Mathematicians often solve problems using wordless mental processes

Sleep research published in 2021 confirmed that there is a 'creative sweet spot' when people are drifting off to sleep, and that if they wake up at that point to work, it can boost their creative problem-solving.

Sleeping on a problem does seem to help—in fact research published in 2026 suggests scientists can seed people's dreams with sound cues about a difficult puzzle during REM sleep and they'll be better able to solve it.

How a brain injury created a mathematical mind

In 2002, two men savagely attacked Jason Padgett outside a karaoke bar, leaving him with a severe concussion--and turning Padgett into a mathematical genius.

Early 20th-century debates about horses with 'paranormal mathematical abilities' helped shape modern approaches to psychological research.

What is the brain up to when the mind wanders?

As you stare into the distance while washing your home's seemingly never-ending pile of dishes, the brain activates its default mode network. This collection of "task-negative" brain regions associated with memory, reward, action consideration, and prediction alight with activity when humans mentally turn inward, possibly giving rise to the brain's ability to wander and humans' ability to construct internal narratives about the world.

Sleep is no longer considered a binary state—scientists are seeking to map a variety of liminal states between sleep and waking, including the hypnagogic state associated with boosts of creativity.

Mathematical breakthroughs influenced Shakespeare’s stories

New ideas in math were transforming Europe at the time Shakespeare penned his plays, including newly popular conceptions of zero via philosophers Descartes and Leibniz.

How metaphors shape the way we think

A metaphor is often viewed as a form of art, deployed only in special circumstances to help an audience grasp something complex or unique. But metaphors are far more fundamental than that, shaping much of how we think about our relationship to others and the world. This article explores how a focus on the metaphors we use can clarify and strengthen our thinking. Learn more here.

The sophisticated math jokes of 'The Simpsons'

"The Simpsons" has always been a slyly sophisticated show and has featured more than a few references to complex mathematical concepts. That's largely due to its writers' room, which has featured more than one writer who studied math before transitioning to comedy.

In 2000, the Clay Mathematics Institute placed million-dollar bounties on seven unsolved problems in mathematics—the Millennium Prize Problems—in the hopes of spurring solutions.

The science of optical illusions

The human brain has developed cognitive shortcuts to help us navigate in a world where our visual senses are constantly bombarded of information. But these shortcuts can also be exploited, tricking the brain into telling us something is happening when it really isn't. This short video unpacks the neuroscience behind some of the most common (but perplexing) optical illusions.

The dreaming brain, as understood by scientists

Technology has advanced to where the seemingly separate fields of psychology and neuroscience are asking the same questions about dreams, such as whether they're more closely related to perception or visualization. Some field-intersecting findings include research that shows people with impaired face perception don't dream of faces, those with brain legion-related cognition difficulties deal with similar issues in dreams, and children's dreams change in parallel to their cognitive development.

How entanglement lies at the heart of quantum computing

Theoretical physicist and mathematician Frank Wilczek boils down the complex notion of quantum entanglement into this reasonably understandable summary. Wilczek discusses the concepts of quantum entanglement and how it relates to other ideas like the Many-Worlds Interpretation (MWI) of quantum mechanics. What sounds a bit "science-fictiony" can have actual real-world and concrete applications.

Read the writing advice of science-fiction legend Ray Bradbury.

A "no math" primer on quantum mechanics

Quantum mechanics may not immediately grasped, but it can be slowly understood through the experiments and discussions offered here by writer Dr. Miguel F. Morales. A professor of physics at the University of Washington in Seattle, Dr. Morales makes understanding the basics of quantum mechanics fun with a promised "no math" approach.

During dream sleep, the brain connects new information learned during the day with information already stored in the mind, creating a revised mind-wide web of associations.

'Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world.'

- Albert Einstein

How your subconscious influences your decisions

How much of your desires and daily choices are truly yours? One of the most effective ways to influence someone's subconscious is when they're asleep. To study this possibility, researchers trained participants to be able to dream lucidly, a phenomenon that occurs when a dreamer knows they're dreaming.

How the brain processes a calendar year

When most of us picture a calendar year, we have a foggy image in our minds of the months, in rough, calendar-like grids. But for some folks, that picture is a whole lot more clear. Around 1 percent of the population has a strange ability known as "calendar synaesthesia," which means they're able to visualize in crisp detail an elaborate calendar of the months of the year. Learn how it works here.

Explore over 30,000 Olympiad-level math problems from around the world.

Some people really cannot imagine. Literally.

There are many people who are astonished to discover that their complete lack of ability to picture visual imagery is different from the norm.

Explaining the many-worlds interpretation

A mind-bending, jargon-free account of the popular interpretation of quantum mechanics, helping to explain how something can be in two states at once.

How math whizzes exploit sportsbooks

Chris Dirkus is a professional sports bettor who approaches the hobby as a math problem, not from a fan's perspective. He focuses on mispriced odds in popular games where many fans are betting for high liquidity. He's one of a rare segment of sports bettors exploiting slow models and human mistakes.

As of 2025, the College Board has banned the use of calculators with computer algebra systems on the SAT and PSAT.

Ray Dalio credits much of his success to his transcendental meditation practice

Dalio took up transcendental meditation after learning that The Beatles practiced it on a trip to India in 1968. He says the practice allows him to clear his mind, be more creative, and unlock his subconscious.

How art helps us process our feelings

The internet is chock full of hacks and apps that supposedly help you optimize yourself into a more productive person. But Aparna Chivukula argues that spending so much mental energy on "wellness" misses a larger philosophical issue, one that can only be explored through art.

The history of the 10 axioms of Zermelo–Fraenkel set theory with choice, regarded as the most universal foundations of math

Like postulates in geometry—accepted truths that form the logical building blocks for formal proofs of more complex theorems—axioms cannot be proven, but set the underlying logic and rules for systems. The ZFC axioms resulted from painstaking work to identify the underlying rules of mathematics and are built on the premise of sets, or well-defined collections of elements.

