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This methodology involves one party proving it has information it claims is true and a second party that wants to verify that the first party's information is indeed true. With a zero-knowledge proof system, the proving party does not transmit any secretive information that could substantiate whether what it claims is true.

– SearchSecurity

A zero-knowledge proof requires no real knowledge or secret information to prove the claim.

Zero-knowledge proofs are used in modern cybersecurity in situations where one system claims to possess sensitive data yet does not want to transmit that data to prove it to another system.

The true work of the mathematician is not experienced until the later parts of graduate school, when the student is challenged to create knowledge in the form of a novel proof. It is common to fill page after page with an attempt, the seasons turning, only to arrive precisely where you began, empty-handed — or to realize that a subtle flaw of logic doomed the whole enterprise from its outset. The steady state of mathematical research is to be completely stuck. It is a process that Charles Fefferman of Princeton, himself a onetime math prodigy turned Fields medalist, likens to ‘‘playing chess with the devil.’’ The rules of the devil’s game are special, though: The devil is vastly superior at chess, but, Fefferman explained, you may take back as many moves as you like, and the devil may not. You play a first game, and, of course, ‘‘he crushes you.’’ So you take back moves and try something different, and he crushes you again, ‘‘in much the same way.’’ If you are sufficiently wily, you will eventually discover a move that forces the devil to shift strategy; you still lose, but — aha! — you have your first clue.

Tao has emerged as one of the field’s great bridge-­builders. At the time of his Fields Medal, he had already made discoveries with more than 30 different collaborators. Since then, he has also become a prolific math blogger with a decidedly non-­Gaussian ebullience: He celebrates the work of others, shares favorite tricks, documents his progress and delights at any corrections that follow in the comments. He has organized cooperative online efforts to work on problems. ‘‘Terry is what a great 21st-­century mathematician looks like,’’ Jordan Ellenberg, a mathematician at the University of Wisconsin, Madison, who has collaborated with Tao, told me. He is ‘‘part of a network, always communicating, always connecting what he is doing with what other people are doing.’’

As a group, the people drawn to mathematics tend to value certainty and logic and a neatness of outcome, so this game becomes a special kind of torture. And yet this is what any ­would-be mathematician must summon the courage to face down: weeks, months, years on a problem that may or may not even be possible to unlock. You find yourself sitting in a room without doors or windows, and you can shout and carry on all you want, but no one is listening.

For their work, Tao and Green salvaged a crucial bit from an earlier proof done by others, which had been discarded as incorrect, and aimed at a different goal. Other maneuvers came from masterful proofs by Timothy Gowers of England and Endre Szemeredi of Hungary. Their work, in turn, relied on contributions from Erdos, Klaus Roth and Frank Ramsey, an Englishman who died at age 26 in 1930, and on and on, into history. Ask mathematicians about their experience of the craft, and most will talk about an intense feeling of intellectual camaraderie. ‘‘A very central part of any mathematician’s life is this sense of connection to other minds, alive today and going back to Pythagoras,’’ said Steven Strogatz, a professor of mathematics at Cornell University. ‘‘We are having this conversation with each other going over the millennia.’’

Most mathematicians tend to specialize, but Tao ranges widely, learning from others and then working with them to make discoveries. Markus Keel, a longtime collaborator and close friend, reaches to science fiction to explain Tao’s ability to rapidly digest and employ mathematical ideas: Seeing Tao in action, Keel told me, reminds him of the scene in ‘‘The Matrix’’ when Neo has martial arts downloaded into his brain and then, opening his eyes, declares, ‘‘I know kung fu.’’ The citation for Tao’s Fields Medal, awarded in 2006, is a litany of boundary hopping and notes particularly ‘‘beautiful work’’ on Horn’s conjecture, which Tao completed with a friend he had played foosball with in graduate school. It was a new area of mathematics for Tao, at a great remove from his known stamping grounds. ‘‘This is akin,’’ the citation read, ‘‘to a leading English-­language novelist suddenly producing the definitive Russian novel.’’

An effort to prove that 1 equals 0 is not likely to yield much fruit, it’s true, but the hacker’s mind-set can be extremely useful when doing math. Long ago, mathematicians invented a number that when multiplied by itself equals negative 1, an idea that seemed to break the basic rules of multiplication. It was so far outside what mathematicians were doing at the time that they called it ‘‘imaginary.’’ Yet imaginary numbers proved a powerful invention, and modern physics and engineering could not function without them.

Early encounters with math can be misleading. The subject seems to be about learning rules — how and when to apply ancient tricks to arrive at an answer. Four cookies remain in the cookie jar; the ball moves at 12.5 feet per second. Really, though, to be a mathematician is to experiment. Mathematical research is a fundamentally creative act. Lore has it that when David Hilbert, arguably the most influential mathematician of fin de siècle Europe, heard that a colleague had left to pursue fiction, he quipped: ‘‘He did not have enough imagination for mathematics.’’

