Yes, you are correct that there are limitations to the positivist approach, and many critics argue that it is not possible to completely divorce empirical evidence from prior assumptions, values, and beliefs. In fact, this issue has been a topic of debate among philosophers for centuries, and many have argued that knowledge cannot be derived solely from observation and experience.

In response to your question, some positivists do acknowledge that the process of acquiring knowledge is not entirely based on empirical evidence. They argue that certain assumptions and basic principles are necessary to interpret empirical data and to make sense of it. However, they still believe that empirical evidence is the primary source of knowledge and that other sources, such as intuition or metaphysics, should be avoided.

It's worth noting that there are different forms of positivism, and not all of them hold the same views. For example, logical positivism, which emerged in the early 20th century, was more radical in its rejection of metaphysics and other non-empirical sources of knowledge. Later forms of positivism, such as critical positivism, were more nuanced in their approach and acknowledged the limitations of the empirical method.

In any case, the question of how we acquire knowledge, including mathematical knowledge, is a complex and contested issue that continues to be debated by philosophers and scholars.
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