Lessons from Terence Tao

On Learning and Understanding

"Learn and relearn your field, but don't be afraid to learn things outside your field." [1]
- Learning: This advice emphasizes the importance of continuous learning, not just within one's specialty but also in broader areas to foster new connections and insights.
- Source: What's new - Career advice, terrytao.wordpress.com [1]
"Just because you know a statement and proof of Fundamental Lemma X, you shouldn't take that lemma for granted; instead, you should dig deeper until you really understand what the lemma is all about." [2]
- Learning: True understanding goes beyond memorization. It involves questioning, exploring alternative proofs, understanding hypotheses, and seeing the bigger picture.
- Source: What's new - Learn and relearn your field, terrytao.wordpress.com [2]
"If you can't explain the solution of a problem to a classmate, then you haven't really understood the solution yourself."
- Learning: The ability to teach a concept is a powerful test of one's own comprehension. It forces clarity and exposes gaps in understanding.
- Source: This is a widely attributed paraphrase of his advice, rooted in his emphasis on clear communication. A similar sentiment can be found in his advice on problem-solving.
"Ask yourself dumb questions – and answer them!" [1]
- Learning: Simple or "dumb" questions can often lead to the most profound insights by challenging foundational assumptions and revealing a deeper structure.
- Source: What's new - Career advice, terrytao.wordpress.com [1]
"Mathematics is not a spectator sport. You have to actively engage with the concepts to truly understand them."
- Learning: Passive learning is insufficient in mathematics. Active problem-solving and engagement are crucial for building genuine understanding.
- Source: This is a popular quote encapsulating his philosophy. The core idea is frequently expressed in his writings on mathematical education.
"It is particularly useful to lecture on your field, or write lecture notes or other expository material, even if it is just for your own personal use." [2]
- Learning: The act of organizing and presenting information for others is a powerful tool for deepening one's own knowledge.
- Source: What's new - Learn and relearn your field, terrytao.wordpress.com [2]
"Don't expect to understand 100% of any given talk, especially if it is in a field you are not familiar with; as long as you learn something, the effort is not wasted."
- Learning: Attending talks outside one's immediate area of expertise is valuable for exposure to new ideas, even if comprehension is partial.
- Source: This advice is often given to graduate students and is part of his blog's career advice.
"The accuracy of Wikipedia can be dodgy in some places, but in maths, it's really quite good."
- Learning: Acknowledging the utility of modern, accessible resources like Wikipedia for learning mathematical concepts.
- Source: Attributed to various interviews and discussions.

On Problem Solving

"You want to get to the top of the cliff. But that's not what you focus on immediately. You focus on the next ledge just beyond your reach."
- Learning: Break down large, intimidating problems into smaller, manageable steps. Progress is made incrementally.
- Source: This analogy is used in his MasterClass and various interviews.
"It can often be profitable to try a technique on a problem even if you know in advance that it cannot possibly solve the problem completely."
- Learning: Partial progress is valuable. Applying a technique, even if it fails to provide a full solution, can yield significant insights and pave the way for a complete solution.
- Source: A-Z Quotes, summarizing his approach to research. This idea is a recurring theme in his blog.
"The journey of solving a problem is often more rewarding than the solution itself."
- Learning: The process of exploration, discovery, and overcoming challenges holds immense value and satisfaction, often more so than the final answer.
- Source: A widely cited quote that reflects his philosophy on the nature of mathematical research.
"The act of solving problems is what mathematics is all about. Mathematics is not about answers, it is about the process of finding the answers."
- Learning: This quote emphasizes that the core of mathematics lies in the methodology and creative process of problem-solving, not just the final results.
- Source: Bookey, summarizing his views on mathematics.
"In your research, be both flexible and patient." [1]
- Learning: Research requires the ability to adapt to unexpected turns and the patience to persevere through long periods without major breakthroughs.
- Source: What's new - Career advice, terrytao.wordpress.com [1]
"Be sceptical of your own work, and don't be afraid to use the wastebasket." [1]
- Learning: Critical self-evaluation is essential. It's important to be willing to discard ideas or approaches that are not working.
- Source: What's new - Career advice, terrytao.wordpress.com [1]
"What's really interesting are the problems just at the boundary between what we can do relatively easily and what are hopeless." [3]
- Learning: The most fruitful problems are often those that are challenging but not impossible, where existing techniques can be pushed to their limits.
- Source: Lex Fridman Podcast #472, youtube.com [3]
"By moving the inessential components of a problem, you can focus on what's really going on."
- Learning: Abstraction is a key mathematical tool for simplifying problems to their core essence.
- Source: MasterClass article on mathematical problem-solving.

