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Posts Tagged ‘knowing

“Mathematics is the music of reason”*…

An illustration of a mathematician engaged in work, drawing geometric shapes and formulas on paper, with a three-dimensional geometric object and interconnected lines of mathematical concepts in the background.

New technologies, most centrally AI, are arming scientists with tools that might not just accelerate or enhance their work, but altogether transform it. As Jordana Cepelewicz reports, mathematicians have started to prepare for a profound shift in what it means to do math…

Since the start of the 20th century, the heart of mathematics has been the proof — a rigorous, logical argument for whether a given statement is true or false. Mathematicians’ careers are measured by what kinds of theorems they can prove, and how many. They spend the bulk of their time coming up with fresh insights to make a proof work, then translating those intuitions into step-by-step deductions, fitting different lines of reasoning together like puzzle pieces.

The best proofs are works of art. They’re not just rigorous; they’re elegant, creative and beautiful. This makes them feel like a distinctly human activity — our way of making sense of the world, of sharpening our minds, of testing the limits of thought itself.

But proofs are also inherently rational. And so it was only natural that when researchers started developing artificial intelligence in the mid-1950s, they hoped to automate theorem proving: to design computer programs capable of generating proofs of their own. They had some success. One of the earliest AI programs could output proofs of dozens of statements in mathematical logic. Other programs followed, coming up with ways to prove statements in geometry, calculus and other areas.

Still, these automated theorem provers were limited. The kinds of theorems that mathematicians really cared about required too much complexity and creativity. Mathematical research continued as it always had, unaffected and undeterred.

Now that’s starting to change. Over the past few years, mathematicians have used machine learning models (opens a new tab) to uncover new patterns, invent new conjectures, and find counterexamples to old ones. They’ve created powerful proof assistants both to verify whether a given proof is correct and to organize their mathematical knowledge.

They have not, as yet, built systems that can generate the proofs from start to finish, but that may be changing. In 2024, Google DeepMind announced that they had developed an AI system that scored a silver medal in the International Mathematical Olympiad, a prestigious proof-based exam for high school students. OpenAI’s more generalized “large language model,” ChatGPT, has made significant headway on reproducing proofs and solving challenging problems, as have smaller-scale bespoke systems. “It’s stunning how much they’re improving,” said Andrew Granville, a mathematician at the University of Montreal who until recently doubted claims that this technology might soon have a real impact on theorem proving. “They absolutely blow apart where I thought the limitations were. The cat’s out of the bag.”

Researchers predict they’ll be able to start outsourcing more tedious sections of proofs to AI within the next few years. They’re mixed on whether AI will ever be able to prove their most important conjectures entirely: Some are willing to entertain the notion, while others think there are insurmountable technological barriers. But it’s no longer entirely out of the question that the more creative aspects of the mathematical enterprise might one day be automated.

Even so, most mathematicians at the moment “have their heads buried firmly in the sand,” Granville said. They’re ignoring the latest developments, preferring to spend their time and energy on their usual jobs.

Continuing to do so, some researchers warn, would be a mistake. Even the ability to outsource boring or rote parts of proofs to AI “would drastically alter what we do and how we think about math over time,” said Akshay Venkatesh, a preeminent mathematician and Fields medalist at the Institute for Advanced Study in Princeton, New Jersey.

He and a relatively small group of other mathematicians are now starting to examine what an AI-powered mathematical future might look like, and how it will change what they value. In such a future, instead of spending most of their time proving theorems, mathematicians will play the role of critic, translator, conductor, experimentalist. Mathematics might draw closer to laboratory sciences, or even to the arts and humanities.

Imagining how AI will transform mathematics isn’t just an exercise in preparation. It has forced mathematicians to reckon with what mathematics really is at its core, and what it’s for…

Absolutely fascinating: “Mathematical Beauty, Truth, and Proof in the Age of AI,” from @jordanacep.bsky.social‬ in @quantamagazine.bsky.social‬. Eminently worth reading in full.

James Joseph Sylvester

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As we wonder about ways of knowing, we might spare a thought for a man whose work helped trigger an earlier iteration of this enhance/transform discussion and laid the groundwork for the one unpacked in the article linked above above: J. Presper Eckert; he died on this day in 1995. An electrical engineer, he co-designed (with John Mauchly) the first general purpose computer, the ENIAC (see here and here) for the U.S. Army’s Ballistic Research Laboratory. He and Mauchy went on to found the Eckert–Mauchly Computer Corporation, at which they designed and built the first commercial computer in the U.S., the UNIVAC.