Many people think that substantial progress on Navier-­Stokes may be impossible, and years ago, Tao told me, he wrote a blog post concurring with this view. Now he has some small bit of hope. The twin-prime conjecture had the same feel, a sense of breaking through the wall of intimidation that has scared off many aspirants. Outside the world of mathematics, both Navier-­Stokes and the twin-prime conjecture are described as problems. But for Tao and others in the field, they are more like opponents. Tao’s opponent has been known to taunt him, convincing him that he is overlooking the obvious, or to fight back, making quick escapes when none should be possible. Now the opponent appears to have revealed a weakness. But Tao said he has been here before, thinking he has found a way through the defenses, when in fact he was being led into an ambush. ‘‘You learn to get suspicious,’’ Tao said. ‘‘You learn to be on the lookout.’’

The human problem-solvers — 609 high school students from 108 countries — won 58 gold medals, 123 silver and 145 bronze. The A.I. performed at the level of a silver medalist, solving four out of six problems for a total of 28 points. It was the first time that A.I. has achieved a medal-worthy performance on an Olympiad’s problems.

Nonetheless, Dr. Kohli described the result as a “phase transition” — a transformative change — “in the use of A.I. in mathematics and the ability of A.I. systems to do mathematics.”

Dr. Gowers added in an email: “I was definitely impressed.” The lab had discussed its Olympiad ambitions with him a couple of weeks beforehand, so “my expectations were quite high,” he said. “But the program met them, and in one or two instances significantly surpassed them.” The program found the “magic keys” that unlocked the problems, he said.

Haojia Shi, a student from China, ranked No. 1 and was the only competitor to earn a perfect score — 42 points for six problems; each problem is worth seven points for a full solution. The U.S. team won first place with 192 points; China placed second with 190.

The Google system earned its 28 points for fully solving four problems — two in algebra, one in geometry and one in number theory. (It flopped at two combinatorics problems.) The system was allowed unlimited time; for some problems it took up to three days. The students were allotted only 4.5 hours per exam.

“The fact that we’ve reached this threshold, where it’s even possible to tackle these problems at all, is what represents a step-change in the history of mathematics,” he added. “And hopefully it’s not just a step-change in the I.M.O., but also represents the point at which we went from computers only being able to prove very, very simple things toward computers being able to prove things that humans can’t.”

“Mathematics requires this interesting combination of abstract, precise and creative reasoning,” Dr. Davies said. In part, he noted, this repertoire of abilities is what makes math a good litmus test for the ultimate goal: reaching so-called artificial general intelligence, or A.G.I., a system with capabilities ranging from emerging to competent to virtuoso to superhuman

One approach was an informal reasoning system, expressed in natural language. This system leveraged Gemini, Google’s large language model. It used the English corpus of published problems and proofs and the like as training data.

Dr. Hubert’s team developed a new model that is comparable but more generalized. Named AlphaProof, it is designed to engage with a broad range of mathematical subjects. All told, AlphaGeometry and AlphaProof made use of a number of different A.I. technologies.

In January, a Google DeepMind system named AlphaGeometry solved a sampling of Olympiad geometry problems at nearly the level of a human gold medalist. “AlphaGeometry 2 has now surpassed the gold medalists in solving I.M.O. problems,” Thang Luong, the principal investigator, said in an email.

The informal system excels at identifying patterns and suggesting what comes next; it is creative and talks about ideas in an understandable way. Of course, large language models are inclined to make things up — which may (or may not) fly for poetry and definitely not for math. But in this context, the L.L.M. seems to have displayed restraint; it wasn’t immune to hallucination, but the frequency was reduced.

Another approach was a formal reasoning system, based on logic and expressed in code. It used theorem prover and proof-assistant software called Lean, which guarantees that if the system says a proof is correct, then it is indeed correct. “We can exactly check that the proof is correct or not,” Dr. Hubert said. “Every step is guaranteed to be logically sound.”

Another crucial component was a reinforcement learning algorithm in the AlphaGo and AlphaZero lineage. This type of A.I. learns by itself and can scale indefinitely, said Dr. Silver, who is Google DeepMind’s vice-president of reinforcement learning. Since the algorithm doesn’t require a human teacher, it can “learn and keep learning and keep learning until ultimately it can solve the hardest problems that humans can solve,” he said. “And then maybe even one day go beyond those.”

Dr. Hubert added, “The system can rediscover knowledge for itself.” That’s what happened with AlphaZero: It started with zero knowledge, Dr. Hubert said, “and by just playing games, and seeing who wins and who loses, it could rediscover all the knowledge of chess. It took us less than a day to rediscover all the knowledge of chess, and about a week to rediscover all the knowledge of Go. So we thought, Let’s apply this to mathematics.”

Dr. Gowers doesn’t worry — too much — about the long-term consequences. “It is possible to imagine a state of affairs where mathematicians are basically left with nothing to do,” he said. “That would be the case if computers became better, and far faster, at everything that mathematicians currently do.”

“There still seems to be quite a long way to go before computers will be able to do research-level mathematics,” he added. “It’s a fairly safe bet that if Google DeepMind can solve at least some hard I.M.O. problems, then a useful research tool can’t be all that far away.”

