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ianwrightsite.wordpress.com | ||
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www.quantamagazine.org
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| | | | | Number theorist Andrew Granville on what mathematics really is - and why objectivity is never quite within reach. | |
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njwildberger.com
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| | | | | There are several approaches to the modern theory of "real numbers". Unfortunately, none of them makes complete sense. One hundred years ago, there was vigorous discussion about the ambiguities with them and Cantor's theory of "infinite sets". As time went by, the debate subsided but the difficulties didn't really go away. A largely unquestioning uniformity... | |
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daniellefong.com
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| | | | | The following occurred to me on a run about two years ago: It's not given much press, but the the Halting Problem is intimately related to Gödel's First Incompleteness Theorem. Indeed it produces it as a correllary. Historically, Gödel's incompleteness results were proved by hacking arithmetic into a Turing complete system, and this is still... | |
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algorithmsoup.wordpress.com
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| | | The ``probabilistic method'' is the art of applying probabilistic thinking to non-probabilistic problems. Applications of the probabilistic method often feel like magic. Here is my favorite example: Theorem (Erdös, 1965). Call a set $latex {X}&fg=000000$ sum-free if for all $latex {a, b \in X}&fg=000000$, we have $latex {a + b \not\in X}&fg=000000$. For any finite... | ||
