4 ms·

Vectors from Leibniz to Einstein

58 points by jamespropp 3y ago

jamespropp 3y ago
This essay tells the story of how the modern theory of vectors was gradually discovered, and covers the colorful yet often forgotten late-19th century quarrel between quaternionists and vectorialists.
- 082349872349872 3y ago
  A tangent, and (for Dr Propp) suggestion and question: - Related: https://mathenchant.wordpress.com/2023/06/16/what-is-a-matrix/ https://mathenchant.wordpress.com/2023/06/16/what-is-a-matri... - May I suggest adding a link to https://mathenchant.wordpress.com/feed/ https://mathenchant.wordpress.com/feed/ somewhere more easily found than via "View Source"? - Since you (a) mentioned a working title "... Adventures of Plus and Times" and (b) have classroom experience, do you have any opinions on the following: when overloading the "high school" arithmetic operators for ring- and lattice-like structures, is it easier for people to learn (a) overloads with traditional use (ie × ↦ meet, + ↦ xor in Electrical Engineering), (b) overloads with mnemonic stories: "product makes things larger; modulo makes things smaller", or (c) any consistent overloads, because just making the abstraction leap is much more difficult than remembering any operator assignments?
  - jamespropp 3y ago
    I don’t have classroom experience with this kind of teaching. I may write a companion to “When 1+1=0” called “When 1+1=1”, at which point I’ll have to think harder about these issues. (Are we using a different 0 and a different 1 here? Or a different +? Or a different =?)
- deleted 3y ago
  [deleted]
Beijinger 3y ago
OT: As a high school student I read a lot of philosophy books, but I always stayed away from Leibniz's Monadology, it sounded too bizarr and esoteric. https://en.wikipedia.org/wiki/Monadology https://en.wikipedia.org/wiki/Monadology But Lee Smolin come up with an interesting interpretation: https://www.quantamagazine.org/how-to-understand-the-universe-when-youre-stuck-inside-of-it-20190627/ https://www.quantamagazine.org/how-to-understand-the-univers...
- lqet 3y ago
  > Looking back historically, people like Einstein and Bohr and John Wheeler all took philosophy very seriously; it directly influenced their physics. I only recently learned that Einstein had three large portraits in his Berlin office: Michael Faraday, James Clerk Maxwell, and Arthur Schopenhauer. Einstein considered Schopenhauer a genius. Friends and colleagues frequently found him reading random passages of "The World as Will and Representation". He even seems to have copied Schopenhauer's hairstyle in later years. Schopenhauer was also very important to Wolfgang Pauli and Erwin Schrödinger [0]. [0] https://en.wikipedia.org/wiki/Arthur_Schopenhauer#Influence_and_legacy https://en.wikipedia.org/wiki/Arthur_Schopenhauer#Influence_...
  - crabbone 3y ago
    Schopenhauer has few things that could inspire, not necessary his philosophy. He was like the only philosopher contemporary of Hegel who didn't even care what Hegel had to say. While Hegel was wildly popular and created a lot of following, Schopenhauer was just a bitter eccentric dude sitting in his castle... but he wasn't stupid / unable to appreciate Hegel's ideas. He really had a good point. In a sense, he was kind of like that poor doctor who was ridiculed by his peers for suggesting doctors need to wash their hands (and who later committed suicide because of all the bullying). Sorry, cannot recall his name at the moment. So, it could've been admiration of determination and confidence in one's own beliefs. More so than the admiration of the beliefs themselves.
    - Beijinger 3y ago
      Ignaz Semmelweis
hackandthink 3y ago
Grassmann should get more credit. Lawvere admired Grassmann. https://ncatlab.org/nlab/show/Ausdehnungslehre https://ncatlab.org/nlab/show/Ausdehnungslehre
- rubberpoliceman 3y ago
  Also, http://www.numdam.org/item/SPHM_1979___2_A1_0.pdf http://www.numdam.org/item/SPHM_1979___2_A1_0.pdf
- jamespropp 3y ago
  I’ll look into this. Thanks!

