Mathematics as a Language

I had an idea. A profound one, I thought. About how reality works, how decisions unfold, how the infinite arises from the finite. I wanted to express it mathematically because mathematics felt like the right language for cosmic truths.

So I wrote:

±Δ⁸=∞

It looked right. It felt right. The symbols captured something true about my intuition. Plus-or-minus for duality and uncertainty. Delta for change. Eight elevated to represent states of being. Infinity as the outcome.

But here’s the thing about mathematics: it’s not just symbols that look cool. It’s a language, and languages have grammar. What I wrote was the equivalent of arranging beautiful Mandarin characters in a pattern that pleased my eye—without actually forming a sentence that means anything in Mandarin.

In standard mathematical notation, ±Δ⁸=∞ would be read as: “Plus or minus delta-to-the-eighth-power equals infinity.”

This doesn’t parse. Any finite value (delta, whatever it represents) raised to any finite power (8) produces another finite value—not infinity. The equals sign claims an identity that can’t exist. A mathematician would look at this and see grammatical nonsense, like writing “The blue runs quickly sandwich.”

The symbols individually have meanings. But strung together this way, they don’t form a valid mathematical statement.

I had a real idea. I just didn’t know the language to express it.

Let me explain the idea in plain language first. Then we’ll translate it into proper mathematics.

Will I go to the park this weekend?

Right now, that question has no definite answer. I might go. I might not. Something could come up. I could change my mind. I could—let’s be honest—die before the weekend arrives. The future exists as a cloud of possibilities, not a fixed point.

In quantum mechanics, this is called superposition. A particle doesn’t have a definite position until it’s measured. Before measurement, it exists in all possible positions simultaneously, each weighted by probability. My weekend plans are in superposition: all possibilities coexist until the moment of decision collapses them into actuality.

Then the weekend arrives. Someone observes me. The superposition collapses. I’m either at the park or I’m not—but more than that, I’m in a particular emotional state.

Let’s say there are eight fundamental emotional states I could be in. Why eight? Because 8 = 2³—three binary choices, like quantum bits, can produce eight distinct outcomes. It’s also a number that appears across traditions: eight trigrams in the I Ching, eight-fold path in Buddhism, eight directions in Hindu cosmology.

But here’s where it gets interesting. Even though I’ve collapsed into “joy,” my experience of joy is not identical to your experience of joy, or to my experience of joy yesterday. The texture of that emotional state varies infinitely.

Joy while watching a sunset. Joy while holding a child. Joy mixed with relief. Joy tinged with melancholy because you know it won’t last. These are all “joy”—the same discrete state—but the felt experience has infinite gradations.

Superposition → Discrete States → Infinite Variation

Now let’s translate this idea into proper mathematical notation—the kind that a physicist or mathematician would recognize as grammatically valid.

|Ψ⟩ — In quantum mechanics, |Ψ⟩ (pronounced “psi” or “ket psi”) represents a quantum state—a mathematical object that encodes all the possible outcomes of a measurement and their probabilities. Before observation, the system exists as |Ψ⟩, containing all possibilities in superposition. The vertical bar and angle bracket aren’t decoration—they’re Dirac notation, invented by physicist Paul Dirac specifically to handle quantum states.

S⁸ — This represents a space with eight distinct states. The superscript isn’t an exponent in the usual sense—it’s a dimension indicator—telling us this space has eight possible positions.

→ — The arrow indicates a process or transformation. It doesn’t claim equality. It says: “this becomes that.” It’s the mathematical equivalent of “and then.”

Putting it together:

|Ψ⟩ → S⁸ → ∞

“Superposition transitions to eight-state space, which opens to infinity.”

A quantum state (undetermined possibilities) collapses upon observation into one of eight discrete states, each of which contains infinite continuous variation.

Here’s what I learned through this process—something that changed how I think about mathematics entirely:

Mathematics is a language for expressing ideas, not just a tool for counting things.

Most of us learn math as arithmetic. Addition, subtraction, multiplication, division. We think of it as “working with numbers”—a way to calculate quantities.

But that’s like thinking of English as “a way to label things.” Yes, you can use English to label objects: “table,” “chair,” “lamp.” But English can also express love, argue philosophy, construct narratives, make promises, declare war, compose poetry. The labeling function is the smallest part of what language does.

Mathematics is the same. Numbers and arithmetic are real, but they’re the equivalent of labeling objects. The deeper function of mathematics is to express relationships, transformations, structures, and patterns that can’t be expressed any other way.

When you learn a foreign language, you go through predictable stages:

Stage 1: You know individual words but can’t form sentences. You point at things and say their names.

Stage 2: You can form simple sentences, but you make grammatical errors constantly. Native speakers understand you, but you sound like a tourist.

Stage 3: Your grammar improves. You can express complex ideas, though you still make mistakes.

Stage 4: Fluency. The language becomes a natural vehicle for thought.

My original equation—±Δ⁸=∞—was Stage 1 mathematics. I knew some symbols, and I arranged them in a pattern that felt meaningful. But I hadn’t learned the grammar yet.

The corrected expression—|Ψ⟩ → S⁸ → ∞—is me reaching toward Stage 2. The grammar is correct. A mathematician can parse it.

You might ask: why bother? If I could explain the idea in plain English, why struggle to express it mathematically?

Because mathematics can express things that English can’t.

When Maxwell wrote his equations for electromagnetism, he unified electricity, magnetism, and light into a single framework. Those equations predicted radio waves before anyone had ever detected them. The mathematics contained more truth than Maxwell himself knew how to interpret.

When Dirac wrote his equation for the electron, it predicted antimatter—a form of matter no one had imagined, which was later discovered in cosmic rays. The mathematics knew something the physicist didn’t.

Mathematics doesn’t just describe reality. It discovers reality.

There’s something important in admitting: “I got it wrong.”

I could have insisted that my original expression was fine—that it was “symbolic” or “poetic” or “beyond conventional mathematics.” People do this all the time with pseudo-profound statements. They shield their confusion behind claims of transcendence.

But that would have been intellectually dishonest. My idea deserved proper expression. The concept was real; only my articulation was flawed.

This is what education is. Not the accumulation of facts, but the acquisition of languages—including the language of mathematics—that allow us to think thoughts we couldn’t think before.

What I wanted to express: Reality begins in superposition—all possibilities coexisting. Observation collapses possibility into discrete states. Within each state, infinite variation unfolds. The finite gives birth to the infinite through the act of becoming definite.

What I wrote:

±Δ⁸=∞

What I should have written:

|Ψ⟩ → S⁸ → ∞

The idea was always valid. Only the expression needed to grow.

The mistake is not the failure. The mistake is refusing to learn the language.