Posts Tagged ‘truth’
“Mathematics is the music of reason”*…
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.
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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.
“The pure and simple truth is rarely pure and never simple”*…
An all-too-timely 2016 piece from philosophy professors Scott Aikin and Robert Talisse…
So much Political commentary seems to proceed by means of debate rather than report. This is an understandable consequence of new technology which makes engagement easy. Our heightened exposure to debate is a good thing, too. Open debate is democracy’s lifeblood. Yet popular political disagreement has taken on an odd hue. Rather than presenting facts and professing a view, commentators present views concerning the views of their opponents. And often, it’s not only views about opponents’ views, many go straight to views about opponents. Despite heated disagreements over Big Questions like healthcare, stem-cell research, abortion, same-sex marriage, race relations and global warming, we find a surprising consensus about the nature of political disagreement itself: All agree that, with respect to any Big Question, there is but one intelligent position, and all other positions are not merely wrong, but ignorant, stupid, naïve. And as a consequence, those who cling to these views must be themselves either ignorant or wicked. Or both.
A minute in the Public Affairs section of any bookstore confirms this: Conservatives should talk to liberals “only if they must” because liberalism is a “mental disorder.” Liberals dismiss their Conservative opponents, since they are “lying liars” who use their “noise machine” to promote irrationality.
Both views betray a commitment to the Simple Truth Thesis, the claim that Big Questions always admit of a simple, obvious, and easily-stated solution. The Simple Truth Thesis encourages us to hold that a given truth is so simple and so obvious that only the ignorant, wicked, or benighted could possibly deny it. As our popular political commentary accepts the Simple Truth Thesis, there is a great deal of inflammatory rhetoric and righteous indignation, but in fact very little public debate over the issues that matter most. Consequently, the Big Questions over which we are divided remain unexamined, and our reasons for adopting our different answers are never brought to bear in public discussion.
This brings us back to our original observation – there seems to be so much debate. Yet what passes for public debate is in fact no debate at all. No surprise, really. Debate or discussion concerning a Big Question can be worthwhile only when there is more than one reasonable position regarding the question; and this is precisely what the Simple Truth Thesis denies.
It would be a wonderful world were the Simple Truth Thesis true. Our political task simply would be to empower those who know the simple truth, and rebuke the fools who do not. But the Simple Truth Thesis is not true. In fact, it’s a fairytale—soothing, but ultimately unfit for a serious mind. For any Big Question, there are several defensible positions; it is precisely this feature that makes them big. Of course, to say that a position is defensible is not to say that it’s true. To oppose the Simple Truth Thesis is not to embrace relativism (which is itself a version of the Simple Truth view), nor is it to give up on the idea that there is truth; it is rather to give up on the view that the truth is always simple.
This intellectual distance is difficult because we feel invested in our own Big Answers. But it’s a fantasy to think that the billions of people with whom we disagree have all simply failed to appreciate the facts. This fantasy is easily dissolved once we come to realize that those who reject our own Big Answers often give good reasons for their views and against ours. We might not find ourselves convinced by their reasons, of course, but we can no longer see them as ignorant or foolish.
The lesson to draw is that there is a difference between being stupid and being wrong; the most important truths are often the most difficult to discern, even by the most careful and sincere inquirers. This lesson dismantles the Simple Truth Thesis and leads us to acknowledge that although there may be but one correct answer to each Big Question, there are several defensible views concerning which of the going answers is, indeed, correct. So if the Big Questions matter to us, we should be most eager to hear the reasons of our opponents. We should pursue real disagreement, with real interlocutors, not the cooked-up arguments against caricatured opposition on offer from the political commentary industry.
Democracy is the proposition that a just, peaceful, and morally decent society is possible among equals who disagree over Big Questions. Democracy tries to enable such a society by maintaining the conditions under which citizens could reason together, and, despite ongoing disagreement, come to see each other as reasonable. Citizens who see each other in this way can agree to share in the task of collective self-government despite ongoing and even growing discord over Big Questions. The Simple Truth Thesis repudiates this ideal. Accordingly, as our politics become more argumentative, they become less concerned with actual argument. Yet if we lose our capacity to argue with each other—to confront openly each other’s reasons—we will lose our capacity to see each other as equal partners in self-government, and thus we will lose our democracy…
If only: “The Myth of Simple Truths,” in @3QD.
(Image above: source)
* Oscar Wilde
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As we dig Diogenes, we might send exciting birthday greetings to Otto Binder; he was born on this date in 1911. An author of science fiction and non-fiction books and stories, and comic books, he is best known as the co-creator of Supergirl and for his many scripts for Captain Marvel Adventures and other stories involving the entire superhero Marvel Family. He is credited with writing over 4,400 stories across a variety of publishers under his own name, as well as more than 160 stories under the pen-name Eando Binder.
