Psychology of Philosophy: The Strong Program
Why do you endorse Platonism about numbers, or idealism, or skepticism, or consequentialism, or any other a priori justified (you think) philosophical opinion? I suggest that you look not to the philosophical truth of that view, with which you are somehow better in touch with than those with whom you disagree. Rather, look to your own biases, personality, and background.
The "strong program" in sociology of science holds as a methodological precept that in explaining the rise and fall of scientific theories one may not appeal to the truth or falsity of those theories. So also, I recommend, in thinking about the origins of your own and others' philosophical views, you would do well not to think about the truth of those views. You would also do well not to think in other terms that imply success or insight, such as "seeing the weaknesses" in a predecessor's view, or "recognizing" troubles or advantages of a substantive, philosophical sort, or "proving" or "establishing" anything.
Where philosophy blurs into science or math, this constraint becomes blurry. I say this because I think -- unlike the cartoon version of sociologists of science -- that in science and math features of the world (or of mathematical structures) start to apply strong pressures on views. The farther toward a priori philosophy, the less influence from such pressures. The farther toward concrete science or formal mathematics, the more the influence from such pressures.
No a priori claim is so patently false that no philosopher would endorse it. No constellation of views is so bizarre and self-contradictory that no philosopher would sign up for those views, were her background motivations and culturally-given assumptions right. If there are any a priori philosophical truths among the factors influencing our philosophical theorizing, their influence is modest.