A really adept tool might make mathematics accessible to more people, speed up the research process, nudge mathematicians outside the box. Eventually it might even pose novel ideas that resonate.

In England, the abolition of trial by ordeal led to the adoption of trial by jury for criminal cases. This required a new doctrine of evidence and a new method of inquiry, and led to what the historian Barbara Shapiro has called “the culture of fact”: the idea that an observed or witnessed act or thing—the substance, the matter, of fact—is the basis of truth and the only kind of evidence that’s admissible not only in court but also in other realms where truth is arbitrated. Between the thirteenth century and the nineteenth, the fact spread from law outward to science, history, and journalism.

Lynch isn’t terribly interested in how we got here. He begins at the arrival gate. But altering the flight plan would seem to require going back to the gate of departure.

Lynch thinks we are frighteningly close to this point: blind to proof, no longer able to know. After all, we’re already no longer able to agree about how to know. (See: climate change, above.)

Empiricists believed they had deduced a method by which they could discover a universe of truth: impartial, verifiable knowledge. But the movement of judgment from God to man wreaked epistemological havoc.

For the length of the eighteenth century and much of the nineteenth, truth seemed more knowable, but after that it got murkier. Somewhere in the middle of the twentieth century, fundamentalism and postmodernism, the religious right and the academic left, met up: either the only truth is the truth of the divine or there is no truth; for both, empiricism is an error.

That epistemological havoc has never ended: much of contemporary discourse and pretty much all of American politics is a dispute over evidence. An American Presidential debate has a lot more in common with trial by combat than with trial by jury,

came the Internet. The era of the fact is coming to an end: the place once held by “facts” is being taken over by “data.” This is making for more epistemological mayhem, not least because the collection and weighing of facts require investigation, discernment, and judgment, while the collection and analysis of data are outsourced to machines

“Most knowing now is Google-knowing—knowledge acquired online,”

We now only rarely discover facts, Lynch observes; instead, we download them.

“The Internet didn’t create this problem, but it is exaggerating it,”

nothing could be less well settled in the twenty-first century than whether people know what they know from faith or from facts, or whether anything, in the end, can really be said to be fully proved.

In his 2012 book, “In Praise of Reason,” Lynch identified three sources of skepticism about reason: the suspicion that all reasoning is rationalization, the idea that science is just another faith, and the notion that objectivity is an illusion. These ideas have a specific intellectual history, and none of them are on the wane.

Their consequences, he believes, are dire: “Without a common background of standards against which we measure what counts as a reliable source of information, or a reliable method of inquiry, and what doesn’t, we won’t be able to agree on the facts, let alone values.

When we Google-know, Lynch argues, we no longer take responsibility for our own beliefs, and we lack the capacity to see how bits of facts fit into a larger whole

Essentially, we forfeit our reason and, in a republic, our citizenship. You can see how this works every time you try to get to the bottom of a story by reading the news on your smartphone.

what you see when you Google “Polish workers” is a function of, among other things, your language, your location, and your personal Web history. Reason can’t defend itself. Neither can Google.

rump doesn’t reason. He’s a lot like that kid who stole my bat. He wants combat. Cruz’s appeal is to the judgment of God. “Father God, please . . . awaken the body of Christ, that we might pull back from the abyss,” he preached on the campaign trail. Rubio’s appeal is to Google.

Is there another appeal? People who care about civil society have two choices: find some epistemic principles other than empiricism on which everyone can agree or else find some method other than reason with which to defend empiricism

Lynch suspects that doing the first of these things is not possible, but that the second might be. He thinks the best defense of reason is a common practical and ethical commitment.

That, anyway, is what Alexander Hamilton meant in the Federalist Papers, when he explained that the United States is an act of empirical inquiry: “It seems to have been reserved to the people of this country, by their conduct and example, to decide the important question, whether societies of men are really capable or not of establishing good government from reflection and choice, or whether they are forever destined to depend for their political constitutions on accident and force.”

if there’s more than even a scant 5 percent possibility that an event occurred by chance, scientists will reject the causal claim.

correlation is not necessarily causation, because we need to rule out the possibility that we are just observing a coincidence.

. In the 18th and 19th centuries, this conservatism generally took the form of a demand for a large amount of evidence; in the 20th century, it took on the form of a demand for statistical significance

The 95 percent confidence level is generally credited to the British statistician R. A. Fisher, who was interested in the problem of how to be sure an observed effect of an experiment was not just the result of chance.

the 95 percent level has no actual basis in nature. It is a convention, a value judgment.

scientists place the burden of proof on the person making an affirmative claim.

It places the burden of proof on the victim rather than, for example, on the manufacturer of a harmful product.

it might be reasonable to accept a lower statistical threshold when examining effects in people, because you already have reason to believe that the observed effect is not just chance.