Indeed, it was as Eando that he wrote “I, Robot” is a scifi short story , part of a series about a robot named Adam Link, that was published in the January 1939 issue of Amazing Stories. Very innovative for its time, “I, Robot” was one of the first robot stories to break away from Frankenstein clichés. It was reprised in two different comic series, and adapted into episodes of The Outer Limits.
Isaac Asimov— who is famous for his own I, Robot and the series of novels that followed from it, was heavily influenced by the Binder short story. In his introduction to the story in Isaac Asimov Presents the Great SF Stories (1979), Asimov wrote: “It certainly caught my attention. Two months after I read it, I began ‘Robbie’, about a sympathetic robot, and that was the start of my positronic robot series. Eleven years later, when nine of my robot stories were collected into a book, the publisher named the collection I, Robot over my objections. My book is now the more famous, but Otto’s story was there first.”
“We are not what we know but what we are willing to learn”*…
Abigail Tulenko argues that folktales, like formal philosophy, unsettle us into thinking anew about our cherished values and views of the world…
The Hungarian folktale Pretty Maid Ibronka terrified and tantalised me as a child. In the story, the young Ibronka must tie herself to the devil with string in order to discover important truths. These days, as a PhD student in philosophy, I sometimes worry I’ve done the same. I still believe in philosophy’s capacity to seek truth, but I’m conscious that I’ve tethered myself to an academic heritage plagued by formidable demons.
The demons of academic philosophy come in familiar guises: exclusivity, hegemony and investment in the myth of individual genius. As the ethicist Jill Hernandez notes, philosophy has been slower to change than many of its sister disciplines in the humanities: ‘It may be a surprise to many … given that theology and, certainly, religious studies tend to be inclusive, but philosophy is mostly resistant toward including diverse voices.’ Simultaneously, philosophy has grown increasingly specialised due to the pressures of professionalisation. Academics zero in on narrower and narrower topics in order to establish unique niches and, in the process, what was once a discipline that sought answers to humanity’s most fundamental questions becomes a jargon-riddled puzzle for a narrow group of insiders.
In recent years, ‘canon-expansion’ has been a hot-button topic, as philosophers increasingly find the exclusivity of the field antithetical to its universal aspirations. As Jay Garfield remarks, it is as irrational ‘to ignore everything not written in the Eurosphere’ as it would be to ‘only read philosophy published on Tuesdays.’ And yet, academic philosophy largely has done just that. It is only in the past few decades that the mainstream has begun to engage seriously with the work of women and non-Western thinkers. Often, this endeavour involves looking beyond the confines of what, historically, has been called ‘philosophy’.
Expanding the canon generally isn’t so simple as resurfacing a ‘standard’ philosophical treatise in the style of white male contemporaries that happens to have been written by someone outside this demographic. Sometimes this does happen, as in the case of Margaret Cavendish (1623-73) whose work has attracted increased recognition in recent years. But Cavendish was the Duchess of Newcastle, a royalist whose political theory criticises social mobility as a threat to social order. She had access to instruction that was highly unusual for women outside her background, which lends her work a ‘standard’ style and structure. To find voices beyond this elite, we often have to look beyond this style and structure.
Texts formerly classified as squarely theological have been among the first to attract significant renewed interest. Female Catholic writers such as Teresa of Ávila or Sor Juana Inés de la Cruz, whose work had been largely ignored outside theological circles, are now being re-examined through a philosophical lens. Likewise, philosophy departments are gradually including more work by Buddhist philosophers such as Dignāga and Ratnakīrti, whose epistemological contributions have been of especial recent interest. Such thinkers may now sit on syllabi alongside Augustine or Aquinas who, despite their theological bent, have long been considered ‘worthy’ of philosophical engagement.
On the topic of ‘worthiness’, I am wary of using the term ‘philosophy’ as an honorific. It is crucial that our interest in expanding the canon does not involve the implication that the ‘philosophical’ confers a degree of rigour over the theological, literary, etc. To do so would be to engage in a myopic and uninteresting debate over academic borders. My motivating question is not what the label of ‘philosophy’ can confer upon these texts, but what these texts can bring to philosophy. If philosophy seeks insight into the nature of such universal topics as reality, morality, art and knowledge, it must seek input from those beyond a narrow few. Engaging with theology is a great start, but these authors still largely represent an elite literate demographic, and raise many of the same concerns regarding a hegemonic, exclusive and individualistic bent.
