I went to an in-person Q&A featuring a Fields medalist. The audience was a collection of undergraduate and high school math students, with a few professors in attendance.
One of the young students asked exactly this question to which everyone in the audience collectively groaned. The Fields medalist gave a short answer, something along the lines of "I don't know a single mathematician that thinks it's invented."
He was being polite, but you could tell he didn't think there was anything else interesting to say.
It's both. The axioms are invented, the corpus of theorems is discovered. As once the axioms are chosen the provable theorems are already fixed.
But the axioms are a choice, and we can pick different ones. The common choice of axioms is utilitarian, they lead to interesting math that helps us describe the universe.
Proving a theorem given a set of axioms is a search problem. Given a set of axioms you can apply rules of inference to generate the graph of all provable theorems. Proving a theorem is about finding a path from the axioms to the vertex which is your theorem.
But you can make the same case for axioms - that they are not invented but discovered through a process of search in the space of axioms.
I'm not sure I see why the axioms were not also discovered though? Choice between irreducible assumptions does not seem to make them any more 'invented'.
Without entering into an endless debate about semantics and metaphysics I would simply say that if you want to use the word discovered for the axioms then you must acknowledge that the theorems are not the same kind of discovered.
Are the theories beyond axiom fundamentally different if axioms are changed, though? And if not, aren't then axioms merely props or placeholders for invariants?
I think it's one of the those things where it doesn't matter what the answer is because it doesn't provide a useful lens for advancing your mathematical thinking.
This is a really bad article. There is a nugget of truth, but it's never properly articulated because the author himself doesn't actually understand the distinction.
If you've already decided you want to build a unicorn and are willing to make take that <1% chance of success, YC will help you. If you want a lifestyle business, you could probably skip YC (though you probably wouldn't get accepted even if you applied with that idea anyways).
this type of discourse is very common in physics, where the theoretical physicists propose a testable hypothesis and then the experimentalists verify or disprove it.
This might be a dumb question, but would you use the deployed version of the admin interface for writing your blog? And what if you weren't online but wanted to work on a new essay or make edits to an existing post?
Well, taoofmac.com has 20+ years, 9000-odd pieces of content (that translate into 47.000 blob storage items), multiple navigation constructs (archives, backlinks - it’s a wiki - and even a 3D sitemap), and an incremental build for a post that links to 5-6 others (and resizes images, updates the home page, archives, linked page footers and backlinks) takes ~10s, including uploading the results (and an updated SQLite database) to Azure.
I use SQLite FTS for full-text search (it’s the only non-static endpoint).
A full site re-render on a Raspberry Pi takes 5 minutes, and a full reindexing - FTS plus linkmap - plus publishing around 10, but I only do that yearly or when I update things like CSS, layout, etc.
It all runs off a Git hook, uses SQLite to hold all the FTS, base HTML and metadata, and is as asynchronous as can be (including my own asyncio blob upload library). Costs me effectively zero.
I want to be able to hit edit on the post, do it right there, save and see my change immediately. I could set that up with a static site system but.... the admin I made for editing the post would remain the same. Furthermore the editor I have has a bind for CTRL+V that checks for images and automatically uploads them but also inserts an html tag for the media I just uploaded. This is something that is a terrible experience when editing a git repo having to manually link images and other media.
This doesn't provide a counterpoint to the original comment's point.
IMHO, it's true that a static site generator should be the way to go in 2023, instead of using a web framework with a db.
I am personally running a hugo blog, on netlify, with netlify CMS. I have 0 costs, great performances, everything needed out of the box. What else to ask for ?
In my blog, comments are not supported, and it would be easy to add if I developed my own blog with Django. But I feel that this single feature does not overweight the simplicity and velocity I have with such setup.
In regards to the requirement to know Git: It is just necessary in the setup phase, the blog posts creation and edition are controlled through Netlify CMS for me.
