Posts Tagged ‘education’
“A proof tells us where to concentrate our doubts”*…
Number theorist Andrew Granville on what mathematics really is, on why objectivity is never quite within reach, and on the role that AI might play…
… What is a mathematical proof? We tend to think of it as a revelation of some eternal truth, but perhaps it is better understood as something of a social construct.
Andrew Granville, a mathematician at the University of Montreal, has been thinking about that a lot recently. After being contacted by a philosopher about some of his writing, “I got to thinking about how we arrive at our truths,” he said. “And once you start pushing at that door, you find it’s a vast subject.”
“How mathematicians go about research isn’t generally portrayed well in popular media. People tend to see mathematics as this pure quest, where we just arrive at great truths by pure thought alone. But mathematics is about guesses — often wrong guesses. It’s an experimental process. We learn in stages…
Quanta spoke with Granville about the nature of mathematical proof — from how proofs work in practice to popular misconceptions about them, to how proof-writing might evolve in the age of artificial intelligence…
[excerpts for that interview follow…]
How mathematicians go about research isn’t generally portrayed well in popular media. People tend to see mathematics as this pure quest, where we just arrive at great truths by pure thought alone. But mathematics is about guesses — often wrong guesses. It’s an experimental process. We learn in stages…
The culture of mathematics is all about proof. We sit around and think, and 95% of what we do is proof. A lot of the understanding we gain is from struggling with proofs and interpreting the issues that come up when we struggle with them…
The main point of a proof is to persuade the reader of the truth of an assertion. That means verification is key. The best verification system we have in mathematics is that lots of people look at a proof from different perspectives, and it fits well in a context that they know and believe. In some sense, we’re not saying we know it’s true. We’re saying we hope it’s correct, because lots of people have tried it from different perspectives. Proofs are accepted by these community standards.
Then there’s this notion of objectivity — of being sure that what is claimed is right, of feeling like you have an ultimate truth. But how can we know we’re being objective? It’s hard to take yourself out of the context in which you’ve made a statement — to have a perspective outside of the paradigm that has been put in place by society. This is just as true for scientific ideas as it is for anything else…
[Granville runs through a history of the proof, from Aristotle, through Euclid, to Hilbert, then Russel and Whitehead, ending with Gödel…]
To discuss mathematics, you need a language, and a set of rules to follow in that language. In the 1930s, Gödel proved that no matter how you select your language, there are always statements in that language that are true but that can’t be proved from your starting axioms. It’s actually more complicated than that, but still, you have this philosophical dilemma immediately: What is a true statement if you can’t justify it? It’s crazy.
So there’s a big mess. We are limited in what we can do.
Professional mathematicians largely ignore this. We focus on what’s doable. As Peter Sarnak likes to say, “We’re working people.” We get on and try to prove what we can…
[Granville then turns to computers…]
We’ve moved to a different place, where computers can do some wild things. Now people say, oh, we’ve got this computer, it can do things people can’t. But can it? Can it actually do things people can’t? Back in the 1950s, Alan Turing said that a computer is designed to do what humans can do, just faster. Not much has changed.
For decades, mathematicians have been using computers — to make calculations that can help guide their understanding, for instance. What AI can do that’s new is to verify what we believe to be true. Some terrific developments have happened with proof verification. Like [the proof assistant] Lean, which has allowed mathematicians to verify many proofs, while also helping the authors better understand their own work, because they have to break down some of their ideas into simpler steps to feed into Lean for verification.
But is this foolproof? Is a proof a proof just because Lean agrees it’s one? In some ways, it’s as good as the people who convert the proof into inputs for Lean. Which sounds very much like how we do traditional mathematics. So I’m not saying that I believe something like Lean is going to make a lot of errors. I’m just not sure it’s any more secure than most things done by humans…
Perhaps it could assist in creating a proof. Maybe in five years’ time, I’ll be saying to an AI model like ChatGPT, “I’m pretty sure I’ve seen this somewhere. Would you check it out?” And it’ll come back with a similar statement that’s correct.
And then once it gets very, very good at that, perhaps you could go one step further and say, “I don’t know how to do this, but is there anybody who’s done something like this?” Perhaps eventually an AI model could find skilled ways to search the literature to bring tools to bear that have been used elsewhere — in a way that a mathematician might not foresee.