WHY don’t scientists pick the standard that is appropriate to the case at hand, instead of adhering to an absolutist one?

the history of science in relation to religion.

long tradition in the history of science that valorizes skepticism as an antidote to religious faith

scientists consciously rejected religion as a basis of natural knowledge, they held on to certain cultural presumptions about what kind of person had access to reliable knowledge.

they do practice a form of self-denial, denying themselves the right to believe anything that has not passed very high intellectual hurdles.

vigorously denying its relation to religion, modern science retains symbolic vestiges of prophetic tradition, so many scientists bend over backward to avoid these associations.

the Common Core standards used by a majority of K-12 programs in the country require that students be able to “distinguish among fact, opinion, and reasoned judgment in a text.”

So what’s wrong with this distinction and how does it undermine the view that there are objective moral facts?

For example, many people once thought that the earth was flat. It’s a mistake to confuse truth (a feature of the world) with proof (a feature of our mental lives)

Furthermore, if proof is required for facts, then facts become person-relative. Something might be a fact for me if I can prove it but not a fact for you if you can’t. In that case, E=MC2 is a fact for a physicist but not for me.

worse, students are taught that claims are either facts or opinions. They are given quizzes in which they must sort claims into one camp or the other but not both. But if a fact is something that is true and an opinion is something that is believed, then many claims will obviously be both

How does the dichotomy between fact and opinion relate to morality

Kids are asked to sort facts from opinions and, without fail, every value claim is labeled as an opinion.

Here’s a little test devised from questions available on fact vs. opinion worksheets online: are the following facts or opinions? — Copying homework assignments is wrong. — Cursing in school is inappropriate behavior. — All men are created equal. — It is worth sacrificing some personal liberties to protect our country from terrorism. — It is wrong for people under the age of 21 to drink alcohol. — Vegetarians are healthier than people who eat meat. — Drug dealers belong in prison.

The answer? In each case, the worksheets categorize these claims as opinions. The explanation on offer is that each of these claims is a value claim and value claims are not facts. This is repeated ad nauseum: any claim with good, right, wrong, etc. is not a fact.

In summary, our public schools teach students that all claims are either facts or opinions and that all value and moral claims fall into the latter camp. The punchline: there are no moral facts. And if there are no moral facts, then there are no moral truths.

It should not be a surprise that there is rampant cheating on college campuses: If we’ve taught our students for 12 years that there is no fact of the matter as to whether cheating is wrong, we can’t very well blame them for doing so later on.

If it’s not true that it’s wrong to murder a cartoonist with whom one disagrees, then how can we be outraged? If there are no truths about what is good or valuable or right, how can we prosecute people for crimes against humanity? If it’s not true that all humans are created equal, then why vote for any political system that doesn’t benefit you over others?

the curriculum sets our children up for doublethink. They are told that there are no moral facts in one breath even as the next tells them how they ought to behave.

Our children deserve a consistent intellectual foundation. Facts are things that are true. Opinions are things we believe. Some of our beliefs are true. Others are not. Some of our beliefs are backed by evidence. Others are not.

Value claims are like any other claims: either true or false, evidenced or not.

The hard work lies not in recognizing that at least some moral claims are true but in carefully thinking through our evidence for which of the many competing moral claims is correct.

Moral truths are not the same as scientific truths or mathematical truths. Yet they may still be used a guiding principle for our individual lives as well as our laws.But there is equal danger of giving moral judgments the designation of truth as there is in not doing so. Many people believe that abortion is murder on the same level as shooting someone with a gun. But many others do not. So is it true that abortion is murder?Moral principles can become generally accepted and then form the basis for our laws. But many long accepted moral principles were later rejected as being faulty. "Separate but equal" is an example. Judging homosexual relationships as immoral is another example.

Whoa! That Einstein derived an equation is a fact. But the equation represents a theory that may have to be tweaked at some point in the future. It may be a fact that the equation foretold the violence of atomic explosions, but there are aspects of nature that elude the equation. Remember "the theory of everything?"

Here is a moral fact, this is a sermon masquerading as a philosophical debate on facts, opinions and truth. This professor of religion is asserting that the government via common core is teaching atheism via the opinion vs fact.He is arguing, in a dishonest form, that public schools should be teaching moral facts. Of course moral facts is code for the Ten Commandments.

As a fourth grade teacher, I try to teach students to read critically, including distinguishing between facts and opinions as they read (and have been doing this long before the Common Core arrived, by the way). It's not always easy for children to grasp the difference. I can only imagine the confusion that would ensue if I introduced a third category -- moral "facts" that can't be proven but are true nonetheless!

horrible acts occur not because of moral uncertainty, but because people are too sure that their views on morality are 100% true, and anyone who fails to recognize and submit themselves are heathens who deserve death.I can't think of any case where a society has suffered because people are too thoughtful and open-minded to different perspectives on moral truth.In any case, it's not an elementary school's job to teach "moral truths."

The characterization of moral anti-realism as some sort of fringe view in philosophy is misleading. Claims that can be true or false are, it seems, 'made true' by features of the world. It's not clear to many in philosophy (like me) just what features of the world could make our moral claims true. We are more likely to see people's value claims as making claims about, and enforcing conformity to, our own (contingent) social norms. This is not to hold, as Mr. McBrayer seems to think follows, that there are no reasons to endorse or criticize these social norms.