As Hernandez quips: ‘[W]e know white, Western men have not cornered the market on deeply human, philosophical questions.’ And furthermore, ‘we also know, prudentially, that philosophy as a discipline needs to (and must) undergo significant navel-gazing to survive … in an ever-increasingly difficult time for homogenous, exclusive academic disciplines.’ In light of our aforementioned demons, it appears that philosophy is in urgent need of an exorcism.
I propose that one avenue forward is to travel backward into childhood – to stories like Ibronka’s. Folklore is an overlooked repository of philosophical thinking from voices outside the traditional canon. As such, it provides a model for new approaches that are directly responsive to the problems facing academic philosophy today. If, like Ibronka, we find ourselves tied to the devil, one way to disentangle ourselves may be to spin a tale…
Wisdom is where we find it: “Folklore is philosophy,” in @aeonmag. Eminently worth reading in full.
Apposite: “Syncretic Past.”
* Mary Catherine Bateson
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As we update our understanding of understanding, we might send thoughtful birthday greetings to Michael Sandel; he was born on this date in 1953. A philosopher and professor of government theory at Harvard Law School (where his course Justice was the university’s first course to be made freely available online and on television, seen so far by tens of millions of people around the world), he is probably best known for his critique of John Rawls‘ A Theory of Justice (in Sandel’s book, Liberalism and the Limits of Justice).
Sandel subscribes to a certain version of communitarianism (although he is uncomfortable with the label), and in this vein he is perhaps best known for his critique of John Rawls’s A Theory of Justice. Rawls’s argument depends on the assumption of the veil of ignorance, which Sandel argues commits Rawls to a view of people as “unencumbered selves”. Sandel’s view is that we are by nature encumbered to an extent that makes it impossible even hypothetically to have such a veil. Some examples of such ties are those with our families, which we do not make by conscious choice but are born with, already attached. Because they are not consciously acquired, it is impossible to separate oneself from such ties. Sandel believes that only a less-restrictive, looser version of the veil of ignorance should be postulated. Criticism such as Sandel’s inspired Rawls to subsequently argue that his theory of justice was not a “metaphysical” theory but a “political” one, a basis on which an overriding consensus could be formed among individuals and groups with many different moral and political views.
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“The truth is rarely pure and never simple”*…
For a century, the idea of truth has been deflated, becoming terrain from which philosophers fled. Crispin Sartwell argues that they must return – urgently…
It is often said, rather casually, that truth is dissolving, that we live in the ‘post-truth era’. But truth is one of our central concepts – perhaps our most central concept – and I don’t think we can do without it. To believe that masks prevent the spread of COVID-19 is to take it to be true that they do. To assert it is to claim that it is true. Truth is, plausibly, central to thought and communication in every case. And, of course, it’s often at stake in practical political debates and policy decisions, with regard to climate change or vaccines, for example, or who really won the election, or whom we should listen to about what.
One might have hoped to turn to philosophy for a clarification of the nature of truth, and maybe even a celebration of it. But philosophy of pragmatist, analytic and continental varieties lurched into the post-truth era a century ago. If truth is a problem now for everyone, if the idea seems empty or useless in ‘the era of social media’, ‘science denialism’, ‘conspiracy theories’ and suchlike, maybe that just means that ‘everyone’ has caught up to where philosophy was in 1922…
[Sartwell sketches the last 100 years of philosophy, and it’s undermining of the very idea of truth.]
I don’t think, despite all the attacks on the notion by all sorts of philosophers for a good century, that we’re going to be able to do without truth. In a way, I don’t think all those attacks touched truth at all, which (we’re finding) is necessary, still the only possible cure…
As a first step… we might broaden the focus from the philosophical question of what makes a sentence or proposition true or false to focus on some of the rich ways the concept of truth functions in our discourse. That love is true does not mean that it is a representation that matches up to reality. It does not mean that the love hangs together with all the rest of the lover or lovee’s belief system. It doesn’t mean that the hypothesis that my love is true helps us resolve our problems (it might introduce more problems). It means that the love is intense and authentic, or, as I’d like to put it, that it is actual, real. That my aim is true does not indicate that my aim accurately pictures the external world, but that it thumps the actual world right in the centre, as it were.
Perhaps what is true or false isn’t only, or even primarily, propositions, but loves and aims, and the world itself. That is, I would like to start out by thinking of ‘true’ as a semi-synonym of ‘real’. If I were formulating in parallel to Aristotle, I might say that ‘What is, is true.’ And perhaps there’s something to be said for Heidegger’s ‘comportment’ after all: to know and speak the real requires a certain sort of commitment: a commitment to face reality. Failures of truth are, often, failures to face up. Now, I’m not sure how much that will help with mathematics, but maths needs to understand that it is only one among the many forms of human knowledge. We, or at any rate I, might hope that an account that addresses the traditional questions about propositional truth might emerge from this broader structure of understanding. That is speculative, I admit.
