The beautiful Harz Mountains sit right at the center of the modern Germany. Its calm ancient forests, its dense trees, its silky fogs and its altitude so close to the sky make it a paradise to clear the mind and cleanse the soul. City dwellers,…
Gaston Darboux dans ses belles années : Formation d’un scientifique engagé This is a presentation that I am going to give on Princeton Research Day (9 May, 2019). If you have any comments, questions or suggestions, please write to hainaw@princeton.edu. 🙂 PART 1 Introduction “We…
This is the second part of a presentation that I am going to give on “1866-1874: the beautiful years of Gaston Darboux”. Because I write the parts backwards, please find the third part here and expect the first part to come out later… Today, in a…
Extension fields and automorphisms are the most amazing thing I have ever learned. Essentially, when you think of them together, they are like a flower whose petals communicate with each other, but the center remains fixed. I’m still reading the part on extension fields and…
When we look at modern mathematics, it seems that both concepts of sets and maps are equally important. But strangely, in computer science, maps (functions, subroutines, methods) are made extremely important and diverse in appearance, whereas the not-so-map-related sets (classes, directories) are relegated to some “bag of random things” kind of meaning,…
In my previous article I mentioned that Elise Riemann contributed to the last part of Bernhard Riemanns Lebenslauf. Now I am almost sure that she actually contributed more than 60% to the beginning paragraphs, too, i.e. about the early years of Bernhard Riemann. I have just read a…
This is not really a pleasant topic for me due to the sad life of Bernhard Riemann. Riemann lost his mother in his teenage years and lost his father in 1855, when he was 29. In the 10 years that followed, four of Bernhard’s five siblings died…
So learning to count is just the beginning of appreciating what is finite and what is infinite: all understanding is impossible without the ability to construct functions. “Functions are the sole thing that makes thinking possible.” This may be exaggerated, but when it comes to set theory, it’s true….
Mathematics, the study of numbers and shapes, is generally considered profound, but it originated from acts as simple as counting and generalising. In ancient Greece, some people were intrigued by the patterns that pebbles gave: three pebbles formed a triangle, four pebbles, a square; six,…