On Talent, Hard Work, and Genius

"Talent is important, but how one develops and nurtures it is even more so." [4]
- Learning: Innate ability is only one part of the equation. Consistent effort, good habits, and a supportive environment are crucial for realizing one's potential.
- Source: What's new - Does one have to be a genius to do maths?, terrytao.wordpress.com [4]
"Does one have to be a genius to do mathematics? The answer is an emphatic NO." [4]
- Learning: Success in mathematics is more about hard work, persistence, and collaboration than a mythical "genius gene."
- Source: What's new - Does one have to be a genius to do maths?, terrytao.wordpress.com [4]
"Mathematical breakthroughs are not powered solely (or even primarily) by 'Eureka' moments of genius, but are in fact largely a product of hard work." [5]
- Learning: The romanticized image of sudden inspiration is misleading. Real progress comes from sustained effort and dedication.
- Source: What's new - Work hard, terrytao.wordpress.com [5]
"Relying on intelligence alone to pull things off at the last minute may work for a while, but, generally speaking, at the graduate level or higher it doesn't." [5]
- Learning: Advanced mathematics requires deep, systematic work that cannot be accomplished through last-minute bursts of intelligence.
- Source: What's new - Work hard, terrytao.wordpress.com [5]
"I don't have any magical ability. I look at a problem, play with it, work out a strategy."
- Learning: Demystifying his own process, he emphasizes a methodical and playful approach over any supernatural talent.
- Source: BrainyQuote, widely attributed from interviews.
"In some cases, an abundance of raw talent may end up (somewhat perversely) to actually be harmful for one's long-term mathematical development." [4]
- Learning: If solutions come too easily, one might not develop the resilience and work ethic needed for more challenging problems later on.
- Source: What's new - Does one have to be a genius to do maths?, terrytao.wordpress.com [4]

On Career and Collaboration

"Don't base career decisions on glamour or fame." [1]
- Learning: Choose a path based on genuine interest and enjoyment of the work, rather than external validation or prestige.
- Source: What's new - Career advice, terrytao.wordpress.com [1]
"To really get anywhere in mathematics requires hard work. If you don't enjoy what you are doing... it will be difficult to put in the sustained amounts of energy required to succeed."
- Learning: Intrinsic motivation and enjoyment are key to sustaining the effort required for a successful career in a demanding field.
- Source: Words of Wisdom from Terence Tao, summarizing his blog posts.
"Collaboration is very important for me, as it allows me to learn about other fields, and, conversely, to share what I have learned about my own fields with others."
- Learning: Collaboration is a powerful way to broaden one's knowledge, gain new perspectives, and make work more enjoyable.
- Source: Interview with the Clay Mathematics Institute.
"Don't prematurely obsess on a single 'big problem' or 'big theory'." [1]
- Learning: It's important to build a broad base of knowledge and work on a variety of problems rather than fixating on a single, high-stakes goal too early.
- Source: What's new - Career advice, terrytao.wordpress.com [1]
"Write down what you've done, and make your work available." [1]
- Learning: Sharing your work, even in preliminary stages, is crucial for getting feedback and contributing to the mathematical community.
- Source: What's new - Career advice, terrytao.wordpress.com [1]
"The objective in mathematics is not to obtain the highest ranking... instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students)." [4]
- Learning: The goal of a mathematical career should be about contribution and understanding, not competition.
- Source: What's new - Does one have to be a genius to do maths?, terrytao.wordpress.com [4]

