One Object, Many Languages

1. The Algebra of Translation: A Periodic Table

Any translation between mathematical theories is characterized by two main coordinates: information preservation (rows 1, 2, 3) and the direction of the arrows (columns A, B). Inside every cell a third dimension appears (I, II, III) — namely, exactly what we work with: formalism (equations, axioms) or semantics (spaces, the object's reality).

2. The Translation Method (Pipeline)

Any coordinate from the table above is used to carry problems between theories. The working scheme has four steps:

This four-step scheme is an ideal. Its weak point is step 4 (back-translation). For it to work, language \(\mathcal D\) must not forget what was said in language \(\mathcal C\). For the first three kinds of translation (row 1: bijections, equivalences, dualities) reversibility is guaranteed, and the scheme works flawlessly. But losses grow as we move down the list: embedding into a specific class (row 2) introduces untranslatable "extra" structure, and at the level of invariants (row 3) step 4 disappears completely — once the structure has been discarded, there is no going back.

3. Catalog of Translations (Examples)

Below is a survey of classical mathematical results, each tagged with its exact coordinates in the Periodic Table.

• Arithmetization of syntax (Gödel, 1931). Formulas, proofs, and rules are coded by numbers; metamathematics becomes arithmetic, and \(\mathsf{PA}\) gains the ability to talk about itself.
• Arithmetic and finite set theory. Let \(\mathsf{HF}\) be the theory of hereditarily finite sets (no axiom of infinity, but with transitive closure). Sets are coded by numbers via Ackermann's encoding (through binary digits), and numbers, in turn, by ordinals. The theories are bi-interpretable: the round-trip composition of translations is, within the theory, provably isomorphic to the identity. This is, quite literally, one object in two languages.

Objects of different nature turn out to be one and the same object (isomorphism), or a single structure is given by two vocabularies at once (definitional equivalence).

We stay within the text of the axioms, but reverse the order. Every true statement rewrites into its dual.

• The Boolean principle (De Morgan). The map \(x\mapsto\neg x\) swaps \(0\) and \(1\), \(\land\) and \(\lor\). The list of Boolean algebra axioms maps to itself. De Morgan's laws and the interdefinability of the quantifiers (\(\forall x\,\varphi\leftrightarrow\neg\exists x\,\neg\varphi\)) are a direct consequence.
• Axioms of the projective plane. "Two points determine a line" ↔ "two lines meet in a point" — the axiom list is self-dual in exactly the same way Boolean algebra is, except the role of negation is played by polarity with respect to a fixed conic.

Polarity with respect to the conic itself sends a point \(P\) on the conic to the tangent \(t_P\) at that point, and sends the conic to itself. So the inscribed hexagon turns into the circumscribed one, the side \(P_iP_{i+1}\) turns into the vertex \(Q_i=t_i\cap t_{i+1}\), the intersection point of opposite sides turns into the diagonal joining opposite vertices, and "three points are collinear" turns into "three lines are concurrent."

The same polarity works one dimension up — no longer with respect to a conic in the plane, but with respect to the sphere itself. On the unit sphere \(S^2\subset\mathbb R^3\), the role of "lines" is played by great circles — the sections of the sphere by planes through its center.

Just as with the conic, this polarity carries an entire figure to its dual. Take a spherical triangle \(ABC\) (sides are arcs of great circles) and its polar triangle \(A^{*}B^{*}C^{*}\), where \(A^{*}\) is the pole of side \(a=BC\), and so on. The pole of a side of the original triangle turns out to be a vertex of the dual one — exactly the same "side ↔ vertex" transformation as with Pascal and Brianchon.

The highest art of translation. Structures of different nature are linked functorially, with an exact reversal of order.

Formalism embeds into formalism, preserving derivability, but without any way of strictly returning to the original.

• \(\mathsf{ZF}\) and \(\mathsf{ZFC}\). Zermelo–Fraenkel set theory (without choice) and the same theory with choice are mutually interpretable. \(\mathsf{ZFC}\) interprets \(\mathsf{ZF}\) trivially, while \(\mathsf{ZF}\) interprets \(\mathsf{ZFC}\) within itself via Gödel's constructible universe \(L\). Yet they are not bi-interpretable (Ali Enayat, 2016). The second translation returns us to a narrowed, "purified" copy of the original theory (\(L\)), not to the original universe of sets.

The abstract is realized concretely. The translation carries information into a more tangible medium, but adds "scaffolding" to the object (a basis, coordinates, the specifics of a set).

