Earlier this week, President Obama stated explicitly what everyone already knew: The U.S. is not prepared to go to war with Russia over its annexation of Crimea. We'll impose some tough sanctions, we'll say some mean things about Vladimir Putin, and John McCain will fulminate for a while longer on Sunday-morning talk shows. But the current situation on the ground isn't going to change — because Putin successfully showed that the American president never had any intention of backing up his warnings and threats with military force.

Since I don't think the United States has much of a strategic interest in preventing Russia from swallowing parts of eastern Ukraine — any more than Russia would be especially concerned if we annexed a chunk of northern Mexico — I have a hard time getting worked up about recent developments. But that doesn't mean the events of the past few weeks won't have dangerous geopolitical consequences.

Every time the president allows a stated line to be crossed — as he did in Syria last year over Bashar al-Assad's use of chemical weapons, and now again with Russia's actions in Ukraine — he signals that American security commitments may be hollow.

The overall importance of such signaling in international relations is a contentious topic among those who study foreign affairs. But there is one potential theater of conflict in the world where we can be quite certain that America's recent actions — or rather, inactions — have been very closely noted: the Taiwan Straits.

Ever since Chiang Kai-shek's nationalist government fled Mao Zedong's communist takeover of mainland China in 1949, relocating to the island of Formosa (henceforth renamed Taiwan), the United States has tacitly guaranteed the island's security. The arrangement became more explicit with the Taiwan Relations Act of 1979, which included a commitment to "resist any resort to force or other forms of coercion" against the island. Over China's strenuous objections — the People's Republic considers Taiwan to be its sovereign territory — we've backed up that pledge by selling the Taiwanese government significant numbers of weapons over the years, most recently a $5.8 billion package of military hardware in 2011.

So what happens if China, having noted our reluctance to stand up to Assad and complete unwillingness to challenge Moscow militarily, decides to test us by taking Taiwan?

On one level, going to war with a major world power of 1.35 billion people in order to defend an island on the other side of the planet sounds crazy — a quintessential act of imperial overreach. That's certainly my initial reaction, since I'd prefer to see the U.S. playing a more modest role in world affairs more generally. What better way to downscale our global commitments than to back away from this East Asian relic of Cold War brinksmanship?

But before we make that move, we need to be clear about the stakes — and the likely consequences. The United States has made a lot of blunders — and done a fair amount of geopolitical mischief — over the years. But all told and weighed against the realistic alternatives, our military hegemony since the end of World War II has been salutary, minimizing cross-border conflicts and enforcing order across large swaths of the globe. The Pax Americana isn't just a propaganda slogan.

If we allowed China to take direct control of Taiwan (even if the mainland promised to treat it as a semi-autonomous region, like Hong Kong), it would signal once and for all the end of this American-dominated era and the start of another. Coupled with our passivity in the face of Russia's recent actions, we would swiftly find ourselves in a world where nations revert to acting as they have for much of human history: freely invading each other's borders and fighting wars in the pursuit of national self-interest, with no overseeing hyperpower imposing global peace and order from above.

How likely is it that China will make a move against Taiwan?

At the moment, not very. Relations between Beijing and Taipei have improved since Taiwan elected Ma Ying-jeou as president in 2008. On the other hand, China has recently started flexing its military muscles in unprecedented ways. And earlier this week hundreds of Taiwanese university students stormed and occupied the national legislature in Taipei to protest a trade pact with the mainland, a reminder that nationalist sentiment remains strong in Taiwan. How might Beijing respond if the opposition gains the upper hand, threatening to scuttle its plans for slow-motion reunification with an island it still considers a breakaway province?

The fact is that we just don't know. (Last spring no one would have predicted that a year later Crimea would be a part of Russia.)

What we do know is that, if China does make a move in the Taiwan Straits, the future of the geopolitical order will depend on how America responds.

Or doesn't.