Questions tagged [alternative-proof]
Looking for a proof different from the standard proof(s) of a result
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Proofs without words
Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I ...
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Quick proofs of hard theorems
Mathematics is rife with the fruit of abstraction. Many problems which first are solved via "direct" methods (long and difficult calculations, tricky estimates, and gritty technical theorems) later ...
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Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional
A very important theorem in linear algebra that is rarely taught is:
A vector space has the same dimension as its dual if and only if it is finite dimensional.
I have seen a total of one proof of ...
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What are some proofs of Godel's Theorem which are *essentially different* from the original proof?
I am looking for examples of proofs of Godel's (First) Incompleteness Theorem which are essentially different from (Rosser's improvement of) Godel's original proof.
This is partly inspired by ...
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Proofs of theorems that proved more or deeper results than what was first supposed or stated as the corresponding theorem
Recently, I figured out that a colleague of mine has had published during recent years a proof of a theorem in which he was actually proving a deeper result which we both thought to be still open. ...
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Examples of advance via good definitions
In my research I came across a case where I could derive a known theorem with rather straightforward way by choosing "non-standard" definitions using my knowledge from a related field. This particular ...
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Different way to view action of fundamental group on higher homotopy groups
There are a couple of ways to define an action of $\pi_1(X)$ on $\pi_n(X)$. When $n = 1$, there is the natural action via conjugation of loops. However, the picture seems to blur a bit when looking at ...
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Which theorems have Pythagoras' Theorem as a special case?
Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...
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Easy proof of the fact that isotropic spaces are Euclidean
Let $X$ be a finite-dimensional Banach space whose isometry group acts transitively on the set of lines (or, equivalently, on the unit sphere: for every two unit-norm vectors $x,y\in X$ there exist a ...
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Where to publish a new proof of an old theorem?
A few months ago I came up with a proof for an old theorem. After being excited for a moment, I then tried to find my proof in the literature. Since I did not find it, then I started to wonder if it ...
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Is Cauchy induction used for proofs other than for AM–GM?
The proof by Cauchy induction of the arithmetic/geometric-mean inequality is well known. I am looking for a further theorem whose proof is much neater by this method than otherwise.
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Noncombinatorial proofs of Ramsey's Theorem?
I know of 2(.5) proofs of Ramsey's theorem, which states (in its simplest form) that for all $k, l\in \mathbb{N}$ there exists an integer $R(k, l)$ with the following property: for any $n>R(k, l)$, ...
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Collecting alternative proofs for the oddity of Catalan
Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am looking for your help in my attempt to collect alternative proofs of the following fact: $C_n$ is odd if and ...
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Deuring's result on elliptic curves. Any proof reference
I have heard of this result from Deuring 1941 paper: Given $\mathbb F_p$ ($p$ prime number) and any number $n$ in the Hasse interval $[p+1-2\sqrt p, p+1+2\sqrt p]$ there is an elliptic curve over $\...
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Alternative proof of unique factorization for ideals in a Dedekind ring
I'm writing some commutative algebra notes, but I'm facing a difficulty in organizing the order of the topics. I'd like to have the topics about factorization before speaking of integral closure. This ...
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Has Goedel's Second Incompleteness Theorem been proven using Lawvere's Fixed Point Theorem?
This question is a request for assistance in surveying the existing literature on applications of Lawvere's Fixed Point Theorem (LFPT).
Yanofsky [0] has demonstrated several applications of LFPT to ...
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Link between the hairy ball theorem and the fundamental theorem of algebra
I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...
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Euler's rotation theorem revisited - Elementary geometric proofs
This is a very elementary topic but I thought it might be worth giving it a try here, I would be very interested in any comments - I originally posted it to Maths SE.
Euler's Rotation Theorem, proved ...