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