58 points by jamespropp 3y ago

This essay tells the story of how the modern theory of vectors was gradually discovered, and covers the colorful yet often forgotten late-19th century quarrel between quaternionists and vectorialists.

082349872349872 3y ago
A tangent, and (for Dr Propp) suggestion and question: - Related: https://mathenchant.wordpress.com/2023/06/16/what-is-a-matrix/ https://mathenchant.wordpress.com/2023/06/16/what-is-a-matri... - May I suggest adding a link to https://mathenchant.wordpress.com/feed/ https://mathenchant.wordpress.com/feed/ somewhere more easily found than via "View Source"? - Since you (a) mentioned a working title "... Adventures of Plus and Times" and (b) have classroom experience, do you have any opinions on the following: when overloading the "high school" arithmetic operators for ring- and lattice-like structures, is it easier for people to learn (a) overloads with traditional use (ie × ↦ meet, + ↦ xor in Electrical Engineering), (b) overloads with mnemonic stories: "product makes things larger; modulo makes things smaller", or (c) any consistent overloads, because just making the abstraction leap is much more difficult than remembering any operator assignments?
- jamespropp 3y ago
  I don’t have classroom experience with this kind of teaching. I may write a companion to “When 1+1=0” called “When 1+1=1”, at which point I’ll have to think harder about these issues. (Are we using a different 0 and a different 1 here? Or a different +? Or a different =?)
deleted 3y ago
[deleted]

Beijinger 3y ago

OT: As a high school student I read a lot of philosophy books, but I always stayed away from Leibniz's Monadology, it sounded too bizarr and esoteric. https://en.wikipedia.org/wiki/Monadology https://en.wikipedia.org/wiki/Monadology But Lee Smolin come up with an interesting interpretation: https://www.quantamagazine.org/how-to-understand-the-universe-when-youre-stuck-inside-of-it-20190627/ https://www.quantamagazine.org/how-to-understand-the-univers...

lqet 3y ago
> Looking back historically, people like Einstein and Bohr and John Wheeler all took philosophy very seriously; it directly influenced their physics. I only recently learned that Einstein had three large portraits in his Berlin office: Michael Faraday, James Clerk Maxwell, and Arthur Schopenhauer. Einstein considered Schopenhauer a genius. Friends and colleagues frequently found him reading random passages of "The World as Will and Representation". He even seems to have copied Schopenhauer's hairstyle in later years. Schopenhauer was also very important to Wolfgang Pauli and Erwin Schrödinger [0]. [0] https://en.wikipedia.org/wiki/Arthur_Schopenhauer#Influence_and_legacy https://en.wikipedia.org/wiki/Arthur_Schopenhauer#Influence_...
- crabbone 3y ago
  Schopenhauer has few things that could inspire, not necessary his philosophy. He was like the only philosopher contemporary of Hegel who didn't even care what Hegel had to say. While Hegel was wildly popular and created a lot of following, Schopenhauer was just a bitter eccentric dude sitting in his castle... but he wasn't stupid / unable to appreciate Hegel's ideas. He really had a good point. In a sense, he was kind of like that poor doctor who was ridiculed by his peers for suggesting doctors need to wash their hands (and who later committed suicide because of all the bullying). Sorry, cannot recall his name at the moment. So, it could've been admiration of determination and confidence in one's own beliefs. More so than the admiration of the beliefs themselves.
  - Beijinger 3y ago
    Ignaz Semmelweis

hackandthink 3y ago

Grassmann should get more credit. Lawvere admired Grassmann. https://ncatlab.org/nlab/show/Ausdehnungslehre https://ncatlab.org/nlab/show/Ausdehnungslehre

rubberpoliceman 3y ago
Also, http://www.numdam.org/item/SPHM_1979___2_A1_0.pdf http://www.numdam.org/item/SPHM_1979___2_A1_0.pdf
jamespropp 3y ago
I’ll look into this. Thanks!