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

Infinity can come in different sizes and be ordered by these sizes

Russian mathematician Georg Cantor demonstrated that certain sets of numbers (e.g., the natural numbers) form countable infinities, while others cannot be counted (e.g., the real numbers) and are therefore uncountable, representing larger versions of infinity. Through the axiom of choice, German mathematician Ernst Zermelo proved this concept of ordering infinities.

The simple grandeur of atomic theory

Is the universe made up of a continuous substance, or a finite amount of tiny, indivisible atoms? The atomic hypothesis, largely vindicated in the 20th century, was forgotten for nearly 2,000 years after first being formulated by the Greek philosopher Democritus. Learn more about the theory and its implications for science and human life with this reflective essay.

Axions, a hypothetical subatomic particle, are a leading dark matter candidate

Among the possibilities for what makes up dark matter, some physicists champion axions, which could clump into galaxy-scale quantum states. These states would act as massive, unified superparticles that could explain the observational evidence for the universe's missing matter.

Mathematicians often solve problems using wordless mental processes

It's a common technique among expert mathematicians, where thinkers rely on intuition, blurry perceptions and subconscious work—no writing, no formulas. The state is distinct from daydreaming, and often relies on sudden illuminations stemming from extended periods of subconscious reflection.

Time may be an emergent, rather than a fundamental, property of the universe

Some theories of everything—mathematical models that try to explain all aspects of the universe, including quantum phenomena and gravity—do not incorporate time. If a time-neglecting version of such a theory were ever validated, it might suggest that time is a feature that emerges from our limited perception of the universe.

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.

The work of Claude Shannon, for whom Anthropic's AI chatbot is named

Known as the father of information theory, Claude Shannon applied Boolean algebra—a subset of mathematics that quantifies true or false statements as 1 or 0, respectively—to circuit design to develop the concept of a binary digit. His paper "A Mathematical Theory of Communication" (1948) laid the groundwork for all modern digital systems.

The Ishango bone suggests mathematics was born about 20,000 years ago

Ancient civilizations developed mathematical systems for various purposes, including the Egyptians for the construction of the pyramids and the Ancient Greeks for logical reasoning in theorems. However, these systems were predated by the Ishango bone—found in the Democratic Republic of the Congo—whose notches suggest a base ten system, prime numbers, and simple mathematical operations.

Scientific progress works toward absolute truth, but can never reach it

Through the acquisition of scientific knowledge, scientists construct explanations of the world, which are rigorously tested and verified until they narrow down to a leading theory of the natural world. However, because of how science works, these ideas will never become factual, as they must always remain open to be iteratively revision based on new data.

Four video explainers on quantum fundamentals

Ready to level up your working knowledge of quantum mechanics? Check out these four TED-Ed Lessons written by Chad Orzel, associate professor in the Department of Physics and Astronomy at Union College

Explore Albert Einstein's Zürich notebook and review his notes on relativity

Within the pages are calculations, sketches and his first attempts at deriving what would be the field equations for gravity. It also documents some dead ends and the difficulties Einstein encountered as he describes part of the mathematics of what would become general relativity, describing it as "too involved."

How a competition honoring a king's birthday uncovered history's first chaotic system

A committee established a contest for King Oscar II's 60th birthday, featuring four unsolved problems in mathematics, including one on proving the stability of a system of more than two objects held together by gravity. Focusing on three celestial objects, Henri Poincaré won the prize for his proof, which he later learned was incorrect and, upon correcting it, instead proved these systems to be unstable.

Understanding college basketball's guru, UCLA's John Wooden

John Wooden coached UCLA to an unprecedented 10 NCAA titles, including seven straight from 1967 to 1973. He studied the characteristics that made people successful and, in 1948, created his iconic triangular diagram with 25 common behaviors. This article unpacks the 25-block framework that makes the "Pyramid of Success" and shows how Wooden's off-court wisdom shaped future champs like Bill Walton and Steve Kerr.

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...

Scholars trace existential anxiety back to early Hebrew scriptures

Paul Megna, a scholar of literature and philosophy, notes that the existentialist ideas of Kierkegaard (and, to a lesser extent, Sarte and Camus) had their roots in much earlier texts, which posited that "fear of the Lord" was "the beginning of wisdom."

Listen to Ray Dalio break down his most valuable principles in life and work

Dalio's now-famous principles started out as a collection of rules the finance behemoth set for himself. Since then, the list has expanded, becoming required reading for all Bridgewater Associates employees. In 2017, Dalio published an expanded version of these rules called "Principles: Life & Work," which has sold more than 5 million copies worldwide.

When Marcus Aurelius created endowed philosophy chairs in Athens

The Roman Emperor Marcus Aurelius is known as one of the most famous Stoic writers or thinkers, penning his classic opus ‘Meditations.’ In 176 CE, he endowed four philosophy chairs in Athens: one for Stoicism, and one for three of Stoicism’s rival schools: Aristotelianism, Platonism, and Epicureanism.

A look at the principles guiding the Met's collection practices

While there are serious questions about the provenance of pieces that the Met has acquired in the past, the museum says its collecting practices today are all guided by three principles: research, transparency, and collaboration. A particularly interesting facet of their approach to research is looking into how pieces might have changed hands during the Nazi occupation of Europe and to whom (and where) they truly belong.

The 1941 Atlantic Charter was a World War II alliance and model for the UN

The Atlantic Charter was a pivotal 1941 agreement between Franklin D. Roosevelt and Winston Churchill, outlining shared goals for a post-World War II world rooted in freedom and self-determination. These ideals later inspired the founding principles of the United Nations.

