"Do we want to keep mathematics a dark secret, or what?" Apparently, yes, we do. Obviously, the definition of a topos is the same for all of us. But the way we use it, the way we attach it to other mathematical structures, is part of our own secrets. And there are apparently many such secrets. I might even believe that Grothendieck's own perception, of what toposes really are, changed over time, and indeed in dialogue with the scientific community, a community which is not a closed one, but very much part of society. Clearly, there may remain parts of "aus liebe zur Kunst" in topos theory, as for any part of mathematical theories for that matter, but generally speaking, there are always objectives, and requirements for theories to be applicable, applicability in a broader sense. Are there real-world applications of toposes? Journalists would love to know, I guess. Best, Patrik On 2024-09-02 08:32, Vaughan Pratt wrote:
"I'm relieved the journalist didn't try to explain what a topos was, or indeed anything mathematical."
Why would anyone object to journalists doing exactly those things? Do we want to keep mathematics a dark secret, or what?
A topos is simply one of many possible generalization of sets and their functions that allows many other mathematical objects besides sets to be imbued with some of the essential properties that make sets so valuable in mathematics.
For example graphs and their maps form a topos with very similar properties to sets and their functions, such as having the notion of a power set. But not all properties, for example the law of the excluded middle, which holds for sets but not graphs.
Vaughan Pratt
On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com> wrote:
Thanks - I saw this! I'm relieved the journalist didn't try to explain what a topos was, or indeed anything mathematical.
Sent
On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu> wrote:
An article about Alexander Grothiendieck has just appeared in the Guardian online newspaper. Be warned, it contains some seriously weird stuff! Toposes get a mention, though "not as we know them", along with Huawei, AI and Olivia Caramello. Beyond that, I'm not going to comment!
Since Microsoft mangles web addresses, here is the address with the punctuation removed:
www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence
Paul Taylor.
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