Bad Company Objection

Understanding the Bad Company Objection

The bad company objection is a significant critique leveled against mathematical abstractionism. It raises concerns about the methodology used to form abstract mathematical objects, particularly when dealing with fundamental laws that can lead to contradictions.

Core of the Objection

The central issue is the difficulty in distinguishing between legitimate and illegitimate forms of abstraction. Critics argue that if a principle of abstraction, like Frege’s Basic Law V, leads to paradoxes, it casts doubt on the validity of the entire abstractionist program.

Frege’s Basic Law V and Paradoxes

Gottlob Frege’s attempt to ground arithmetic in logic relied on Basic Law V, which states that the extension of a concept is determined by the objects falling under it. However, this law, when applied to certain concepts (like the concept of ‘concepts that do not contain themselves’), leads to the infamous Russell’s Paradox, a fundamental contradiction.

Implications for Abstractionism

The paradoxes generated by such foundational laws pose a serious challenge. If the very tools used to construct mathematical objects are flawed, then the existence or nature of those objects becomes questionable. This ‘bad company’ (the paradoxical law) is seen to corrupt the abstract object (the extension) it purports to define.

Distinguishing Legitimate Abstraction

A key challenge for abstractionists is to provide a clear criterion for what constitutes a valid abstraction. How can one ensure that the objects defined through abstraction are coherent and free from paradoxes, especially when starting from potentially problematic principles?