However, I don’t understand how ChatGPT can go beyond a certain level to do proofs in a way that outstrips us. ChatGPT and other machine learning programs are not thinking. They are using word associations based on many examples. So it seems unlikely that they will transcend their training data. But if that were to happen, what will mathematicians do? So much of what we do is proof. If you take proofs away from us, I’m not sure who we become…
Eminently worth reading in full: “Why Mathematical Proof Is a Social Compact,” in @QuantaMagazine.
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As we add it up, we might send carefully calculated birthday greetings to Edward G. Begle; he was born on this date in 1914. A mathematician who was an accomplished topologist, he is best remembered for his role as the director of the School Mathematics Study Group (SMSG), the primary group credited for developing what came to be known as The New Math (a pedagogical response to Sputnik, taught in American grade schools from the late 1950s through the 1970s)… which will be well-known to (if not necessarily fondly recalled by) readers of a certain age.
“Poverty is the worst form of violence”*…
Two economic historians, Peter A. Coclanis and Louis M. Kyriakoudes, on why about 20% of counties in the U.S. South are marked by “persistent poverty”…
For a brief moment in the summer of 2023, the surprise No. 1 song “Rich Men North of Richmond” focused the country’s attention on a region that often gets overlooked in discussions of the U.S. economy. Although the U.S. media sometimes pays attention to the rural South — often concentrating on guns, religion and opioid overdoses — it has too often neglected the broad scope and root causes of the region’s current problems.
As economic historians based in North Carolina and Tennessee, we want a fuller version of the story to be told. Various parts of the rural South are struggling, but here we want to focus on the forlorn areas that the U.S. Department of Agriculture refers to as “rural manufacturing counties” — places where manufacturing is, or traditionally was, the main economic activity.
You can find such counties in every Southern state, although they were historically clustered in Alabama, Georgia, North and South Carolina, and Tennessee. And they are suffering terribly.
First, let’s back up. One might be tempted to ask: Are things really that bad? Hasn’t the Sun Belt been booming? But in fact, by a range of economic indicators — personal income per capita and the proportion of the population living in poverty, for starters – large parts of the South, and particularly the rural South, are struggling.
Gross domestic product per capita in the region has been stuck at about 90% of the national average for decades, with average income even lower in rural areas. About 1 in 5 counties in the South is marked by “persistent poverty” — a poverty rate that has stayed above 20% for three decades running. Indeed, fully 80% of all persistently poor counties in the U.S. are in the South.
Persistent poverty is, of course, linked to a host of other problems. The South’s rural counties are marked by low levels of educational attainment, measured both by high school and college graduation rates. Meanwhile, labor-force participation rates in the South are far lower than in the nation as a whole.
Unsurprisingly, these issues stifle economic growth.
Meanwhile, financial institutions have fled the region: The South as a whole lost 62% of its banks between 1980 and 2020, with the decline sharpest in rural areas. At the same time, local hospitals and medical facilities have been shuttering, while funding for everything from emergency services to wellness programs has been cut.
Relatedly, the rural South is ground zero for poor health in the U.S., with life expectancy far lower than the national average. So-called “deaths of despair” such as suicides and accidental overdoses are common, and rates of obesity, diabetes, hypertension, heart disease and stroke are high – much higher than in rural areas in other parts of the U.S. and in the U.S. as a whole…
Although some people think that these areas have forever been in crisis, this isn’t the case. While the South’s agricultural sector had fallen into long-term decline in the decades following the Civil War — essentially collapsing by the Great Depression — the onset of World War II led to an impressive economic growth spurt.
War-related jobs opening up in urban areas pulled labor out of rural areas, leading to a long-delayed push to mechanize agriculture. Workers rendered redundant by such technology came to constitute a large pool of cheap labor that industrialists seized upon to deploy in low-wage processing and assembly operations, generally in rural areas and small towns.
Such operations surged between 1945 and the early 1980s, playing a huge role in the region’s economic rise. However humble they may have been, in the South — as in China since the late 1970s — the shift out of a backward agricultural sector into low-wage, low-skill manufacturing was an opportunity for significant productivity and efficiency gains.
This helped the South steadily catch up to national norms in terms of per-capita income: to 75% by 1950, 80% by the mid-1960s, over 85% by 1970, and to almost 90% by the early 1980s…
By the early 1980s, however, the gains made possible by the shift out of agriculture began to play themselves out. The growth of the rural manufacturing sector slowed, and the South’s convergence upon national per capita income norms stopped, remaining stuck at about 90% from then on.
