Circular answer because the question asks why cos and sin are defined by the complex exponential in that way, so Euler's identity doesn't answer the question...
@RRL You voted to delete the question or answer? I don't have a problem with the question, though it's surely a duplicate. But it's the answer that cannot make it.
Besides, the so-called "simple manipulations" aren't even right.
@user21820: OK I see what you are saying -- although the question already has collected 5 downvotes. With question on hold I don't have a way to vote on answer deletion.
@RRL Actually that's not the reason. Answers with non-negative score cannot be deleted, and many upvoters carelessly upvote things that they think are good, so that's why we can't delete it.
@Holo You're not sure about (1), right? Well, it's using the fact that (a−b) divides a polynomial if the polynomial is zero when substituting a=b. But that's no easier than solving the original problem, and worse still the answer doesn't even make the use of said fact clear.
Actually it's interesting that the factor theorem is not enough to prove this instance of a generalization: If P(a,b,c,d,...) is a polynomial in arbitrary variables including a,b,c,d, and ( ab = cd implies P(a,b,c,d,...) = 0 ), then (ab−cd) is a (polynomial) factor of P(a,b,c,d,...).
The benefits of demanding justification for every technique used cannot be overstated:
By the way, that pdf links to this paper that explains how Cauchy made the same mistake of differentiating under the integral sign in some cases where it was invalid, leading to divergent integrals making their way into a 'respectable' table of convergent integrals! — user218201 min ago
Lol that arXiv paper has a curious id, but the previous one doesn't look quite as interesting (because I don't understand it).
4 hours later…
user131753
2:22 PM
Not sure whether I have mentioned this earlier in this room. But anyone willing to undelete this answer? @gimusi, @BillDubuque, @user334732, @AlexanderGruber or anyone else?
@AlexanderGruber It is total gibberish. It calls ZF "craps", and claims that "Russell is the only mathematician I ever see that actually explains mathematics in a precise formal way", and makes the nonsensical assertion "There are also another proves that go against Gödel's theorem, such as Tarski's". You can check with @AndrésE.Caicedo or @CarlMummert for corroboration.
It also claims Russell proved Principia consistent, which of course can only be circular or relative to some other system, contrary to the wishful thinking of a lot of Godel-deniers.
user131753
Admittedly there are some parts of the post that are objectionable which I, personally would like to be clear about. But I think that the post answers the question in the post. In other words, even though in its current form the question indeed is highly objectionable, I think that it can be edited to retain the essential information that indeed answers the question.
@user170039 Actually, it's not an option because you're not supposed to edit a post to make its meaning significantly different from the originally intended one.
@user170039 Most logicians will agree on the mathematical uselessness of that answer even with the wrong or offensive bits removed, based on our current understanding of logic. It is sheer cheek for Ricardo to call those logicians who do not think Principia Mathematica is worth studying today amateurs, given his own poor understanding of logic. Advice by arrogant ignorant people is harmful to sincere students of mathematics.
The author clearly wishes to disparage modern mathematics despite being a victim of the Dunning-Kruger effect in the field of logic. However much I do not like ZFC set theory, I think it is patently unfair to call it "craps".
user131753
5:34 PM
Oh. I see. So I have mentioned it earlier in this room. Sorry for mentioning it again.
@user21820 @user170039 if this is a frequently wrong opinion then perhaps there is a case to see it published and downvoted, and have comments attached stating what the errors and misconceptions are, else the frequent erroneous conclusions will quite reasonably be arrived at again by others and posted again as if it's the first time, only to require further moderation.
If it stays, it is there along with its critique for those who might think that way to see that others did the same and were corrected.
@MikeMiller "common" refers to @user21820 "contrary to the wishful thinking of a lot of Godel-deniers."
user131753
@user334732 To be fair, I think I agree with user21820 regarding this. It is really a bit too much to call $\sf{ZFC}$ "craps" and that too without any justification. But that being said I do think that the post indeed answers the question. Whether we can edit it and if we can whether we are willing to do it is a different issue of course.
@user334732 That would be like trying to move a boulder. They have made their decisions. And the "lot" here is less a corollary of the proportion of people who disbelieve Godel and more the massive number of people there are.
