5 ms·

Einstein’s First Proof

86 points by jeo1234 11y ago

ck2 11y ago
Have any of Einstein's relatives proved remotely as clever as him? aha, of course there's an article https://en.wikipedia.org/wiki/Einstein_family https://en.wikipedia.org/wiki/Einstein_family
- deleted 11y ago
  [deleted]
- Jun8 11y ago
  In that regard it's hard to beat the Curies (barring the older Bernoulli family, of course): https://www.reddit.com/r/todayilearned/comments/3laog8/til_that_%C3%A8ve_curie_unlike_her_mother_father/ https://www.reddit.com/r/todayilearned/comments/3laog8/til_t.... That's 5 Nobel Prizes (or 4 if you disregard the Peace Prize that the son-in-law earned).
deleted 11y ago
[deleted]
vitthal 11y ago
I don't remember how to prove it anymore... This can be a good problem to solve...
Jtsummers 11y ago
The second moment occurred soon after he turned twelve, when he was given “a little book dealing with Euclidean plane geometry.” The book’s “lucidity,” he wrote—the idea that a mathematical assertion could “be proved with such certainty that any doubt appeared to be out of the question”—provoked “wonder of a totally different nature.” Mathematical proofs, first really introduced in Geometry classes for most US students around 9th or 10th grade, are what really hooked me on math. I understood everything up to that point (and after), but it was a structural understanding (the rules and syntax, essentially). Once we arrived at logic, I (as a very bored precocious student) spent most of my class time proving everything from the axioms we were given (for algebra or geometry) that I could. Giving students the tools to play and explore freely (we had proof assignments, of course, but I took it far beyond that) is what really hooks them on a subject. Mandate that they must read these books, they'll hate it. Require them to read a book and a few excerpts of others, and give them access to a large library, and they'll read forever. Same with every other subject. That's where many of the real geniuses[0] of a field come from. [0] Whether technically geniuses by IQ or just acknowledged as such for their understanding and creations.
- jwdunne 11y ago
  I notice this learning maths as an adult. A good intro to logic and proofs opened up a new world. After struggling with a proof, I came to the conclusion that I needed to study the axioms I was obviously expected to know futher - most importantly, I wanted to. How did I want to study and understand them? By proof. I remember a similar course with programming, actually. One curiosity led to the next once I equipped myself with the tools to think and experiment with computers.
  - Jtsummers 11y ago
    I will also say, and meant to mention in my original comment, good textbooks are beautiful things. Most textbooks by the time I got to HS (mid-1990s) were meh. And they seemed to be getting worse each year. Compared to the books my parents and previous generations experienced. (Specifically for math.) Older books tended to be small, concise and precise. Think Sipser's automata text, but for everything in math. Being a third generation mathematician certainly helped by providing me access to an older library of books than a lot of my high school and college peers realized were available.
    - jwdunne 11y ago
      Thanks for the tip :) I'll explore that. Do you have any recommendations for textbooks to sink my teeth into?
      - Jtsummers 11y ago
        Unfortunately, not much. I was pondering that after my post. It's been a long time since I was a math student, and professionally it's had zero to do with my career. A lot of books ended up boxed up at my parents' home as I moved around a number of times right after college. I'll check my own home tonight to see what I still have, but my shelves these days are mostly filled with fiction, programming, RPG, and history books. EDIT: Off the top of my head, for CS: Introduction to Algorithms: https://mitpress.mit.edu/books/introduction-algorithms https://mitpress.mit.edu/books/introduction-algorithms Introduction to the Theory of Computation: http://www.amazon.com/Introduction-Theory-Computation-Michael-Sipser/dp/113318779X http://www.amazon.com/Introduction-Theory-Computation-Michae... Math: Calculus Made Easy: http://www.amazon.com/gp/product/0312185480?ref_=cm_lmf_tit_5 http://www.amazon.com/gp/product/0312185480?ref_=cm_lmf_tit_... - I'm really not sure how good this one is for a beginner, I picked it up while assisting my sister in refreshing her calculus skills for grad school (Aerospace Engineering) I can't remember the algebra and geometry textbooks (my dad's or my grandfather's) that I used, in addition to the assigned text, in high school. Anything by Knuth. Seriously, one summer a professor and I just picked up copies of Concrete Mathematics and worked through large portions of it for fun. Technically I got some math credits for it, but it was really just because we wanted to. Actually, this one helped me a lot with understanding calculus. Somehow, up to that point while I knew calculus, may brain had never made the connection that integration was summation until I saw the discrete counterpart to continuous integration. I had a mechanical understanding, but no deep understanding until that moment.
Jun8 11y ago
BTW, the book that was briefly mentioned in the article, Fractals, Chaos, and Power Laws: Minutes from an Infinite Paradise by Schroeder (http://www.amazon.com/Fractals-Chaos-Power-Laws-Infinite/dp/0716721368 http://www.amazon.com/Fractals-Chaos-Power-Laws-Infinite/dp/...) is truly excellent, it is written in the form of short pieces collected into chapters investigating a very wide-ranging set of phenomena, ranging from Brownian Motion to self-similarity, Cantor sets and cellular automata. For $10-$15 on Amazon, it's a great bargain!
blahblah3 11y ago
here's a proof that the ratio of the triangle area to the square is scale invariant: imagine covering a geometric shape with a bunch of little squares, each with side length s. you could imagine a limit argument, with the squares getting small enough to arbitrarily approximate the area of the shape. uniform scaling increases the distance between any two points by some factor k. the area of each of the original squares is s^2, and the new area is (ks)^2 = k^2*s^2 thus the area of the triangle increases by a factor of k^2. the same argument will apply to the square drawn on the hypotenuse. thus this k^2 term will cancel out when you take the ratio. (we could've used the area formula for a triangle here, but this argument applies more generally: for example we can deduce the area of a circle is proportional to radius^2 without deriving the formula)
Dave_Rosenthal 11y ago
My physics professor showed me that exact proof 20 years ago. I remember being amazed. He used alpha instead of F as the scale factor :) Still, it's easier for me to believe that it was attributed to Einstein at some point in the past 60 years than it originated with him.
amai 11y ago
"Arthur Eddington ... was asked if it was really true that only three people in the world understood the theory, he said nothing. “Don’t be so modest, Eddington!” his questioner said. “On the contrary,” Eddington replied. “I’m just wondering who the third might be.” The third person was https://en.wikipedia.org/wiki/Karl_Schwarzschild https://en.wikipedia.org/wiki/Karl_Schwarzschild .

