ABOUT EINSTEIN'S BOOK "RELATIVITY : THE SPECIAL AND GENERAL THEORY". https://www.gutenberg.org/ebooks/5001 --- Chapter 32 is titled "The Structure of Space According to the General Theory of Relativity". However, the point of the previous chapters is precisely to show (very convincingly in my humble opinion) that the word "space" has no absolute meaning. Was relativity too revolutionary for Einstein himself?
A QUOTE OF WITTGENSTEIN. 'p' is true = p. Ludwig Wittgenstein, Remarks on the foundations of mathematics, Part 1, Appendix 1, Section 6. (This point occurs already in Section 4.442 of the Tractatus Logico-Philosophicus. See the phrase Frege’s "judgement stroke" "⫞" is logically quite meaningless. --- Links to the Tractatus Logico-Philosophicus: https://people.umass.edu/klement/tlp/ --- https://www.wittgensteinproject.org/w/index.php/Project:All_texts )
Here is the whole of Section 6:
6. For what does a proposition's 'being true' mean? 'p' is true = p . (That is the answer.)
So we want to ask something like: under what circumstances do we assert a proposition? Or: how is the assertion of the proposition used in the language-game? And the 'assertion of the proposition' is here contrasted with the utterance of the sentence e.g. as practice in elocution,---or as part of another proposition, and so on.
If, then, we ask in this sense: "Under what circumstances is a proposition asserted in Russell's game?" the answer is: at the end of one of his proofs, or as a 'fundamental law' (Pp.). There is no other way in this system of employing asserted propositions in Russell's symbolism.
Link: https://archive.org/details/remarksonfoundat0000witt_c4e5/page/n7/mode/2up
THE MONTY HALL PROBLEM. Then Monty Hall problem is very famous. A reminder is given below. Here is my preferred solution.
Suppose we have two players, called Switch and No-switch. Switch always switches, No-switch never does. Then Switch wins if and only if she picks a door hiding a goat, which is a probability 2/3 event, whereas No-switch wins if and only if she picks the door hiding the car, which is a probability 1/3 event.
Here is the reminder:
Setting: We have three doors, one door hides a car, each of the other doors hides a goat. Player wins if she reveals the car.
See: Wikipedia entry and A text of Andrew Vazsonyi.
NEW TEXTS
Existence of a least increasing surjection from an artinian poset onto a set of ordinals. We prove that, given an artinian poset X, there is a least increasing surjection f from X onto a set of ordinals. https://github.com/Pierre-Yves-Gaillard/Artin-poset-to-ordinals
Transfinite Recursion --- The only purpose of this short text is to point out that the Transfinite Recursion Theorem is a particular case of the Transfinite Induction Theorem https://github.com/Pierre-Yves-Gaillard/The-Transfinite-Recursion-Theorem
Apparent speed in different inertial frames in special relativity - https://zenodo.org/records/14632096 - In the setting of special relativity, let p be a particle moving at constant velocity. We compare the apparent speeds of p in two inertial frames.
About the Drinker Paradox - https://zenodo.org/records/14599048
Recovering a quadratic form from its orthogonal group --- We prove the following statement: Let K be a field of characteristic different from 2, let V be a K-vector space of finite dimension n > 1, let q and r be two quadratic forms on V, let O(q) and O(r) be the respective orthogonal groups, and let Q, R : V × V → K be the respective associated symmetric bilinear forms. Then we have O(q) = O(r) ⇔ Kq = Kr. https://zenodo.org/records/14513720
Self-dual vector spaces - Let V be a vector space and G its automorphism group. Say V is *self-dual* if the dual V^* of V is G-isomorphic to V. We answer the question: Which vector spaces are self-dual? https://zenodo.org/records/14361088
A few comments about "Principles of Mathematical Analysis" by Rudin https://zenodo.org/records/13955297
The Jordan-Hölder Theorem - https://zenodo.org/records/13870026
Sylow's Theorems - https://zenodo.org/records/13854863
Simplicity of A_n https://zenodo.org/records/13844112
Zorns's Lemma. We prove the following statement. Let P be a poset all of whose well-ordered subsets have an upper bound. Then P has a maximal element. https://zenodo.org/records/13791363
The dimension of a finitely generated vector space https://zenodo.org/records/13740597
A QUESTION: Do we always have dim End_{End(V)}(V^{**}/V) = 1 ? Explanation: V is a K-vector space and V^{**} is its second dual. In particular V^{**} is an End(V)-module, and so is the quotient V^{**}/V. As such this quotient has a K-algebra A of endomorphisms. And the question is: Is A always one-dimensional? The answer is trivially yes if V is finite dimensional. More on this: https://mathoverflow.net/q/162162/461
OLD TEXTS
Optimal moves in tic-tac-toe - Zenodo https://doi.org/10.5281/zenodo.7940866
The first move in a drawing tic-tac-toe is always optimal https://doi.org/10.5281/zenodo.10252432
The Fundamental Theorem of Galois Theory: We give a short and self-contained proof of the Fundamental Theorem of Galois Theory for finite degree extensions. https://vixra.org/abs/1207.0051
Motivating the Gauss sum proof of the quadratic reciprocity https://doi.org/10.5281/zenodo.6642138
A formula for the Jacobi symbol https://doi.org/10.5281/zenodo.6721050
Pointless moves in tic-tac-toe https://doi.org/10.5281/zenodo.7940866
Natural Ways of Mapping Subsets to Subsets 10.5281/zenodo.6564983 --- viXra
Finite faithful G-sets are asymptotically free 10.5281/zenodo.6573066 --- viXra
Each topological space X is of the form Aut(Y)\Y --- pdf file --- tex file --- (Eric Wofsey found a simpler argument. His argument shows that Y can be required to be T_0. See https://math.stackexchange.com/a/3992100/660)
Any finite connected poset is isomorphic to Aut(X)\X for some finite poset X --- pdf file --- tex file
The Burnside Q-algebras of a monoid --- pdf file --- tex file
The universal profinitization of a topological space --- pdf file --- tex file
A text about the book Introduction to Commutative Algebra by Michael Atiyah and Ian G. MacDonald (including solutions to some exercises) --- https://zenodo.org/record/6498378
A text about the book Categories and Sheaves by by Masaki Kashiwara and Pierre Schapira --- https://zenodo.org/record/6498631 --- https://vixra.org/abs/1602.0067
WORKS IN PROGRESS
A few comments about ``Topology'' by Munkres https://www.overleaf.com/read/kdwwjvqjrzwb#9fe3a6 - https://github.com/Pierre-Yves-Gaillard/About-Topology-by-Munkres
A few comments about ``Linear Algebra Done Right'' by Axler https://zenodo.org/records/14028516 - https://github.com/Pierre-Yves-Gaillard/About-LADR-by-Axler - https://www.overleaf.com/read/fbvxfjsvrgcv#afc0a5
Email: pierre.yves.gaillard at gmail.com
A mini mirror of this site is maintained by my daughter Sophie Gaillard