Three men interacting with a large vintage computer console, with tape reels in the background.
Eckert (standing and gesturing) and Mauchy (at the console), demonstrating the UNIVAC to Walter Cronkite (source)

“Wonder is the beginning of wisdom”*…

An argument for curiosity, openness, and the humility that underlies them both…

Philosophers aren’t the only ones who love wisdom. Everyone, philosopher or not, loves her own wisdom: the wisdom she has or takes herself to have. What distinguishes the philosopher is loving the wisdom she doesn’t have. Philosophy is, therefore, a form of humility: being aware that you lack what is of supreme importance. There may be no human being who exemplified this form of humility more perfectly than Socrates. It is no coincidence that he is considered the first philosopher within the Western canon.

Socrates did not write philosophy; he simply went around talking to people. But these conversations were so transformative that Plato devoted his life to writing dialogues that represent Socrates in conversation. These dialogues are not transcripts of actual conversations, but they are nonetheless clearly intended to reflect not only Socrates’s ideas but his personality. Plato wanted the world to remember Socrates. Generations after Socrates’s death, warring philosophical schools such as the Stoics and the Skeptics each appropriated Socrates as figurehead. Though they disagreed on just about every point of doctrine, they were clear that in order to count themselves as philosophers they had to somehow be working in the tradition of Socrates.

What is it about Socrates that made him into a symbol for the whole institution of philosophy? Consider the fact that, when the Oracle at Delphi proclaims Socrates wisest of men, he tries to prove it wrong. As Plato recounts it in the Apology:

I went to one of those reputed wise, thinking that there, if anywhere, I could refute the oracle and say to it: “This man is wiser than I, but you said I was.” Then, when I examined this man—there is no need for me to tell you his name, he was one of our public men—my experience was something like this: I thought that he appeared wise to many people and especially to himself, but he was not. I then tried to show him that he thought himself wise, but that he was not. As a result he came to dislike me, and so did many of the bystanders. So I withdrew and thought to myself: “I am wiser than this man; it is likely that neither of us knows anything worthwhile, but he thinks he knows something when he does not, whereas when I do not know, neither do I think I know; so I am likely to be wiser than he to this small extent, that I do not think I know what I do not know.”

If Socrates’s trademark claim is this protestation of ignorance, his trademark activity is the one also described in this passage: refuting the views of others. These are the conversations we find in Plato’s texts. How are the claim and the activity related? Socrates denies that his motivations are altruistic: he says he is not a teacher, and insists that he is himself the primary beneficiary of the conversations he initiates. This adds to the mystery: What is Socrates getting out of showing people that they don’t know what they take themselves to know? What’s his angle?

Over and over again, Socrates approaches people who are remarkable for their lack of humility—which is to say, for the fact that they feel confident in their own knowledge of what is just, or pious, or brave, or moderate. You might have supposed that Socrates, whose claim to fame is his awareness of his own ignorance, would treat these self-proclaimed “wise men” (Sophists) with contempt, hostility, or indifference. But he doesn’t. The most remarkable feature of Socrates’s approach is his punctilious politeness and sincere enthusiasm. The conversation usually begins with Socrates asking his interlocutor: Since you think you know, can you tell me, what is courage (or wisdom, or piety, or justice . . .)? Over and over again, it turns out that they think they can answer, but they can’t. Socrates’s hope springs eternal: even as he walks toward the courtroom to be tried—and eventually put to death—for his philosophical activity, he is delighted to encounter the self-important priest Euthyphro, who will, surely, be able to say what piety is…

Socrates seemed to think that the people around him could help him acquire the knowledge he so desperately wanted—even though they were handicapped by the illusion that they already knew it. Indeed, I believe that their ill-grounded confidence was precisely what drew Socrates to them. If you think you know something, you will be ready to speak on the topic in question. You will hold forth, spout theories, make claims—and all this, under Socrates’s relentless questioning, is the way to actually acquire the knowledge you had deluded yourself into thinking you already had…

It’s one thing to say, “I don’t know anything.” That thought comes cheap. One can wonder, “Who really and truly knows anything?” in a way that is dismissive, uninquisitive, detached. It can be a way of saying, “Knowledge is unattainable, so why even try?” Socratic humility is more expensive and more committal than that. He sought to map the terrain of his ignorance, to plot its mountains and its rivers, to learn to navigate it. That, I think, is why he speaks of knowledge of his own ignorance. He’s not just someone who acknowledges or admits to his ignorance, but someone who has learned to dwell within it.

Admittedly, this may seem like a paradoxical project. It’s one thing to be missing your wallet—you will know it once you’ve found it. But suppose you’re missing not only your wallet, but also the knowledge that you ever had a wallet, and the understanding of what a wallet is. One of Socrates’s interlocutors, Meno, doubts whether it’s possible to come to know anything if you know so little to begin with. If someone doesn’t know where she’s going, it doesn’t seem as though she can even take a first step in the right direction. Can you map in total darkness?

