Atiscant

Seems somewhat related to https://en.wikipedia.org/wiki/Appropriate_technology

I mean, even if could produce generic metal would it produce Igorrr? Meshugga? Tim Henson? Baby Metal? All of these are driven by other things then just producing metal. I agree pure AI music would properly rejected unless there was some point to it. I could see it have some part, but then as a weird instrument. Take a model for music, randomly mutate internal weights and then let it produce a drum beat. Keep doing that unless you hit some limit and perhaps that is interesting.

It is kind of ironic that the AI building tool is so hostile to AI. Copilot studio really is a hot mess, at least for me.

My wife and I are very different people. She is intuitive, involved in people, in the moments. I am slow, considering, post-hoc. We converge on Pratchett. We both read all the books, and been through all audiobooks (both new anf old). It’s become shared metaphors and a common frame of reference. We both have have a tendency to return to one or another of the series once or twice a year. I would have loved more, but I am deeply grateful for what we got.

I had a similar experience explaining logic, especially nested expressions, with cats and boxes. Also for showing syntactic versus semantic. We _can_ use cats if we wanted and retain the semantics. Also my proudest moment as a teacher was students producing a meme based on some of the discrete mathematics on graphs. They understood the point well enough to make a joke of it.

(Same reply as to another comment in this thread)

In Denmark the official identification app does basically this. When you to officially verify yourself for e.g. the bank, government sites or whatever you type a “username” (identity string that officially should not be linkable to you but in practice often is). The site then displays a QR code that you scan with phone and then approve with a slider. It is not perfect but it is fairly easy for everybody.

In Denmark the official identification app does basically this. When you to officially verify yourself for e.g. the bank, government sites or whatever you type a “username” (identity string that officially should not be linkable to you but in practice often is). The site then displays a QR code that you scan with phone and then approve with a slider. It is not perfect but it is fairly easy for everybody.

Would you mind adding some details about how this is actually setup?

It is interesting enough, but the report kind of feels very AI generated and generic. Most of the questions present the choices in a good vs bad way, i.e it sounds bad saying I disagree and sounds good when I say I agree. Other test usually have postive versions of both ends of the spectrum which is missing here. I also agree that there needs to be some validation of why these dimensions, how the correlated internally etc.

Great toy. https://michae2.github.io/c-turtl/?dna=cfllfbpfrbcfbbp&scale...

For most of my computer science PhD the “trick” was just to get the inductive definition to work, and then how to tweak it for the next paper. Or, get enough structuret we can do an “abstract nonsense” proof[0].

[0] https://ncatlab.org/nlab/show/category+theory#AbstractNonsen...

As noted in another reply, the natural numbers example is contrived, but illustrative. Nevertheless, if you have a set theoretical foundation, e.g. ZF/C, at some point you need to define what you are doing in that foundation. Most mathematicians do not and happily ignore the problem. That works until it dont. The whole reason Vladimir Voevodsky started work on HoTT and univalent foundations was that he believed we in fact DO need to be able to pull definitions back to foundations and to verify mathematics all the way down.

Sure you can work around it most of the time, but some times you cant. The whole point is that isomorphic is not equality in set theory, and sometimes proofs does not transfer along isomorphism because they refer to implementations. I agree that it is much preferable to work with abstract structure, but that not always what happens in practice. The natural number example is contrived but easy to see. My point of view is also that I do not like the Lean approach. It would actually like no junk theorems to exist in my theory. I am much more partial to the univalent approach and in particular univalent implementation that compute e.g. cubical. Regarding how easy it is to formalize, you are right. Lots of good work happens with set theory based type theory. My point was also that set theoretic foundations themselves are very hard to formalize, e.g ZFC + logic is very difficult to work from. A pure type theoretical foundations is much easier to get of the ground from. To prove that plus commutes directly from ZFC is a nightmare.

As one of those that do not like the “sets at the bottom” approach I just want to highlight why. For me, mathematics built on sets have leaky abstractions. Say I want natural numbers, I need to choose a concrete implementation in set theory e.g. Von Neumann, but there are multiple choices. For all good definitions, so get Peano arithmetic and can work with, but the question “Is 1 and member of 3” depends on your chosen implementation. Even though it is a weird question, it is valid and not isomorphic under implementations. That is problematic, since it is hidden in how we do mathematics mostly. Secondly, it is hard to formalize, and I think mathematics desperately needs to be formalized. Finally, I do not mind sets, they are great, and a very useful tool, I just do not like they as the foundation. I firmly believe we should teach type theoretic or categorical foundations in mathematics and be less dependent on sets.

Absolutely great. Thank you for sharing.

