#73 Steven Strogatz: Exploring Curiosities
Mathematician Steven Strogatz, professor of applied mathematics at Cornell University, explores how math helps understand the world, from HIV treatment to GPS. He discusses fostering curiosity and problem-solving skills in math education, along with insights on decision-making and the evolution of cooperation.
Deep Dive Analysis
16 Topic Outline
Early Mathematical Encounters and Influences
Pivotal High School Geometry Problem
College Struggles and Teaching Philosophy Development
Why Math is Often Perceived as Difficult
Teaching Math Through Exploration and Safe Spaces
Parental Role in Fostering Math Curiosity
The Essence and Impact of Calculus
Calculus's Application in HIV Treatment
Calculus and Relativity in GPS Technology
Comparing Math Accuracy in Physics vs. Social Sciences
Emergence of Order and Morality from Chaos
The Prisoner's Dilemma and Evolutionary Game Theory
Strategic Quitting and Problem Selection in Research
The Small-World Network Discovery
Attributes Beyond Raw Skill in Scientific Success
The Social Enterprise of Science and Idea Communication
10 Key Concepts
Angle Bisector
A line segment that divides an angle of a triangle into two equal angles, extending from the vertex to the opposite side.
Isosceles Triangle
A triangle in which at least two sides are of equal length, and consequently, the angles opposite those sides are also equal.
Math as a Web vs. Tower
This mental model suggests that math is a network of interconnected ideas where one can enter at various points and navigate, rather than a rigid, linear structure where missing a step means falling off completely.
Safe Space (Mathematical Confusion)
An educational environment where students feel secure to express confusion, ask questions, and take intellectual risks without fear of being judged or appearing unintelligent, recognizing confusion as a normal part of exploring the unknown.
Calculus
The branch of mathematics focused on the study of change. It provides tools to quantify and understand things that change, especially in dynamic and continuously evolving ways, making it essential for describing a world in flux.
Rates of Change (Derivative)
A fundamental concept in calculus that measures how one quantity changes in response to changes in another. Examples include speed (rate of change of position over time) or marginal utility in economics.
Prisoner's Dilemma
A classic game theory scenario modeling situations where two individuals, acting in their own self-interest, might choose to 'defect' (betray) each other, even though mutual cooperation would lead to a better collective outcome for both.
Tit for Tat
A simple and effective strategy in repeated Prisoner's Dilemma games, where a player initially cooperates and then, in subsequent turns, mirrors the opponent's previous action (cooperating if the opponent cooperated, defecting if the opponent defected).
Strategic Quitting
The deliberate decision to abandon a research problem or endeavor that is not yielding progress, cutting losses to pursue more promising avenues. This contrasts with the sunk cost fallacy, where one continues due to past investment rather than future potential.
Small-World Network
A type of network characterized by short average path lengths between any two nodes, meaning that any two individuals or points are connected by a relatively small number of steps or 'handshakes,' even in very large systems.
10 Questions Answered
His initial interest was sparked by competitive success against friends in junior high, but a pivotal moment came in high school when a teacher challenged the class with a geometry problem he himself couldn't solve, leading Strogatz to an obsession with solving it.
Math is often taught as a rigid 'tower' where each concept builds on the last, making it easy to fall behind. The traditional 'curriculum dump' approach stifles exploration and makes the subject seem dry, rather than a creative 'web' of interconnected ideas.
Parents should not be afraid to admit when they don't know an answer, instead approaching it as a joint problem-solving opportunity ('Let's figure it out'). Utilizing online resources like YouTube channels (e.g., Three Blue One Brown, Khan Academy) can also be highly effective for modern learners.
Calculus is the mathematics of change, designed to quantify things that change, especially in dynamic and continuously evolving ways. It's crucial because everything in the world is in flux, making it a fundamental tool for understanding and describing the dynamics of various systems.
In the mid-90s, calculus was used to analyze the exponential drop in viral load after protease inhibitor treatment, revealing that the body was producing and clearing billions of virus particles daily. This understanding led to the development of the effective three-drug therapy, which made it much harder for the virus to mutate and develop resistance.
GPS uses calculus to measure rates of change (like position over time for velocity) and relies on precise timekeeping from atomic clocks on satellites. These clocks require Einstein's relativity corrections because their speed and distance from Earth affect the rate at which time passes, without which the system would fall apart in about 20 minutes.
Physics deals with simpler, inanimate systems (like the moon) that don't react to being measured and have uniform components (e.g., all electrons are identical). Biology and economics involve intrinsically complex, diverse, and dynamic systems with feedback loops, ethical limitations on experimentation, and significant randomness, making them much harder to quantify and predict.
