Thinking fast slow and slower

Based on the sources provided, the following is the full text of the article "Thinking fast, slow, and super slow" by David Bessis, which explores how mathematicians train their intuition.

Thinking fast, slow, and super slow: How mathematicians train their intuition

A ball and a bat cost a total of $1.10. The bat costs $1 more than the ball. How much does the ball cost?. Borrowed from Daniel Kahneman’s Thinking, Fast and Slow, most people incorrectly answer 10¢ because if the ball cost 10¢, the bat would cost $1.10, totaling $1.20. Even when corrected, people often find excuses not to calculate the right answer, which is 5¢.

This problem illustrates cognitive biases and Kahneman's theory of two cognitive systems. System 1 provides immediate, instinctive, and sometimes incorrect responses, while System 2 is used for rigorous calculations but is tiresome and consumes significant mental energy. Biologically, humans have a preference for intellectual laziness, often relying on System 1 without verification. Kahneman recommends fighting this inclination by memorizing cognitive biases and forcing the use of System 2.

David Bessis suggests a better way. When he first took the test, he instinctively answered “5¢” without conscious calculation. His friend, a cognitive science student, called this "cheating" because he was a mathematician, implying it was impossible to see the answer immediately. Bessis was surprised that others struggled to see the answer as "visually evident" and investigated why. He found that most non-mathematicians would choose intuition over reason in a personal conflict, yet their intuitions failed them on this simple math problem.

Bessis disagrees with Kahneman’s assumption that the "intuitive answer" is necessarily false and that we must simply "resist" it. He argues that top students at elite universities have faulty intuitions because they have not trained them, whereas students who "see" the answer have a massive competitive advantage. He critiques the idea that intuition is hardwired and unchangeable, comparing it to the belief that ancient Romans couldn't mentally represent large numbers.

Instead of just resisting intuition, Bessis proposes System 3, which involves introspection and meditation techniques to establish a dialogue between intuition and rationality. This system focuses on resolving the dissonance between one's gut feeling and logic. In practice, this means:

• Translating intuition into a simple, intelligible story.
• Picturing logical reasoning to experience it in the body.
• Acting as a referee to understand where the misalignment occurs.

While logic is "inert like a pebble," intuition is organic, living, and growing. Intuition is the tangible manifestation of synaptic connections—a network of mental associations containing all your experience. Bessis argues that an error in intuition is not a sign of intellectual inferiority but an opportunity for mental representations to reconfigure.

To solve the ball and bat problem intuitively, Bessis visualizes prices as lengths. By seeing the bat as a "ball plus $1" and placing them together, it becomes visually obvious that two balls plus $1 equals $1.10, meaning the ball must be 5¢. He recommends that others reprogram their intuition by constructing pictures that work for them. His key principles for this are:

1. Reprogram your intuition.
2. Use misalignment as an opportunity to create new ways of seeing.
3. Allow the process to happen at an organic pace.
4. Don’t force it; play with what you already understand.
5. Strengthen intuitive capacities through short, regular practice.

Bessis concludes that nothing is counterintuitive by nature; it is only temporarily so until you find a way to make it intuitive. Understanding is the act of making a concept intuitive for yourself.

Summary of the Three Systems:

Feature	System 1	System 2	System 3
Name	Intuition	Rationality	Thought / Reflection / Meditation
Verb	See	Follow the rules	Reflect / Meditate
Adjective	Instinctive	Procedural	Introspective
Output	Mental image	Calculated value	Updating System 1
Speed	Fast (Immediate)	Slow (Seconds/Minutes)	Super slow (Days/Months/Years)
Metaphor	Electrical	Mechanical	Organic
Benefits	Speed, sincerity	Accuracy	Strength, self-confidence
Limitations	Imprecise	Not human	Asynchronous

System 2 reasoning often feels like acting like a robot, which humans are biologically poor at sustaining. System 3 is a meditation constrained by the principle of noncontradiction, aimed at revising System 1. Just as surfers habituate System 1 to Newtonian physics, practitioners of System 3 habituate System 1 to logical consistency. For those who use System 3, math does not feel like work; they are simply seeing pictures in their heads and asking naive questions.

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This post is adapted from Chapter 11 of "Mathematica: A Secret World of Intuition and Curiosity" by David Bessis.