This is nonsense. Giving kids the tools to distinguish between fact and opinion is hard enough in an age when Republicans actively deny reality on Fox News every night. The last thing we need is to muddy their thinking with the concept of "moral facts."A fact is a belief that everyone _should_ agree upon because it is observable and testable. Morals are not agreed upon by all. Consider the hot button issue of abortion.

Truthfully, I'm not terribly concerned that third graders will end up taking these lessons in the definition of fact versus opinion to the extremes considered here, or take them as a license to cheat. That will come much later, when they figure out, as people always have, what they can get a way with. But Prof. McBrayer, with his blithe expectation that all the grownups know that there moral "facts"? He scares the heck out of me.

I've long chafed at the language of "fact" v. "opinion", which is grounded in a very particular, limited view of human cognition. In my own ethics courses, I work actively to undermine the distinction, focusing instead on considered judgment . . . or even more narrowly, on consideration itself. (See http://wp.me/p5Ag0i-6M )

The real waffle here is the very concept of "moral facts." Our statements of values, even very important ones are, obviously, not facts. Trying to dress them up as if they are facts, to me, argues for a pretty serious moral weakness on the part of those advancing the idea.

Our core values are not important because they are facts. They are important because we collectively hold them and cherish them. To lean on the false crutch of "moral facts" to admit the weakness of your own moral convictions.

I would like to believe that there is a core of moral facts/values upon which all humanity can agree, but it would be tough to identify exactly what those are.

For the the ancient philosophers, reality comprised the Good, the True, and the Beautiful (what we might now call ethics, science and art), seeing these as complementary and inseparable, though distinct, realms. With the ascendency of science in our culture as the only valid measure of reality to the detriment of ethics and art (that is, if it is not observable and provable, it is not real), we have turned the good and the beautiful into mere "social constructs" that have no validity on their own. While I am sympathetic in many ways with Dr. McBrayer's objections, I think he falls into the trap of discounting the Good and The Beautiful as valid in and of themselves, and tries, instead, to find ways to give them validity through the True. I think his argument would have been stronger had he used the language of validity rather than the language of truth. Goodness, Truth and Beauty each have their own validity, though interdependent and inseparable. When we artificially extract one of these and give it primacy, we distort reality and alienate ourselves from it.

Professor McBrayer seems to miss the major point of the Common Core concern: can students distinguish between premises based on (reasonably construed) fact and premises based on emotion when evaluating conclusions? I would prefer that students learn to reason rather than be taught moral 'truth' that follows Professor McBrayer's logic.

Moral issues cannot scientifically be treated on the level that Prof. McBrayer is attempting to use in this column: true or false, fact or opinion or both. Instead, they should be treated as important characteristics of the systematic working of a society or of a group of people in general. One can compare the working of two groups of people: one in which e.g. cheating and lying is acceptable, and one in which they are not. One can use historical or model examples to show the consequences and the working of specific systems of morals. I think that this method - suitably adjusted - can be used even in second grade.

Relativism has nothing to do with liberalism. The second point is that I'm not sure it does all that much harm, because I have yet to encounter a student who thought that he or she had to withhold judgment on those who hold opposing political views!

One Republican talking point is that voter fraud is widespread. For example, on April 7, 2014 Dick Morris claimed that over 1 million people voted twice in 2012. If this was true, then it would obviously be a serious matter: widespread voter fraud could change the results of elections and rob the legitimate voters of their right to decide

Settling this matter requires looking at the available facts. In regards to Dick Morris’ claim (which made the rounds as a conservative talking point), the facts show that it is false.

Republicans have argued for voter ID laws by contending that they will prevent fraud. However, investigation of voter fraud has shown only 31 credible cases out of one billion ballots. As such, this sort of fraud does occur—but only at an incredibly low rate.

One rather important matter is the moral issue of whether it is more important to prevent fraud or to prevent disenfranchisement.

In the United States, there is a presumption of innocence on the moral grounds that it is better that a guilty person goes free than an innocent person is unjustly punished.

Keith Bentele and Erin E. O’Brien published a study entitled “Jim Crow 2.0? Why States Consider and Adopt Restrictive Voter Access Policies.” Based on their analysis of the data, they concluded “the Republican Party has engaged in strategic demobilization efforts in response to changing demographics, shifting electoral fortunes, and an internal rightward ideological drift among the party faithful.”

One of the best-known methods proposed to counter voter fraud is the voter ID law. While, as shown above, the sort of fraud that would be prevented by these laws seems to occur 31 times per 1 billion ballots, it serves to disenfranchise voters. In Texas 600,000-800,000 registered voters lack such IDs with Hispanics being 40-120% more likely to lack an ID than whites.

It would seem that the laws and policies allegedly aimed at voter fraud would not reduced the existing fraud (which is already miniscule) and would have the effect of suppressing voters. As such, these laws and proposals fail to protect the rights of voters and instead are a violation of that basic right. In short, they are either a misguided and failed effort to prevent fraud or a wicked and potentially successful effort to suppress minority voters. Either way, these laws and policies are a violation of a fundamental right of the American democracy.

