HN Theater @HNTheaterMonth

My example was so simple that it would be taken as obvious for an experienced reader. With any nontrivial problem, the proof would have much more information than the construction. For example, the proof might require the construction of auxiliary objects not used in the main construction.

This is beautifully showcased by a 3Blue1Brown video about an extremely clever proof of the equivalence of three constructions of an ellipse: https://www.youtube.com/watch?v=pQa_tWZmlGs.

⬐ b33j0r

I’m not Godel, but my ad absurdum was meant to be slightly weirder than the objection that seems to have come across. I’ll try this way.

I had a flash of a question about whether the external information required in either case offsets the information provided by the objects in the proof (implicitly, explicitly, or artificially). I was thinking in terms of both formal logic and entropy, loosely.

The (usefully simple) construction here implies use of a compass or string. In a sense, the physical constraints of a compass encode the same information as the lemmas and theorems do abstractly.

“Brah, you should google metaphysics and Bertrand Russell,” is probably about right But, I’m sure there is a term that I just don’t know or can’t recall.

⬐ anderskaseorg

Perhaps an answer to one version of your question is that the Tarski–Seidenberg theorem implies that Euclidean geometry is decidable: there exists an algorithm that finds a proof for any theorem of Euclidean geometry. This algorithm, however, is too slow to be practical in general (double exponential time). The proofs it finds definitely don’t correspond one-to-one with the constructions in any reasonable sense.

The compass encodes a constant-radius constraint, and the string encodes a sum-of-distances constraint, but it’s not at all obvious why these two constraints turn out to be the same under a uniform stretching. There are plenty of similar-looking hypotheses that turn out to be false (for example, a curve of constant offset to an ellipse looks a lot like an ellipse but isn’t one).

One of the most incredible feelings is when you make that new connection of understanding on an idea. Sometimes it's when two things you knew get connected in a way you didn't think existed, and sometimes it's when a complicated idea all fits into place in your mind. It's probably the drug that keeps people programming despite all the configuration hell we have to deal with on any non-trivial project (and even most trivial ones).

Every single one of 3Blue1Brown's has given me a big hit of that new brain connection drug. If you enjoy this video, I recommend checking out 3Blue1Brown's video "What does genius look like in math? Where does it come from? (Dandelin spheres)", which deals with how and why ellipses and conic sections are related. https://www.youtube.com/watch?v=pQa_tWZmlGs

⬐ wruza

Moreover, I would recommend to watch most of their videos, at least to myself. No one could introduce me to so many ideas in so little hours without leaving any open question. This channel and its companions are fantastic if you somehow missed deep math sense at school. It is not “lets learn another seemingly useless theorem” study, it is kickstart introduction to selected interesting topics.

⬐ FiveDegrees

You may enjoy the game The Witness. The entire game is basically built around inducing that feeling.

⬐ etherealG

Seconding that. This was one of my favourite games ever because of how well it manages to achcieve that feeling.

⬐ laythea

I agree. This guy is the best math teacher I've heard.

⬐ knrz

I love the feeling you're describing! The understanding of an abstraction (or in other words, a mental model) that links two parallel thought processes in an unexpected, and fun way.

I like the term Kensho [0], thought you might too. Interested in thoughts from other people as well.

[0]: https://www.lesswrong.com/posts/tMhEv28KJYWsu6Wdo/kensho

⬐ Jarwain

I like the concept of kensho. I find the idea that certain ideas can only be accurately conveyed through experience to be interesting as well.

I'm working my way through a book called "The Book Of Secrets", which has 114 different meditative techniques for different kinds of minds. One of which, is practically certain to work for any given individual.

⬐ ylbss

3Blue1Brown saved me when I started university comp sci math 10 years after highschool. One of my favorite youtube channels. His Fourier Transform videos are the best explanations I've seen.

⬐ eboyjr

Sidenote: 3Brown1Blue has the most intuitive explanations for linear algebra and matrix operations I have found thus far. Check his playlists series.

⬐ phkahler

I don't watch YouTube much but I have see a few of these videos. This one was recommended to me last night and I watched it. Now it's here on HN the next morning. This is certainly not a coincidence.

Favorite quote "You can often view glimpses of ingeniousness... not as inexplicable miracles, but as the residue of experience." Did he pen that one or borrow it from someone else?

⬐ vole

vorite quote "You can often view glimpses of ingeniousness... not as inexplicable miracles, but as the residue of experience." Did he pen that one or borrow it from someone else?

Although I can't say it hasn't been said before, that's a common theme in his videos.

⬐ JadeNB

Google doesn't recognise "residue of experience" as part of a familiar quote, but the phrase itself seems to be rather frequent, at least in academese: https://www.google.com/search?q=%22residue+of+experience%22.

⬐ amch

Funnily enough, as of this writing, the top ranked google result (at least for me) is this post.

⬐ curiousgal

I'm so meta even this acronym

⬐ techbio

While Redditisms leaking into HN comments are usually downvoted, I actually got this for the first time just now. Probably because while on HN I wear my thinking cap, and on Reddit I read mainly for amusement and don't spend the time to look as closely. So a qualified thank you, that was clever.

⬐ sp332

It was popularized by Douglas Hofstadter, the guy who wrote Gödel, Escher, Bach among other things. https://en.wikipedia.org/wiki/Douglas_Hofstadter

⬐ trukterious

'Reside of experience'. Yes, more especially the residue of imaginative experience.

Daydreaming can be dysfunctional & defensive, as in:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5962718/

Yet some people are free in their thoughts at least some of the time and thus are able to daydream productively.

(One difference I think is that they are obsessed with problems).

⬐ egonschiele

3Blue1Brown is amazing -- one of the best math explainers I have seen. He also had a good video on other math channels he likes (https://www.youtube.com/watch?v=VcgJro0sTiM). It's amazing to see how things have changed over the last couple of years. Earlier, searching for math explanations on YT, I felt like I mostly saw hard-to-follow lectures. Now there's tons of content created specifically for YT and it is really well done.