Level A1 · Elementary

Survival

"The most incomprehensible thing about the world is that it is comprehensible." Albert Einstein

Before you can read a mathematical text, you need an alphabet — and a rule for how its symbols combine into ever more complex expressions. Level A1 is that alphabet: it teaches you to see the internal architecture of any formula, independently of whether you yet understand what it means.

What you'll learn

—Why the language of mathematics, $\Math$, has both alphabetic and ideographic symbols, and how graphemes and rigid graphemes differ.
—The Matryoshka principle: how every lexeme is built by nesting simpler lexemes into the placeholders of a dependent grapheme — and how to reverse the process with a parse tree.
—Why syntax and semantics are separate layers, and how a formal system can be reused for entirely unrelated subject matter without changing a single symbol.
—What reflection is, why it produces paradoxes like the heterological one, and how Gödel turned the same mechanism into a proof technique.
—The difference between a designation (a natural-language name) and a term, and how bound and free variables keep formulas from becoming ambiguous.

1. An Illustrative Tale

The chapter opens with a thought experiment: an AI-equipped probe lands on an exoplanet inhabited by "Mathematicians" who communicate through intricate symbols drawn directly in the air. The probe's task is to reverse-engineer their language from scratch — with no dictionary, no Rosetta Stone, only observed text.

It first notices the language is both alphabetic and ideographic: some symbols repeat constantly, others are rare but almost always decomposable into longer alphabetic sequences. Then it notices something more important — every complex expression has the internal structure of a nested doll: simple symbols slot into templates, which slot into larger templates, without any ceiling on depth. Once the probe can parse this structure, it moves to semantics: proposing interpretations, testing them for consistency, and building translators in both directions — exactly the arc this chapter then walks the reader through, symbol by symbol, for real mathematical notation.

2. The Matryoshka Principle, precisely

A lexeme of $\Math$ is defined recursively, in exactly two clauses:

Definition

(i) any independent grapheme — including the symbols of the alphabet — is a lexeme; and

(ii) the result of substituting lexemes into the placeholders of a dependent grapheme is a lexeme.

Watch it in action. Take the addition symbol — a dependent grapheme with two numeric placeholders — and substitute the alphabetic symbols $a$ and $b$:

\[a + b\]

Do the same with multiplication to get $c\cdot d$. Now take the addition symbol again, and substitute the two lexemes just built into its placeholders:

\[(a+b) + (c\cdot d)\]

This is the Matryoshka principle: complexity grows purely by nesting, with no upper bound — the same mechanism that lets you write a five-story power tower like $5^{4^{3^{2^1}}}$, or Cardano's formula for the cubic, and still have every expression parse back down, unambiguously, into a tree of the templates that built it. That unambiguity is not a convenience; it is the reason a compiler — or, today, the syntax a language model has to reconstruct implicitly from statistics — can recover a formula's structure at all.

The chapter also draws the line between a predicate (a lexeme denoting a relation — "$k$ has colour $c$") and a term (a lexeme denoting an object — an arithmetic expression), then shows how the very same formula, unchanged symbol for symbol, can describe crocodiles in Loch Ness or neutron stars in the Milky Way. Once a formal system is born from some semantics, it detaches from it — which is exactly why the same theorem, proved once, stays true everywhere its axioms happen to hold.

3. Try it yourself

The first five exercises from the chapter's problem set, from the English edition.

Draw a parse tree for the formula of the roots of a quadratic equation: $x = \dfrac{-b + \sqrt{b^2 - 4ac}}{2a}$. Indicate the hierarchy of templates used (fraction, addition, subtraction, square root, power, multiplication).
In mathematics, parentheses are often omitted based on operator precedence. Restore all omitted parentheses in the following expressions to make the structure unambiguous:
1. $a \cdot b + c \cdot d$;
2. $x^2 + 2xy + y^2$;
3. $a / b / c$ (assume standard left-to-right associativity);
4. $e^{x^2 + y}$.
Draw (or describe) the parse tree for Euler's identity, identifying the hierarchy of templates used: $e^{i\pi} + 1 = 0$.
Deconstruct the following nested expression into layers. What is the outermost "shell" (template), and what are the inner components? $\sqrt{1 + \sqrt{1 + \sqrt{1 + x}}}$
In the expression below, identify which symbols act as objects (nouns/variables/constants) and which act as actions/relations (verbs/operators): $\dfrac{d}{dx}\left(x^3+3x\right) = 3x^2+3$

Level A1 has 16 exercises in total, each with a full worked solution in the book's appendix.

The rest of Level A1 — and four more levels

This digest covers a fraction of what's in the chapter: reflection and the liar/barber/heterological paradoxes, the full treatment of designations and terminology, variable binding with worked examples from convolution to Plato's Academy, seven named features that separate $\Math$ from natural language, and sixteen exercises with complete solutions. Levels A2 through C1 take the same rigor all the way to the Axiom of Choice and the von Neumann universe.

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