Hilbert’s Early Math Foundations: More Than Just Models?

Hey there! Let’s talk about something super cool, something that sits right at the heart of how we think about mathematics itself. We’re diving into the world of David Hilbert, a giant figure whose work around the turn of the last century really shook things up in the foundations of math and logic. You know, the guy who gave us those famous problems that set the agenda for 20th-century mathematics?

His early stuff, especially on geometry and numbers, is legendary for bringing in the formal axiomatic method. Instead of thinking of axioms as obvious truths about space or numbers, Hilbert treated them more like rules of a game, defining the concepts implicitly. It was a real game-changer, moving towards a more abstract, structural view of mathematics. But here’s where it gets interesting: how exactly did Hilbert think about these axiomatic systems and what they meant? That’s been a bit of a puzzle, and there’s a standard way people have seen it for years.

The Standard Story: Hilbert as a Model Theory Pioneer

For a long time, the general consensus, what folks in the know call the “received view,” is that Hilbert’s early work was a major stepping stone towards modern model theory. You know, the part of logic where you study mathematical structures (the “models”) and whether logical sentences are “true” or “satisfied” in them. The argument goes that when Hilbert wanted to prove his axioms were consistent (didn’t lead to contradictions) or independent (you couldn’t derive one axiom from the others), he basically built “models.”

For instance, to show his geometry axioms were consistent, he’d use analytic geometry over a specific number field. He’d say, “Look, if you interpret points and lines as coordinates and equations in this number system, all the geometry axioms hold true. So, if you found a contradiction in geometry, you’d find one in the arithmetic of this number field too.” Since we trust that arithmetic is consistent (mostly!), this gives us confidence in the geometry.

Similarly, for independence, he’d construct a weird “analytic model” where most axioms held, but the one he wanted to prove independent failed. This technique, using these “analytic models,” looked a lot like the model-theoretic approach we use today. People like Jaakko Hintikka have strongly argued that Hilbert’s Foundations of Geometry was a key entry point for model-theoretic thinking into 20th-century logic and philosophy. It seemed like a done deal: Hilbert was a proto-model theorist.

This received view often rests on a few key ideas about Hilbert’s approach:

• The Subject-Matter Thesis: Axioms define *classes* of structures or systems. The real subject isn’t one specific thing (like intuitive space) but the abstract structure shared by all systems satisfying the axioms.
• The Invariance Thesis: All systems satisfying the axioms are equally valid interpretations. Isomorphic systems (those with the same structure, even if the “things” are different) are indistinguishable from the theory’s perspective.
• The Reinterpretation Thesis: The language used for the theory is “formal” in the sense that its basic terms (like “point” or “line”) don’t have a fixed meaning beforehand but get their meaning *from* the axioms and can be reinterpreted in different models.

This picture paints Hilbert as someone already thinking about mathematical languages whose terms are like placeholders, waiting to be assigned meaning in different structures. It’s a very modern, model-theoretic way of looking at things.

Hitting the Books: An Alternative View Emerges

But hold on! While the received view is widespread, some recent scholarly work has started poking holes in it. Turns out, when you look really closely at Hilbert’s actual words from that period, things aren’t quite so clear-cut. Some scholars, like Eder and Schiemer, have pointed out that even though Hilbert used terms like ‘true’ or ‘satisfied,’ his use wasn’t always as precise or as clearly model-theoretic as modern logicians might assume. Dean has even suggested reconstructing Hilbert’s geometry results using *syntactic* notions, focusing on proof and interpretation *between theories* rather than just models.

These alternative readings suggest that classifying Hilbert’s early methods as *prototypically* model-theoretic might be debatable. There are important conceptual nuances that need more attention. And that’s exactly what we’re digging into here. Our aim is to contribute to this debate and offer a different way to look at Hilbert’s early metatheory, one that might fit the historical evidence a bit better.

Our main idea is this: alongside the standard model-theoretic interpretation, there’s a perfectly good alternative reconstruction of Hilbert’s approach using the modern ideas of syntactic translations between languages and interpretability between theories. We think this view isn’t just consistent with what little textual evidence we have from Hilbert’s early work, but it might actually explain some of his later scattered remarks on modeling methods more adequately. Plus, it lines up nicely with how mathematicians in the late 19th century were already thinking about relating different mathematical theories – often using the idea of “translation.”

Translating Ideas: A Different Kind of Connection

So, how does this “translation” idea work? Instead of seeing Hilbert as primarily focused on interpreting a single, formal language in multiple structures (the model-theoretic way), we propose he was thinking more about relating *different mathematical languages*, each perhaps already having a fixed, intended interpretation. And the way you relate them is through a kind of translation.

Think about it. Mathematicians back then often talked about showing that one theory was “equivalent” to another by setting up a correspondence between their elements and relations. Henri Poincaré, for example, talked about a “translation dictionary” to show how hyperbolic geometry could be modeled within Euclidean geometry. Richard Dedekind, another giant, used the idea of a “translation mapping” in his famous proof that all systems of natural numbers are essentially the same (isomorphic).

These examples suggest that the idea of “translation” was floating around as a way to understand the relationship between different mathematical systems or theories. We want to clarify this idea and show how it might apply to Hilbert. We’ll distinguish between a couple of types of translations relevant to his work.

Dedekind’s Blueprint: Translational Isomorphism

To really get a handle on this, let’s take a quick detour to Dedekind’s work on natural numbers from 1888. Dedekind defined natural numbers abstractly as “simply infinite systems.” This was a structural definition, much like Hilbert’s later axiomatic systems. A simply infinite system is basically a set with a starting element and a successor function that satisfies certain properties.