Truth may not be the eternal unchanging Form that Plato thought it was, but that doesn’t mean it can be destroyed by a few malevolent politicians, tech moguls or linguistic philosophers, though the tech moguls and some of the philosophers (David Chalmers, for instance) might be trying to undermine or invent reality, as well. Until they manage it, the question of truth is as urgent, or more urgent, than ever, and I would say that despite the difficulties, philosophers need to take another crack. Perhaps not at aletheia as a joy forever, but at truth as we find it, and need it, now…
On why philosophy needs to return of the question of truth: “Truth Is Real,” from @CrispinSartwell in @aeonmag.
Source of the image above, also relevant: “The difference between ‘Truth’ and ‘truth’.”
* Oscar Wilde
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As we wrestle with reality, we might recall that it was on this date in 1986 that Geraldo Rivera opened “Al Capone’s Vault”…
Notorious and “most wanted” gangster, Al Capone, began his life of crime in Chicago in 1919 and had his headquarters set up at the Lexington Hotel until his arrest in 1931. Years later, renovations were being made at the hotel when a team of workers discovered a shooting-range and series of connected tunnels that led to taverns and brothels making for an easy escape should there be a police raid. Rumors were spread that Capone had a secret vault hidden under the hotel as well. In 1985, news reporter Geraldo Rivera had been fired from ABC after he criticized the network for canceling his report made about an alleged relationship between John F. Kennedy and Marilyn Monroe. It seemed like a good time for Rivera to scoop a new story to repair his reputation. It was on this day in 1986 that his live, two-hour, syndicated TV special, The Mystery of Al Capone’s Vault aired. After lots of backstory, the time finally came to reveal what was in that vault. It turned out to be empty. After the show, Rivera was quoted as saying “Seems like we struck out.”
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“Lying is done with words, and also with silence”*…
The redoubtable John Bayley deconstructs deceit in his 1990 review of Philip Kerr‘s The Penguin Book of Lies…
Kipling’s unspoken criterion, that if a thing is well done enough it must be the case, has caught on in a big way. It is all made up by the media, and by those faithful camp-followers of art who tell us on radio or TV what they were ‘trying to do’ when they wrote, painted or composed. The fashionably perceptive novel, like Julian Barnes’s memorable Flaubert’s Parrot, cannot find out what it is looking for, or what is going on inside itself. The fashionable thriller does not know the answers, or which side is which. These reactions against realism discard, often to great effect, the idea of truth as solution. So, fortuitously, did Henry James in The Turn of the Screw, and Stendhal’s Fabrice when he could not find the battle of Waterloo, or be sure that he had taken part in it. That truth is a lie shows that life is stupid: once art could fix that, but now this is not so certain, even of the best art.
As Philip Kerr has perceived, and embodied in his choice of extracts, a self-consciousness about truth goes with scepticism about it to produce the modern science of propaganda. Truth is the first casualty in war, whether hot or cold; when Churchill remarked that truth in wartime was ‘so precious that she should always be attended by a bodyguard of lies’ he was not just being cynical, like Bacon’s jesting Pilate, but assuming that truth, like right, must be on one’s own side: enemy truth, attended by her SS bodyguard, must be a false creature. As Orwell saw, faith in the truth leads to lying. Since the Party is always right, the concoction of deliberate lies is a moral duty. The victims of the show-trials in Moscow and Prague found relief in the lies they were made to utter. In 1940 Orwell supposed – which is interesting in the light of what happened later in the Soviet Union, and indeed in 1984 – that ‘already there are countless people who would think it scandalous to falsify a scientific textbook, but would see nothing wrong in falsifying a historical fact.’ Today science can appear just as much ‘the father of lies’ as Herodotus himself: not because of our appetite for wonders, but because the scientist has become much more aware of the possibilities of propaganda, whether personally or ideologically motivated…
More on the forms of fabrication: “Art’ll fix it,” in @LRB
(Image above: source)
* Adrienne Rich, Women and Honor: Some Notes on Lying
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As we ponder prevarication, we might rejoice in the naively and nobly inventive: it was on this date in 1605 that El Ingenioso Hidalgo Don Quijote de la Mancha (or The Ingenious Hidalgo Don Quixote of La Mancha— aka Don Quixote), the masterwork of Miguel de Cervantes (and of the Spanish Golden Age) and a founding work of Western literature was first published. Widely considered the first modern novel (published in the Western world), it is also considered by many (still) to be the best; it is in any case the second most translated work in the world (after the Bible).
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