Introduction
Hello! Welcome to the first lesson of our new module, "Formal Systems and Their Philosophical Limits." In our last discussion, we evaluated the challenges to neo-logicism, particularly how Hume's Principle functions as an implicit definition of the concept of number. This idea—that a set of statements can define a concept by specifying its structural role—is a powerful and controversial one, and it serves as a perfect entry point into today's topic.
Our learning outcome is to analyse the Frege-Hilbert controversy on the nature of axioms as truths versus implicit definitions. This famous dispute, which took place through a series of letters and publications around 1900, represents a fundamental clash between two visions of what a mathematical theory is and what its foundations should look like.
The central question is this: Are the axioms of a mathematical theory, like geometry, a set of fundamental truths about a specific, pre-existing subject matter (Frege's view)? Or are they a set of stipulations that implicitly define an abstract structure, where the primitive terms are merely placeholders for any objects that satisfy the axiomatic relations (Hilbert's view)? This isn't just a technical disagreement; it's a profound philosophical debate about the nature of mathematical truth, meaning, and existence.
Hilbert's Modern View: Axioms as Implicit Definitions
The controversy was sparked by David Hilbert's groundbreaking work, Grundlagen der Geometrie (Foundations of Geometry), published in 1899. Hilbert's goal was to provide a fully rigorous axiomatisation for Euclidean geometry, but his method was revolutionary. He treated the axioms not as self-evident truths, but as a formal system whose logical properties—like consistency and the independence of its axioms—could be rigorously investigated.
To understand Hilbert's approach, please read the following introduction to his project.
Frege-Hilbert controversy - Stanford Encyclopedia of Philosophy
This section from the Stanford Encyclopedia of Philosophy entry 'The Frege-Hilbert Controversy' explains the innovative technique Hilbert used in his Foundations of Geometry.
Please read Section 2, 'Hilbert’s Foundations of Geometry'. Focus on understanding the core of Hilbert's method: using the reinterpretation of geometric terms to prove relative consistency and independence.
As you've just read, Hilbert's key technique was reinterpretation. To prove that the axiom of parallels is independent of the other axioms, he would construct a "model" or interpretation where the other axioms hold true, but the parallel axiom is false. He famously did this by reinterpreting "points", "lines", etc., in terms of concepts from real analysis (algebraic numbers and functions).
This method reveals Hilbert's radical conception of axioms. If the axioms were truths about a unique, intended domain of geometric objects, then reinterpreting them to be about numbers would be irrelevant. For Hilbert, the specific meaning of terms like "point" and "line" did not matter for the logical investigation. The axioms merely define a "scaffolding or schema of concepts." This is the essence of his view of axioms as implicit definitions. In a famous letter to Frege, he wrote:
"it is surely obvious that every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes. If in speaking of my points I think of some system of things, e.g., the system: love, law, chimney-sweep … and then assume all my axioms as relations between these things, then my propositions, e.g., Pythagoras’ theorem, are also valid for these things."
The axioms don't describe points; they define what it is to be a system of points. Any collection of objects that satisfies the axioms is a valid instance of the defined structure.
Formalising Hilbert's Idea: Structural Definitions
The term "implicit definition" can be ambiguous. Modern logicians and philosophers have refined the terminology to better capture Hilbert's idea. What Hilbert was pioneering is now often called a structural definition. Let's make this concept more precise.
[PDF] What are Implicit Definitions?
The following text from a paper by Giovannini and Schiemer provides a clear, formal distinction between two kinds of implicit definition. This will help us precisely locate Hilbert's view within modern terminology.
Please read Section 2, 'What Axiom Systems Define', and Section 3.1, 'Structural Definitions of Primitive Terms'. \nIn Section 2, focus on the distinction between implicit definitions in the strict sense (which uniquely define one term relative to others) and structural definitions (which holistically define all primitive terms of a theory). \nIn Section 3.1, note how Hilbert's own words are used to illustrate his view that axioms bestow meaning on primitive terms by establishing their relations within a system.
This reading clarifies the heart of Hilbert's position. He is not providing an "implicit definition in the strict sense," where one new term is uniquely fixed by old ones. Instead, an entire axiom system acts as a structural definition. It holistically specifies a type of structure (e.g., a Euclidean space, a complete ordered field) by laying down the relations that must hold between its components. The consistency of the axioms is the guarantee that such a structure is possible—that the definition is not incoherent.
Frege's Classical View: Axioms as True Thoughts
Frege, coming from a very different philosophical tradition, found Hilbert's approach to be a source of profound confusion. For Frege, the goal of logic and mathematics was to secure objective truth.
A central tenet of Frege's philosophy is his staunch anti-psychologism. The laws of logic are not "laws of thought" describing how humans happen to reason; they are objective, timeless truths.
Frege, Russell, & Modern Logic - A. J. Ayer & Bryan Magee (1987)
In this brief clip from a discussion with A. J. Ayer, the importance of Frege's anti-psychologism is highlighted. This commitment to objectivity is crucial for understanding his reaction to Hilbert.