86 points by jeo1234 11y ago

ck2 11y ago

Have any of Einstein's relatives proved remotely as clever as him? aha, of course there's an article https://en.wikipedia.org/wiki/Einstein_family https://en.wikipedia.org/wiki/Einstein_family

deleted 11y ago
[deleted]
Jun8 11y ago
In that regard it's hard to beat the Curies (barring the older Bernoulli family, of course): https://www.reddit.com/r/todayilearned/comments/3laog8/til_that_%C3%A8ve_curie_unlike_her_mother_father/ https://www.reddit.com/r/todayilearned/comments/3laog8/til_t.... That's 5 Nobel Prizes (or 4 if you disregard the Peace Prize that the son-in-law earned).

deleted 11y ago

[deleted]

vitthal 11y ago

I don't remember how to prove it anymore... This can be a good problem to solve...

Jtsummers 11y ago

The second moment occurred soon after he turned twelve, when he was given “a little book dealing with Euclidean plane geometry.” The book’s “lucidity,” he wrote—the idea that a mathematical assertion could “be proved with such certainty that any doubt appeared to be out of the question”—provoked “wonder of a totally different nature.” Mathematical proofs, first really introduced in Geometry classes for most US students around 9th or 10th grade, are what really hooked me on math. I understood everything up to that point (and after), but it was a structural understanding (the rules and syntax, essentially). Once we arrived at logic, I (as a very bored precocious student) spent most of my class time proving everything from the axioms we were given (for algebra or geometry) that I could. Giving students the tools to play and explore freely (we had proof assignments, of course, but I took it far beyond that) is what really hooks them on a subject. Mandate that they must read these books, they'll hate it. Require them to read a book and a few excerpts of others, and give them access to a large library, and they'll read forever. Same with every other subject. That's where many of the real geniuses[0] of a field come from. [0] Whether technically geniuses by IQ or just acknowledged as such for their understanding and creations.