Computer tournaments of the Prisoner's Dilemma showed that strategies that were 'nice' (never defecting first), 'forgiving' (not retaliating forever), 'retaliatory' (punishing unprovoked defections), and 'clear' (predictable) tended to thrive. This suggests that principles resembling morality (like 'eye for an eye') can emerge from self-interested interactions in repeated games.
It's a difficult decision balancing sunk cost fallacy with opportunity cost. Factors include lack of progress, insufficient technical tools, the problem not being as interesting as hoped, time horizons (e.g., needing to graduate), and willingness to take risks. Quitting allows for pursuing more tractable or promising new ideas.
Courage, good judgment, taste (knowing what problems are interesting and potentially impactful), and the ability to communicate why one's work matters are crucial. Broad interests and the ability to apply mental models from different disciplines can also lead to novel insights and 'easy problems' that others overlook.
35 Actionable Insights
1. Prioritize Win-Win Relationships
In long-term relationships, prioritize win-win outcomes and avoid taking advantage with promises to ‘make it up later,’ as this often leads to inaccurate scorekeeping and relationship breakdown.
2. Go Positive, Forgive Unless Malicious
In human interactions, a good life strategy is to ‘go positive and go first’ by initiating cooperation, and to forgive others unless their actions are clearly malicious.
3. Adopt ‘Tit for Tat’ Strategy
In repeated interactions, adopt a strategy of being nice (never defect first), forgiving (don’t retaliate forever), retaliatory (punish unprovoked defection), and clear (avoid confusing complexity) to thrive.
4. Practice Strategic Quitting
Recognize when to strategically quit a problem you’re stuck on, weighing sunk costs against opportunity costs to avoid wasting time and open up possibilities for new progress.
5. Cultivate Broad Interests
Cultivate broad, interdisciplinary interests (e.g., humanities, philosophy, sociology) to identify unique problems and gain an edge by applying diverse perspectives.
6. Create Safe Space for Confusion
Establish an environment where confusion is normalized and intellectual risk-taking is safe, fostering trust and collaboration without fear of looking stupid.
7. Be Vulnerable in Collaboration
In collaborative research, cultivate vulnerability and a safe relationship with collaborators to openly admit confusion, ask for clarification, and suggest ideas without fear.
8. Admit Not Knowing, Then Learn
When faced with something you don’t know, especially with children, be strong enough to admit it and then actively work to figure it out together, either by thinking or looking it up.
9. Don’t Pass on Anxieties
Avoid passing on your own anxieties or negative mindsets (e.g., about math) to your children, as this is unhelpful for their learning.
10. Set High Expectations
Expect challenging things from students, especially younger ones, as they will often rise to the level of expectation and astonish you with what they can do.
11. Reject ‘Weeding Out’ Mentality
Do not assume students are hopeless or lack potential based on superficial analysis, as people often have more potential than initially perceived.
12. Make Learning Engaging
To make a subject exciting, show pictures, provide intuition, connect it to the real world, share its history, and bring it alive, rather than presenting it dryly.
13. Uncover Material, Don’t Cover
Instead of merely covering curriculum, strive to ‘uncover’ material by removing fog and misunderstanding, allowing for discovery and revealing its essence.
14. Embrace Exploration & Problem-Solving
Cultivate the skill of exploring and making progress when lost or uncertain, as this applies to all aspects of life and fosters the thrill of problem-solving.
15. Foster Problem-Solving with Puzzles
Teach by giving puzzles and guiding students when they are stuck, helping them develop problem-solving strategies and cope with frustration.
16. Prioritize Honesty Over Authority
In intellectual matters, prioritize honesty by admitting when you don’t know something, rather than pretending to maintain an authority figure image, as you’ll likely be found out anyway.
17. Utilize Online Math Resources
Parents can improve their own math understanding and help their kids by using excellent online resources like YouTube channels (e.g., three blue, one brown, Mathologer) and Khan Academy.
18. Tailor Teaching to Child
Adapt teaching approaches and strategies based on the individual child’s reaction to math, whether they are bored, anxious, or gifted.
19. Communicate Work’s Importance
To make a discovery truly great, communicate it and help others understand why it matters, as science is a social enterprise.
20. Understand Audience Value
In research, understand what others find valuable and interesting, as science is a social enterprise where communicating the importance of your work is as crucial as the work itself.
21. Prioritize Problem Selection
For intellectual pursuits like a PhD, the first and most critical step is selecting a problem that allows you to discover something new, interesting, and uniquely yours, fostering innovation.
22. Target ‘Second Hardest’ Problems
When selecting problems, consider targeting the ‘second hardest’ rather than always the most ambitious, as it can be a more effective path to success and building up to greater challenges.
23. Leverage Comparative Advantage
When choosing what to work on, identify and leverage your unique strengths or comparative advantages that give you an edge over others.
