How did logical positivists explain the status of mathematics and logic?

Logical positivists explained the status of logic and mathematics by categorizing them as a priori disciplines, distinct from the empirical sciences. While the positivists maintained that meaningful statements about the world must be verifiable through observation and experiment, they recognized that mathematical and logical propositions could not be tested in this way.12

To resolve this, they argued that logic and mathematics consist of analytic propositions. These statements are considered true or false solely by virtue of the meanings of the terms they contain and the conventions underlying the use of symbols. In this view, mathematical truths—such as "2 + 3 = 5"—are necessary and universal because their denial would result in a self-contradiction.12

Influenced by the work of Ludwig Wittgenstein, many logical positivists, particularly Rudolf Carnap, viewed these necessary truths as tautologies derived from the rules of a linguistic system. Consequently, logic and mathematics do not describe the world; rather, they function as formal frameworks that are valid independently of empirical experience.321