foundations of mathematics – Quanta Magazine https://www.quantamagazine.org Illuminating science Mon, 16 Sep 2024 14:10:47 -0400 en-US hourly 1 https://wordpress.org/?v=6.2 How the Square Root of 2 Became a Number https://www.quantamagazine.org/how-the-square-root-of-2-became-a-number-20240621/ https://www.quantamagazine.org/how-the-square-root-of-2-became-a-number-20240621/#respond Fri, 21 Jun 2024 13:50:15 +0000 https://www.quantamagazine.org/?p=138611 The post How the Square Root of 2 Became a Number first appeared on Quanta Magazine

]]> The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them — fractions, or what we now call rational numbers. But this aspiration was undermined when they considered a square with sides of length 1, only to find that the length of its diagonal couldn’t possibly be written as a fraction. The first proof of this...

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]]> https://www.quantamagazine.org/how-the-square-root-of-2-became-a-number-20240621/feed/ 0 The Year in Math https://www.quantamagazine.org/the-biggest-discoveries-in-math-in-2023-20231222/ https://www.quantamagazine.org/the-biggest-discoveries-in-math-in-2023-20231222/#respond Sat, 23 Dec 2023 01:33:58 +0000 https://www.quantamagazine.org/?p=132809 The post The Year in Math first appeared on Quanta Magazine

]]> Mathematical truths are often born of the conflict between order and disorder. Mathematicians discover patterns, and, to better understand the mysterious forces at play, they look for countervailing impulses that disrupt those patterns. That tension came up repeatedly in our coverage this past year. We covered breakthroughs in graph theory, combinatorics, number theory and geometry — areas where...

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]]> https://www.quantamagazine.org/the-biggest-discoveries-in-math-in-2023-20231222/feed/ 0 The Deep Link Equating Math Proofs and Computer Programs https://www.quantamagazine.org/the-deep-link-equating-math-proofs-and-computer-programs-20231011/ https://www.quantamagazine.org/the-deep-link-equating-math-proofs-and-computer-programs-20231011/#respond Wed, 11 Oct 2023 14:30:52 +0000 https://www.quantamagazine.org/?p=131227 The post The Deep Link Equating Math Proofs and Computer Programs first appeared on Quanta Magazine

]]> Some scientific discoveries matter because they reveal something new — the double helical structure of DNA, for example, or the existence of black holes. However, some revelations are profound because they show that two old concepts, once thought distinct, are in fact the same. Take James Clerk Maxwell’s equations showing that electricity and magnetism are two aspects of a single phenomenon...

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]]> https://www.quantamagazine.org/the-deep-link-equating-math-proofs-and-computer-programs-20231011/feed/ 0 Alan Turing and the Power of Negative Thinking https://www.quantamagazine.org/alan-turing-and-the-power-of-negative-thinking-20230905/ https://www.quantamagazine.org/alan-turing-and-the-power-of-negative-thinking-20230905/#respond Tue, 05 Sep 2023 14:44:25 +0000 https://www.quantamagazine.org/?p=129730 The post Alan Turing and the Power of Negative Thinking first appeared on Quanta Magazine

]]> Algorithms have become ubiquitous. They optimize our commutes, process payments and coordinate the flow of internet traffic. It seems that for every problem that can be articulated in precise mathematical terms, there’s an algorithm that can solve it, at least in principle. But that’s not the case — some seemingly simple problems can never be solved algorithmically.

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]]> https://www.quantamagazine.org/alan-turing-and-the-power-of-negative-thinking-20230905/feed/ 0 Why Mathematical Proof Is a Social Compact https://www.quantamagazine.org/why-mathematical-proof-is-a-social-compact-20230831/ https://www.quantamagazine.org/why-mathematical-proof-is-a-social-compact-20230831/#respond Thu, 31 Aug 2023 14:04:58 +0000 https://www.quantamagazine.org/?p=129663 The post Why Mathematical Proof Is a Social Compact first appeared on Quanta Magazine

]]> In 2012, the mathematician Shinichi Mochizuki claimed he had solved the abc conjecture, a major open question in number theory about the relationship between addition and multiplication. There was just one problem: His proof, which was more than 500 pages long, was completely impenetrable. It relied on a snarl of new definitions, notation, and theories that nearly all mathematicians found...

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