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This is a nice overview of some of the fundamental aspects of modern mathematics and how they fit together. It's definitely targeted at an advanced reader thoug

This is a nice overview of some of the fundamental aspects of modern mathematics and how they fit together. It's definitely targeted at an advanced reader though. A book describing the large-scale structure of modern mathematics and its fundamental concepts at an undergraduate level is Mathematics, Form and Function by Saunders Mac Lane. If you like this paper but feel like it's currently out of your reach, you might like this book. Something deep about mathematics that I feel has only begun to be explored in the last few decades or so is how intimately connected computation/logic is to algebra and geometry. Connes doesn't really touch on that here but research programs like Geometry of Interaction and Geometric Complexity Theory are really exciting to me.

Myrmornis 9y ago
Looks really interesting, I'm going to try to read it, and thanks for your suggestion of the MacLane book. Another really excellent resource along those lines is the Princeton Companion to Mathematics, in particular the introductory essay by Tim Gowers (who is also the editor of the book). By along those lines, I mean pointing to advanced mathematics, from the point of view of someone who is kind of OK with introductory undergrad math.
- stablemap 9y ago
  Gowers also has an entry in the Very Short Introduction series, which you might like: https://global.oup.com/academic/product/mathematics-a-very-short-introduction-9780192853615 https://global.oup.com/academic/product/mathematics-a-very-s...
gtani 9y ago
some other survey /panorama books for various levels: What is Mathematics, Courant and Robbins (I haven't read the 2nd ed, edited by Ian Stewart) https://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071 https://www.amazon.com/All-Mathematics-You-Missed-Graduate/d...
- hackermailman 9y ago
  The revised Stuart edition nothing is changed except Stewart has added a preface, and a 37 page chapter "Recent Developments". John Stillwell also has a good survey book "Elements of Mathematics From Euclid to Gödel"
  - goialoq 9y ago
    Stillwell is a great author. Naive Lie Theory is a great read, with a big section on quaternions.
- gtani 9y ago
  also for all in 1 math for physics books, there's a bunch: Arfken et al., Mary Boas, Riley/Hobson /Bence etc: https://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Comprehensive/dp/0123846544 https://www.amazon.com/Mathematical-Methods-Physicists-Seven... and http://www.goldbart.gatech.edu/PostScript/MS_PG_book/bookmaster.pdf http://www.goldbart.gatech.edu/PostScript/MS_PG_book/bookmas... These are sort of orthogonal to the OP's link, they're not covering analysis, abstract algebra and topology, instead they're covering dif eq, spectral analysis, probability/stats, linear algebra.
Myrmornis 9y ago
We should also mention Penrose's "The Road To Reality" in this subthread. While it is essentially insane as a commercial/popular publishing project, I suspect it is also one of the best attempts to survey all of maths and mathematical physics in one location. Unless you have a PhD in mathematical physics, you will not be able to read all of it profitably. However, I suspect it might be a useful set of signposts for someone attempting to learn mathematical physics, to come back to again and again (over the course of, say, a lifetime). I'm sure it has some idiosyncracies (his graphical tensor notation I don't think has caught on that much in the field, and is likely to cause some surprise to anyone who hasn't seen it before) and controversial treatments of some areas, but in general I think (as someone firmly in early undergrad math territory) it is a fantastic book. The illustrations. Those have to be the best collection of informal attempts to illustrate physics-oriented maths that exist.

Myrmornis 9y ago

Looks really interesting, I'm going to try to read it, and thanks for your suggestion of the MacLane book. Another really excellent resource along those lines is the Princeton Companion to Mathematics, in particular the introductory essay by Tim Gowers (who is also the editor of the book). By along those lines, I mean pointing to advanced mathematics, from the point of view of someone who is kind of OK with introductory undergrad math.

stablemap 9y ago
Gowers also has an entry in the Very Short Introduction series, which you might like: https://global.oup.com/academic/product/mathematics-a-very-short-introduction-9780192853615 https://global.oup.com/academic/product/mathematics-a-very-s...

gtani 9y ago

some other survey /panorama books for various levels: What is Mathematics, Courant and Robbins (I haven't read the 2nd ed, edited by Ian Stewart) https://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071 https://www.amazon.com/All-Mathematics-You-Missed-Graduate/d...

hackermailman 9y ago
The revised Stuart edition nothing is changed except Stewart has added a preface, and a 37 page chapter "Recent Developments". John Stillwell also has a good survey book "Elements of Mathematics From Euclid to Gödel"
- goialoq 9y ago
  Stillwell is a great author. Naive Lie Theory is a great read, with a big section on quaternions.
gtani 9y ago
also for all in 1 math for physics books, there's a bunch: Arfken et al., Mary Boas, Riley/Hobson /Bence etc: https://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Comprehensive/dp/0123846544 https://www.amazon.com/Mathematical-Methods-Physicists-Seven... and http://www.goldbart.gatech.edu/PostScript/MS_PG_book/bookmaster.pdf http://www.goldbart.gatech.edu/PostScript/MS_PG_book/bookmas... These are sort of orthogonal to the OP's link, they're not covering analysis, abstract algebra and topology, instead they're covering dif eq, spectral analysis, probability/stats, linear algebra.

Myrmornis 9y ago

We should also mention Penrose's "The Road To Reality" in this subthread. While it is essentially insane as a commercial/popular publishing project, I suspect it is also one of the best attempts to survey all of maths and mathematical physics in one location. Unless you have a PhD in mathematical physics, you will not be able to read all of it profitably. However, I suspect it might be a useful set of signposts for someone attempting to learn mathematical physics, to come back to again and again (over the course of, say, a lifetime). I'm sure it has some idiosyncracies (his graphical tensor notation I don't think has caught on that much in the field, and is likely to cause some surprise to anyone who hasn't seen it before) and controversial treatments of some areas, but in general I think (as someone firmly in early undergrad math territory) it is a fantastic book. The illustrations. Those have to be the best collection of informal attempts to illustrate physics-oriented maths that exist.