The syntactic structure of sentences exhibits a striking regularity: dependencies tend to not cross when drawn above the sentence. We investigate two competing explanations. The traditional hypothesis is that this trend arises from an independent principle of syntax that reduces crossings practically to zero. An alternative to this view is the hypothesis that crossings are a side effect of dependency lengths, i.e. sentences with shorter dependency lengths should tend to have fewer crossings. We are able to reject the traditional view in the majority of languages considered. The alternative hypothesis can lead to a more parsimonious theory of language.