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- Intuition pt2
GM, today we’ll go over some ideas that can help you start out in your journey of learning mathematics. Lets jump into it.
Intuition in math comes with time and practice, but everyone needs a starting place. The resources below offer just that. They’re not perfect, they’re not the best, but they’re a solid place to begin. Below is the breakdown.
Developing Your Intuition For Math
The right perspective makes math click.
Math is about ideas — formulas are just a way to express them.
Strategy for Developing Insight
Step 1: Find the central theme of a math concept.
Where was the idea first used? What was the discoverer doing when they discovered it? This use may be different from our modern interpretation and application.
Linear algebra example: Its central theme is to transform and manipulate space.
Step 2: Explain a property/fact using the theme.
Use the theme to make an analogy to the formal definition.
Linear algebra example: Think of a matrix as a set of instructions for transforming a space, like rotating, scaling, or shifting a geometric object.
Once you have an analogy or interpretation that works, see if it applies to other properties. Sometimes it will, sometimes it won’t (and you’ll need a new insight), but you’d be surprised what you can discover.
Linear algebra example: Investigate how these transformations interact with each other, how they can be combined, and what their cumulative effects are.
First Principles: The Building Blocks of True Knowledge
First principles is all about breaking down complex ideas to their most basic, foundational elements.
When you reason from first principles, you strip away assumptions, conventions, and analogies. What remains is the essence of the problem or concept. This mental model is incredibly powerful because it allows you to see beyond conventional wisdom and discover what’s truly possible.
Why Use First Principles?
For Fresh Starts: When you’re tackling something completely new, starting from first principles ensures you're not influenced by preconceived notions or biases.
Handling Complexity: In complex situations, reducing the problem to its basic components can make it more manageable and clear.
Understanding Difficult Situations: When you're struggling to grasp or resolve an issue, breaking it down to its core elements can provide clarity and direction.
Tools for First-Principles Thinking
Socratic Questioning: Continuously ask probing questions to deconstruct and explore the underlying assumptions and logic.
The 5 Whys: Keep asking "Why?" until you reach the root cause of a problem or concept.
Caution: When Intuition Falls Short
Intuition is often built on familiar experiences and analogies. But when you’re dealing with problems that have never been solved before, relying solely on intuition can be misleading. First-principles thinking helps you step back from the known and approach the unknown with a clear, unbiased perspective.
Embracing first-principles reasoning cuts through the noise and helps you see the world as it truly is. It opens up possibilities you might have otherwise overlooked. So next time you’re faced with a challenge, consider stripping it down to its first principles
Rethinking Arithmetic: A Visual Guide
Arithmetic is about the transformation of numbers. Smoosh, slide, and stretch. These changes help us understand and work with the world.
Addition: Bringing Things Together
Accumulate: Adding is like counting up. You’re gathering more.
Combine: You can also make something new by joining two numbers together.
Slide: Imagine shifting a number along a scale. Addition moves it forward.
Negatives: Undoing and Reversing
Inverses: Negatives undo what was done. They reverse addition.
Opposites: Negatives also change direction, like going back on a path.
Multiplication: Growing and Shrinking
Repetition: Multiplication is like adding the same number many times.
Scaling: It stretches a number bigger or smooshes it smaller, all at once.
Transforming Numbers
In arithmetic, you bend numbers into new forms. Each change has a meaning. You’re not just moving numbers; you’re shaping them.
The Feynman Learning Technique
“The person who says he knows what he thinks but cannot express it usually does not know what he thinks.”
—Mortimer Adler
The Feynman Technique
Pretend to teach a concept you want to learn about to a student in the sixth grade.
Identify gaps in your explanation. Go back to the source material to better understand it.
Organize and simplify.
Transmit (optional).
When you’re just starting out, the key is to dive in and get moving. So take these links, explore them, and get cracked.
Catch you later,
-Chris
