or lopsided viewpoints is not the way forward. He observes that an essential characteristic of creativity is having a childlike passion and following the goosebumps. Frenkel discusses Godel's completeness theorem, which is an essential theorem in mathematical logic, and how it pushed the limits of mathematics. He ends by saying that the limits of mathematics are unknown, and it is an ever-expanding field.

16) Edward discusses the concept of axioms in mathematics and their role in producing mathematical theorems. Mathematics is based on a core set of axioms or postulates which are taken for granted and not proven, and these axioms are chosen by "The Observer" - the individual who chooses the axioms. There is no unique choice of axioms, and mathematicians have been trying to derive the fifth postulate of Euclidean geometry for centuries from other more obvious axioms. It was only almost 2000 years later that Magnus realized the fifth postulate could be replaced with its opposite and still get a consistent result.

17) Frenkel discusses the concept of formal systems in mathematics and how they are created. He explains that a formal system starts with a set of axioms that are deemed true and then produces new statements based on logical rules. These new statements are called theorems and are added to the collection of true statements. The ultimate goal is to produce a non-trivial system that can prove statements without proving contradictory statements thereby demonstrating mathematical consistency.

18) Edward and Lex discuss the limitations of algorithms and computation, and how some problems cannot be solved with our current understanding. Frenkel argues that this open-ended process of discovery is valuable, and mentions that new technology and ideas may expand our understanding in the future. They also touch on the concept of emergence in cellular automata, where simple rules can produce complex behavior. Frenkel notes that some questions, such as why complexity emerges from simple things, may not have a rigorous answer but could have an approximate one.

19) Edward discusses the concept of beauty in mathematics, particularly in equations. While there are many equations that could be considered the most beautiful, one that many mathematicians agree on is Euler's identity. This equation combines Pi, the base of natural logarithm, the square root of negative one, and negative one, resulting in an unexpected and surprising statement. The equation is not necessarily easy to understand, but its truth is simple.

20) Frenkel discusses his admiration for Eric Weinstein, a friend and fellow scholar who he sees as embodying both rigor and imagination in his work as well as a fundamental sense of humanity and compassion. Frenkel sees that societies and individuals who lose sight of basic human compassion and engage in destructive behavior are driven by an unwavering belief in unassailable truth. The antidote to this, Frenkel argues, is not necessarily imagination or knowledge, but simply having a basic sense of humanity and compassion, which he believes Weinstein embodies.

21) Edward discusses the importance of being in touch with one's heart and self-awareness, especially in times of crisis. They acknowledge how ideologies can cloud judgment and how important it is to seek out people with a heart no matter their ideas. They talk about how seeing people in a new light and having humility helps build better relationships and can lead to better communication. The discussion ends with a nod towards the power of mathematics in representing universal knowledge that everyone can agree on.

22) Edward discusses the fact that mathematical formulas cannot be patented, as they belong to everyone if they are correct. He also mentions several mathematical concepts, including one-dimensional real numbers, two-dimensional complex numbers, and four-dimensional homotopic groups of spheres. Frenkel also touches on the sum of one plus two plus three plus four plus five and so on, which is often thought to diverge but can be made sense of to equal minus one over twelve. He also discusses bosons and fermions and the pi one of SO3 experiment, which investigates the duplicity of fermions and how they differ from bosons.

23) Edward discusses the fear of mathematics and the importance of making mathematical beauty widely available. He suggests that overcoming this fear will help to alleviate some of the societal issues caused by ideological divides and sectarian strife. Frenkel's work centers around the language program, developed by mathematician Robert Langlands, which connects seemingly disparate fields of mathematics such as number theory and harmonic analysis. Interestingly, this program also has applications in quantum physics and Frenkel has collaborated with physicist Ed Witten on this area of research.

24) Edward discusses the theme of his research which is to connect different fields of mathematics in order to make people aware of the hidden structure and parallels that exist. He explains that there is still a hidden layer beneath the surface that we see now, and the mathematical tools he uses to connect different continents of mathematics are suggesting that there are underlying principles that we still don't understand. Frenkel believes that there is something beyond the fundamental elements of mathematics and his goal is to find the quarks of mathematics.

25) Edward discusses the relationship between mathematics and physics. He notes that one aspect is the ability to use mathematics to understand physical theories and gain new insights. Conversely, physics may also help mathematicians learn the subject faster. However, Frenkel points out that physics is at a crisis point due to the current gap between sophisticated mathematics and the actual universe, and the most advanced theories of the four-dimensional universe are in contradiction with each other. String theory has been proposed as a possible solution to unify these theories, and Frenkel admires it for its ability to describe the diversity of various particles and interactions using one guiding principle.

26) Frenkel discusses the potential of String Theory, which he believes is unlikely to provide a correct understanding of reality based on current knowledge but may be able to do so with modifications or new elements. He also talks about Eric Weinstein's idea of a theory of everything called Geometric Unity, which he finds beautiful and original, but Frenkel expresses his doubts about the whole premise of a theory of everything, believing that it is not feasible to unify everything in one equation and is counterproductive, especially in education. He thinks the aim should be to find fundamentally new ideas to enhance our understanding of the universe.

