Mariano Suárez-Alvarez
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 Sep 30 comment Motivation for definition of quotient map and “passing to the quotient” Well, once you come to accept that, then a good plan is to wait a bit, and see for what the notion is used. Soon you will see the notion at play in various contexts, and you'll not need anymore to be given a motivation for it for you'll have seen it yourself: to do what you will be doing. The way we teach math these days requires a generous dose of "suspension of disbelief" on the part of the student. An abstract explanation of what "passing to the quotient" is is worth 0.0000000000% of actually seeing it done in a real life context, when the need for it arises for you. Sep 30 comment Motivation for definition of quotient map and “passing to the quotient” Nowadays students are presented with a definition of what a group is and then examples and motivations, while in real life people had been using groups and being motivated looooong before anyone thought to define what a group is. Successful definitions were, in history, successful because they captured a notion that had already been in the minds of mathematicians, and for which they felt ---usually after the fact--- a need. Sep 30 comment Motivation for definition of quotient map and “passing to the quotient” Definitions arise from the observation of what we do: we use definitions to capture useful notions, and they are useful before we turn them into definitions. This is how mathematics is done in real life. The idea that we define things out of some mythical «motivation» is an outgrowth of the way we teach, and of a mistaken and quite damaging image that mathematics has outside of its circle of practitioners (in great part due to the fault of some of its practitioners, alas) Sep 30 comment Motivation for definition of quotient map and “passing to the quotient” The motivation for doing it is that we do it all the time. Sep 29 answered Spanish translation for the term operad? Sep 29 comment A question about Homotopy (Michael Harris's recent book) You seem to be grandly underestimating the effort needed to write down a minimally sensible answer to what you want. Sep 29 comment Finding holes in a non-simply connected open subset of $\mathbb{C}$ Among the good textbooks on complex analysis are the ones by Remmert, Conway, and Lang. Sep 29 answered Finding holes in a non-simply connected open subset of $\mathbb{C}$ Sep 29 comment How can I find all the subfields of Q? Your definition of ring includes having a unit element? Sep 29 comment How to compute the Endomorphism ring EndR(M) @MarianoVelasquez, please move what is useful in these comments to the body of the question itself. Comments are mostly for other people to make comments :-) Sep 28 comment Basic tensor derivation @L.Henry, I have reverted your edit — which turned the question incomprehensible. Please do not do that. Sep 28 revised Basic tensor derivation rolled back to a previous revision Sep 28 comment A symmetric idempotent matrix Then edit the question and mention it! :-) Sep 28 comment A symmetric idempotent matrix The matrix is real? Sep 28 revised Divergence equation on noncompact manifold. deleted 50 characters in body Sep 28 comment Functors from $\mathsf{Set}$ to $\mathsf{Mon}$? The free abelian monoid, the free abelian monoid of exponent $n$, the composition of any functor from set to something with a functor to Mon (the functor mapping a set $X$ to Brauer group of the field of rational functions $\mathbb C(X)$, say, to give an pretty absurd example). There are many! Sep 28 comment Compute an Endomorphism ring and that a module is indecomposable I wouldn't have written what I wrote if I did not think so and, in fact, I know the two problems have nothing to do with each other. Please split them into two separate questions. Also, you should read meta.math.stackexchange.com/questions/9959/… and provide some context —most importantly, what your background is and what youhave already tried. Sep 28 comment Subgroup generated by $\langle a,b \rangle$ where $ab=ba$ This does not have anything to do with rigor, really. What you wrote originally was wrong, not unrigorous, and what you wrote in the comment was meaningless (the "$a^n=e$" on the left of the colon in the set cannot be there), not unrigorous. Sep 28 comment Subgroup generated by $\langle a,b \rangle$ where $ab=ba$ No, that is slightly worse, as it does not even make sense :-) You should have writtten something like «$\langle a\rangle=\{a^n:n\in\mathbb Z\}$». Sep 28 comment Subgroup generated by $\langle a,b \rangle$ where $ab=ba$ «it suffices to shows that S is a subgroup of G» and that it contains both $a$ and $b$, actually.