24. Quit Strategically, Then Pivot
Don’t be afraid to quit a project when it’s not panning out, but immediately pivot to something new rather than giving up entirely.
25. See Wonder in the Obvious
Develop the skill of recognizing the mysterious and wonderful in things that are ‘right under everybody’s noses,’ as this can lead to significant discoveries.
26. Assess Quitting Based on Factors
When considering strategic quitting, assess factors like frustration levels, remaining time, real-world obligations (e.g., family, job), desired payoff, and willingness to gamble, similar to investment decisions.
27. Match Risk to Personality
Tailor project risk levels to individual personality traits, such as fearlessness, and be prepared to work secretly and without immediate results on high-risk, high-reward endeavors.
28. Value Courage, Judgment, Taste
Recognize that beyond technical skill, attributes like courage, good judgment, and ’taste’ (knowing what will be interesting or ‘cool’) are crucial for success in any discipline.
29. Adapt Strategy to Relationship
Adjust business or interaction strategies based on whether you are in a one-shot transaction (e.g., tourist trap) or a long-term relationship, as different approaches are viable.
30. Build Meaningful Relationships
Actively cultivate meaningful relationships that are not solely dependent on professional position or work-related trade-offs, to avoid finding yourself isolated after retirement.
31. Avoid Opportunistic Exploitation
Do not attempt to occasionally take advantage of established trust, as even small opportunistic actions can lead to a breakdown of trust that is very difficult to restore.
32. Sympathize with Struggling Students
When teaching, have sympathy for struggling students and avoid assuming they are hopeless, as their difficulties may not reflect their true potential.
33. Focus on Order from Chaos
Cultivate an intellectual interest in understanding how order and self-organization spontaneously emerge from chaotic systems.
34. Employ Multi-Drug Therapy
When dealing with rapidly mutating pathogens, employ multi-drug therapies (e.g., three drugs) to significantly reduce the odds of simultaneous mutations and maintain long-term effectiveness.
35. Tailor Treatment to Disease Model
Adjust treatment strategies (e.g., when to administer drugs) based on the current understanding or model of how a disease behaves, to optimize effectiveness and prevent issues like drug resistance.
9 Key Quotes
A great discovery that no one appreciates is not really a great discovery because science is a social enterprise. It's not just enough to do the work. You have to communicate it and help other people understand why it matters.
Steven Strogatz
The students will rise to the level that the teacher expects.
Jaime Escalante (quoted by Steven Strogatz)
Don't try to cover the material, try to uncover it.
A teacher (quoted by Steven Strogatz)
Confusion is the normal state of affairs when you're trying something really hard and when you're exploring the unknown.
Steven Strogatz
Calculus is the math for describing a world in flux. And since everything is in flux, you could see that it's bound to be pretty useful to have the ability to do that.
Steven Strogatz
It was a furious battle of attrition, an all-out war that was being held to a standstill. The immune system was holding HIV at bay. Until it gets exhausted and... That's it.
Steven Strogatz
It's not some kind of universal best way to behave. It turns out it's a little too ungenerous. It's a little bit stern.
Steven Strogatz
Sometimes you have to quit when it's not panning out. But then don't just give up. You have to come up with something else.
Steven Strogatz
Deep is our highest compliment, actually, in math. That person is deep. That work is deep. That theorem is deep. That's what, that's the standard of excellence, but not really for me. I mean, whether it's just because I'm not capable of it, I don't know, but I kind of like shallow and broad.
Steven Strogatz
3 Protocols
Math Explorations Course Teaching Method
Steven Strogatz- Give students puzzles to solve.
- Avoid lecturing or directly providing answers.
- Encourage students to work in groups and share ideas to figure out solutions.
- When students are stuck, guide them to develop problem-solving strategies and cope with frustration.
- Cultivate a 'safe space' where mathematical confusion is normalized, and intellectual risk-taking is encouraged without fear of judgment.
Parental Engagement in Math
Steven Strogatz- Do not be afraid to admit when you don't know the answer to a math problem.
- Approach math problems as a joint effort with your child, saying, 'Let's figure it out.'
- Skillfully utilize online resources like YouTube videos (e.g., Three Blue One Brown, Khan Academy) for learning and exploration.
Successful Strategies in Prisoner's Dilemma (Axelrod's Tournaments)
Steven Strogatz (describing Robert Axelrod's findings)- Be Nice: Never be the first to defect; always begin by cooperating.
- Be Forgiving: Don't retaliate forever if the other player cheats occasionally; let bygones be bygones after a while.
- Be Retaliatory: Inflict punishment if someone defects without provocation.
- Be Clear: Keep your strategy simple and predictable so others can understand and build a relationship with you.