The motto works in two directions.
From liberty, diversity
When people are genuinely free to build their own communities and live by their own terms, real variety follows. Not variety imposed by a quota, but variety that grows naturally when no single authority is forcing everyone into the same mold.
That kind of diversity is wider than most political debate allows for. Diversity has been narrowed, in common usage, to mean race and ancestry, as if that is the only difference between people worth naming. But genuine variety includes the communities that form around shared work, shared land, shared tradition. Regional dialects that carry centuries of local history. Crafts, festivals, customs that differ from one province to the next. Rural and urban ways of life that have almost nothing in common. These are real differences, and they emerge when people have the freedom to develop them rather than having a single culture pressed down from above.
Globalization shows what happens when that freedom erodes. Distinct places become interchangeable: the same chains, the same architecture, the same habits everywhere. People still travel to find difference. What they are looking for already tells you that real diversity has value, and that it does not survive without the conditions that allow it.
From diversity, liberty
The second direction is less obvious but just as important.
When communities are genuinely different from each other, each one has every reason to want the others left alone. A group that has built its own way of life does not want another group’s rules imposed on it. That mutual interest in non-interference is what makes liberty durable. It is not just a principle: it is the only arrangement that actually works when real differences are on the table.
Communities that coexist without absorbing each other need a framework that protects that coexistence. Not a framework that homogenizes them, but one that keeps them from overriding each other. Diversity, in other words, creates a structural need for liberty.
Live and let live is the conclusion both directions arrive at.