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One of the most beautiful things in mathematics are theorems of conjectures that are very simple to state and understand (e.g. for K-12, lower undergrad levels), but extremely hard to prove.
This is in contrast to conjectures in certain areas where you'd have to study for a few months just to precisely understand all the definitions and the interest of the problem statement.
Bibliography:
- mathoverflow.net/questions/75698/examples-of-seemingly-elementary-problems-that-are-hard-to-solve
- www.reddit.com/r/mathematics/comments/klev7b/whats_your_favorite_easy_to_state_and_understand/
- mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts this one is for proofs for which simpler proofs exist
- math.stackexchange.com/questions/415365/it-looks-straightforward-but-actually-it-isnt this one is for "there is some reason it looks easy", whatever that means
Good place to hunt for the beauty of mathematics.
Main article: Section "Brain-computer interface".
This is one of the deep tech bets that Ciro Santilli would put his money in as of 2020.
How hard could it be? You just have to learn the encoding of the neural spine/eyes/ear, add an invasive device that multiplexes it, and then the benefits could be mind blowing.
Interestingly and obviously, the initial advances in the area are happening for people that have hearing or vision difficulties. Since they already have a deficient sense, you don't lose that much by a failed attempt.
Hearing is likely to be the first since it feels the simplest. Ciro heard there are even already clinical applications there. TODO source.
It boggles Ciro Santilli's mind that people use mailing list to collaborate on projects!
The only explanation is that the dinosaurs who created the projects are unable to adapt to new superior technologies.
Yes, Ciro is talking to you, big fundamental projects from last century: Linux kernel, GNU Compiler Collection (gcc.gnu.org/lists.html), Binutils (sourceware.org/binutils/), etc.
Some of you are already using Bugzilla for the bugs, so kudos. But if you've seen their benefit, why you still use the mailing list for patches?
Advantages of mailing lists:
- threaded replies, which almost no issue tracker has. GitHub feature request: github.com/isaacs/github/issues/837
Disadvantages: everything else:
- cannot subscribed to a single thread. Which forces you to create an email filter for each one of them you subscribe to.
- no metadata, notably the notion of closing / merging, but also upvotesYou have to read thirty messages before you can know if the bug was solved or not.
- it is insanely hard to reply to messages from before you were subscribed: webapps.stackexchange.com/questions/23197/reply-to-mailman-archived-message/115088#115088This forces everyone to subscribe to all lists, and then set up email filters to not be flooded with emails.
- hard to apply patches locally to test them out: stackoverflow.com/questions/5062389/how-to-use-git-am-to-apply-patches-from-email-messages/49082916#49082916Unless they use Patchwork, which adds one more website on top of the mess.And then Gmail corrupts your patches, and you are forced to use
git send-email, which does not work on some network configurations: stackoverflow.com/questions/28038662/how-to-solve-unable-to-initialize-smtp-properly-when-using-using-git-send-ema or setup ThunderBird. - often have to subscribe to post at all, thus cluttering your inbox further
- you can edit posts to make them clearer.Yes, people could vandalize their answers when they get mad, and threads might stop making sense after edits. But this can be solved with an undeletable post history like Stack Overflow has (but not any other tracker does).Or archive.org :-)In any case, what do you think will happen more often and have greater impact:
- people vandalize their posts
- people fix their silly typos and improve content
- searchable by author, keyword, etc. without Google. Yes, mailing list trackers could have decent implementations to overcome that. But no, GNU Mailman which everyone uses does not have it. Google barely indexes it.And I don't think Google properly indexes many of the mailing list archives for some reason: I never get hits for my own posts a week later, while I often do on GitHub issues.
- people have to learn about top posting vs inline posting, and this requires infinite education of new users
- Line comments in code reviews like GitHub and GitLab.On mailing lists: either put a comment in the middle of a huge patch and let other people find it, or (more likely) copy paste the part of the patch that you are talking about.
- most mail web UIs suck.OK, this is not an unsolvable or intrinsic problem, but still a problem.E.g.:
ezmlmit is not possible to see the entire content in a single page: gcc.gnu.org/ml/gcc/2015-07/threads.html.Unless you like reading threads backwards and with 4 levels of>quotations.The alternative: do like LLVM and send attachments. Yes, I we all love opening up attachments on our browsers.The real solution: everyone can create branches and pull requests. Also has the benefit of running CI on the pull requests.
Not sure:
- you can have infinitely many trackers to replicate data in case apocalypse happens in some part of the world.Although I'm not sure this is an advantage, as you don't know anymore which one is the canonical trackers an advantage, as you don't know anymore which one is the canonical tracker.And all web interfaces already have an API to export messages, and someone has already scripted it to import from any web UI to any web UI for you.And GitHub offers infinite precise history transparently on its API.
Multi-user:
They do have some really good ones.
It is interesting that in different episodes they often switch the dominant/passive roles, so it's not fixed as in Laurel and Hardy.
Are we the Baddies? by Mitchell and Webb
. Source. See also: cirosantilli.com/china-dictatorship/nazi.Discoverer by Mitchell and Webb
. Source. Makes fun of the many terrible naming choices British navigators have made while discovering/rediscovering new lands.Homeopathic A&E by Mitchell and Webb
. Source. Anti-Piracy Ad by The IT Crowd (2007)
Source. The fundamental concept of calculus!
The reason why the epsilon delta definition is so venerated is that it fits directly into well known methods of the formalization of mathematics, making the notion completely precise.
Generalize function to allow adding some useful things which people wanted to be classical functions but which are not,
It therefore requires you to redefine and reprove all of calculus.
For this reason, most people are tempted to assume that all the hand wavy intuitive arguments undergrad teachers give are true and just move on with life. And they generally are.
One notable example where distributions pop up are the eigenvectors of the position operator in quantum mechanics, which are given by Dirac delta functions, which is most commonly rigorously defined in terms of distribution.
Distributions are also defined in a way that allows you to do calculus on them. Notably, you can define a derivative, and the derivative of the Heaviside step function is the Dirac delta function.
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