
Einstein’s First Proof - jeo1234
http://www.newyorker.com/tech/elements/einsteins-first-proof-pythagorean-theorem?intcid=mod-latest
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Jtsummers

      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
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
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
Thanks for the tip :) I'll explore that. Do you have any recommendations for
textbooks to sink my teeth into?

~~~
Jtsummers
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-
Michae...](http://www.amazon.com/Introduction-Theory-Computation-Michael-
Sipser/dp/113318779X)

Math:

Calculus Made Easy:
[http://www.amazon.com/gp/product/0312185480?ref_=cm_lmf_tit_...](http://www.amazon.com/gp/product/0312185480?ref_=cm_lmf_tit_5)
\- 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
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/...](http://www.amazon.com/Fractals-Chaos-Power-Laws-
Infinite/dp/0716721368)) 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
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)

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amai
"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)
.

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ck2
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)

~~~
Jun8
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_t...](https://www.reddit.com/r/todayilearned/comments/3laog8/til_that_%C3%A8ve_curie_unlike_her_mother_father/).

That's 5 Nobel Prizes (or 4 if you disregard the Peace Prize that the son-in-
law earned).

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Dave_Rosenthal
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.

