If he were elected president, former Massachusetts governor Mitt Romney says, he would cut companies’ tax bills. But former House speaker Newt Gingrich (Ga.) would cut them more than twice as much.

Romney would eliminate some people’s capital-gains taxes. But Gingrich would eliminate everybody’s.

And Romney would build more Navy ships to project American influence into the Middle East. Gingrich would build a moon colony, projecting it into outer space.

In dozens of campaign promises, the two GOP front-runners have provided a kind of aspirational self-portrait. These are the ideas that got them in the race and that would guide them in the day-to-day work of governing.

Romney’s big ideas, while sometimes at odds with those of past GOP nominees, are largely within tinkering distance of the status quo. They mirror the man who came up with them: wonky and technical, but generally tailored for real-world application.

Gingrich’s proposals go much further, requiring massive overhauls of the tax code, Social Security and the judicial branch — all at the same time. They are something like the candidate: epic, crowd-pleasing and ultimately a long shot.

Last week, Gingrich told a Florida audience that he would establish a lunar colony by the end of his second term in office. Some scientists say that might be possible; it would require mining ice at the moon’s poles and using it to make drinking water, oxygen and hydrogen-based rocket fuel.

“You could have a sustainable moon colony with a dozen folks who were there and doing scientific research, engineering, planning, building roads,” said Jack O. Burns, a professor at the University of Colorado.

Could this colony’s population reach 13,000? Gingrich gave that number as the threshold for the moon to petition for statehood.

“Sorry, did you say 13,000? Wow,” Burns said, laughing. “No. Certainly, that’s not going to be true by 2020.”

By now, Romney and Gingrich have piled up dozens of promises. They are still coming: With Florida’s Republican primary happening Tuesday, Gingrich has vowed to overthrow the communist government in Cuba and ban embryonic stem-cell research.

In some areas, the two men actually agree.

Both want to repeal two key laws passed under President Obama: the health-care overhaul and the Dodd-Frank financial regulations. Both want to repeal the estate tax (the “death tax,” they call it) and get tougher with Iran. And both would amend the Constitution to ban same-sex marriage.

All of these are GOP orthodoxy: For a mainstream candidate, they are choices as obvious as getting out of the rain.

But in the differences between Romney and Gingrich, it is possible to see the gulf between the pragmatic and ideological wings of the Republican Party — and between two very unlike minds.

It starts with taxes, a subject dear to both men.

“With Romney’s plan, it’s much more a plan that works within the confines of the existing tax systems,” said Curtis Dubay, a tax specialist at the Heritage Foundation. “Gingrich’s, to work, would have to blow the whole thing up and start from scratch.”

Romney’s plan would lower a corporation’s tax rate from 35 percent to 25 percent. For individual taxpayers making less than $200,000 per couple, he would end taxes on capital gains and interest. The cost? Probably about $180 billion less in government revenue per year, according to the nonpartisan Tax Policy Center.

But Gingrich would do far more. He would cut the corporate rate to 12.5 percent. He would eliminate all taxes on capital gains and interest. And he would allow individuals to opt out of the current tax code, giving them the option of a 15 percent “flat tax” with limited deductions.

The price tag? The Tax Policy Center estimated it at $850 billion a year. That would create a sinkhole in the federal budget bigger than the cost of the Defense Department.

“Gingrich’s plan, obviously, is more aggressive and bolder than what Romney’s doing,” said Daniel J. Mitchell of the libertarian Cato Institute. “But unless he somehow — I don’t know, adopts Ron Paul’s spending plan [to cut a large amount] — I’m not sure it’s a serious proposal.”

On Social Security, the two men exhibit a similar pattern. Romney would raise the age at which future retirees could collect benefits, adding “a year or two,” he said in a debate. And he would tinker with formulas so that wealthier retirees would see their benefits grow less over time.

Gingrich, by contrast, wants to create a new kind of Social Security. He would keep the old system but allow people to opt into a separate system, in which they could keep their money in private, personal accounts. The same private-account plan has worked in Chile, Gingrich is fond of saying.

But it was installed while Chile was run by a military dictator.

“You don’t have problems with Congress” in that situation, said the Heritage Foundation’s David John. A President Gingrich probably would, since the plan would be likely to add cost and complexity in the short run.

On immigration, Romney and Gingrich have plans that could be hard to pass through Congress.

Both have called for a stepped-up crackdown on employers, who would have to use a system of identification cards to check that every worker is here legally. Buta similar plan is already languishing on the Hill: The agriculture industry thinks it could decimate the country’s farm workforce.

Romney predicts that those who couldn’t work would simply “self-deport.” But Gingrich has a more complicated vision. He has advocated letting some longtime illegal residents remain in the country — and letting new citizen councils choose which ones.

“It would never pass the Congress. And . . . if it ever passed Congress, it would be litigated for decades,” because of the chance that immigrants could get unequal treatment in different towns, said Frank Sharry of the liberal group America’s Voice.

Romney and Gingrich are to the right of President George W. Bush and 2008 GOP nominee John McCain, who both proposed that millions of illegal residents be given a path to legal status. Still, Sharry applauded Gingrich for proposing that some subset of illegal residents might stay.

“As a plan, it makes no sense,” Sharry said. “As a political move, it’s gutsy.”

Romney’s most grandiose plan might be his call to increase the construction of naval ships, from nine to about 15 every year. Gingrich, by contrast, has spoken about the need to “recapitalize our military infrastructure,” without many details.

“The difference is, Romney is a marginalist, and Gingrich is more revolutionary,” said Roberton Williams, a senior fellow at the Tax Policy Center. He said that Romney’s approach, though less ambitious, seems more likely to win approval. “The problem with the Gingrich approach is that he would go in and throw the bomb, and then maybe there would be pieces that you would pick up afterwards.”