Science tells me there are 2, and they are determined at conception by chromosomes.
No it doesn't; since intersex people exist and they are neither male or female and while traditionally there'd be surgical intervention to create what wikipedia calls "more socially acceptable sex characteristics" (which I think is a really funny term for some reason) but that's really quite controversial: there's no real proof that doing so actually benefits intersex people and the rate of gender dysphoria amongst the intersex community is much, much higher than it is amongst the general population, which is only to be expected really.
There's been a move away from surgical intervention at birth by default unless it is urgently needed in favour of leaving it up to the individual later on in life to decide if they want surgery, and if they do what type they would like: indeed its now illegal in Malta and "suspended" in Chile and there was a court case in Germany where a person who was operated on without being given the full information in the 70s (she had ambiguous genitals and was assigned male and had early male puberty: then at 14 during another procedure they found out that she had a full set of functioning female organs and a chromosomal pattern typical for that of a women: but didn't tell her the last part but instead told her that they also found testicular tissue when they hadn't and operated, and then later on in life she wanted to transition to live as a women and then found all of this stuff in it) which led to the person involved being awarded €100,000 in damages. Its a very touchy issue and proof really that there isn't two biological sexes either.