• Cayley's theorem. Every group (a semantic object given by axioms) embeds into the symmetric group \(\operatorname{Sym}(G)\) via left translations. The same holds for linear representations (embedding into matrices). The abstract becomes concrete.
• The Cayley graph. The same group \(G\), with a fixed set of generators \(S\), becomes a labeled graph: vertices are the elements of \(G\), edges are multiplication by a generator. Algebra turns into geometry — a metric appears on the group, and one can study its growth or build expanders from it. The translation depends on the choice of \(S\) — the same "extra" structure as in Cayley's theorem above.

Words become objects. We take formalism and turn it into a semantic object.

• Free algebras. We take pure formalism (letters of an alphabet and operation symbols, out of which we assemble "words" — terms) and look at them as mathematical objects: trees. Factoring these by the desired axioms yields a free algebra.
• Henkin models. The apotheosis of this approach in logic. In proving the completeness theorem, the elements of the model are declared to be the closed terms of the formal language itself, and predicates are defined by their provability. Syntax literally turns into semantics.

Antitone translations on a single ontological level.

• The Galois correspondence for fields. The correspondence between intermediate fields \(K \subset M \subset L\) and subgroups of \(\operatorname{Gal}(L/K)\). The larger the field, the smaller the group of automorphisms that fix it pointwise.
• The Krasner correspondence (Automorphism groups). The more symmetries (automorphisms) a structure has, the fewer predicates (definability lattices) can be placed on it without parameters.

The same antitonicity as above, but now between syntax and semantics.

• Syntax and models (\(\mathrm{Th}\) and \(\mathrm{Mod}\)). More axioms, fewer models. The closed elements on the left are the deductively closed theories; on the right, the axiomatizable classes of models.
• The Nullstellensatz. The geometric twin of Th/Mod: a Galois connection between ideals of polynomials (formalism) and algebraic sets (semantics) over an algebraically closed field.

Information loss as a working method. The assignment is not reversible, but what gets discarded does not get in the way of computation.

• The fundamental group \(\pi_1(X)\) and homology. A topological invariant discards all metric information, leaving only the algebraic "shadow" of the space. Back-translation is impossible (spaces with the same \(\pi_1\) need not be homeomorphic), but if the groups differ, the spaces are guaranteed not to be homeomorphic. The Euler characteristic compresses the homology groups themselves down to a single number.

Compressing formalism into a mathematical measuring object.

• Proof-theoretic ordinals. Every formal theory can be assigned an ordinal (for Peano arithmetic it is \(\epsilon_0\)). We discard the entire deductive richness of the theory, keeping only its semantic "shadow," which measures its strength. A translation with colossal information loss (different theories can share the same ordinal), but if the ordinal of \(T\) is strictly smaller than the ordinal of \(S\), then \(T\) cannot prove the consistency of \(S\).

It sounds paradoxical (we compress information, yet the arrows reverse), but behind it lies a geometric idea: observation through functions.

• Cohomology and the restriction of functions. If a space \(X\) is embedded in an ambient space \(Y\) (\(X \subset Y\)), any function on all of \(Y\) can be restricted to \(X\). The embedding of spaces generates a map of function algebras running the other way (\(F(Y) \to F(X)\)). De Rham cohomology integrates differential forms ("functions") over cycles: the space is compressed into an algebraic invariant, but the arrows of the morphisms reverse.

4. Layering: Fourier, Laplace, and Series

The most powerful tools in engineering and mathematics work because they punch through several cells of our table at once.

Why are generating functions, Fourier series, and the Laplace transform so effective? In all of these translations the invariant core is the same: convolution turns into pointwise multiplication, and differentiation turns into multiplication by the variable. A differential equation becomes algebraic, is solved by division, and back-translation (Step 4) returns the answer.

This trick is a layer cake of three translations from the Periodic Table:

• 1A-II The algebraic layer (Isomorphism). The ring of formal power series uses convolution by definition. Passing from a sequence to a series is just relabeling. The power is that nobody knows how to work with convolution by hand, while everybody can work with polynomials.
• 1B-II The dual layer. The Fourier series works because it rests on Pontryagin duality between a function on the circle \(\mathbb T\) and a sequence on \(\mathbb Z\).
• 2A-II The semantic representation. A space with convolution is a Banach algebra. Its Gelfand representation (translating an abstract element into a function on the spectrum) is exactly the Fourier transform! "Convolution became multiplication" simply because the Gelfand representation preserves the operations.

Where Translation Breaks

A summary of the hypotheses most often dropped from statements. Each row is a place where the translation stops working once the condition is lifted.