Madison's ideas of "democracy" and "mob rule" apply to no regime today, and no modern self-styled "democracy" fits Madison's definition.
mises.org
"virtually no one argues for the sort of democracy denounced by Madison. Wanting to abolish the electoral college, or even abolishing the Senate, does not make one into an advocate of Madison's version of a democracy.
In fact, every single regime on earth today that calls itself "a democracy" clearly qualifies as a republic according to Madison's definition. All the countries that are described as democracies in contemporary discourse use representative schemes of government, and all have a system which at least in part "derives all its powers directly or indirectly from the great body of the people." They also all elect representatives instead of employing direct lawmaking.
Consequently, in contemporary usage, there is no relevant difference between the words "republic" and "democracy." Thus, claiming a preference for a republic over a democracy communicates essentially zero information unless one precisely defines the two terms in a way that departs significantly from Madison's definitions.
Not surprisingly, of course, those who bring up the Founding Fathers and their republic-not-a-democracy claim rarely bother to define the actual difference between the two. If these modern republicans were to use Madison's definition, of course, they would quickly find that warnings about Madison's sort of democracy are irrelevant in the modern world.
Now, this isn't to say that one cannot argue against the excesses of majoritarian government — or even against majoritarian government
in toto. There are many
plausible and respectable arguments against it.
But, if anyone wants to argue against majoritarianism, he should simply do so. There is no need to rely on a half-baked usage of the writings of "the Founding Fathers" who clearly supported a political system in which majority votes play a big part in selecting elected officials, and which is obviously a democracy according to the modern usage of the term.