Challenges to Neo-Logicism

1. The Julius Caesar Problem: What is a Number?

Frege himself was the first to identify this problem. It was his primary reason for abandoning Hume's Principle in favour of the ill-fated Basic Law V. The problem is subtle but profound.

Recall the form of Hume's Principle:
$\#(F)=\#(G)\leftrightarrow F\approx G$
This principle gives us a criterion for the identity of two objects when both are presented to us as numbers (i.e., via the '#' operator). It tells us that the number of moons of Mars is identical to the number of my hands if and only if the set of Martian moons can be put in one-to-one correspondence with the set of my hands.

But what about an identity statement where only one side is a number-term? For example:
$\#(F)=\text{Julius Caesar}$
Does Hume's Principle tell us whether this is true or false? It does not. The principle is silent. This is the Julius Caesar Problem.

A proper definition of a concept, Frege argued, must provide criteria that allow us to determine, for any object whatsoever, whether it falls under that concept. If HP is the definition of cardinal number, it should give us the means to decide whether Julius Caesar is a number or not. Since it fails to do so, Frege concluded it was an inadequate definition.

To get a clearer grasp of this objection, please watch the following video.

As Potter explains, the problem is general. For any abstraction principle, it provides identity conditions within the domain of the new abstracts, but fails to demarcate that domain from everything else.

Neo-Logicism's Responses

Unlike Frege, neo-logicists cannot simply abandon Hume's Principle. So, how do they respond to the Caesar Problem?

As you've just read, the main responses are:

1. The Structuralist Response: Argue that the problem doesn't matter. As long as an object plays the role of the number 2 within the arithmetical structure generated by HP, its "real" identity is irrelevant. Arithmetic is about structure, not the intrinsic nature of the objects that instantiate it.
2. The Reformulation Response: Try to build a solution into the abstraction principle itself, for instance by using more complex logical tools like modal operators.
3. The Metaphysical Response: This is the most influential approach, championed by Crispin Wright and Bob Hale. They argue that HP alone isn't meant to solve the problem. It must be supplemented with general metaphysical principles. They argue that 'number' and 'person' are different fundamental categories of object (maximal sortal concepts). A basic metaphysical truth, they claim, is that objects from different categories cannot be identical. Thus, Julius Caesar cannot be a number for the same reason a symphony cannot be a prime number—they belong to different kinds of thing.

The Caesar problem, and its more general version concerning identities between abstracts from different principles (the $\mathbb{C}\text{-}\mathbb{R}$ Problem mentioned in the subsequent section of that IEP article), reveals that abstraction principles do not operate in a philosophical vacuum. Their adequacy as definitions seems to depend on a wider metaphysical framework.

2. The Bad Company Objection

The second major challenge is perhaps even more damaging to the philosophical claims of neo-logicism. The core of the neo-logicist project is that HP has a special epistemic status: it is an analytic truth, an implicit definition that we are justified in accepting without further proof.

The Bad Company Objection challenges this claim. It asks: If we can grant this special status to HP, why not to other abstraction principles of the exact same logical form? The problem is that HP keeps some very "bad company."

• Inconsistent Company: Frege's own Basic Law V has the same abstractive form as HP, but as Russell's paradox showed, it is inconsistent. If we can't tell the good from the bad on form alone, on what grounds do we anoint HP as special?
• Incompatible Company: More troubling is the existence of abstraction principles that are individually consistent but mutually inconsistent. That is, we can't hold them all to be true at the same time.

This means the neo-logicist needs a principled, non-arbitrary criterion to separate the "good" principles (like HP) from the "bad" ones. The search for this criterion is the central task posed by the Bad Company objection.

Let's summarise the dialectic you just read:

1. The Problem: We can't just accept any principle of the form $@(\alpha )=@(\beta )\leftrightarrow E(\alpha ,\beta )$ because some are inconsistent (e.g., Basic Law V).
2. Attempted Solution 1: Only accept consistent abstraction principles.
3. Refutation 1: This fails. The Nuisance Principle is consistent, but it is incompatible with Hume's Principle. HP requires an infinite domain to model it, while the Nuisance Principle only has finite models. We cannot accept both, so consistency alone is not enough to select the "good" principles.
4. Attempted Solution 2: Only accept principles that are conservative. The idea is that a legitimate abstraction principle should only introduce its new objects; it shouldn't have surprising consequences for other, unrelated domains. For example, a principle about numbers shouldn't suddenly prove new facts about electrons or zebras.
5. Refutation 2: This also fails. As the article explains, there are pairs of abstraction principles that are both consistent and conservative, yet are still incompatible with each other.

The unresolved nature of the Bad Company objection strikes at the heart of the neo-logicist claim. If there is no clear, logical criterion for distinguishing HP from its "bad" or "incompatible" companions, then the decision to accept HP seems arbitrary or, at best, pragmatic. It undermines the argument that HP is a truth of logic or analyticity, suggesting it is more like a powerful mathematical axiom we choose to adopt for its fruitful consequences—a choice that requires justification, rather than being a justification in itself.

Conclusion

Today we have grappled with the two most significant philosophical hurdles for neo-logicism. While the program has an undeniable technical elegance, these problems challenge its claim to have provided a secure philosophical foundation for arithmetic.

Key Takeaways:

• The Julius Caesar Problem questions whether Hume's Principle can function as a complete definition of number. It provides identity conditions for numbers but fails to demarcate them from non-numbers (like Julius Caesar). Neo-logicists typically appeal to a wider metaphysical framework of categories to resolve this.
• The Bad Company Objection questions the special epistemic status of Hume's Principle. There is no obvious, formal criterion that distinguishes HP from other, incompatible abstraction principles. This undermines the claim that HP is a self-evident analytic truth, making its adoption seem more like an axiomatic choice.
• Together, these challenges suggest that the reduction of arithmetic to "logic + definitions" is not as straightforward as neo-logicism hopes. The "definition" part (HP) seems to require justification that may not be purely logical.

Preview of the Next Lesson:

Our discussion of Hume's Principle as an "implicit definition" of number provides a perfect bridge to our next topic. We will step back to the turn of the 20th century to examine a major dispute that revolved around this very concept. In the next module, we will begin by analysing the Frege-Hilbert controversy over the nature of axioms: are they, as Frege insisted, statements of truth about a specific subject matter, or are they, as Hilbert argued, implicit definitions that define a structure? This debate raises fundamental questions about the nature of mathematical theories and truth itself.