Why did logical positivists consider arithmetic truths a priori and analytic?

Logical positivists classified arithmetic truths as a priori and analytic because they viewed mathematics as a system of definitions and rules governing language, rather than empirical observations about the world.12

They argued that arithmetic statements are analytic because their truth is derived solely from the meanings of the terms involved—such as the definitions of numbers and operators—making them true by virtue of those meanings. Consequently, these propositions are a priori because they are valid independently of experience; their denial would result in a self-contradiction.12

By following the work of Gottlob Frege and Bertrand Russell, logical positivists rejected the idea that mathematics is empirical. They maintained that because these truths are established through the "relation of ideas," they are necessary and universal, meaning no physical observation could ever refute them. This perspective aligns with the view that statements of necessity are simply by-products of the rules governing language use.12