Philosophy of mathematics | Discrete geometry
In philosophy, the Weyl's tile argument, introduced by Hermann Weyl in 1949, is an argument against the notion that physical space is "discrete", as if composed of a number of finite sized units or tiles. The argument purports to show a distance function approximating Pythagoras' theorem on a discrete space cannot be defined and, since the Pythagorean theorem has been confirmed to be approximately true in nature, physical space is not discrete. Academic debate on the topic continues, with counterarguments proposed in the literature. A demonstration of Weyl's argument proceeds by constructing a square tiling of the plane representing a discrete space. A discretized triangle, n units tall and n units long, can be constructed on the tiling. The hypotenuse of the resulting triangle will be n tiles long. However, by the Pythagorean theorem, a corresponding triangle in a continuous space—a triangle whose height and length are n—will have a hypotenuse measuring units long. To show that the former result does not converge to the latter for arbitrary values of n, one can examine the percent difference between the two results: Since n cancels out, the two results never converge, even in the limit of large n. The argument can be constructed for more general triangles, but, in each case, the result is the same. Thus, a discrete space does not even approximate the Pythagorean theorem. In response, Kris McDaniel has argued the Weyl tile argument depends on accepting a "size thesis" which posits that the distance between two points is given by the number of tiles between the two points. However, as McDaniel points out, the size thesis is not accepted for continuous spaces. Thus, we might have reason not to accept the size thesis for discrete spaces. (Wikipedia).
In this video, I present a very classical example of a duality argument: Namely, I show that T^T is one-to-one if and only if T is onto and use that to show that T is one-to-one if and only if T^T is onto. This illustrates the beautiful interplay between a vector space and its dual space,
From playlist Dual Spaces
Set Theory (Part 2): ZFC Axioms
Please feel free to leave comments/questions on the video and practice problems below! In this video, I introduce some common axioms in set theory using the Zermelo-Fraenkel w/ choice (ZFC) system. Five out of nine ZFC axioms are covered and the remaining four will be introduced in their
From playlist Set Theory by Mathoma
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This is a short animation showing one way to think of the difference of squares formula (at least when the two numbers involved are positive). #manim #math #proofwithoutwords #visualproof
From playlist Algebra
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What is the symmetric difference of 2 sets? In this video we go over the symmetric difference of sets, explaining it in a couple ways including what is probably the briefest way. The symmetric difference of two sets A and B is (A union B)-(A intersect B). If you need to know what the defin
From playlist Set Theory
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From playlist Set theory
This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.
From playlist Proofs
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From playlist Dual Spaces
Proof: A'-B' = B-A (Double Inclusion) | Set Theory
We prove A'-B'=B-A. That is, the complement of A minus the complement of B equals B minus A. We prove this using double inclusion, meaning we prove both sets are subsets of each other, which by definition establishes set equality. This will require basic applications of set theory definiti
From playlist Set Theory
O'Reilly Webcast: Part 2 - The Best of Velocity: Mobile Web Application Performance
Part2 Mobile browser performance is challenged by bandwidth, battery, and memory constraints. Slow loading and reacting sites create bad user experiences. Sites that drain batteries or crash the browser are infuriating. Porting a web application designed and developed for desktop devices—
From playlist O'Reilly Webcasts 2
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From playlist ICTS at Ten
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From playlist Mathematics
Weyl groups, and their generalizations in, enumerative geometry II - Okounkov
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From playlist Hermann Weyl Lectures
Weyl groups, and their generalizations, in enumerative geometry I - Andrei Okounkov
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From playlist Hermann Weyl Lectures
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From playlist Fall 2017
To learn to think like a scientist check out http://Brilliant.org/SpaceTime PBS Member Stations rely on viewers like you. To support your local station, go to: http://to.pbs.org/DonateSPACE Check out the new Space Time Merch Store! https://pbsspacetime.com/ Support Space Time on Patreo
From playlist Understanding the Holographic Universe
The intersection of two open sets is open -- Proofs
This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.
From playlist Proofs
Weyl Anomalies and Cosmology by Atish Dabholkar
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From playlist String Theory: Past and Present
Quantum Anomalies and magneto-transport in Weyl semimetals by Amit Agarwal
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From playlist Novel Phases of Quantum Matter 2019
David Zywina, Computing Sato-Tate and monodromy groups.
VaNTAGe seminar on May 5, 2020. License: CC-BY-NC-SA Closed captions provided by Jun Bo Lau.
From playlist The Sato-Tate conjecture for abelian varieties