https://jill-184.szhdyy.com.cn/qf/K6728h3
121 notes
·
View notes
In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history.
Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths.
But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency.
His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.
Natalie Wolchover, How Gödel’s Proof Works, Quanta Magazine, July 14, 2020
31 notes
·
View notes
Introduction to Topology Master Post
Metric Spaces
Topological Spaces and Continuity
Closed Sets and Limit Points
Hausdorffness
Connectedness
Path Connectedness
Compactness
Bases and Second Countability
Product Spaces
The Heine-Borel Theorem
Quotient Spaces
Important Examples
Conclusions and Remaining Questions
As of making this post, the entire series isn't out yet so some links won't be here yet (24/04/2024)
37 notes
·
View notes
58K notes
·
View notes
You guys are never gonna believe what the name of this sculpture is
24K notes
·
View notes
My latest New Scientist cartoon
12K notes
·
View notes
There’s a lot of excellent examples of the difference between a million and a billion, but here’s my new personal favorite from a conversation I had today:
A million minutes ago was April 2021, the height of the COVID pandemic.
A billion minutes ago was November 121 CE, the height of the Roman Empire.
69K notes
·
View notes
9K notes
·
View notes
People on this site will put together polls like "The Banach-Tarski Paradox versus Camembert Cheese", then act like the results prove that they're surrounded by idiots.
9K notes
·
View notes
28K notes
·
View notes
Grad school quote of the day:
“Lovely man. They shouldn't let him teach.”
9K notes
·
View notes
In the remote Arctic almost 30 years ago, a group of Inuit middle school students and their teacher invented the Western Hemisphere’s first new number system in more than a century. The “Kaktovik numerals,” named after the Alaskan village where they were created, looked utterly different from decimal system numerals and functioned differently, too. But they were uniquely suited for quick, visual arithmetic using the traditional Inuit oral counting system, and they swiftly spread throughout the region. Now, with support from Silicon Valley, they will soon be available on smartphones and computers—creating a bridge for the Kaktovik numerals to cross into the digital realm.
Today’s numerical world is dominated by the Hindu-Arabic decimal system. This system, adopted by almost every society, is what many people think of as “numbers”—values expressed in a written form using the digits 0 through 9. But meaningful alternatives exist, and they are as varied as the cultures they belong to.
Continue Reading
30K notes
·
View notes
He's right tho
9K notes
·
View notes