# Kant on Analytic vs. Synthetic Propositions
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Published: 2025-05-01
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*Written in Spring 2025 for [PHIL 630](https://catalog.ku.edu/liberal-arts-sciences/philosophy/#courseinventory) (Philosophy of Mathematics)*

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##### 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*.

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**Reference**
1. Shapiro, Stewart. "Chapter 4." _Thinking about Mathematics: The Philosophy of Mathematics_, Oxford University Press, 2000, pp. 76–93.