Public fascination with the Supreme Court, shrouded in mystery and robed with power, lies in its operatic authority not just to dispense justice but also to do so with dramatic flair. Someone always wins, someone always loses. This is precisely what makes the court too blunt an instrument for resolving many conflicts of rights.

The court heard arguments Tuesday for a case that illustrates this perfectly. It involves a baker who refused on religious grounds to produce a cake for a gay couple’s wedding. The dispute is described as a landmark test of LGBT rights. It’s also being heralded as a moment to protect religious liberty.

In reality, this case is testing the limits of the courts’ ability to resolve social disputes. No matter which party prevails, there is no winning scenario.

Left to the political process — or, even better, to informal mechanisms of society — the conflict almost surely could be resolved without forcing a choice between anti-discrimination laws and religious freedom. Surely no one believes same-sex couples actually want the services of a baker they consider a bigot hostile to their rights. The object of the case is not to secure Masterpiece Cakeshop’s services. It is to dragoon its owner, Jack Phillips, into compliance with their views.

The problem is Phillips can’t be forced to agree with those views. He can only be made to deliver a cake, but that outcome would almost surely set the lesbian, gay, bisexual and transgender rights movement back by stoking resentment from its opponents. That is exactly what happened in the late 1990s and early 2000s when court rulings sparked a wave of state constitutional amendments defining marriage heterosexually.

Court cases — or in the Colorado bakery instance, civil rights complaints resolved by administrative judges — tend to do that. While the political process is rooted in persuasion and compromise, rights claims signal to their targets that complainants do not believe they should have to engage in either.

Legal scholar Mary Ann Glendon calls this “rights talk” or the substitution of rights claims for reason-giving. Whereas reason-giving — ordinary dialogue and persuasion, essentially — brings people together, rights talk splits them further apart. Where compromise bridges conflicts, rights talk actually accentuates them.

In this case, rights talk is forcing a conflict between two fundamental values. Of course, sometimes values collide and society must choose. But the prudent first course should be to avert collision.

The most obvious option is for a couple to obtain their wedding cake from a baker who is happy to supply it and from whom they are pleased to purchase it. Masterpiece Cakeshop is outside Denver. The supply of bakers there is ample. Common sense — or common courtesy — provides supple tools to resolve the dispute.

Those who believe in tougher measures against discrimination also have tools: There is a growing market for same-sex weddings that Masterpiece Cakeshop is losing — a more severe punishment than courts are capable of meting out.

In either case, the challenge for rights advocates is implementing long-term changes. The solution is erecting a deep foundation of support for them. James Madison recognized as much. He derided bills of rights as “parchment barriers” easily overrun by majorities who did not genuinely support them. When he came around to endorsing a Bill of Rights, Madison said one of its foremost purposes would be creating a basis for educating and appealing to the public.

This political model of rights has given way to a judicial ethic that explicitly aims to spare citizens the slower yet surer work of persuading one another. It has exempted elected officials from the responsibility of balancing competing values.

In this case, LGBT advocates can hope for a pyrrhic victory at best. Conscientious objectors to same-sex weddings may be pressed into service, but only at the long-range cost of intensifying their opposition and resentment. A vindication of religious liberty, meanwhile, would tarnish that value, however unfairly, with the taint of discrimination.

These are unfortunate choices, all the more so because they are unnecessary. Diverting the case to court only means that, by forcing a winner, everyone loses.

Greg Weiner is an associate professor of political science at Assumption College. He is author of “Madison’s Metronome: The Constitution, Majority Rule and the Tempo of American Politics.”