An alternative algorithm is known that could have let the conventional computer be more competitive, or even win, by exploiting what Neven called a “bug” in D-Wave’s design. Neven said the test his group staged is still important because that shortcut won’t be available to regular computers when they compete with future quantum annealers capable of working on larger amounts of data.

“For a specific, carefully crafted proof-of-concept problem we achieve a 100-million-fold speed-up,” said Neven.

“the world’s first commercial quantum computer.” The computer is installed at NASA’s Ames Research Center in Mountain View, California, and operates on data using a superconducting chip called a quantum annealer.

Google is competing with D-Wave to make a quantum annealer that could do useful work.

Martinis is also working on quantum hardware that would not be limited to optimization problems, as annealers are.

Government and university labs, Microsoft (see “Microsoft’s Quantum Mechanics”), and IBM (see “IBM Shows Off a Quantum Computing Chip”) are also working on that technology.

“it may be several years before this research makes a difference to Google products.”

Several cities reported an increase in hate crimes against Asian people in 2020. One survey reported incidents of harassment and discrimination related to the coronavirus.

After showing his scars, Wong sat back down and buttoned up his shirt. "Prejudice is hate," he told the gathered crowd. "And that hate can be changed. We are human. We need to be kinder, gentler to one another. Because we are all the same. We are one human being on this earth."

Our brains are statistical organs that are built simply to predict what will happen next

This makes sense because, in prehistoric times, faulty predictions could lead to some very unpleasant surprises — like a tiger eating you or sinking in quicksand.

So-called prediction errors (like finding salmon instead of turkey on your plate on Thanksgiving) send us into a tizzy because our brains interpret them as a potential threat.

Keep doing what you’ve been doing, because you did it before, and you didn’t die.

all essentially subconscious efforts to make your world more predictable, orderly and safe.

Routines and rituals also conserve precious brainpower

It turns out our brains are incredibly greedy when it comes to energy consumption, sucking up 20 percent of calories while accounting for only 2 percent of overall body weight.

Our brains are literally overburdened with all the uncertainty caused by the pandemic.

Not only is there the seeming capriciousness of the virus, but we no longer have the routines that served as the familiar scaffolding of our lives

“It’s counterintuitive because we think of meaning in life as coming from these grandiose experiences

grandiose

grandiose

Of course, you can always take routines and rituals too far, such as the extremely controlled and repetitive behaviors indicative of addiction, obsessive-compulsive disorder and various eating disorders.

it’s mundane routines that give us structure to help us pare things down and better navigate the world, which helps us make sense of things and feel that life has meaning.”

In the coronavirus era, people may resort to obsessive cleaning, hoarding toilet paper, stockpiling food or neurotically wearing masks when driving alone in their cars. On the other end of the spectrum are those who stubbornly adhere to their old routines because stopping feels more threatening than the virus.

You’re much better off establishing a new routine within the limited environment that we find ourselves in

Luckily, there is a vast repertoire of habits you can adopt and routines you can establish to structure your days no matter what crises are unfolding around you

The point is to find what works for you. It just needs to be regular and help you achieve your goals, whether intellectually, emotionally, socially or professionally. The best habits not only provide structure and order but also give you a sense of pleasure, accomplishment or confidence upon completion.

It could be as simple as making your bed as soon as you get up in the morning or committing to working the same hours in the same spot.

Pandemic-proof routines might include weekly phone or video calls with friends, Taco Tuesdays with the family, hiking with your spouse on weekends, regularly filling a bird feeder, set times for prayer or meditation, front yard happy hours with the neighbors or listening to an audiobook every night before bed.

The truth is that you cannot control what happens in life. But you can create a routine that gives your life a predictable rhythm and secure mooring.

This frees your brain to develop perspective so you’re better able to take life’s surprises in stride.

It wasn’t that the meal itself was bad. In fact, the meal was outstanding. The problem was that it wasn’t the meal everyone was expecting.

Too many people are still longing for their old routines. Get some new ones instead.

I attended a Thanksgiving dinner several years ago where the hostess, without warning family and friends, broke with tradition and served salmon instead of turkey, roasted potatoes instead of mashed, raspberry coulis instead of cranberry sauce and … you get the idea.

When there are discrepancies between expectations and reality, all kinds of distress signals go off in the brain.

It doesn’t matter if it’s a holiday ritual or more mundane habit like how you tie your shoes; if you can’t do it the way you normally do it, you’re biologically engineered to get upset.

