In Arcadia, you are what you think. Different opinions on the same subject illustrate more fundamental aspects of the characters. For example, even something as straightforward as geometry elicits a variety of responses. Lady Croom's approval of geometry displays her general appreciation of order and symmetry; Hannah's similar view shows that she values logic over emotion. Thomasina's dissatisfaction with geometry is part of her larger rebellion against what everyone assumes to be true: she thinks the world is more complicated than what geometry can handle.
Arcadia must be a very sexy place, if the goings-on in gazebos and elsewhere are any indication. But not everyone in the play thinks about sex the same way. Thomasina's youth shows itself when she's both fascinated and repelled by the tidbits Septimus drops about "carnal embrace." Septimus's own fling with the witless Mrs. Chater as a relief for his longer crush on the witty Lady Croom suggests that he values intelligence more than, ahem, "tropical humidity" (1.1). Bernard traces Hannah's caution to her repressed sexuality, while Chloë attributes Bernard's energy to his lack of sexual repression. Sex not only signals how the characters think, it's also part of how they evaluate one another.
Arcadia's kind of like a university in miniature: in attendance across the two centuries are two literary scholars, a mathematician, a polymath tutor, a precocious student, and at least one poet (Mr. Chater's claim to the title is dubious). The bias towards the scholarly comes out in the obsession nearly every major character has with knowledge and discovery. Some of the characters fit their career profile fairly well: Valentine the mathematician thinks about reality in terms of numbers, Bernard the literary scholar wants to tell exciting stories in his research. Comparing Thomasina's and Hannah's varied and differing approaches against Valentine's and Bernard's, however, suggests that occupation doesn't always tell the whole story when it comes to character – and that the science/humanities divide isn't as strong as it might seem on the surface.