in responce to your "things i did'nt like" comment on wondering how real it was for the Doctor to not be cautious after Rose,and jumping into another Doctor/companion "relationship",one might say,i believe i can shed some light on this.
As a person with Type 1 Bi-polar disorderi relate to the character of the Doctor in many ways, the Doct has always been a VERY Manic character.
Manic people are IMPULSIVE,and they DO jump from one relationship into the next,feet first not worrying about consequences,or being cautious.
And,like the Doctor, Do not mistake mania for foolishness, being impulsive does not equate to intelligence. it is consistent for the Doctor to get on with life, after all,hurt by Rose's departure or not, his 900+ years of wisdom have sunk in time and again,that the only way to get over a loss is to move forward.
And i completely agree..I to, do not see Martha as a "replacement" for Rose, but a New companion.
George S- usa
There isn't really a clear explanation in my mind for why the Doctor is acquiring yet another companion.
Because he's lonely. Simple as that.
Also, he is an attention whore and likes having somebody around to go "ooh!" and "ahh!" when he does something clever.
Anyway, have you forgotten "The Runaway Bride"? He offered a berth in the Tardis to Donna after he'd known her for about two hours. And Donna pointed out to him that he really needed to have a companion so that he wouldn't run off the rails and start destroying planets: "You need someone to tell you to stop." Assume that "Smith and Jones" takes place quite soon after "Runaway Bride" (at least from the Doctor's perspective) and it makes sense that he'd be trying to find someone to do just that.
I don't really feel that the Doctor's desire for a companion needs explanation, as such, to be honest. Not when the episode makes it clear that he's a showoff and also likes being with people -- the right kind of people, anyway. Previous Doctors were more emotionally self-sufficient so they sort of needed to be pushed together with their companion; this one is kinda needy.
As for the doctor's quick enlistment of Martha, I liked Katherine's comment that he *wants* someone to temper his actions. I'd think the Doctor's godlike existence could get kinda bored and he LIKES a challenging companion, either smart or better yet, bright and adventurous...which leads into my second comment. He spends a lot of time accessing Martha's response to the unimaginable situation she's in. She's cool headed, calm and logical throughout most of the ordeal. The Doctor seems very impressed with her response. I think this, above all else, motivated him to pick Martha as a traveling companion. She will provide the stability he needs in his adventures. Rose really didn't challenge him so much as use her adventures to *grow up*. Many of the doctor's enemies and even Sarah Jane has taken him to task regarding his casual attitude (even indifference) towards those he interacts with. Rose didn't. Martha likely will. Even a god sometimes wants boundaries.