Two factors were largely responsible: new technologies, which reduced the number of workers needed in manufacturing, and globalization, which greatly increased competition. This latter point became increasingly important, since the South, a low-cost manufacturing region in the U.S., is a high-cost manufacturing region when compared to, say, Mexico.
Like Mike Campbell’s bankruptcy in Hemingway’s “The Sun Also Rises,” the rural South’s collapse came gradually, then suddenly: gradually during the 1980s and 1990s, and suddenly after China’s entry into the World Trade Organization in December 2001…
A sobering read: “Poor men south of Richmond? Why much of the rural South is in economic crisis.”
* Mahatma Gandhi
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As we dive into the dynamics of development, we might recall that it was on this date in 1718 that the famous pirate Edward Teach– better known as Blackbeard– was killed off the coast of North Carolina.
Edward Teach, also known as Blackbeard, is killed off North Carolina’s Outer Banks during a bloody battle with a British navy force sent from Virginia.
Believed to be a native of England, Edward Teach likely began his pirating career in 1713, when he became a crewman aboard a Caribbean sloop commanded by pirate Benjamin Hornigold. In 1717, after Hornigold accepted an offer of general amnesty by the British crown and retired as a pirate, Teach took over a captured 26-gun French merchantman, increased its armament to 40 guns, and renamed it the Queen Anne’s Revenge.
During the next six months, the Queen Anne’s Revenge served as the flagship of a pirate fleet featuring up to four vessels and more than 200 men. Teach became the most infamous pirate of his day, winning the popular name of Blackbeard for his long, dark beard, which he was said to light on fire during battles to intimidate his enemies. Blackbeard’s pirate forces terrorized the Caribbean and the southern coast of North America and were notorious for their cruelty.
In May 1718, the Queen Anne’s Revenge and another vessel were shipwrecked, forcing Blackbeard to desert a third ship and most of his men because of a lack of supplies. With the single remaining ship, Blackbeard sailed to Bath in North Carolina and met with Governor Charles Eden. Eden agreed to pardon Blackbeard in exchange for a share of his sizable booty.
At the request of North Carolina planters, Governor Alexander Spotswood of Virginia dispatched a British naval force under Lieutenant Robert Maynard to North Carolina to deal with Blackbeard. On November 22, Blackbeard’s forces were defeated and he was killed in a bloody battle of Ocracoke Island. Legend has it that Blackbeard, who captured more than 30 ships in his brief pirating career, received five musket-ball wounds and 20 sword lacerations before dying…
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“Any fool can know. The point is to understand.”*…
… and, Rachael Scarborough King and Seth Rudy argue, to serve a clear purpose…
Right now, many forms of knowledge production seem to be facing their end. The crisis of the humanities has reached a tipping point of financial and popular disinvestment, while technological advances such as new artificial intelligence programmes may outstrip human ingenuity. As news outlets disappear, extreme political movements question the concept of objectivity and the scientific process. Many of our systems for producing and certifying knowledge have ended or are ending.
We want to offer a new perspective by arguing that it is salutary – or even desirable – for knowledge projects to confront their ends. With humanities scholars, social scientists and natural scientists all forced to defend their work, from accusations of the ‘hoax’ of climate change to assumptions of the ‘uselessness’ of a humanities degree, knowledge producers within and without academia are challenged to articulate why they do what they do and, we suggest, when they might be done. The prospect of an artificially or externally imposed end can help clarify both the purpose and endpoint of our scholarship.
We believe the time has come for scholars across fields to reorient their work around the question of ‘ends’. This need not mean acquiescence to the logics of either economic utilitarianism or partisan fealty that have already proved so damaging to 21st-century institutions. But avoiding the question will not solve the problem. If we want the university to remain a viable space for knowledge production, then scholars across disciplines must be able to identify the goal of their work – in part to advance the Enlightenment project of ‘useful knowledge’ and in part to defend themselves from public and political mischaracterisation.
