I overheard part of a conversation at the con - forgive me, whoever was participating, as I've entirely forgotten! - affirming that although the Doctor could obviously be played by a Black actor, he could never be played by an actress, as this would make no scientific sense. The more I think about this simple remark, the deeper it gets.
Until quite recently, race was considered to be a real, natural category - something obvious and commonsense, decreed by biology or God. For decades, science's self-correcting mechanism has been chipping away at this illusion. We now understand that, biologically, there's no such thing as race - which means it's a cultural idea, something we invented and imposed. The evidence of our own eyes, which once seemed to confirm the illusion, now challenges it, as we encounter more and more people of mixed race. Sadly, the scientific evidence doesn't mean an end to racism, any more than scientific evidence means an end to Creationism; but we can hope to see this imaginary system of categories gradually blur away to nothing.
Physical sex, bodily gender, are another matter. Nature seems to confirm the obvious, solid categories of male and female with every birth: there are plenty of children of mixed race, but there are no children of mixed sex.
Except, obviously, that there are: millions of them. Transgender people, intersex people, people with any of
a great many genetic syndromes which complicate our common sense, either/or picture of gender (and who may suffer very much as a result of our insistence on it). This powerfully challenges the idea that, while race is not a natural category, sex is, and it therefore wouldn't make sense for the Doctor to change sexes.
Despite this complexity, though, it's not usual for humans to change sex without medical assistance. (It's not impossible: there's a family in the Dominican Republic where some of the girls become boys at puberty.) Outside the human race, though, there are species
across the animal kingdom who naturally change sex - to have as many offspring as possible, or to keep their social hierarchies organised. Some
fish do it, some
sea cucumbers do it, shrimp, sea snails, bristle worms, frogs... some species change sex just once, some can switch back and forth as necessary.
These natural sex changes are a matter of rebuilding the body, something which is difficult for adult human beings to do without help - but clearly trivial for Time Lords, who do precisely that at every regeneration. (Hmm, I wonder if this is why men and women on Gallifrey wear the same clothes?)
In fine: whatever objections one might have to casting an actress as the Doctor, science doesn't really lend them any support. Without making any judgements on anyone, I think those objections have more to do with our ideas about sex and gender than the concrete, fleshly reality of sex and gender: and I think those ideas are worth examining.