The term 'algorithm' originated from the Latinized name of al-Khwārizmī, a 9th-century mathematician whose works introduced Arabic numerals and the decimal-based numeral system to Europe and whose treatise 'Al-Jabr' gave algebra its name.

A mathematical and visual analysis distinguishing chaos from randomness

Random systems lack any underlying pattern or causal relationships, while chaos is mathematically deterministic because the behavior follows established laws of nature. In 3D plots, chaotic systems produce outputs that approximate a mathematical function, while random points appear as statistical noise.

Wilhelm Wundt's place in the history of psychology

While ancient philosophers considered the central problems of psychology, it was Wilhelm Wundt who first called himself one. Here's a quick video overview of his contribution.

Many of the founding fathers were deists—believers in a God of reason

A uniquely American iteration of the Enlightenment movement thrived during the Revolution. This selection from a longer write-up on the American Enlightenment highlights six core ideas: deism, liberalism, republicanism, conservatism, toleration, and scientific progress.

Linguistics professors believe memes are a language

It's generally agreed upon that languages exhibit at least some of the following characteristics: compositionality, social transmission, arbitrariness of form and meaning, and accomplishment of linguistic functions like asking questions and expressing emotions, among other qualities. Memes fulfill these basic requirements, giving them linguistic legitimacy.

The science of cryptography, including methods to encrypt data and how to break encryption

Monoalphabetic ciphers use a set of rules to replace each letter with another consistently, making them susceptible to brute force attacks that try every kind of replacement. While more complex ciphers can transform text in less obvious ways, because certain letters are used more often than others, an analysis of letter frequency may reveal patterns that help uncover the rules that make up a cipher.

Mark Zuckerberg credits some of Facebook's success to constant evolution

Before becoming the head of OpenAI, Sam Altman was the president of Y Combinator—a startup accelerator company that supported companies like Airbnb, DoorDash, and Coinbase. In 2016, Altman interviewed Zuckerberg about creating Facebook and its growth over time. Zuckerberg said that, unlike other companies (think: CourseMatch), Facebook kept growing and adding new features, which prevented the app from becoming obsolete.

Apeirophobia is an overwhelming fear of living forever, perhaps stemming from the brain's inability to conceptualize an infinite life.

Chaos theory—informally known as the butterfly effect—is the reason why weather forecasts are presented with probabilities rather than certainties.

Trace how the Bill of Rights, inspired by centuries of political theory, took shape between 1774 and 1789.

While atoms are normally made of protons, neutrons, and electrons, exotic ones have been built with antimatter—alternate versions of normal matter that live for fractions of a second and are about a billion times as rare.

SpongeBob exemplifies Aristotle's philosophy for living well because he understands that happiness doesn't come from external factors like money, recognition, or power.

Clever Hans was a 1900s horse that was believed to be able to solve math equations, tell time, and read and comprehend German, until a psychology student discovered how he got his answers.

As of 2025, the most cited papers in science include a review of graphene, methods to improve image recognition, streamlining electron density calculations, and optimizing protein studies.

The first robot in literature may have been a 'mechanical man' in 1907's 'Ozma of Oz,' a book in L Frank Baum's 'Wizard of Oz' series.

Warren Buffett's former professor Benjamin Graham is known as the 'father of value investing,' a strategy in which investors seek to buy stocks trading below their intrinsic worth.

Most of the observable universe's volume consists of voids and supervoids—enormous regions of space with little to no galaxies, stars, or light.

The Brothers Grimm were lawyers before they were storytellers, and became interested in fairy tales as they began investigating how rules and customs had been passed down through cultures.

20,000 letters to the advice columnist 'Dear Abby' provide insight into prevailing sex, religion, and LGBTQ fears, anxieties, and concerns from the 1950s onward.

Many legends about George Washington, including his supposed wooden teeth and the story of cutting down the cherry tree, lack evidence.

Famed poet TS Eliot rejected the manuscript for George Orwell's political fable 'Animal Farm,' writing that its criticism of authoritarianism was not 'the right point of view.'

The name 'Silicon Valley' arose from the founding of Shockley Semiconductor Laboratory in Fairchild Semiconductor in 1956-57, founded near Stanford University in Palo Alto

How Amazon operates a 'flywheel' business model, a self-reinforcing cycle that drives continuous growth

See how each of the more than two million known species relates to each other—and to common ancestors—in this zoomable tree of life.

Researchers believe the brain consolidates memories, processes emotions, and plans for the future during dream states

Legendary architect Frank Lloyd Wright designed the SC Johnson headquarters in Wisconsin, aiming to make a cathedral-like workplace

Georgia O'Keeffe's flower paintings are often assumed to be sexual, though the artist claimed she was simply interested in highlighting the beauty of flowers that most people neglect

See a 3D reconstruction of Tenochtitlan, the Aztec capital that was home to an estimated population of 200,000 to 400,000.

A small portion of The Winstons' 'Amen, Brother,' an obscure '60s gospel record, is the basis for thousands of hip-hop tracks, making it one of the most sampled records ever that's informally known as 'the Amen break.'

The first steps to rebuilding your credit score include getting to know exactly what's on your credit report and figuring out how far you are from the '700 club,' or 'good score' range.

A 2019 Facebook event called on attendees to "Storm Area 51, They Can't Stop All of Us," resulting in countless memes, the creation of a music festival, and eventually the arrests of those who did try to approach the area.