Our volume The Ends of Knowledge: Outcomes and Endpoints Across the Arts and Sciences (2023) asks how we should understand the ends of knowledge today. What is the relationship between an individual knowledge project – say, an experiment on a fruit fly, a reading of a poem, or the creation of a Large Language Model – and the aim of a discipline or field? In areas ranging from physics to literary studies to activism to climate science, we asked practitioners to consider the ends of their work – its purpose – as well as its end: the point at which it might be complete. The responses showed surprising points of commonality in identifying the ends of knowledge, as well as the value of having the end in sight…
Read on for a provocative case that academics need to think harder about the purpose of their disciplines and a consideration of whether some of those should come to an end: “The Ends of Knowledge,” in @aeonmag.
* Albert Einstein
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As we contemplate conclusions, we might recall that it was on this date in 1869 that the first issue of the journal Nature was published. Taking it’s title from a line of Wordsworth’s (“To the solid ground of nature trusts the Mind that builds for aye”), its aim was to “provide cultivated readers with an accessible forum for reading about advances in scientific knowledge.” It remains a weekly, international, interdisciplinary journal of science, one of the few remaining that publish across a wide array of fields. It is consistently ranked the world’s most cited scientific journal and is ascribed an impact factor of approximately 64.8, making it one of the world’s top academic journals.
“The gods do not visit you to remind you what you know already”*…
Or do they? As the estimable Emily Watson explains, one woman– Edith Hamilton— had a great deal to do with our acquaintance with the mythology of ancient Greece and Rome, and in a way that had as much to do with the present as the past…
The discipline of “classical” literature has long been associated with social gatekeeping. The mastery of Latin and ancient Greek—or at least enough of an acquaintance to be able to trot out a well-worn tag from Horace and prompt knowing chuckles over the brandy—has often provided a useful qualification for passing as a gentleman and keeping out the plebs (Latin for “common people”) or hoi polloi (ancient Greek for “the many”). It is understandable that George Eliot’s Maggie Tulliver eagerly whizzes through the Latin textbooks neglected by her idle brother Tom, or that Thomas Hardy’s Jude, hoping in vain to escape the obscurity of provincial poverty, slogs through his Greek dictionary until late into the night. For these fictional characters, like many of their real-life equivalents, ancient languages and literature provided the most visible bar against entry into a “higher” social class.
Even in the 20th and 21st centuries, the subject has maintained a close association with systems of exclusion based on income, education, race, and gender…
Over the past century, there have been numerous attempts to provide wider access to the supposed treasures of Greek and Roman antiquity to those once excluded from its riches. In the early 20th century, translations of ancient texts became more widely available, ranging from Gilbert Murray’s wonderfully ornate and virtuosic renditions of Euripides to Hilda Doolittle’s much starker modernist free verse. The Loeb Classical Library was founded in 1911 and featured fairly inexpensive editions of these ancient texts in their original languages, with an English translation on the facing page. These translations heralded a new age in which once-inaccessible works of classical literature became more accessible to a far wider range of people. As Virginia Woolf noted in 1917, it was the Loeb Library editions that helped make it “respectable” for the “amateur” (including female ones) to muddle through Aeschylus.
After the First World War, an ever-larger number of colleges and universities in the United States began to offer classes on ancient texts studied in translation, with no expectation that students would be able to read even a little of the originals. By the early 20th century, the study of ancient literature and history was often considered a prerequisite for understanding contemporary issues in Europe and the United States—regions that were now often lumped together under the term “the West.” The Columbia Core, the first such course in the United States, was developed in 1919 with the explicit goal of showing students the “unique features of the western world,” a world that apparently had begun in ancient Greece and that had now reached its apex in the American present.
The goal of connecting US citizens to a long, largely fabricated notion of “Western civilization” seemed increasingly urgent in the aftermath of a war that had torn the nations of Europe apart. The fantasy of a common “Western” heritage shared by white Europeans and North Americans appeared as a prophylactic against future wars, at least between those who could qualify as “Westerners.” But it also did something else. By excluding the numerous surviving ancient texts and cultural artifacts from the rest of the world, these new courses on “Western civilization” suggested that premodern “civilization” was the exclusive property of the “West”—enabling a kind of mythical/historical justification for continued domination of those peoples deemed to have come from outside this exclusive group, whether it was Black and Asian Americans in the United States or the millions still living under imperial and colonial rule in Asia and Africa.