Socrates’s answer was no. Or at least: you can’t do it alone. The right response to noticing one’s own ignorance is to try to escape it by acquiring someone else’s knowledge. But the only way to do that is to explain to them why you aren’t yet able to accept this or that claim of theirs as knowledge—and that is what mapping one’s ignorance amounts to. Socrates stages an exhibition of this method for Meno by demonstrating how much geometrical progress he can make with a young slave boy by doing nothing but asking questions that expose the boy’s false assumptions. It is when he refutes others’ claims to knowledge that Socrates’s own ignorance takes shape, for him, as something he can know. What appears as a sea of darkness when approached introspectively turns out to be navigable when brought into contact with the knowledge claims of another…

Socrates saw the pursuit of knowledge as a collaborative project involving two very different roles. There’s you or I or some other representative of Most People, who comes forward and makes a bold claim. Then there’s Socrates, or one of his contemporary descendants, who questions and interrogates and distinguishes and calls for clarification. This is something we’re often still doing—as philosophers, as scientists, as interviewers, as friends, on Twitter and Facebook and in many casual personal conversations. We’re constantly probing one another, asking, “How can you say that, given X, Y, Z?” We’re still trying to understand one another by way of objection, clarification, and the simple fact of inability to take what someone has said as knowledge. It comes so naturally to us to organize ourselves into the knower/objector pairing that we don’t even notice we are living in the world that Socrates made. The scope of his influence is remarkable. But equally remarkable is the means by which it was achieved: he did so much by knowing, writing, and accomplishing—nothing at all.

And yet for all this influence, many of our ways are becoming far from Socratic. More and more our politics are marked by unilateral persuasion instead of collaborative inquiry. If, like Socrates, you view knowledge as an essentially collaborative project, you don’t go into a conversation expecting to persuade any more than you expect to be persuaded. By contrast, if you do assume you know, you embrace the role of persuader in advance, and stand ready to argue people into agreement. If argument fails, you might tolerate a state of disagreement—but if the matter is serious enough, you’ll resort to enforcing your view through incentives or punishments. Socrates’s method eschewed the pressure to persuade. At the same time, he did not tolerate tolerance. His politics of humility involved genuinely opening up the question under dispute, in such a way that neither party would be permitted to close it, to settle on an answer, unless the other answered the same. By contrast, our politics—of persuasion, tolerance, incentives, and punishment—is deeply uninquisitive…

Knowing takes radical collaboration: an openness to being persuaded as much as an eagerness to persuade: “Against Persuasion,” from Agnes Callard (@AgnesCallard)

* Socrates

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As we listen and learn, we might recall that it was on this date in 1966 that the Eleventh Plenum of the Eighth Central Committee of the Chinese Communist Party was hastily convened to advance Mao Zedong’s by then decidedly radical agenda for China– teeing up “Red August,” a series of purges of reactionary or otherwise impure thinkers. According to official statistics published in 1980, from August to September in 1966, a total of 1,772 people—including teachers and principals of many schools—were killed in Beijing by Red Guards; 33,695 homes were ransacked and 85,196 families were forced to leave the city. (1985 statistics, which included the areas immediately around Beijing, put the death toll at around 10,000.)

Red August kicked off the Great Proletarian Cultural Revolution, or as we tend to know it, The Cultural Revolution, which lasted until Mao’s death in 1976. Its stated goal was to preserve Chinese communism by purging alternative thought– remnants of capitalist and traditional elements– from Chinese society, and to impose Mao Zedong Thought (known outside China as Maoism) as the dominant ideology in the PRC. A selection of Mao’s sayings, compiled in Little Red Book, became a sacred text in what was, essentially a personality cult.

Estimates of the death toll from the Cultural Revolution, including civilians and Red Guards, vary greatly, ranging from hundreds of thousands to to 20 million. The exact figure of those who were persecuted or died during the Cultural Revolution, however, may never be known, as many deaths went unreported or were actively covered up by the police or local authorities. Tens of millions of people were persecuted (especially members of ethnic minorities): senior officials were purged or exiled; millions were accused of being members of the Five Black Categories, suffering public humiliation, imprisonment, torture, hard labor, seizure of property, and sometimes execution or harassment into suicide; intellectuals were considered the “Stinking Old Ninth” and were widely persecuted—notable scholars and scientists were killed or committed suicide. Schools and universities were closed. And over 10 million urban “intellectual youths were sent to rural areas in the Down to the Countryside Movement.

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Written by (Roughly) Daily

August 1, 2021 at 1:00 am