Conflicting theoriesgeocentric view of Aristotle and Ptolemy was inconsistent with the results of Tycho Brahe, Johannes Kepler, Giordano Bruno and Galileo Galilei. Those dogmas were considered to be true by the Papacy and by much of the dominant scholarship of the time. The new theories conflicted with the old. Some scientists and other thinkers once claimed that the speed of the first automobiles (and railroads too) would kill their passengers; others disagreed. Some observers felt they had reason to believe that it would be impossible for anyone to leave the Earth and walk on the Moon. There were others who disagreed. In both cases, the question would appear to have been settled.
The human sciences are the most favourable environment for the genesis of conflicting theories. Frequently two fierce camps can be distinguished in basic issues of anthropology, ancient history, Biblical studies, theology or linguistics. Usually one side condemns the theories of the other side as unscholarly and unscientific.
The rhetoric, fair or unfair, can be quite heated. For instance, Bruno was burned at the stake for disagreement with orthodox religious views and Galileo was forced to recant by the Catholic church. The burning issue of global warming likewise awaits definitive settlement; meanwhile, government funded scientists like Richard Lindzen and Sallie Baliunas who challenge the main stream of government-funded research feel marginalized. And many biologists and geneticists in Soviet Russia were exiled or even executed for disagreeing with Stalin's favored biologist (ie, T. Lysenko) and with his variant of the inheritance of acquired traits. Supporters of such theorists as von Däniken or Velikovsky often feel put upon by their entrenched, orthodox, opposition. The example of Galileo seems to be expecially inspirational; conferring a sense of legitimacy. Unfortunately, orthodoxy is neither always wrong, nor always right; and mere opposition by the establishment does not indicate that a theory is right.
In the experimental sciences, conflicting theories are, and have often been, also present. In many cases, it has been possible to devise some experiment which demonstrates that one theory is better (ie, better accounts for the observed phenomena) than another. A classic example is that of Spallanzani who demonstrated by experiment that maggots do not arise spontaneously (as in rotting meat), and by Pasteur who similarly demonstrated that bacteria don't either (though in nutrient broth in this case). Similarly, the Michelson-Morley_experiment at Western Reserve University essentially ended any possibility that the theory of the ether could be correct.
In the absence of a way to decide between conflicting theories (eg, experimental demolition of one or the other -- or both), the conflict cannot be resolved. Adherents of one may die off, political or social conditions can so change that the conflict is no longer relevant, etc. But, vanquished theories have regularly returned from the ideological discard pile -- generally with some cosmetic surgery.