This in part explains people’s grief and longing for the routines that were the background melodies of their lives before the pandemic

An aperiodic tiling displays no such “translational symmetry,” and mathematicians have long sought a single shape that could tile the plane in such a fashion. This is known as the einstein problem.

black and white squares also can make weird nonperiodic patterns, in addition to the familiar, periodic checkerboard pattern. “It’s really pretty trivial to be able to make weird and interesting patterns,” he said. The magic of the two Penrose tiles is that they make only nonperiodic patterns — that’s all they can do.“But then the Holy Grail was, could you do with one — one tile?” Dr. Goodman-Strauss said.

now a new paper — by Mr. Smith and three co-authors with mathematical and computational expertise — proves Mr. Smith’s discovery true. The researchers called their einstein “the hat,

“The most significant aspect for me is that the tiling does not clearly fall into any of the familiar classes of structures that we understand.”

“I’m always messing about and experimenting with shapes,” said Mr. Smith, 64, who worked as a printing technician, among other jobs, and retired early. Although he enjoyed math in high school, he didn’t excel at it, he said. But he has long been “obsessively intrigued” by the einstein problem.

Sir Roger found the proofs “very complicated.” Nonetheless, he was “extremely intrigued” by the einstein, he said: “It’s a really good shape, strikingly simple.”

The simplicity came honestly. Mr. Smith’s investigations were mostly by hand; one of his co-authors described him as an “imaginative tinkerer.”

When in November he found a tile that seemed to fill the plane without a repeating pattern, he emailed Craig Kaplan, a co-author and a computer scientist at the University of Waterloo.

“It was clear that something unusual was happening with this shape,” Dr. Kaplan said. Taking a computational approach that built on previous research, his algorithm generated larger and larger swaths of hat tiles. “There didn’t seem to be any limit to how large a blob of tiles the software could construct,”

The first step, Dr. Kaplan said, was to “define a set of four ‘metatiles,’ simple shapes that stand in for small groupings of one, two, or four hats.” The metatiles assemble into four larger shapes that behave similarly. This assembly, from metatiles to supertiles to supersupertiles, ad infinitum, covered “larger and larger mathematical ‘floors’ with copies of the hat,” Dr. Kaplan said. “We then show that this sort of hierarchical assembly is essentially the only way to tile the plane with hats, which turns out to be enough to show that it can never tile periodically.”

some might wonder whether this is a two-tile, not one-tile, set of aperiodic monotiles.

Dr. Goodman-Strauss had raised this subtlety on a tiling listserv: “Is there one hat or two?” The consensus was that a monotile counts as such even using its reflection. That leaves an open question, Dr. Berger said: Is there an einstein that will do the job without reflection?

“the hat” was not a new geometric invention. It is a polykite — it consists of eight kites. (Take a hexagon and draw three lines, connecting the center of each side to the center of its opposite side; the six shapes that result are kites.)

“It’s likely that others have contemplated this hat shape in the past, just not in a context where they proceeded to investigate its tiling properties,” Dr. Kaplan said. “I like to think that it was hiding in plain sight.”

Incredibly, Mr. Smith later found a second einstein. He called it “the turtle” — a polykite made of not eight kites but 10. It was “uncanny,” Dr. Kaplan said. He recalled feeling panicked; he was already “neck deep in the hat.”

Dr. Myers, who had done similar computations, promptly discovered a profound connection between the hat and the turtle. And he discerned that, in fact, there was an entire family of related einsteins — a continuous, uncountable infinity of shapes that morph one to the next.

this einstein family motivated the second proof, which offers a new tool for proving aperiodicity. The math seemed “too good to be true,” Dr. Myers said in an email. “I wasn’t expecting such a different approach to proving aperiodicity — but everything seemed to hold together as I wrote up the details.”

Mr. Smith was amazed to see the research paper come together. “I was no help, to be honest.” He appreciated the illustrations, he said: “I’m more of a pictures person.”

What happened to me demands explanation. Modern physics tells us that the universe is a unity—that it is undivided. Though we seem to live in a world of separation and difference, physics tells us that beneath the surface, every object and event in the universe is completely woven up with every other object and event. There is no true separation. Before my experience these ideas were abstractions. Today they are realities. Not only is the universe defined by unity, it is also—I now know—defined by love. The universe as I experienced it in my coma is—I have come to see with both shock and joy—the same one that both Einstein and Jesus were speaking of in their (very) different ways.

Today many believe that the living spiritual truths of religion have lost their power, and that science, not faith, is the road to truth. Before my experience I strongly suspected that this was the case myself. But I now understand that such a view is far too simple. The plain fact is that the materialist picture of the body and brain as the producers, rather than the vehicles, of human consciousness is doomed. In its place a new view of mind and body will emerge, and in fact is emerging already. This view is scientific and spiritual in equal measure and will value what the greatest scientists of history themselves always valued above all: truth.

In this past weekend’s Styles section of the New York Times, Siegel uses “The Lonely Crowd” to analyze the putative “Yelpification” of contemporary life: according to Siegel, Riesman’s view was that “people went from being ‘inner-directed’ to ‘outer-directed,’ from heeding their own instincts and judgment to depending on the judgments and opinions of tastemakers and trendsetters.” The “conformist power of the crowd” and its delighted ability to write online reviews led Siegel down a sad path to a lackluster expensive dinner.

