In her study of Husserl's logic, Suzanne Bachelard writes
[F]or the logician, the problem of psychologism is not just one problem among other problems: it is a determining problem; for either the logician welcomes psychological justifications, or he considers such recourse to psychology a radical vice. The choice is decisive for the logician.(1968: iii)
The choice is decisive for Wittgenstein in Philosophical Investigations. It is a book of logic (a “conceptual investigation” or “grammatical investigation”) and from front to back in it, Wittgenstein considers psychologism a radical vice. And “vice” is the exact word: psychologism is not treated as a false theory but as a failure more radical – a failure so much as to recognize logic at all. But the fact that psychologism is a radical vice does not mean that combating it is easier than it would be were it a false theory. It means that combating it is more complicated and requires more patience than would combating a false theory. Wittgenstein not only has to struggle against certain beliefs in his reader but also against his reader's philosophical character.
Some background: Gottlob Frege
Before looking at Wittgenstein's treatment of psychologism, it will help to consider Frege's description of psychologism and his response to it. The locus classicus is the beginning of Frege's paper, “Thoughts”. Frege explains that
People may very well interpret the expression “law of thought” by analogy with “law of nature” and then have in mind general features of thinking as a mental occurrence. […]