William of Ockham's case for nominalism rests on the principle now named for him, that entities should not be multiplied beyond necessity: if every particular red object, a rose, a ruby, a sunset, can be adequately explained and classified without positing a further, separately existing universal, redness, that all of them somehow share or participate in, then that further entity is explanatorily idle and should be eliminated from one's ontology, leaving only the particular things themselves and the general words and concepts the mind uses to sort them into groups based on resemblance. This was a direct challenge to the realist tradition running from Plato's theory of Forms through much of medieval scholastic philosophy, which held that universals are genuinely real, existing either independently of particulars, as Plato argued, or at least really present in particulars themselves, as Aristotle's more moderate realism held.
Ockham's nominalism did not deny that things resemble one another, or that general words are useful and even necessary for thought and language, only that this resemblance and usefulness require a separately existing universal entity to explain them, arguing instead that resemblance among particulars is itself a brute, unanalyzed fact, or at most a matter of particulars sharing similar, but still entirely particular, properties, a further refinement called trope theory that later nominalists developed to avoid resting everything on unexplained resemblance alone.