Were Burke and Wills trying to prove or disprove anything?

Answer 1

The objective of Burke and Wills' expedition was not to prove anything: it was more to win the race across the continent.

The Burke and Wills expedition had the potential to be very important to the state of Victoria. Burke and Wills, like John McDouall Stuart in South Australia, were seeking a suitable route for a telegraph line from the southern regions of Australia to the north. A telegraph link to the rest of the world was available off Java, but it had to be linked to a centre in the north, then an overland telegraph line constructed to carry messages down to the main cities in the south.

This would provide a very important link for Australia to the rest of the world, and Victoria wanted to be the first state to receive overseas news, as this would be a psychological advantage for the state. It also could have been the winning card to tip the authorities in favour of making Melbourne Australia's national capital in the future, over the larger and older Sydney (as it was, neither Melbourne nor Sydney was chosen).

Both South Australia and Victoria were vying to be the first to make a successful crossing of the continent, and both states offered sizable rewards. Burke and Wills represented the Victorian bid to cross the continent. In the end, due to their premature deaths, the Burke and Wills expedition sadly had no lasting importance or significance for Australia, except in history books.