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Magic was an umbrella term for the abilities of many beings and life-forms to influence reality in manners that did not respect the conventional laws of physics. As such, belief in "magic" was considered the opposite of rationality.
During the Dark Times of the Doctor's universe, magic was widespread and dominant. The early Time Lords led by Rassilon championed rationality against the forces of magic, (PROSE: Gallifrey: A Rough Guide) and prevailed, installing Laws of Time and Physics based on their own understanding of reality, and making magic supposedly "impossible". (PROSE: The Book of the War, Christmas on a Rational Planet)
However, magic continued to exist on a smaller scale in many corners of the universe; (PROSE: The Celestial Toyshop, The Scarlet Empress etc.) as Merlin boasted, "the supernatural" continued to exist well into the 20th century. However, Merlin also acknowledged that his black magic was useless against the Laws of Time. (COMIC: Dr. Who's Time Tales) Interestingly, some described some of the Time Lords' exclusive abilities as "magic", including regeneration (TV: The Witch's Familiar) and TARDISes. (TV: The Big Bang)
Some universes which existed "sideways in time" were also magic-based. (TV: Battlefield)
History
In the Dark Times
Magic was initially a strong force in the Doctor's universe during the Dark Times, with Gallifrey being ruled not by technocratic politicians but by the Pythia and her priestesses. However, as the Time Lords rose, they shaped the universe to become a clockwork set of rules which they could predict and control, with any irrational or supernatural principles replaced by those of science to underpin reality. (PROSE: Cat's Cradle: Time's Crucible, 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. (PROSE: Cold Fusion)
The Kotturuh were thought to use magic to destroy worlds before the Tenth Doctor discovered that they were using a fast acting retrovirus. (PROSE: The Knight, The Fool and The Dead)
Within the rational universe
The Time Lords retained the use of arts described as "magic". Regeneration, the source of their near-immortality, (TV: The Witch's Familiar) which they were speculated to have obtained from the Dark Times force the Yssgaroth, (PROSE: Goth Opera, The Book of the War) was acknowledged as "the ancient magic of the Time Lords" by the scientist Davros, (TV: The Witch's Familiar) and described as "witchcraft" by the Sycorax leader. (TV: The Christmas Invasion) The Eleventh Doctor also described his TARDIS as a "magic box". (TV: The Big Bang)
Psionics was originally a form of magic, but survived the rationalisation of the universe through being sufficiently grounded in science. Monitoring the network of connections that psionic abilities established across time allowed the Time Lords to possess an "early warning system" against outbursts of wilder magics breaking through into their ordered cosmos. (PROSE: So Vile a Sin)
When the ancient entity known as the Carnival Queen was accidentially released back into the Doctor's universe, she caused irrationality and magic to be reintroduced throughout space-time while destroying the laws of physics and nature in favour of the supernatural. The Seventh Doctor later helped repair the damage she caused, although the Carnival Queen continued to influence reality to a degree from her prison outside the universe. (PROSE: Christmas on a Rational Planet)
Hyspero was one of the few places in N-Space where magic was commonplace, because it was located near the edge of the universe (PROSE: The Scarlet Empress, The Blue Angel) and was a gateway into the irrational Obverse. (PROSE: Enter Wildthyme)
One account suggested that the population of sentient Androzani trees encountered by the Eleventh Doctor (TV: The Doctor, the Widow and the Wardrobe [+]Loading...["The Doctor, the Widow and the Wardrobe (TV story)"]) constituted a genuinely "magical" forest, collectively known as the Treeborg when on the ground and as the Arboreal Migration when travelling between planets. (GAME: Worlds Apart [+]Loading...["Worlds Apart (video game)"])
Belief by the Doctor
Some incarnations of the Doctor claimed not to believe in magic. (TV: The Dæmons, PROSE: The Sorcerer's Apprentice, TV: The Robots of Death) Indeed, most instances of the supernatural encountered by the Doctor's first nine incarnations eventually stood 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)
Additionally, the Doctor was often wont to sternly chastise others for their belief in magic and taking the opportunity to educate them. For instance, in response to Estinee viewing the Kotturuh as supernatural beings who used magic spells, the Tenth Doctor told her they were not supernatural and that all they did was deploy a powerful retro-virus. (PROSE: The Knight, The Fool and The Dead)
The Fourth Doctor was the most vocal of all his incarnations in not believing in the existence of magic and bluntly called it "superstitious rubbish.". (TV: The Talons of Weng-Chiang) His disbelief also carried over to his companion Leela who 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)
However, the Second Doctor believed that there was room for magic in an infinite universe. (PROSE: The Celestial Toyshop) The Seventh Doctor was not entirely inflexible on the matter, as evidenced by the corollary he added to Clarke's Law stating that any sufficiently advanced form of magic was indistinguishable from technology. (TV: Battlefield) The Eighth Doctor asserted this corollary as well, (PROSE: The People's Temple) and notably embraced the concept of magic. (PROSE: The Scarlet Empress) The Eleventh Doctor acknowledged that his TARDIS was a "magic box". (TV: The Big Bang) The Twelfth Doctor noted "real" wrestling used, among other things, magic spells. (TV: Thin Ice)
After Gallifrey
Following the destruction of Gallifrey as a consequence of the War in Heaven (PROSE: The Ancestor Cell, The Adventuress of Henrietta Street) the Time Lords' exorcism of the irrational from the universe was revoked and magic began to regain its potency. (PROSE: The City of the Dead, The Adventuress of Henrietta Street)
In the post-Last Great Time War universe, the Tenth Doctor was initially prepared to consider the linguistic-science of the Carrionites to be witchcraft. (TV: The Shakespeare Code) The Sycorax were able to make use of blood control, which was understood by both Daniel Llewellyn and by the Sycorax themselves to be supernatural, and which the Doctor acknowledged as "voodoo", although he deemed it "cheap". He noted that he "hadn't seen blood control in years". (TV: The Christmas Invasion)
On a number of occasions the Doctor appeared to have suggested that a magical and a scientific understanding of the same phenomenon were interchangeable. (TV: The Girl in the Fireplace, Tooth and Claw) Others made the connection too, Elton Pope noting that the sonic screwdriver was essentially a "magic wand". (TV: Love & Monsters) Clara Oswald, Gabby Gonzalez and Alice Obiefune made the same comparison. (COMIC: Four Doctors)
The Tenth Doctor listed spells, rituals and incantations among those things in which he personally did not believe, but remained prepared to consider that they might nevertheless be real. (TV: The Satan Pit) Jack Harkness considered some phenomena, such as fairies, to exist outside the bounds of scientific inquiry. (TV: Small Worlds)
In dreams
In a shared dream, Santa Claus claimed to Shona McCullough that his flying reindeer were a "scientific impossibility", and as a result, he fed them "magic carrots" so they could fly. (TV: Last Christmas)