Discovering the Hidden Structure of Arithmetic
A Journey into Meta-Recursive Mathematics
By Jeremy Nixon
Core insight: What if addition is not just the start—it's the whole show? Every operation in arithmetic can be built from addition through recursion and inversion.
The Spark: What if Addition was Everything?
Imagine you're sitting at a desk, pencil in hand, and someone asks: "What's the most fundamental thing in arithmetic?" Most of us would say addition. But what if I told you that addition is not just the start—it's the whole show? Every operation you know—multiplication, exponentiation, even those mysterious logarithms—can be built from addition, if you're clever about it. The symbols we use in school—+, ×, ÷, ^—are just costumes. Underneath, it's all addition, repeated and inverted in ingenious ways. Our usual notation hides this beautiful pattern. Let's peel back the curtain and see what's really going on.
Building Up: From Addition to Multiplication to...?
Let's start with the basics. Addition is just add(x, y) → z. But what if you repeat addition? That's multiplication! If you add y to itself x times, you get multiplication. In our new notation, we call this +2—addition recursed once. Now, what if you repeat multiplication? That's exponentiation: +3. And if you repeat exponentiation, you get tetration: +4. This pattern goes on forever, each step a new operation, each one just repeated addition at a higher level. It's a ladder, and every rung is built from the one below.
The magic ingredient is recursion: recurse(operation, n) means "repeat this operation n times." With just add and recurse, you can build the whole tower of arithmetic. But there's one more trick: inversion.
Going Down: Negative Meta-Recursion
What if you want to go down the ladder? That's where inverses come in. Subtraction is the inverse of addition: +-1. Division is the inverse of multiplication: +-2. Logarithms are the inverse of exponentiation: +-3. And it keeps going—super roots, super logarithms, all the way down. Every operation has a mirror image, a way to undo it. In this system, negative exponents always mean "do the opposite."
This symmetry is hidden in traditional math, where every inverse gets a new symbol. Here, it's all systematic. The notation +n for any integer (or even fraction!) tells you exactly what operation you're doing, and its inverse is just +-n.
Notation Comparison: Plus Notation vs. Standard Notation
The table below shows how our systematic plus notation relates to traditional mathematical symbols. Notice how the plus notation reveals the hierarchical structure that standard notation hides.
| Level | Plus Notation | Standard Notation | Operation Name | Example |
|---|---|---|---|---|
| 1 | +1 |
+ |
Addition | a +1 b = a + b |
| 2 | +2 |
× |
Multiplication | a +2 b = a × b |
| 3 | +3 |
^ |
Exponentiation | a +3 b = a^b |
| 4 | +4 |
↑↑ |
Tetration | a +4 b = a↑↑b |
| 5 | +5 |
↑↑↑ |
Pentation | a +5 b = a↑↑↑b |
| -1 | +-1 |
- |
Subtraction | a +-1 b = a - b |
| -2 | +-2 |
÷ |
Division | a +-2 b = a ÷ b |
| -3 | +-3 |
log |
Logarithm | a +-3 b = log_b(a) |
| -4 | +-4 |
slog |
Super-logarithm | a +-4 b = slog_b(a) |
The beauty of this notation is its systematicity. Every operation follows the same pattern: +n where n indicates the level of recursion. Positive integers represent repeated application, negative integers represent inverses. The notation itself encodes the mathematical relationships that traditional symbols obscure.
The Problem with Current Mathematical Notation
Traditional mathematical notation is a historical accident, not a systematic design. We use + for addition, × for multiplication, ÷ for division, ^ for exponentiation, √ for roots, log for logarithms—each with its own symbol, each appearing to be fundamentally different. But this masks the deep unity that exists between these operations.
Information Density and Canonicality Issues
Current notation is information-sparse. The symbol × tells you nothing about its relationship to addition. You have to memorize that multiplication is repeated addition, but the notation doesn't encode this fact. Similarly, ^ doesn't reveal that exponentiation is repeated multiplication. Each symbol is an arbitrary choice that carries no intrinsic meaning about the operation's place in the hierarchy.
There's also no canonical representation. The same mathematical concept can be written in multiple ways: 2 × 3, 2 · 3, (2)(3), or even just 2 3 in some contexts. This lack of standardization creates cognitive overhead and makes pattern recognition harder.
Proliferation of Distinct Symbols for Related Operations
Every operation gets its own symbol, creating the illusion that they're fundamentally different. Addition uses +, multiplication uses ×, exponentiation uses ^, tetration has no standard symbol at all. Each inverse operation gets yet another symbol: - for subtraction, ÷ for division, √ for roots, log for logarithms.
This proliferation makes it impossible to see the systematic relationships. You can't glance at +, ×, and ^ and immediately understand that they're all the same operation at different levels of recursion. The notation actively hides the beautiful pattern that connects them.
Lack of Systematic Generation Rules
There are no rules for generating new operations. If you want to invent a new operation, you have to come up with a new symbol and hope it catches on. There's no systematic way to extend the notation to include operations like tetration, pentation, or their inverses. Each new operation requires inventing new notation from scratch.
This lack of systematicity means that mathematical notation doesn't scale. As we discover new operations or extend existing ones to new domains (like complex numbers or matrices), we're forced to invent ad-hoc notation that doesn't fit with the existing system. The result is a patchwork of symbols that becomes increasingly difficult to learn and use.
In contrast, our meta-recursive notation provides systematic generation rules. Want a new operation? Just use +n where n is the level of recursion. Want its inverse? Use +-n. The notation scales naturally and reveals the underlying structure of mathematics.
The Bigger Picture: A New Mathematical Language
With just three primitive operations—add(x, y), recurse(operation, n), and inverse(operation)—you can generate all of arithmetic. This is a minimal basis for mathematics. Every operation, no matter how complex, is just a combination of these three. The notation is canonical: there's one best way to write everything, and the relationships between operations are built right in. Patterns leap out at you. You can see, at a glance, how multiplication, exponentiation, and division are all just different flavors of addition.
This isn't just elegant—it's efficient. Computers love it. You only need to program three functions, and everything else falls out. Long division? That's just an algorithm for +-2. Memoization and recursion become natural. Algorithms and notation finally speak the same language.
Ordering and Non-Commutativity
But there's a twist. Not all operations are created equal. Addition is commutative: a + b = b + a. Subtraction isn't: a - b ≠ b - a. This lack of symmetry isn't a bug—it's a feature. It reveals deeper structure. Our notation preserves these relationships. When order matters, it's explicit. You can study commutativity and non-commutativity as properties, not assumptions. The structure of mathematics becomes transparent, not hidden behind arbitrary symbols.
Philosophical and Educational Implications
Imagine teaching math this way. Students would see the connections between operations immediately. The arbitrary nature of current notation would fall away. Pattern recognition would become second nature. The unity of mathematics would be visible from the start. Research would be more directed, less random. Notation would evolve toward greater efficiency. And maybe, just maybe, we'd see that addition, recursion, and inversion are the true atoms of mathematics.
Open Questions and Future Exploration
- What is the true meaning of
+0? Is it an identity, or something deeper? - What new operations lie between the integers—fractional or complex meta-recursions?
- How can this system be extended to other branches of mathematics?
- What are the computational limits and opportunities of this approach?
- Can this notation help us discover new mathematics, or even new physics?
The journey is just beginning. The structure is there, waiting to be explored. All you need is addition—and a little curiosity.
Acknowledgments
I am deeply grateful to Aaron Mazel-Gee for his thoughtful feedback and insights on these ideas. Many of the connections between recursion, notation, and mathematical structure were sharpened through our discussions.