Dedekind proved a crucial result: any two simply infinite systems are “similar,” which is the same as being isomorphic in modern terms. There’s a one-to-one mapping between them that preserves the starting element and the successor relation. But here’s the cool part: Dedekind explicitly said that this mapping “effects” a “transfer” or “translation” of theorems from one system to another. He even used the phrase “the translation of an arithmetical theorem from one language into another.”

This gives us the idea of a “translational isomorphism.” It’s an isomorphism (a structure-preserving map between systems) that *induces* a translation between the languages used to describe those systems. In a modern reconstruction, you can think of each isomorphic system (like the standard natural numbers N and some other simply infinite system Omega) as having its own “canonical language.” These languages are “fully interpreted” – their terms refer directly to the objects and relations in that specific system. The isomorphism between the systems then provides a rule for translating sentences from the language of N to the language of Omega, preserving truth or theoremhood.

Hilbert’s Own Words: Isomorphism and Translation in Analysis

Now, let’s bring it back to Hilbert. While his early remarks are sparse, there’s a fascinating passage in his 1905 lecture notes that echoes Dedekind’s idea. Talking about his axioms for the real numbers (which define the system uniquely up to isomorphism), Hilbert says that if any system satisfies these axioms, “for us they are none other than the real numbers.” He adds, “How they are named and labeled, it does not matter at all, because the numbers have very different names in the different languages. The essential is that between two of the systems… there is a one-to-one reversible relation… Then we say too, that the things are the same… only the naming is different. [*] [This] means a translation into another language.”

Boom! There it is. Hilbert explicitly connects isomorphism (“one-to-one reversible relation”) with “translation into another language.” This strongly suggests that, at least in the context of theories that uniquely determine their models up to isomorphism (like the real numbers), Hilbert was thinking in terms of translational isomorphisms between *fully interpreted canonical languages*, just like Dedekind. This is different from the received view’s idea of a single formal language reinterpreted in different structures. Here, each structure comes with its own language, and the structures are related via isomorphism, which in turn relates the languages via translation.

So, for analysis, we can reconstruct Hilbert’s view: the axioms describe a structure type. Any system satisfying the axioms is “the same” (isomorphic) as any other. This isomorphism isn’t just a map between abstract structures; it’s a map that lets you translate statements from the language of one system to the language of another. Both languages are seen as fully interpreted, referring to the specific objects and relations in their respective systems.

Geometry’s New Look: Translation and Interpretability

But what about Hilbert’s famous consistency and independence proofs in geometry? These involved constructing “analytic models” like coordinate geometry over specific number fields (like the Pythagorean field). This seems different from the translational isomorphism idea, as the languages involved (geometry and arithmetic) have very different basic terms (points/lines vs. numbers). The structures aren’t simply isomorphic copies of each other.

Here, we can use a more general notion: syntactic translation between languages of different signatures and the related concept of interpretability between theories. A translation in this sense involves defining the basic elements and relations of one theory (geometry) using formulas in the language of another theory (arithmetic). For example, defining points as pairs of numbers, lines as equations, and geometric relations (like incidence or betweenness) using arithmetic formulas.

Once you have this translation, you can show that if the axioms of geometry are translated using these definitions, they become theorems (or true statements) in the arithmetic theory. This method allows you to construct an “inner model” of geometry *within* the arithmetic structure. Points and lines in this geometric model are not the original geometric objects, but specific definable sets or tuples within the number field structure. The geometric relations are the arithmetic formulas that define them.

Hilbert’s consistency proof for geometry relative to arithmetic can be seen as showing that his geometry theory is *interpretable* in the theory of the Pythagorean field. If the arithmetic theory is consistent, then the geometry theory must also be consistent, because any contradiction in geometry would translate into a contradiction in arithmetic via this interpretation. This aligns perfectly with Hilbert’s own description of the “method of arithmetization” in his later work (1934), where he talks about representing geometric objects and relations by numbers and equations, and how axioms “turn out either as arithmetical identities or provable sentences” based on this “translation.”

So, in geometry, the connection isn’t necessarily a direct translational isomorphism between models of the *same* theory, but a translation between the *languages* of two different theories (geometry and arithmetic) that establishes an interpretability relation between them. This translation allows you to build a model of the first theory *inside* a model of the second theory.

Wrapping It Up

Okay, let’s bring it all together. While it’s true that Hilbert used methods that *look* like precursors to model theory (using “models” to prove things about axioms), a closer look suggests his underlying conceptual framework might have been different from the modern one. The standard “received view” sees him working with formal, reinterpretable languages and interpreting them in different structures.

Our alternative reconstruction argues that Hilbert was likely thinking in terms of fully interpreted canonical languages, each tied to a specific system or structure. The relationships between these systems and the theories describing them were understood through syntactic translations. In cases like analysis, where systems are isomorphic, these were “translational isomorphisms” inducing translations between structurally similar languages. In cases like geometry, where one theory is modeled within another, it involved translations between languages of different types, establishing an interpretability relation between the theories.

This view, focusing on translation and interpretability between theories formulated in canonical languages, seems to fit better with Hilbert’s scattered remarks and the mathematical practices of his time. It shows how his work anticipated modern logical concepts, not necessarily by inventing model theory as we know it, but by exploring the powerful idea of relating mathematical systems and theories through systematic linguistic translations. It’s a fascinating glimpse into the evolving ideas that shaped the foundations of modern logic and mathematics!

Source: Springer