Science

Science was the logic-based system for studying the universe.

Cosmological status
The Pythia were the matriarchal leaders of Gallifrey prior to Rassilon, ruling through magic and superstition. Rassilon rejected this, espousing reason and science. This brought about the Intuitive Revelation and the end of the reign of the Pythia. (PROSE: Cat's Cradle: Time's Crucible)

Following Rassilon's lead, the Time Lords cast out the irrational parts of their souls, which congregated as a great cloud in the upper atmosphere of Gallifrey and were banished from the newly rational universe by Rassilon. He imprisoned it in a tiny corner of creation, the Time Vortex, where it became the Carnival Queen. The Time Lords forgot her and did not allow themselves to dream of her. (PROSE: Christmas on a Rational Planet)

Had the Ferutu, rather than the Gallifreyans, become the Lords of Time, the universe would have continued along magical principles instead of those based on science. (PROSE: Cold Fusion)

Within a rational universe
In 1963, Ian Chesterton taught science at Coal Hill School, (TV: "An Unearthly Child”)

The Exxilons decided to pool their knowledge and create the Great City of the Exxilons. It combined all of the science known to them and was intelligent and capable of repairing itself. The City decided that the only flaw left was the Exxilons themselves and it drove them out. The City drained all the energy from the planet, which meant the Exxilons were incapable of developing any technology above the stone age. They slowly descended into a primitive species. (PROSE: Death to the Daleks)

After the Carnival Queen entered the rational universe and was subsequently banished, Chris Cwej believed she could be fit into science as a personification of a potential Kuhn Paradigm shift in the way the consciousness of observers collapsed mixed-state events. (PROSE: The Death of Art)

Most instances of the supernatural encountered by the Doctor's first eight incarnations eventually were revealed as the effects of alien science, be they the psychic powers and technology used by the Dæmon, Azal (TV: The Dæmons), or the nanotechnology of Elbyon. (PROSE: The Sorcerer's Apprentice)

Leela told Adelaide Lessage, "I too used to believe in magic, but the Doctor has taught me about science. It is better to believe in science." (TV: Horror of Fang Rock)

Psionics was originally a form of magic, but survived the rationalisation of the universe through being sufficiently grounded in science. (PROSE: So Vile a Sin)

After Gallifrey
Following the second destruction of Gallifrey in the Last Great Time War, the Tenth Doctor was initially prepared to consider the word-based science of the Carrionites to be witchcraft. (TV: The Shakespeare Code)

On a number of occasions the Doctor appeared to have suggested that a magical and a scientific understanding of the same phenomenon were now interchangeable. (TV: The Girl in the Fireplace, Tooth and Claw)