What Riesman actually suggested was that we think of social organization in terms of a series of “ideal types” along a spectrum of increasingly loose authority

On one end of the spectrum is a “tradition-directed” community, where we all understand that what we’re supposed to do is what we’re supposed to do because it’s just the thing that one does; authority is unequivocal, and there’s neither the room nor the desire for autonomous action

In the middle of the spectrum, as one moves toward a freer distribution of, and response to, authority, is “inner-direction.” The inner-directed character is concerned not with “what one does” but with “what people like us do.” Which is to say that she looks to her own internalizations of past authorities to get a sense for how to conduct her affairs.

Contemporary society, Riesman thought, was best understood as chiefly “other-directed,” where the inculcated authority of the vertical (one’s lineage) gives way to the muddled authority of the horizontal (one’s peers).

The inner-directed person orients herself by an internal “gyroscope,” while the other-directed person orients herself by “radar.”

It’s not that the inner-directed person consults some deep, subjective, romantically sui generis oracle. It’s that the inner-directed person consults the internalized voices of a mostly dead lineage, while her other-directed counterpart heeds the external voices of her living contemporaries.

“the gyroscopic mechanism allows the inner-directed person to appear far more independent than he really is: he is no less a conformist to others than the other-directed person, but the voices to which he listens are more distant, of an older generation, their cues internalized in his childhood.” The inner-directed person is, simply, “somewhat less concerned than the other-directed person with continuously obtaining from contemporaries (or their stand-ins: the mass media) a flow of guidance, expectation, and approbation.

Riesman drew no moral from the transition from a community of primarily inner-directed people to a community of the other-directed. Instead, he saw that each ideal type had different advantages and faced different problems

As Riesman understood it, the primary disciplining emotion under tradition direction is shame, the threat of ostracism and exile that enforces traditional action. Inner-directed people experience not shame but guilt, or the fear that one’s behavior won’t be commensurate with the imago within. And, finally, other-directed folks experience not guilt but a “contagious, highly diffuse” anxiety—the possibility that, now that authority itself is diffuse and ambiguous, we might be doing the wrong thing all the time.

Siegel is right to make the inference, if wayward in his conclusions. It makes sense to associate the anxiety of how to relate to livingly diffuse authorities with the Internet, which presents the greatest signal-to-noise-ratio problem in human history.

The problem with Yelp is not the role it plays, for Siegel, in the proliferation of monoculture; most people of my generation have learned to ignore Yelp entirely. It’s the fact that, after about a year of usefulness, Yelp very quickly became a terrible source of information.

There are several reasons for this. The first is the nature of an algorithmic response to the world. As Jaron Lanier points out in “Who Owns the Future?,” the hubris behind each new algorithm is the idea that its predictive and evaluatory structure is game-proof; but the minute any given algorithm gains real currency, all the smart and devious people devote themselves to gaming it. On Yelp, the obvious case would be garnering positive reviews by any means necessary.

A second problem with Yelp’s algorithmic ranking is in the very idea of using online reviews; as anybody with a book on Amazon knows, they tend to draw more contributions from people who feel very strongly about something, positively or negatively. This undermines the statistical relevance of their recommendations.

the biggest problem with Yelp is not that it’s a popularity contest. It’s not even that it’s an exploitable popularity contest.

it’s the fact that Yelp makes money by selling ads and prime placements to the very businesses it lists under ostensibly neutral third-party review

But Yelp’s valuations are always possibly in bad faith, even if its authority is dressed up as the distilled algorithmic wisdom of a crowd. For Riesman, that’s the worst of all possible worlds: a manipulated consumer certainty that only shores up the authority of an unchosen, hidden source. In that world, cold monkfish is the least of our problems.

In the new study, researchers, including chemist Shankar Balasubramanian, of the University of Cambridge and Cambridge Research Institute, crafted antibody proteins specifically for this type of DNA. The proteins were marked with a fluorescent chemical, so when they hooked up to areas in the human genome packed with G-quadruplexes, they lit up.

Next, they incubated the antibodies with human cells in the lab, finding these structures tended to occur in genes of cells that were rapidly dividing, a telltale feature of cancer cells. They also found a spike in quadruplexes during the s-phase of the cell cycle, or the phase when DNA replicates just before the cell divides.

the researchers think the four-stranded DNA could be a target for personalized medicine in the future

"What makes me personally very happy about this work is that it again demonstrates that mechanisms first described in ciliated protozoa hold also true for other organisms up to human, demonstrating the strength of this model organism," wrote Lipps wrote.

ever-more incompetent infants, requiring ever-brighter parents to ensure they survive childhood.

The self-reinforcing nature of the process would explain why intelligence is so strikingly overdeveloped in humans compared even with chimpanzees.

developed first in primates, a newish branch of the mammals, a group that is itself relatively young.

found that babies born to mothers with higher IQs had a better chance of surviving than those born to low-IQ women, which bolsters the idea that looking after human babies is indeed cognitively taxing.

none of this adds up to definitive proof.

Any such feedback loop would be a slow process (at least as reckoned by the humans themselves), most of which would have taken place in the distant past.