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In 1479, a few months shy of his eleventh birthday, Niccolò Machiavelli left the school where he’d learned to read and write and went to study with a teacher named Piero Maria. The future author of The Prince spent the next twenty-two months mastering Hindu-Arabic numerals, arithmetical techniques, and a dizzying assortment of currency and measurement conversions. Mostly he did word problems like these:

If 8 braccia of cloth are worth 11 florins, what are 97 braccia worth?

20 braccia of cloth are worth 3 lire and 42 pounds of pepper are worth 5 lire. How much pepper is equal to 50 braccia of cloth?

One type of problem reflected the era’s shortage of currency. Goods that would sell for one price in coins cost a premium if the buyer paid with other goods. (These problems assume familiarity with trading conventions and therefore present ambiguities to the modern reader.)

Two men want to barter wool for cloth, that is, one has wool and the other has cloth. A canna of cloth is worth 5 lire and in barter it is offered at 6 lire. A hundredweight of wool is worth 32 lire. For what should it be offered in barter?

Two men want to barter wool and cloth. A canna of cloth is worth 6 lire and in barter it is valued at 8 lire. The hundredweight of wool is worth 25 lire and in barter it is offered at such a price that the man with the cloth finds he has earned 10 percent. At what price was the hundredweight of wool offered in barter?

Others were brain teasers dressed up in ostensibly realistic detail.

A merchant was across the sea with his companion and wanted to journey by sea. He came to the port in order to depart and found a ship on which he placed a load of 20 sacks of wool and the other brought a load of 24 sacks. The ship began its voyage and put to sea.

The master of the ship then said: “You must pay me the freight charge for this wool.” And the merchants said: “We don’t have any money, but take a sack of wool from each of us and sell it and pay yourself and give us back the surplus.” The master sold the sacks and paid himself and returned to the merchant who had 20 sacks 8 lire and to the merchant who had 24 sacks 6 lire. Tell me how much each sack sold for and how much freightage was charged to each of the two merchants?

Along with their famed humanist arts and letters, the mercantile cities of early modern Italy fostered a new form of education: schools known as botteghe d’abaco. The phrase literally means “abacus workshops,” but the instruction had nothing to do with counting beads or reckoning boards. To the contrary, a maestro d’abaco, also known as an abacist or abbachista, taught students to calculate with a pen and paper instead of moving counters on a board.

The schools took their misleading name from the Liber Abbaci, or Book of Calculation, published in 1202 by the great mathematician Leonardo of Pisa, better known as Fibonacci. Brought up in North Africa by his father, who represented Pisan merchants in the customs house at Bugia (now Béjaïa, Algeria), the young Leonardo learned how to calculate using the nine Hindu digits and the Arabic zero. He was hooked.

After honing his mathematical skill as he traveled throughout the Mediterranean, Fibonacci eventually returned to Pisa. There he published the book that enthusiastically introduced the number system we use today.

Fibonacci’s novel methods of pen-and-paper reckoning were ideal for Italian textile merchants, who wrote lots of letters and needed permanent account records. Beginning in Florence in the early fourteenth century, specialized teachers began teaching the new system and producing handbooks in the vernacular. Consistent sellers, the books served simultaneously as children’s textbooks, merchants’ reference tools, and, with their brain-teasing puzzles, recreational materials.

From the abacists’ classrooms, future merchants and artisans typically graduated to apprenticeships and work. But a grounding in commercial math was also common for those like Machiavelli, who were destined for higher education and a career of statesmanship and letters. In a society based on trade, cultural literacy included calculation.

As they drilled generations of children on how to convert hundredweights of wool into braccia of cloth or to allocate the profits from a business venture to its unequal investors, the abacists invented the multiplication and division techniques we still use today. They made small but important advances in algebra, a subject universities scorned as too mercantile, and devised solutions to common practical problems. On the side, they did consulting, mostly for construction projects. They were the first Europeans to make a living entirely from math.

In his seminal 1976 study of nearly 200 abacus manuscripts and books, historian of mathematics Warren Van Egmond emphasizes their practicality—a significant departure from the classical view of mathematics, inherited from the Greeks, as the study of abstract logic and ideal forms. The abacus books treat math as useful.

“When they study arith­metic,” he writes, “it is to learn how to figure prices, compute interest, and calculate profits; when they study geometry it is to learn how to measure buildings and calculate areas and distances; when they study astronomy it is to learn how to make a calendar or determine holidays.” Most of the price problems, he observes, concern textiles.

Compared to scholastic geometry, the abacus manuscripts, with their problems about trading cloth for pepper, are indeed down to earth. But they don’t scorn abstraction. Rather, they wed abstract expression to the physical world. The transition from physical counters to pen-and-ink numerals is in fact a movement toward abstraction. Symbols on a page represent bags of silver or bolts of cloth and the relationships between them.

Students learn to ask the question, How do I express this practical problem in numbers and unknowns? How do I better identify the world’s patterns—the flow of money in and out of a business, the relative values of cloth, fiber, and dyes, the advantages and disadvantages of barter over cash—by turning them into math? Mathematics, the abacists taught their pupils, can model the real world. It does not exist in a separate realm. It is useful knowledge.

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12/3/1996: Printz v. U.S. argued.

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The good news is, everybody lost in the 2020 election—at least a little bit. Donald Trump lost the White House, Democrats lost ground in the House, and the Senate remains in contention with the most likely outcome being Republican control and therefore divided government.

