I thought about writing separate articles about how gender roles are bullshit, how gender is a binary, and how transgenderism is sexist, but I decided to just put all these truths into one article, because all those things are wrong for the same reason: they attempt to make gender an identity.
The reason I'm defining gender as biological sex is because there's nothing else for it to mean. When someone says they "identify" as a gender other than what they biologically are, they're describing their identity as a person in the same terms that biological sex is described in. Why would you do that if you really understand that your identity as a person is completely separate from biology? The concepts of man and woman have nothing to do with personal identity. Personal identity shouldn't be described in those terms.
No one says the solution to racism is to think that anyone can choose to identity as whatever race they want and other people should refer to them as it.
And it's for exactly the same reason that gender roles are illegitimate, because men and women are the same kind of being and are capable of the same tasks.
Before I end this article, I know someone's going to throw some statistics at my last sentence up there and say "but looky here I've got facts and facts trump philosophy". I could just point out that citations have to take a back seat to a priori reasoning, but instead I'm going to explain how your stats might very well be true: it's a self-fulfilling prophecy. We were all taught gender roles as kids, even if only implicitly, and people act the way they're raised to think is normal for them. Thus, any statistics you're using to show that males and females naturally have different psychology is adequately explained by the prevalence of the belief you're using them to argue for. (While many political "liberals" claim to oppose gender roles, they usually still welcome the idea that there are biologically-caused differences in psychology between men and women; so yes, it is a prevalent belief.)