By the 1920s, a sizable market for popular classics books and translations had emerged in the United States. A new publishing firm, W.W. Norton, decided to seize on it and signed up a recently retired Latin teacher and private school headmistress named Edith Hamilton to translate a trio of Greek tragedies and write two surveys of ancient literature, The Greek Way and The Roman Way. Published in quick succession in the 1930s, these volumes proved to be an immediate success. Along with Mythology, her retelling of the Greco-Roman legends, the books made Hamilton a household name. Mythology has never been out of print since then and has remained an extraordinary commercial success, enriching her heirs and publishers to this day. Probably no other single person has had such an impact in shaping the perceptions of classical literature and mythology in the United States for almost a century.
How did a retired Latin schoolteacher (Hamilton was 62 when The Greek Way was published), with limited formal education and almost no scholarly credentials, come to be one of the most influential “classicists” of the 20th century? Victoria Houseman’s annalistic new biography, American Classicist, does not quite see this question as the puzzle it is, in part because Houseman has so much admiration for her subject that Hamilton’s successes are largely taken for granted. But the book still makes for gripping reading, as we trace the trajectory of Hamilton’s life from her activist youth (she was a member of the Baltimore Equal Suffrage League), through her various travels in Europe and Asia, her health troubles (she was a breast cancer survivor), her fascinating romantic partnerships with other women, to her second career as a popularizer of the ancient world and as a public intellectual who became closely associated with wealthy and powerful conservative groups in the US…
Read on for more of Hamilton’s remarkable– and cautionary– story: “Ancient Worlds,” from @EmilyRCWilson (@emilyrcwilson.bsky.social) in @thenation.
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As we contemplate classics, we might send aquatic birthday greetings to Squidward Tentacles; today is his (fictional) birthday. A main characters of the SpongeBob SquarePants franchise, he is SpongeBob’s and Patrick’s grumpy neighbor and the former’s coworker at the Krusty Krab who lives in an Easter Island head. He is a mostly unpleasant artist and musician, and his favorite hobbies are painting self-portraits and playing the clarinet.
Though Squidward’s name contains the word “squid,” he is an octopus.
“Education is the key to unlock a golden door of freedom”*…
Historian (and author) Ada Palmer on what she convincingly argues is an unappreciated aspect of American cultural and political history…
Many shocking, new ideas shaped the American Experiment and related 18th century democratic ventures; as an historian of the period, I often notice that one of the most fundamental of them, and most shocking to a world which had so long assumed the opposite, often goes unmentioned — indeed sometimes denied — in today’s discussions of democracy: the belief that all people are educable. I think it’s urgent that we bring that principle back into the spotlight if we want to defend democracy from one of its common failure modes: pseudo-populist oligarchy.
Within “all men are created equal” lies the sub-principle that all people, or specifically all enfranchised citizens of a state (which often at the time meant white male adults, though some made it broader, or narrower) that all such people are, if given appropriate educational resources, capable of learning, exercising sound judgment, and acting on said judgment, thus that they all people are equally rational and capable of competent self-governance. This thesis does not assume that all people when adults are equally prepared to participate in government, but that all people when born have the capacity to absorb education if given access to it. Rare intellectual disabilities might make the education process challenging for certain individuals, but (the thesis argues) even then the right support and resources make education possible, and such situations are not the default human state. This is the thesis that all people are fundamentally educable.
