Neo-Logicism and Hume's Principle

1. Frege's Fork in the Road

As you'll recall, Frege's goal was to demonstrate that the truths of arithmetic are analytic, meaning they are derivable from pure logic and definitions alone. His strategy in his magnum opus, Grundgesetze der Arithmetik (Basic Laws of Arithmetic), was to define numbers as objects—specifically, as the extensions (classes) of concepts. This path led him to posit the infamous Basic Law V, which governed the identity of these extensions and ultimately proved inconsistent.

However, in his earlier work, Die Grundlagen der Arithmetik (The Foundations of Arithmetic), Frege had explored a different route. He noticed a fundamental connection between the concept of number and the concept of one-to-one correspondence. This insight is the starting point for neo-logicism.

The following video excerpts from a lecture by the influential philosopher Michael Dummett will set the stage. Dummett was a leading scholar of Frege and a key figure in the debates we will be discussing.

The neo-logicist insight, championed by Crispin Wright in the 1980s, is this: What if Frege made a fatal mistake in abandoning that "fundamental equivalence"? What if that principle itself, rather than the disastrous Basic Law V, is the proper logical foundation for arithmetic?

2. Abstraction Principles and Hume's Principle

The principle Dummett called the "fundamental equivalence" is now widely known as Hume's Principle (HP). It belongs to a class of principles called abstraction principles.

An abstraction principle is a statement of the following general form:

Let's break this down:

• $\alpha$ and $\beta$ are variables for entities of a certain type (e.g., lines, concepts).
• $E$ is an equivalence relation on those entities (a relation that is reflexive, symmetric, and transitive).
• @ is an abstraction operator that maps an entity $\alpha$ to a new object, @(\alpha).

The principle states that the abstract object associated with $\alpha$ is identical to the abstract object associated with $\beta$ if and only if $\alpha$ and $\beta$ stand in the equivalence relation $E$. In essence, it provides identity conditions for a new kind of abstract object.

A classic toy example is the Directions Principle:

The direction of line $l_1$ = the direction of line $l_2$ $\leftrightarrow$ line $l_1$ is parallel to line $l_2$.

Here, the equivalence relation is parallelism, and the abstraction operator gives us the concept of direction.

Hume's Principle is a much more powerful abstraction principle where the entities being related are concepts (or properties). It states:

The number of Fs is the same as the number of Gs if and only if the Fs can be put into a one-to-one correspondence with the Gs.

Formally, using second-order logic (where we can quantify over concepts like F and G):

Here:

• F and G are variables for concepts.
• ≈ is the equivalence relation of equinumerosity (one-to-one correspondence), which is definable in pure second-order logic.
• # is the abstraction operator "the number of...".

The following reading provides a formal introduction to abstraction principles and the central role of Hume's Principle.

Frege's Theorem

Why is Hume's Principle so important? Because of a remarkable result known as Frege's Theorem:

All the axioms of second-order Peano Arithmetic can be proven as theorems within second-order logic augmented with Hume's Principle as a single axiom.

Frege himself sketched the proof, and it has since been rigorously formalized. This is a profound result. It shows that the entirety of arithmetic structure flows directly from this one seemingly simple principle of abstraction. It shows that HP is mathematically sufficient for the logicist project. For example, it allows for a "bootstrapping" of the natural numbers:

1. Define 0 as the number of the concept 'is not identical to itself': $0{=}_{df}\#(x\ne x)$. This concept applies to nothing.
2. Define 1 as the number of the concept 'is identical to 0': $1{=}_{df}\#(x=0)$. This concept applies to exactly one thing (the number 0, whose existence was just secured).
3. Define 2 as the number of the concept 'is identical to 0 or is identical to 1': $2{=}_{df}\#(x=0\vee x=1)$. And so on.

The existence of the infinite series of natural numbers is thus derivable from HP plus logic. The question that remains is philosophical: what is the status of Hume's Principle?

3. The Neo-Logicist Philosophical Claim

The neo-logicist, following Crispin Wright, argues that Hume's Principle is not just another axiom. Instead, it should be regarded as an implicit definition of the concept of cardinal number. It has a special status, akin to being analytic, that justifies our accepting it without further proof.

How can a principle that entails the existence of an infinite number of objects (the numbers) be considered a definition? The argument rests on a philosophical view about the relationship between language and reality, sometimes called the syntactic priority thesis.

The core idea you just read is a reversal of the common-sense picture of reference.

• Common Sense: First, there are objects in the world. Then, we invent singular terms (names) to refer to them.
• Syntactic Priority: First, we establish rules for a class of singular terms and criteria for the truth of sentences containing them. If those rules are satisfied and yield true sentences, it follows that the terms refer to objects.

Hume's Principle is proposed as just such a rule. It stipulates how to use terms of the form #(F). It fixes their identity conditions. And because the right-hand side of HP can be known to be true by logical means alone (e.g., it is a logical truth that the concept 'is not self-identical' is equinumerous with itself), the principle guarantees that statements like #(x≠x) = #(x≠x) are true. By the syntactic priority thesis, it follows that the term #(x≠x) (our definition of 0) successfully refers to an object.

On this view, Hume's Principle doesn't describe a pre-existing reality of numbers. It constitutes the concept of number and guarantees the existence of the objects that fall under it.

Conclusion

Today we have explored the revival of Frege's logicism in its modern, neo-logicist form. We have seen how this program seeks to avoid the paradoxes of the original by returning to a principle Frege himself discovered but discarded.

Key Takeaways:

• Core Tenet: Neo-logicism grounds arithmetic in second-order logic plus Hume's Principle (HP), not the inconsistent Basic Law V.
• Hume's Principle: An abstraction principle stating that the number of Fs equals the number of Gs if and only if the Fs and Gs are equinumerous ($\#(F)=\#(G)\leftrightarrow F\approx G$).
• Frege's Theorem: This mathematical result shows that HP is powerful enough to derive all the standard axioms of arithmetic, securing the technical side of the logicist project.
• Philosophical Foundation: Neo-logicists argue that HP has the epistemic status of an analytic, implicit definition of cardinal number. Combined with a "syntactic priority" view of reference, this allows HP to ground our knowledge of numbers and guarantee their existence as logical objects.

Preview of the Next Lesson:

The neo-logicist program offers an elegant and powerful foundation for arithmetic. However, it is not without its own deep philosophical challenges. Does granting Hume's Principle this special status open the door to other, less benign abstraction principles? And does it truly solve the problem of what numbers are? In our next lesson, we will critically evaluate the most significant challenges to neo-logicism, including the "Bad Company" objection and the lingering Caesar Problem.