Kant on Analytic vs. Synthetic Propositions
Written in Spring 2025 for PHIL 630 (Philosophy of Mathematics).
Prompt: Kant claims that $7+5=12$ is not analytic. Explain why he thinks that.
In logic, a proposition is a statement that expresses a verifiable concept. In mathematics, a proposition is a formal statement of a problem: for example, $7+5=12$.
Immanuel Kant claims that $7+5=12$ is not analytic. This essay will attempt three things. It will first explain what Kant means by an analytic proposition. Then, it will show how Kant reframes the mathematical proposition in question to assess its analyticity. Finally, it will trace Kant's reasoning behind the claim.
Kant discusses a specific type of universal, two-part proposition involving two concepts - the subject (A) and the predicate (B) - arranged in the form 'All A are B.' By concept, Kant refers to an idea (of something) formed by combining all its parts. For such two-part propositions, Kant states that they are analytic if the concept of the predicate can be derived from that of the subject. If the predicate cannot be derived from the subject, then the proposition is synthetic - an assignment we will discuss later.
Importantly, Kant implies that this analytic process does not yield any new knowledge: a proposition is true analytic only if the predicate is already contained within the subject. If the predicate is not contained in the subject, the proposition is not analytic.
Clearly, the proposition $7+5=12$ does not immediately conform to the 'All A are B' format described above. To address this and make an assessment on its analyticity, Kant interprets the section - '$7+5$' - before the equals sign (=) as representing the concept of the 'union of $seven$ and $five$' and the part after as concept of $twelve$.
Next, Kant contends that any examination of the 'union of $seven$ and $five$' itself does not yield another thing. There is no reorganization of this union or its parts that could lead us to the concept $twelve$. Therefore, Kant's condition for a proposition to be deemed analytic is not fulfilled - and the proposition is synthetic.
Kant describes that $twelve$ is synthesized only when the $five$ and $seven$ are combined through a process of mental construction he called intuition. One form of Intuition, Kant proposes, allows us consider a $5$ as a collection of dots or, more vividly, as five human fingers - each finger being a part of the whole and a $7$ as five fingers on one hand and two on the other. Then, when these are counted, one after the other, the number $12$ is produced. That this process yielded a new number, which was underivable from the concept of union of '$seven$ and $five$', is another reason why Kant claims that $7+5=12$ was not analytic.
This essay has attempted to trace Kant's justification for his claim. Because intuition is necessary to prove the validity of $7+5=12$, Kant contends that the mathematical proposition is not analytic but synthetic.
Reference
- Shapiro, Stewart. "Chapter 4." Thinking about Mathematics: The Philosophy of Mathematics, Oxford University Press, 2000, pp. 76-93.