Many in the 18th c. who thought democracy was absurd rejected it because they disagreed with this thesis, believing that the majority of people (even of white men) were not educable, i.e. that even with educational resources most people were born incapable of being guided by Reason and making sound political judgments. Those who believed this predicted that government by the people would collapse into absurdity, since it would be led by a parliament of fools…
[Palmer runs through pre-modern and Enlightenment thinking on the subject– most of which dismissed democracy as impractical on grounds that went to the incapacity of the many to understand and make intelligent decisions…]
It took a lot to get even a small radical fringe by 1750 to entertain the notion that all people–or even just all men–were created equally educable. A long survey of the causes would get unwieldy, but they include (among other things) contact with indigenous cultures in the Americas and other regions which had functional governments without European-style systems, revolutions in medicine and the understanding of the sense organs which undermined old hierarchy-enforcing ideas about how cognition and sensation functioned, second-order consequences of the rags-to-riches path opened by Renaissance courts employing scholars from any background so long as they had good Latin, and Protestantism’s second-order implication that, if people didn’t need priests as intermediaries between their prayers and God, perhaps they didn’t need aristocrats as intermediaries between them and power. But by 1750 that fringe existed, and had enough momentum to implement its experiment in the new United States, which most people who were considered sensible at the time thought would quickly degenerate into chaos, because they didn’t think most people were capable of understanding the world enough to vote sensibly, draft legislation, or serve in a congress, and that the tiny wise minority would be drowned out by the majority wanting to vote for dining on the king’s tab and killing all the lawyers…
Democracy can function, says Thomas Paine (to pick a spokesman for the US founders), because human beings are fundamentally educable, and if given a good teacher, a good reading list, and some newspapers, all human beings, or at least the overwhelming majority of them, will become capable of wise judgment and self-rule. One’s civic duty is not to identify the wise minority and put them in power, but to disseminate the tools of education so the majority can become wise. This thesis is opposed to aristocracy, to oligarchy, to timocracy, even to most forms of meritocracy, since education isn’t supposed to prepare people to be sorted out by the exam but to demonstrate that human beings are so excellent that everyone can pass it…
… Today’s America has seen decades of the intentional conservative-led starving and squeezing of public education, efforts to increase the disparity in education quality between public education and private or charter school education, conservative-led homeschool movements which aim to expose people to a narrow range of ideology, and also the devastation of newspapers, journalism, and a vast misinformation campaign. All this adds up to preventing many who are educable from becoming educated. Thomas Paine, and those I’m using him to represent, would recognize this as a sabotage of their system, one they would say might indeed enable Cade-style populism, which (as in Henry VI) is easy for ambitious elites to then harness to their own ends. Thus, Paine would say: of course the democracy isn’t working well if such an essential precondition is being sabotaged.
In sum, we need to talk more about the vital tie between democracy and the conviction that all people are created educable. It helps make clear how strategic the strangulation of educational resources is, and that one of the less loud but most dangerous threats to our confidence in democracy is the project to make it seem like most people can’t make sensible political judgments, reducing people’s confidence in democracy as a system by seeming to prove true conservative principle that there will always be a few who should rule and many who can’t. When I see conservative thinking start to show up in acquaintances (or Silicon Valley leaders) who consider themselves progressive but also consider themselves smart, it often begins with them feeling that most people are stupid and the world would be better off if the smart were in charge. One can often get such people to pause and reflect by bringing up the question of whether they think all people are fundamentally educable, and whether the solution isn’t to put the reins of power into genius hands but to put the Encyclopedia in everyone else’s. Information is key. Those peasants who shared commons maintained them sustainably for centuries because (as we now recognize) they were educated in the ways that mattered, they learned from families and communities to understand what they were doing, using local knowledge of commons, grazing etc. as they made choices. If one’s democratic state is the commons, people will likewise maintain it well, but not if they’re intentionally deprived of access to basic knowledge of how it works and what can harm or heal it, and drowned instead in deliberate falsehoods.
We all know we need to support education & good journalism, and combat misinformation, but revisiting the principle that all people are created educable is a good way to remember that these are not merely invaluable social goods, like sanitation or public parks. They were conceived from the start as essential components of modern democracy, in direct opposition to the many-centuries-old conservative principle that some are best to rule and others to be ruled. Enlightenment-style democracy cannot function without the conviction that all people are created educable. If we forget that, if we doubt it, if we let it shake our confidence in the experiment which didn’t turn into Jack Cade for more than two centuries (bets were not on America surviving for so long in 1776!), we risk opening the gates to the old failure mode of oligarchy rising when democracy wavers…
In support of the engine of the American Experiment: “All People Are Created Educable, a Vital Oft-Forgotten Tenet of Modern Democracy,” from @Ada_Palmer.
* George Washington Carver
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As we enable education, we might recall that it was on this date in 1818 (15 years after the Senate ratified the Louisiana Purchase) that the “Convention respecting fisheries, boundary and the restoration of slaves” (also known as the London Convention, Anglo-American Convention of 1818, Convention of 1818, or most widely and most simply as the Treaty of 1818) was signed by The U.S. and the United Kingdom.
Primarily aimed at settling border disputes, the treaty set the Canada–United States border on the 49th parallel for most of its length. The British ceded all of Rupert’s Land south of the 49th parallel and east of the Continental Divide, including all of the Red River Colony south of that latitude, while the United States ceded the northernmost edge of the Missouri Territory north of the 49th parallel.
The treaty also allowed for joint occupation and settlement of the Oregon Country, known to the British and in Canadian history as the Columbia District of the Hudson’s Bay Company, and including the southern portion of its sister district New Caledonia.
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