 Frege (1884, introduction and section 27; 1893–1903, introduction; 1894; and 1919) gave several compelling arguments against psychologism. First, it seems that psychologism is incapable of accounting for the truth of sentences that are about all natural numbers, because there are infinitely many natural numbers and clearly, there could not be infinitely many number-ideas in human minds. Second, psychologism seems to entail that sentences about very large numbers (in particular, numbers that no one has ever had a thought about) are not true; for if none of us has ever had a thought about some very large number, then psychologism entails that there is no such number and, hence, that no sentence about that number could be true. Third, psychologism turns mathematics into a branch of psychology, and it makes mathematical truths contingent upon psychological truths, so that, for instance, if we all died, ‘4 is greater than 2’ would suddenly become untrue. But this seems wrong: it seems that mathematics is true independently of us; that is, it seems that the question of whether 4 is greater than 2 has nothing at all to do with the question of how many humans are alive. Fourth and finally, psychologism suggests that the proper methodology for mathematics is that of empirical psychology; that is, it seems that if psychologism were true, then the proper way to discover whether, say, there is a prime number between 10,000,000 and 10,000,020, would be to do an empirical study of humans and ascertain whether there is, in fact, an idea of such a number in one of our heads; but of course, this is not the proper methodology for mathematics. As Frege says (1884, section 27), “Weird and wonderful…are the results of taking seriously the suggestion that number is an idea.”