Sharp wave ripples in the brain tag experiences for long-term memory storage

These ripples were known to quickly replay neuronal sequences during sleep for consolidation. While awake, they help the brain prioritize memories to remember, and their disruption can impair memory formation.

The ideas of Einstein and Freud combined in 'The Persistence of Memory'

The Persistence of Memory was painted on a canvas roughly the size of a sheet of paper, depicting a desolate landscape with melting clocks, ants, and a distorted figure. The piece incorporates Freud and Einstein's ideas, resulting in a haunting composition symbolizing time's fluidity and Dalí's personal anxieties.

Aphantasia involves a lack of mental images—but doesn't affect behavior

The condition, generally known about for over a century, first received its name in 2015 by researcher Adam Zeman. An estimated 4% of people lack this ability to perceive mental images, and research suggests this deficiency does not have any practical impact on behavior.

Some founders of quantum mechanics thought consciousness could influence phenomena

Since quantum systems exist as a superposition of states before producing a definitive outcome, some quantum pioneers wondered if the act of conscious observation was what collapsed the wave function of possibilities into one state.

The story of two female programmers running the simulations behind chaos theory's discovery

Margaret Hamilton and Ellen Fetter worked on MIT's 800-pound Royal McBee LGP-30 computer, which was fed binary code on paper tape to perform computations. Among the computations performed were those involving equations describing the motion of gas in a circulating fluid for Edward Lorenz, widely considered the father of chaos theory.

Studies of psychedelic use may help shed light on the nature of consciousness

The nature of consciousness—what it is, where it comes from, and how it manifests as moment-to-moment experience—remains under debate by scientists. Recent studies show psychedelics were found to boost low-frequency brain signals known as delta waves, a behavior also seen during dream states.

Misunderstandings about how quantum computing works may have overhyped their ability

Quantum computers are often viewed as machines that can solve all problems instantly by trying every solution simultaneously. Instead, they amplify correct answers and cancel out wrong ones, which only works for specific kinds of problems with special structures.

Anthropic's interpretability team discusses how AI models' thinking mimics biology

Just as humans engage in complex behaviors to survive and reproduce, teams of experts in neuroscience, virology, mathematics, and other disciplines argue that large language models develop complex mechanisms to achieve their goals. These mechanisms can be manipulated to assess how they modify outputs—much like stimulating individual neurons—to uncover how LLMs "think."

Quantum computers harness weird physics to explore unsolvable problems

Classical computers manipulate bits as either 0 or 1, but quantum computers use qubits to hold combinations of both, enabling new algorithms. This new approach could one day help with breakthroughs in medicine, chemistry, and AI.

Humans experience the world subjectively. What does that mean for scientific research?

When you close your eyes, do you see static or colorful patterns or an "intrinsic grey" known as Eigengrau in German? This variation of our internal experiences is just the tip of the iceberg, one that scientists are increasingly studying and...

Some scientists argue NDEs are constructed by the brain, not actually experienced

Despite the powerful memories associated with NDEs, scientists question the idea that the experiences prove consciousness endures after death, opting instead to find physiological explanations for common reports of out-of-body experiences, euphoric and lucid thinking, and more.

How the number 15 solved a longstanding math challenge

The “packing coloring” problem asks how many numbers are needed to fill an infinite grid so that identical numbers never get too close to one another. A new computer-assisted proof finds a surprisingly straightforward answer—only 15. While...

Two scientists argue consciousness evolved through 'unlimited associative learning,' the ability to learn almost any new pattern during a lifetime.

The most bizarre numbers in the universe

Even for math-y types, some numbers are incomprehensible. This article dives into several kinds of numbers more complicated than even irrational numbers with endless decimals like "pi" or Euler's number: Noncomputable numbers. Read the bizarre...

Individuals with synesthesia experience involuntarily overlapping sensations

Tasting words or seeing colors in music results from the brain retaining extrasensory connections from infancy. While most lose these neural pathways during development, about 4% of people are synesthetes, maintaining them without realizing it's unusual.

Within mathematics, some infinities are bigger than others

While the concept of infinity refers to a countless set of items, these infinite sets can be compared to one another by matching an item from one set to an item in another, one by one. This analysis shows that certain sets must be larger than others because some infinite sets have leftover, unmatched items.

The study of dreams, formally called oneirology, has evolved over the centuries, from anecdotal interpretations to advanced brain imaging.

Dreams largely occur during REM sleep (but not entirely).

Scientists are learning to guide your dreams using sound and touch.

A 14-year-old "human calculator" shatters six mathematical world records in one day

A teenage prodigy from Maharashtra, India named Aaryan Shukla set the record for the fastest time to mentally add 50 five-digit numbers. He did so in just 25.19 seconds–roughly equivalent to one addition every 0.5 seconds. The math whiz said he practices calculations for around five to six hours a day.

Jazz musicians use different melodic approaches to produce improvised performances

While jazz musicians are known for their improvisational skills, they rely on years of training and a complex understanding of compositions to inform their improvised solos, using a song's structure and chords to make their in-the-moment choices. Although jazz artists could, in theory, play whatever they wanted, musicians often employ common melodic approaches—such as scalar improvisation and arpeggios—to generate solos.

Various types of memory, each serving unique functions, create a complex brain system

These forms of memory, which can overlap in daily life, have been sorted into broad categories, such as short-term and long-term memory, and subtypes such as implicit memory, which uses saved information without conscious awareness that it’s being recalled.

Apophenia, the cognitive bias that leads people to find meaning in randomness, helps explain why humans look for meaning in their dreams and even in their tea leaves.