jwdunne 11y ago
I notice this learning maths as an adult. A good intro to logic and proofs opened up a new world. After struggling with a proof, I came to the conclusion that I needed to study the axioms I was obviously expected to know futher - most importantly, I wanted to. How did I want to study and understand them? By proof. I remember a similar course with programming, actually. One curiosity led to the next once I equipped myself with the tools to think and experiment with computers.
- Jtsummers 11y ago
  I will also say, and meant to mention in my original comment, good textbooks are beautiful things. Most textbooks by the time I got to HS (mid-1990s) were meh. And they seemed to be getting worse each year. Compared to the books my parents and previous generations experienced. (Specifically for math.) Older books tended to be small, concise and precise. Think Sipser's automata text, but for everything in math. Being a third generation mathematician certainly helped by providing me access to an older library of books than a lot of my high school and college peers realized were available.
  - jwdunne 11y ago
    Thanks for the tip :) I'll explore that. Do you have any recommendations for textbooks to sink my teeth into?
    - Jtsummers 11y ago
      Unfortunately, not much. I was pondering that after my post. It's been a long time since I was a math student, and professionally it's had zero to do with my career. A lot of books ended up boxed up at my parents' home as I moved around a number of times right after college. I'll check my own home tonight to see what I still have, but my shelves these days are mostly filled with fiction, programming, RPG, and history books. EDIT: Off the top of my head, for CS: Introduction to Algorithms: https://mitpress.mit.edu/books/introduction-algorithms https://mitpress.mit.edu/books/introduction-algorithms Introduction to the Theory of Computation: http://www.amazon.com/Introduction-Theory-Computation-Michael-Sipser/dp/113318779X http://www.amazon.com/Introduction-Theory-Computation-Michae... Math: Calculus Made Easy: http://www.amazon.com/gp/product/0312185480?ref_=cm_lmf_tit_5 http://www.amazon.com/gp/product/0312185480?ref_=cm_lmf_tit_... - I'm really not sure how good this one is for a beginner, I picked it up while assisting my sister in refreshing her calculus skills for grad school (Aerospace Engineering) I can't remember the algebra and geometry textbooks (my dad's or my grandfather's) that I used, in addition to the assigned text, in high school. Anything by Knuth. Seriously, one summer a professor and I just picked up copies of Concrete Mathematics and worked through large portions of it for fun. Technically I got some math credits for it, but it was really just because we wanted to. Actually, this one helped me a lot with understanding calculus. Somehow, up to that point while I knew calculus, may brain had never made the connection that integration was summation until I saw the discrete counterpart to continuous integration. I had a mechanical understanding, but no deep understanding until that moment.

Jun8 11y ago

BTW, the book that was briefly mentioned in the article, Fractals, Chaos, and Power Laws: Minutes from an Infinite Paradise by Schroeder (http://www.amazon.com/Fractals-Chaos-Power-Laws-Infinite/dp/0716721368 http://www.amazon.com/Fractals-Chaos-Power-Laws-Infinite/dp/...) is truly excellent, it is written in the form of short pieces collected into chapters investigating a very wide-ranging set of phenomena, ranging from Brownian Motion to self-similarity, Cantor sets and cellular automata. For $10-$15 on Amazon, it's a great bargain!

blahblah3 11y ago

here's a proof that the ratio of the triangle area to the square is scale invariant: imagine covering a geometric shape with a bunch of little squares, each with side length s. you could imagine a limit argument, with the squares getting small enough to arbitrarily approximate the area of the shape. uniform scaling increases the distance between any two points by some factor k. the area of each of the original squares is s^2, and the new area is (ks)^2 = k^2*s^2 thus the area of the triangle increases by a factor of k^2. the same argument will apply to the square drawn on the hypotenuse. thus this k^2 term will cancel out when you take the ratio. (we could've used the area formula for a triangle here, but this argument applies more generally: for example we can deduce the area of a circle is proportional to radius^2 without deriving the formula)

Dave_Rosenthal 11y ago

My physics professor showed me that exact proof 20 years ago. I remember being amazed. He used alpha instead of F as the scale factor :) Still, it's easier for me to believe that it was attributed to Einstein at some point in the past 60 years than it originated with him.

amai 11y ago

"Arthur Eddington ... was asked if it was really true that only three people in the world understood the theory, he said nothing. “Don’t be so modest, Eddington!” his questioner said. “On the contrary,” Eddington replied. “I’m just wondering who the third might be.” The third person was https://en.wikipedia.org/wiki/Karl_Schwarzschild https://en.wikipedia.org/wiki/Karl_Schwarzschild .