On Mindset and Philosophy

"I tend to view mathematics as a unified subject and am particularly happy when I get the opportunity to work on a project that involves several fields at once."
- Learning: Seeing the interconnectedness of different mathematical areas can lead to novel and powerful results.
- Source: Interview with the Clay Mathematics Institute.
"If I don't understand something properly, every single component, it really bugs me."
- Learning: A relentless desire for complete understanding is a powerful driver of mathematical progress.
- Source: BrainyQuote, from an interview.
"The beauty of mathematics lies in that you can change the rules as you like, which is something that cannot be done in any other field." [6]
- Learning: Mathematics offers a unique freedom to explore abstract structures and create new worlds governed by consistent rules.
- Source: An Exceptionally Long Interview with Terence Tao, 36kr.com [6]
"Ultimately you should follow advice not because someone tells you to, but because it was something that you already knew you should be doing."
- Learning: Good advice resonates with one's own intuition and self-awareness.
- Source: A-Z Quotes.
"My life is more than just my work."
- Learning: Maintaining a healthy work-life balance is important, even for someone deeply passionate about their work.
- Source: Attributed to interviews discussing his personal life.
"I think the most important thing for developing an interest in mathematics is to have the ability and the freedom to play with mathematics."
- Learning: A playful and experimental approach is essential for fostering a love for mathematics.
- Source: Interview with the International Mathematical Union.
"Education is a complex, multifaceted, and painstaking process, and being gifted does not make this less so."
- Learning: Giftedness does not provide a shortcut through the challenges of education; it requires careful and nuanced support.
- Source: A-Z Quotes, likely from his writings on gifted education.
"It is very humbling to receive the Fields Medal."
- Learning: An expression of humility from one of the world's leading mathematicians upon receiving the highest honor in the field.
- Source: BrainyQuote, from statements made after receiving the award.
"Infinity absorbs a lot of sins." [3]
- Learning: A colloquial way of saying that the concept of infinity can simplify many mathematical problems by abstracting away complexities that arise with finite, bounded quantities.
- Source: Lex Fridman Podcast #472, youtube.com [3]
"Science is usually an interaction among three things: the real world, our observations of the real world, and the models of how we think the world works." [6]
- Learning: A concise description of the scientific process, highlighting the interplay between reality, observation, and theoretical modeling.
- Source: An Exceptionally Long Interview with Terence Tao, 36kr.com [6]
"I was never very good at school with... humanities... anything which was more a matter of opinion."
- Learning: A candid admission about his early preferences for subjects with objective truths over those based on interpretation.
- Source: BrainyQuote.
"When you're concentrating hard, hours can fly by, and it's just you and a math problem."
- Learning: Describes the state of deep focus or "flow" that can be achieved when engrossed in a challenging problem.
- Source: BrainyQuote.
"There is a certain way in which mathematicians approach problems. We abstract them. We break them up into pieces. We make analogies. We try to find connections with other problems."
- Learning: A summary of the core strategies of mathematical thinking.
- Source: MasterClass article on mathematical problem-solving.
"The key to mathematics lies not only in finding an effective technical path but also in eliminating wrong answers among dozens of potentially applicable methods." [6]
- Learning: A significant part of mathematical work involves efficiently discarding unproductive lines of inquiry.
- Source: An Exceptionally Long Interview with Terence Tao, 36kr.com [6]
"It would be very pleasant if one could just dream up the grand ideas and let some 'lesser mortals' fill in the details, but, trust me, it doesn't work like that at all in mathematics." [5]
- Learning: Emphasizing that even the most brilliant ideas in mathematics require rigorous and detailed execution by the originator.
- Source: What's new - Work hard, terrytao.wordpress.com [5]
"I tend to have several things to work on at any given time; when I get stuck on one of them... I write up how far I managed to get, and turn attention to something else."
- Learning: A practical strategy for managing frustration and maintaining productivity by rotating between different projects.
- Source: Interview with the American Mathematical Society blog.
"If there is something that I should know how to do but don't, it bugs me. I feel like I have to sit down and work out exactly what the problem is."
- Learning: Highlighting the internal drive and curiosity that fuels his continuous learning.
- Source: Attributed to various interviews.
"It's not just about finding you know taking a technique that is going to work and applying it but you you need to not take the techniques that don't work." [3]
- Learning: Efficiency in research comes from experience in recognizing which approaches are likely to be dead ends.
- Source: Lex Fridman Podcast #472, youtube.com [3]
"Most of what I do does not save lives."
- Learning: A humble and realistic perspective on the immediate real-world impact of pure mathematics research.
- Source: BrainyQuote.
"I remember having this vague idea that what mathematicians did was that some authority, someone, gave them problems to solve, and they just sort of solved them."
- Learning: Reflecting on the common misconception of what a career in mathematics entails, contrasting it with the reality of self-directed research.
- Source: BrainyQuote.
"For me, I guess the main motivation is the satisfaction of finally understanding some tricky mathematical concept or phenomenon and then explaining it to others."
- Learning: The dual joy of personal understanding and sharing that knowledge is a primary motivator.
- Source: BrainyQuote.
"A shiny medal is not going to solve any of the problems I'm currently working on." [7]
- Learning: A statement made after receiving the Fields Medal, underscoring that awards do not change the fundamental nature of the work and the challenges that remain.
- Source: Lex Fridman Podcast Clips, youtube.com [7]

Learn more:

Lessons from Terence Tao

On Learning and Understanding

On Problem Solving

On Talent, Hard Work, and Genius

On Career and Collaboration

On Mindset and Philosophy

Written by Antoine Buteau

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