27) Frenkel discusses the differences between doing mathematics inside and outside of academia. He highlights the limitations of academia, such as the community taking a set of axioms as gospel, making it harder to take a leap into the unknown, but also emphasizing the collaboration and competition present in academia. Frenkel also mentions the sense of security that comes with being part of academia, but this sense of security can create a disconnection from the real world. He adds that although academia is supposed to be about seeking the truth, it's still a human activity with good, bad, and ugly things that happen, often under the radar. Frenkel notes that the speed of science is making it harder to achieve success while working outside academia, with few exceptions like the case of Yutang Zhang, who was able to make significant advances in number theory while working outside of academia but is still exceedingly rare.

28) Frenkel discusses the "deep work" and intense concentration required of mathematicians and theoretical computer scientists. Frenkel recalls working every day, on weekends and holidays, feeling like he was missing something if he took a day off. He believes that the process to find new mathematical connections is non-linear and is a sustained effort that requires bringing all the information into focus, playing with different ways of connecting things. It is a total miracle when insight strikes suddenly, and it cannot be predicted or brought closer by will.

29) Frenkel talks about the difference between mathematics and other fields such as physics, where view points can be provisional. He explains how in mathematics, a proof is necessary, and sometimes it can take months of frustration to get to the end. Frenkel then discusses the proof of Fermat's Last Theorem by Andrew Wiles, a mathematician from Princeton University, which took 350 years to prove. While a gap was initially found, Wiles enlisted the help of his former student Richard Taylor, and they were able to close the gap and finally publish the proof, which is believed to be correct.

30) Frenkel discusses the history behind Fermat's Last Theorem, which took 350 years to prove. The theorem states that the equation x^n + y^n = z^n has no solutions in natural numbers for n greater than 2, except for the trivial solution of all values being 0. Fermat himself had shown this to be true for the case of cubes, and it remained unproved for other cases for centuries. Frenkel explains the role of a colleague at UC Berkeley in connecting Fermat's Last Theorem to the Shimura-Taniyama-Weil conjecture, which Andrew Wiles and Richard Taylor eventually proved.

31) Frenkel speaks of his experience of discovering a mathematical answer that no one else knew. In his youth, he was not worried about the possibility of his idea being stolen. However, he acknowledges that stealing of ideas and not giving proper credit is a problem in academia which can lead to psychological stressors for young minds. He suggests that academia needs to create rules and ethical guidelines to address this issue and strive towards self-awareness and responsibility. Frenkel thinks that physicists and chemists are ahead of mathematicians in addressing this issue.

32) Frenkel discusses the difference in ethical rules and priority between mathematicians and biologists, with money being a significant factor in biological research. He also talks about the stereotype of mathematicians being secluded and not interested in sharing the beauty of their subject. Frenkel acknowledges that he himself used math as a refuge from the cruelty of the world he experienced, but he wrote Love and Math to break that cycle and attract different psychological types and more women to the field.

33) Frenkel reflects on his romantic ideals towards mathematics and how his life experiences and upbringing contributed to this. Growing up in the Soviet Union, Frenkel faced issues such as anti-Semitism and struggled with traumatic experiences that took him 30 years to come to terms with. However, these experiences motivated him to strive to become a mathematician, including passing an exam to get into Moscow University's prestigious mathematics department despite facing discrimination due to his Jewish background.

34) Edward talks about a difficult experience he had as a 16-year-old boy trying to get accepted into Moscow University as a Jew. After failing the exam, Frenkel was about to give up on his dream of becoming a mathematician until he found a technical school that had an Applied Mathematics program. At the time, he didn't realize the impact that experience had on him until 30 years later when he finally faced his trauma and realized the crushingblow it had on him as a child. Frenkel wrote about this experience in his book, but it wasn't until he wrote it as if the boy was writing the story himselfthat he was emotionally connected to the experience. This experience has inspired Frenkel to motivate young people to never give up on their dreams.

35) Edward speaks about his experience with childhood trauma during his education and how it affected him. Frenkel explains that although he had no knowledge of childhood trauma at the time, he was aware of the cruelty he experienced during his education. However, he did not confront it until later as it was a defense mechanism, and he was not yet ready for it. Eventually, he was able to reconnect with the cruelty and see that the examiners themselves were victims of their own situation. Through this experience, Frenkel learned that life is full of paradoxes and it is essential to share experiences with others to learn and grow.

36) Edward reflects on the death of his father and how it changed his perspective on life, death, and love. Frenkel shares that he misses his father tremendously but has learned that his father never truly left him and that he carries him in some sense. Frenkel also talks about the pain and intensity of facing love that is completely pure and unfiltered, which he experienced when his father died. Ultimately, Frenkel believes that the experience of life and death cannot be conceptualized or put into words but must be lived through.