Understanding Reverse Mathematics: Unlocking Foundations of Math 🔍

Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.

This time.
What is...reverse mathematics? Or: Thinking backwards.

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Slides.
http://www.dtubbenhauer.com/youtube.html

Reverse mathematics.
https://en.wikipedia.org/wiki/Reverse_mathematics
https://press.princeton.edu/books/hardcover/9780691177175/reverse-mathematics
https://en.wikipedia.org/wiki/Reverse_Mathematics:_Proofs_from_the_Inside_Out
https://www.ams.org/journals/notices/201809/rnoti-p1098.pdf

Euclidean geometry.
https://en.wikipedia.org/wiki/Euclidean_geometry
https://mathworld.wolfram.com/EuclidsPostulates.html

Triangles, playfair and the parallel postulate.
https://en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle
https://en.wikipedia.org/wiki/Playfair%27s_axiom
https://en.wikipedia.org/wiki/Parallel_postulate
https://mathworld.wolfram.com/ParallelPostulate.html

Non-Euclidean geometry.
https://en.wikipedia.org/wiki/Non-Euclidean_geometry

Spherical geometry.
https://en.wikipedia.org/wiki/Spherical_geometry
https://en.wikiversity.org/wiki/Ideas_in_Geometry/Spherical_Geometry
http://www.users.miamioh.edu/gaddisj/107slides17/22%20-%20Spherical%20Geometry.pdf

Hyperbolic geometry.
https://en.wikipedia.org/wiki/Hyperbolic_geometry
https://mathworld.wolfram.com/HyperbolicGeometry.html
https://mathimages.swarthmore.edu/index.php/Hyperbolic_Geometry

Peano and co.
https://googology.wikia.org/wiki/User_blog:Vel!/Second-order_arithmetic
https://en.wikipedia.org/wiki/First-order_logic
https://en.wikipedia.org/wiki/Second-order_arithmetic

Brouwer and Jordan.
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem

Mathematica.
https://demonstrations.wolfram.com/ThreeProofsOfThePythagoreanTheorem/

Pictures used.
https://modulouniverse.com/2015/02/23/brouwers-fixed-point-theorem/
https://math.stackexchange.com/questions/563359/is-there-a-dissection-proof-of-the-pythagorean-theorem-for-tetrahedra
https://en.wikibooks.org/wiki/Trigonometry/Angles_of_a_triangle_sum_to_180_Degrees
https://www.geogebra.org/m/hdepmnm9
https://googology.wikia.org/wiki/User_blog:Vel!/Second-order_arithmetic
A picture from the book “Reverse Mathematics: Proofs from the Inside Out”
https://brilliant.org/wiki/brouwer-fixed-point-theorem/
https://en.wikipedia.org/wiki/Jordan_curve_theorem#/media/File:Jordan_curve_theorem.svg

Some videos on non-Euclidean geometry.
https://www.youtube.com/watch?v=kFVxzjRckPY
https://www.youtube.com/watch?v=zQo_S3yNa2w
https://www.youtube.com/watch?v=kEB11PQ9Eo8