A point that is maybe not obvious to people who have not done mathematics at a high level or done “new” mathematics, is that often you end of changing your theorem or at least lemmas and definitions while figuring out the proof. That is, you have something you want to prove, but maybe it is easier to proving something more general or maybe your definitions need to change slightly. Anecdotally, during a project I spend perhaps a year figuring out exactly the right definition for a problem to be able to prove it. Of course, this was a very new thing. For well-know areas it is often straight forward, but at the frontier, both definitions and theorems often change as your proceed and understand the problem better.

A dissenting voice that might of interest here is Peter Zeihan. He claims that the US is one of the few places that won’t collapse in the coming decade. I do claim here is right, but his analysis is, if nothing else, interesting. His book, The End of the World Is Just the Beginning, is an easy enough read.

The US. I am from one of the Scandinavian countries with small kids, and trying to align work to move, at least temporarily, in a couple of years when they are able to handle them self in English. My wife was an exchange student many years ago and kept connection with the host family. We’ve visited multiple times. The company I work for has a US branch. It’s the only other country I've ever wanted to live in. Visited many others. It seems to me to be the most “alive” and dynamic western country, even with faults. Scandinavia and western Europe is stagnant, regulated to death and slumbering. It might change, I hope it will, but living here, not on the horizon yet.

Biochar is exactly doing that and is an active area of research in many places. There is several ongoing projects also showing that biochar can improve soil quality and crop yields.

I work with technology and research management at a multi national industrial company. This includes strategy and roadmapping for specific products, but also broadly following research trends and networking with universities. I did a phd in theoretical computer science and stumbled into by accident. It is interesting following research but also see it applied and working with business from the ground up. It is a lot of stakeholder management and knowledge dissemination which I enjoy. Lots of internal networking and relations and informal power. Besides job, I am on the board a handful of volunteer organizations or non-profits. Last year I also taught a semester which I missed a lot from academia. I need many different things to keep my interest so the job works well for me.

As mentioned by another comment, this is a big reason that Vladimir Voevodsky started his Homotopy Type Theory and Univalent Foundations program. He had see first hand a field collapse by a mistake in “first lemma on the first page” of a foundational paper. Arguably, he initial work on UniMath and the special year at IAS ending up in the HoTT book, pushed the whole formalization of mathematics topic forward to where it is today.

Or for the mathematically inclined: How many n x n puzzles with unique solutions exists for a given size n?

n=1 is trivial, and n=2 it small enough to enumerate with 3^4 = 81 solutions, but many of them being degenerate (no solutions), but already n=3 is pretty bad with ~20.000 possible puzzles. I do not see an obvious path to compose solutions either and make use of some kind of structural induction.

At least it is possible to force a single solution (discounting backtraces which is always possible) in 4x4:

  | 2 | 3 | 3 | 3 | 
  | 3 | 3 | 3 | 3 | 
  | 3 | 3 | 3 | 1 | 
  | 3 | 3 | 3 | G | 

With arbitrary generation rules they are surely not. This is a counter example on 4x4:

  | 1 | 1 | 1 | 1 |
  | 1 | 3 | 3 | 3 |
  | 1 | 3 | 2 | 2 | 
  | 1 | 3 | 2 | G |

  | 2 | 2 | 2 | 2 |
  | 2 | 2 | 2 | 1 |
  | 2 | 2 | 2 | 2 | 
  | 2 | 1 | 2 | G |

Edit: formatting

Thanks a lot!

Would you be willing to share your config for this setup?

Learning to write proofs is really learning a new kind of thinking, and that is not easy to do alone. For sure you can learn from books or other resources alone, but as with everything, when you are stuck in some way, good guidance can be very beneficial. I’ve thought proofs and logic at multiple levels and even though people struggle with similar issues, there is almost never a one-size-fits-all way to help an individual progress. It all depends on your context, way of learning, temperament etc etc.

I suggest you start out with a course or book on formal logic in computer science. In particular doing proofs in intuitionistic logic. You can use a simple proof assistant to verify your proofs and it will help you really understand how proofs are done and how they are just a kind of programs in a special programming language called mathematics (given some assumptions). Proofs in higher level books are all informal, so you can’t do them programmatically in most cases, but it really helps with understanding how proofs works. After that, follow what subjects finds your fancy.

I don’t think there is a definitive book which represent exactly what I have in mind. It would depend a lot on your background. If you have a mathematical background with some formal logic, you could read the introduction to the Homotopy Type Theory Book, but it is aiming at something different. This video and accompanying paper could also help https://math.andrej.com/2016/10/10/five-stages-of-accepting-.... If you have no logic background, it would require a bit more.

What is your background and interest? I could perhaps help if you want to learn something like this

(Constructive) Mathematics from the group up using intuitionism, formal logic, Curry-Howard and Type theory.