Please watch from 06:09 to 07:04.
Given this commitment to objective truth, Frege believed that mathematical sentences were meaningful only because they expressed determinate thoughts (what we would now call propositions). A thought has a fixed subject matter and is timelessly true or false.
From this perspective, Hilbert's methods were deeply flawed. Let's explore why.
Frege-Hilbert controversy - Stanford Encyclopedia of Philosophy
This next section from the SEP article details Frege's objections to Hilbert's project.
Please read Section 3, 'Frege—Background And Initial Differences'. Pay close attention to why Frege believes: Hilbert is confusing sentences with the thoughts they express. Reinterpretation simply changes the subject and cannot prove anything about the original theory. Axioms must be true, and their consistency is a consequence of their truth, not something that needs a separate proof.
To summarise Frege's position:
- Axioms must be truths. An axiom is the starting point of a science; it must express a true thought about that science's subject matter. For geometry, axioms must be truths about space.
- Definitions must fix reference. A proper definition must uniquely determine the object or concept being defined. Hilbert's "definitions" leave their terms ("point," "line") completely indeterminate.
- "Reinterpretation" is incoherent. When Hilbert reinterprets geometric axioms using real numbers, Frege argues he is simply changing the topic. Proving that a set of sentences expresses truths about numbers does nothing to establish the consistency of the original geometric thoughts.
- Truth implies consistency. For Frege, the demand to prove the consistency of a set of axioms is misguided. If the axioms are true, they cannot possibly contradict each other. A consistency proof is therefore superfluous. The real task is to secure the truth of the axioms.
Analysing the Core of the Disagreement
The Frege-Hilbert debate is often characterised as a clash between the "old" and the "new" way of doing mathematics. At its heart lies a disagreement about the very nature of a formal theory. Frege represents the contentful or fixed-domain conception, while Hilbert champions the modern structuralist conception.
But was Frege simply failing to grasp Hilbert's modern, formal approach? A more charitable interpretation suggests the disagreement runs even deeper, down to their very conceptions of logic and entailment.
Frege-Hilbert controversy - Stanford Encyclopedia of Philosophy
This final reading unpacks the deeper philosophical disagreement, suggesting that Frege and Hilbert may have been working with fundamentally different notions of logical consequence.
Please read Section 4, 'The Deeper Disagreement'. Focus on the idea that for Frege, logical entailment might be sensitive to the content of the concepts involved, not just their formal structure. The 'nightmare/dream' example is an excellent illustration of this point.
This section offers a compelling way to understand Frege's resistance. For Hilbert, logic is purely formal. If a contradiction isn't derivable from a set of sentences treated as uninterpreted strings, the set is consistent. The meanings of non-logical terms like 'point' or 'nightmare' are irrelevant to the formal proof machinery.
For Frege, however, logical relations are relations between thoughts. The content of the concepts within those thoughts matters. The thought Jones had a nightmare might logically entail Jones had a dream not because of the formal syntax Nj → Dj, but because the very concept of nightmare contains the concept of dream. An analysis of the concepts can reveal logical entailments that are invisible at the purely formal level.
From this perspective, Hilbert's consistency proofs are inadequate. He only shows that a set of sentences is formally consistent. He hasn't shown that the thoughts they express are consistent, because a deeper conceptual analysis might reveal a hidden contradiction.
Conclusion
Today we have dissected one of the most important philosophical turning points in modern logic and mathematics. The Frege-Hilbert controversy forced a clarification of what we mean when we talk about axioms, definitions, and mathematical theories.
Key Takeaways:
- Hilbert's View: Axioms are implicit (or structural) definitions. They are not truths about a specific domain, but schemata that define a type of structure. Their key virtue is consistency, which is demonstrated by finding a model that satisfies them. This view is the forerunner of modern model theory.
- Frege's View: Axioms are foundational truths that express determinate thoughts about a fixed subject matter. Their key virtue is truth, from which consistency follows as a corollary. Reinterpretation is a confusion between a sentence and the thought it expresses.
- The Core Conflict: The debate represents a clash between a contentful, fixed-domain conception of mathematical theories (Frege) and a formal, structuralist conception (Hilbert).
- Deeper Divide: Underlying this is a potential disagreement about the nature of logical consequence itself: Is it purely formal (Hilbert), or is it sensitive to the conceptual content of the terms involved (Frege)?
Ultimately, Hilbert's view became the dominant paradigm for 20th-century mathematics. The practice of defining structures axiomatically and studying their properties via model theory is now standard. However, the philosophical questions Frege raised have not disappeared; they continue to surface in debates about mathematical structuralism and the foundations of mathematics.
Preview of the Next Lesson:
Hilbert's victory in this debate emboldened him to propose a grand vision for securing the foundations of all of mathematics using his formal, axiomatic method. In our next lesson, we will examine the philosophical goals of Hilbert's Programme, his ambitious project to prove the consistency of all of mathematics using only finitistic, secure methods.