Mathematics, Intuition, and Curiosity – David Bessis

Math books function like reference manuals, not novels - they're designed for random access when you need specific information, not linear reading. Bessis explains how he discovered this as a PhD student when he couldn't read past the first page of a recommended book, revealing a fundamental gap between how math is written and how it should be consumed.

• •Math books are like toaster manuals or phone books - you consult them when needed, not read cover-to-cover
• •Words in math books are precisely defined concepts that build on each other, making linear reading overwhelming and boring
• •The correct approach: jump directly to what you need, then backtrack only if necessary - this provides real motivation
• •Most mathematicians learn this implicitly; explicit teaching of this metacognitive skill is rare
• •Even in courses, instructors guide students through textbooks rather than expecting linear reading

Bessis distinguishes between 'official math' (written definitions, theorems, proofs) and 'secret math' - the informal knowledge of how to actually think about mathematics. This includes both mathematical intuition shared at blackboards and deeper metacognitive tricks shared only in informal settings like hiking or drinking, which are considered too 'wacky' to write down but are essential for progress.

• •Mathematical knowledge has two sides: formal corpus and informal metacognitive knowledge rarely written down
• •Daily work in math departments involves sharing intuitions at blackboards, not just formal proofs
• •Deep metacognitive secrets emerge in informal settings - Jean-Pierre Serre's rock climbing metaphor for reversible proofs, cooking as brain relaxation
• •Mathematicians approach their field like a martial art with trade secrets, but these seem too 'new age' to formalize
• •The gap in transmitting this secret knowledge explains much of the inequality in mathematical ability

Fear is the number one inhibitor of mathematical progress at every level. Bessis describes how professional mathematicians experience the same fear as primary school students when facing material too hard for them, but learn to domesticate it rather than flee. He shares a powerful example of breaking a culture of fear by interrupting a seminar talk that nobody understood.

• •Fear of feeling stupid is identical whether you're in primary school or a research department - only the stakes differ
• •Professional mathematicians learn to accept struggle and fight the instinct to avoid difficult material
• •Conference talks are often not understood by anyone in the room, but fear prevents people from asking questions
• •Bessis's breakthrough: interrupting a 90-minute talk to demand real explanations, discovering the entire team had been pretending to understand for years
• •The ability to ask 'stupid questions' repeatedly until concepts become transparent is essential but rarely taught

Thurston, one of the greatest geometers, was born with severe strabismus and couldn't see in 3D. His mother's systematic training to rebuild his vision, combined with his own daily visualization practice from primary school onward, led him to develop the ability to visualize in 4D and 5D - capabilities that seem impossible but produced groundbreaking mathematical insights.

• •Thurston had no natural 3D vision due to strabismus - a massive handicap that became an advantage
• •His mother spent hours with special books and exercises training him to reconstruct 3D vision from 2D slices
• •He made a conscious decision in primary school to practice visualization daily, applying the same technique to higher dimensions
• •This 'n=1' example is powerful because it's documented and shows how idiosyncratic development creates extreme outliers
• •His ability to 'see' in 5D produced theorems and conjectures that look like they came from an alien intelligence

Bessis argues that the extreme inequality in mathematical outcomes (orders of magnitude difference, not the 10% seen in physical traits) cannot be explained by genetic variation alone. He compares it to wealth distribution - a Pareto distribution caused by feedback loops, not a bell curve from polygenic traits. The mechanism is idiosyncratic cognitive development with self-reinforcing effects.

• •Height follows a bell curve (everyone basically the same ±30%) because it's polygenic; math ability shows species-redefining inequality
• •Terry Tao is 10x faster than Fields Medalists at absorbing math - far beyond genetic variation in any physical trait
• •Within-species variability of this magnitude doesn't exist genetically; examples like queen bees are epigenetic, not genetic
• •Human cognition is complex machinery building over decades - small differences in emotional regulation and practice compound massively
• •The argument isn't that genetics plays no role, but that developmental pathway variance exceeds genetic variance

After failing his first PhD, Bessis spent years experimenting with metacognitive techniques. Two breakthrough periods - seven weeks of fear-driven intense focus, then six weeks of 'egoless' epiphany - produced more mathematical output than his previous twelve years combined. He proved a 40-year-old conjecture and experienced mathematics 'thinking through him' involuntarily, ultimately leading him to quit math.

• •First breakthrough: 7 weeks working 10-15 hours daily driven by fear of being scooped, proving all but one case of a major conjecture
• •This intense period opened a new 'mode' in his brain - ability to enter deep meditative states on abstract concepts without fear
• •Second breakthrough 18 months later: 6 weeks of continuous epiphanies, math 'thinking against his will', couldn't turn it off
• •Flipped through a 100-page textbook in the middle of the night and everything was 'obvious like a cartoon'
• •Wrote two papers capturing ~10% of ideas; one took 4 referees and 7 years to publish due to difficulty and scope

Bessis shares concrete techniques he uses with his own children: early introduction to negative numbers through elevator games, encouraging dream recall and articulation to train intuition, and teaching emotional awareness. He emphasizes that pride in mathematical ability creates massive advantages, and these simple interventions create divergent outcomes even if only 20% of parents do them.

• •Introduce advanced concepts playfully: teaching negative numbers via elevator games to a 6-year-old, enabling subtraction far beyond grade level
• •Daily dream recall and articulation trains the back-and-forth between language and imagination, sharpening intuition
• •Teaching children to name and locate emotions develops metacognitive skills essential for both math and professional success
• •Creating pride in mathematical ability is more important than the specific skills learned
• •These interventions are simple but create compound effects - not everyone does them, creating early divergence

Bessis proposes three modest but powerful changes to math education: (1) explicitly celebrate the emotional journey from confusion to transparency at least once per year, (2) dedicate 5-10% of teaching to metacognitive lessons about how math actually works, and (3) implement peer mentoring where students who understand explain to those who don't, creating structured knowledge transfer.

• •Make students aware they don't understand something, then guide them to transparency - celebrate this transformation as the core learning
• •Dedicate one lesson in 15 to metacognition: explaining that math involves 'unseen actions in your head' (many students never learn this)
• •Share the oral tradition explicitly: trust your intuition but correct it, ask stupid questions repeatedly, do mental visualization
• •Split classes in half for peer mentoring sessions - those who understand explain to those who don't (already successful in some classrooms)
• •These shouldn't replace formal teaching but augment it with the 'secret math' that's currently transmitted haphazardly