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The Math Hashmap: How Programmers Already Know Advanced Math

I learned math in five seconds. Or at least, that’s how it felt.

It started when I realized something simple: advanced math is just a different syntax for concepts I already use as a programmer.

The Problem: Why People Think Math Is Hard​

Most people are taught math symbol-first:

  • First, you memorize ∑, ∫, lim, ∀, ∃.
  • Then, you try to remember what each one means while doing problems.
  • You hope intuition magically follows.

But programmers don’t learn this way. We learn concepts first:

  • A for loop before we ever write a for (i=0; ...).
  • A recursive idea before we write the syntax.
  • A hash table concept before we touch an actual dict or map.

So here’s a different approach: treat math symbols as a hashmap.

The Math Hashmap (Symbols → Code)​

(ToDo: Fix -- Currently commented out... build failing)

Key point: In math, the output shape isn’t always explicit:

  • Is this returning a single value (like reduce)?
  • Is it returning a collection of values (like map or explode)?
  • Does the input and output length match (len(input) == len(output)), or is it being collapsed?

This is something programmers are very conscious of, but math traditionally assumes you infer it from context.

Once you see this mapping and think about output shape, you stop panicking at symbols. They’re just compressed syntax for ideas you already know.

Why Symbols Exist (and Why They Look Weird)​

Early on, I used to think: “Why don’t we just use real variable names? Like velocity instead of v?”

Here’s the reason:

  1. Portability – Math notation is independent of spoken language.

    • ∀x works whether you speak English (for all), German (für alle), or Chinese (对所有).
    • Scientists from different countries could collaborate centuries before English became the default scientific language.

  2. Compression – Symbols are short, like pointer variables in code, but also historically shaped by writing technology. With a dip pen or quill, long descriptive names would slow work and blur ink; compact symbols kept math readable and practical.

    • If you wrote forEveryElementInTheDomain, your equation would look like a JSON document.
    • Single-letter variables keep the structure readable and efficient to write.

  3. Universality – Math symbols are the Esperanto of ideas.

    • You don’t need to know Greek to use π.
    • You don’t need to know Hebrew to understand ∑.

Mathematicians invented a language-independent interface for knowledge centuries before we had programming.

The Big Idea​

You don’t need to “become a math person.” You already are one—if you code, you already:

  • Think in terms of iteration (for, while).
  • Think in terms of convergence (loops until stable state).
  • Think in terms of composition and transformation (functions of functions).

Math is just a different syntax for the same mental models.

Learn the concepts first (CS-style), then load the math hashmap of {symbol: meaning}—and also note whether it’s a map or a reduce. Suddenly, those intimidating symbols feel like home.

Takeaway​

If math scared you in school, it might be because you were handed symbols before intuition. Flip it:

  • Understand it like code.
  • Then just map the symbols—and check the output shape.

And who knows—you might end up like me, saying: “Wait, did I just learn calculus in 5 seconds?”