The most sought-after prize in politics is a mandate: a win so big that it justifies ramming through an ambitious political agenda. The idea of being in such a position is so alluring that politicians who just barely managed to eke out a victory—or parties that are barely clinging to a majority—will sometimes still try to claim a mandate. It was a thorn in Trump’s side that he lost the popular vote so spectacularly while winning the presidency in 2016, for instance, because it made claiming widespread popular support that much less plausible.

But divided government makes it difficult to posture in this particular way, and that’s likely to be a good thing for fans of limited government and fiscal discipline during a Joe Biden administration. It’s much harder to go big when you are constantly at risk of being told to go home.

Single party control typically comes with a big price tag, regardless of the party in control. In Trump’s first term, spending went up about 10 percent, according to data from the Office of Management and Budget. Under George W. Bush, that number was 24 percent. Both men enjoyed Republican majorities in Congress for much of their presidencies.

Under Bill Clinton, however, when government was largely divided, spending increased only 3 percent. And during the presidency of Barack Obama it fell 10 percent. This was not because those Democratic presidents were deeply committed to cutting spending—far from it, though previous generations of Democrats were warmer to the idea of fiscal restraint than the current generation.

The same is true, of course, of previous generations of Republicans. Under past administrations, Republicans were well established (rhetorically, if little else) as fiscal hawks. But they have in recent years largely abandoned that banner, choosing to downplay concerns about the long-term consequences of budget imbalances when those imbalances were brought about by spending on GOP priorities, such as defense and tax cuts, with no offsetting spending reductions. The real question is whether the dynamic of gridlock as a check on the growth of spending still holds in a country where the orthodoxy about debt and deficits has shifted.

As we saw during the confirmation battle over Amy Coney Barrett’s nomination to the Supreme Court, leading congressional Republicans do not mind a little hypocrisy in the service of higher goals. So it may be that Republicans snatch up the flag of fiscal conservatism once again, brush off the mud, and pretend as if they haven’t been trampling on it for years.

Another significant difference from the last time there was divided government is the increasing abdication of congressional responsibility in favor of executive power. Gridlock may well exacerbate this already established tendency of presidents to take recourse to executive fiat when the legislative or confirmation process stalls. Biden has made clear that he will reverse several of Trump’s signature executive orders on his first days in office. Given his positions on a path to citizenship for people covered by the Deferred Action for Childhood Arrivals program as well as spending on a border wall and the treatment of asylum seekers, it will become alarmingly traditional for U.S. immigration policy to swing wildly every time power changes hands—a bad thing for Americans and would-be Americans alike.

The same is true of trade policy and foreign policy, where presidents have significant latitude. Wars are expensive—both hot wars and trade wars. Biden didn’t run on either kind of bellicosity, though presidents have a way of coming to terms with debt and deaths after assuming power.

At the same time, some of the most alarming and highest-variance outcomes threatened by Democrats are now unlikely to occur. Court packing will be all but impossible under divided government, for example. Making and implementing massive climate change accords will prove difficult, especially if Republicans do indeed retain control of the Senate and its treaty-approving powers. Biden has already made clear that his agenda for health care reform is less radical than that of much of his party, but the possibilities there will be limited by a lack of congressional cooperation as well.

Gridlock is not the same thing as a libertarian moment. Gummed-up works are a consolation prize when there is so much urgent policy work to be done, including economic deregulation, dismantling of the barriers to free movement of people and goods, spending cuts, criminal justice reform, and more.

The election results did yield some encouraging signs for personal freedom, however. When Americans were asked in their polling places about specific individual freedoms, they chose “leave people alone” in large numbers. Faced with an unusually interesting slate of referenda, voters consistently selected the libertarian option without much fuss. Drug legalization and decriminalization ran the table, with all eight initiatives winning. Affirmative action will not be brought back in California college admissions and government hiring. Uber and Lyft do not have to reclassify their drivers as employees in the Golden State. In Massachusetts, voters approved a “right-to-repair” measure that protects property rights. The list goes on and on.

The Libertarian Party (L.P.) also put in a respectable showing, with its second-best national result ever. The party did well enough in several states to cover the spread between the major party candidates. While this, inevitably, will occasion claims that the L.P. played spoiler, there’s always the hope that instead of angrily demanding fealty from Libertarian supporters, one of the major parties will take the concerns of that largely untapped well of voters and potential voters into account and attempt to appeal to them more systematically in the next election cycle.

Another way to interpret the results is that on a deeper level nothing changed. The country is divided. We knew that before Election Day. Watching day after day of vote counting where totals wavered back and forth over the line of a bare majority drove that fact home more viscerally than ever. It’s pretty much exactly about 50/50 out there, folks. That’s how our parties are built. We’ll have a couple of years in which this particular variation of partisan control in Washington constrains political actors in certain ways, until another election slightly tweaks the balance of power again and we start over—while leaving nearly every incumbent in place and the options for dissatisfied voters almost completely unchanged.

In general, if people are dancing in the streets after an election, it’s a bad sign. It shows that the stakes of politics are too high, that people think too little of the other side (or too highly of their guy), and that voters have overvalued a single race at the top of the ticket. The slow burn of this cycle’s results tempered the feelings of both victory and defeat. Gridlock isn’t anyone’s first choice—not even libertarians’—but everyone losing a little is likely the best case for the next couple of years. Sore winners beget sorer losers, though, and since we have to do this again quite soon it behooves everyone to think about what it will be like when you are, as you inevitably will be, on the other side again.

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