“ These geometrical concepts do not exist in nature. There are no lines and squares. If it's obvious then why did it take 4.5 billion years since the development of life to emerge?”
What makes you think lines and squares don’t exist in nature? And what on earth does that have to do with how long life took to emerge?!
Perfect lines and squares don’t exist as physical objects, sure, but geometry is less about material perfection than it’s about relationships. Nature constantly approximates geometric regularities because physics imposes them: energy minimization gives spheres, space-filling gives hexagons, angular momentum gives spirals.
Life didn’t need 4.5 billion years to “invent” geometry; geometry constrained life from the beginning. We only invented the formal language to describe it.
>> These geometrical concepts do not exist in nature. There are no lines and squares. If it's obvious then why did it take 4.5 billion years since the development of life to emerge?
> And what on earth does that have to do with how long life took to emerge?!
I think you misunderstood that part you quoted. He's not claiming that it had a causative effect on how long life took to develop, he's claiming that it took 4.5 billion years after life first appeared for those geometrical concepts to emerge.
It's the usual "until we prove animals do _X_ we can safely assert only humans do _X_" trope of biology.
As we learn that animals do things like have homosexual relationships, giggle when tickled, and understand basic rules of economics... biologists are learning to phrase it as "until we prove animals do _X_ we cannot be sure if animals do _X_", which is much safer.
(Also, there are trillions of lines in nature - WTF? Squares are somewhat rarer, except on the ground in wombat territory...)
Rooks don't have taxicab geometry. Their metric space is compact even on an infinite board. I think you're thinking of the wazir: https://en.wikipedia.org/wiki/Wazir_(chess)
I would've also appreciated a discussion of how intuition of geometry might apply to chess playing abilities and how it might not be sufficient for playing chess well.
As a side note, I appreciated the small typos as a further signal that this was written by a human.
For anyone actually interested in the question of measuring animal intelligence, I recommend the book 'Are we smart enough to know how smart animals are?' by frans de waal.
(And if you care about measuring artificial intelligence, you should definitely care about what we've learned from trying to measure animal intelligence...)
I thought this article was going to be about how chess at its core is a game about intersecting lines or crosses (+ or x). Also, there are really interesting ideas that could be explored around why a rook on a bare board always controls the same number of squares no matter where it is placed, but for other pieces like the bishop or the knight, they control more squares the closer they are to the center of the board.
If you watch any Hikaru Nakamura content, you will see him draw "classic right angle triangle"s with three pieces, "classic wooden shield"s (a cross showing the scope of a centralized bishop), so he definitely uses some kind of geometry while playing.
Not sure if he just recognizes the shapes as they appear or tries to make them appear, would be nice if he came here to answer.
False.
Crows for example understand geometry. I’d wager there are plenty more.
https://www.science.org/doi/10.1126/sciadv.adt3718
“ These geometrical concepts do not exist in nature. There are no lines and squares. If it's obvious then why did it take 4.5 billion years since the development of life to emerge?”
What makes you think lines and squares don’t exist in nature? And what on earth does that have to do with how long life took to emerge?!
Life didn’t need 4.5 billion years to “invent” geometry; geometry constrained life from the beginning. We only invented the formal language to describe it.
> And what on earth does that have to do with how long life took to emerge?!
I think you misunderstood that part you quoted. He's not claiming that it had a causative effect on how long life took to develop, he's claiming that it took 4.5 billion years after life first appeared for those geometrical concepts to emerge.
Here's Gromov, one of the greatest geometers of the last 50 years, discussing his viewpoint on this.
He also has this series of talks beginning with the question "What is probability, what is randomness?"
https://www.youtube.com/watch?v=aJAQVletzdY&list=PLx5f8IelFR...
I'd go even further and postulate that all intelligence is an understanding of geometry.
You can use things without understanding them. See people asking chatgpt to do sth for them.
As we learn that animals do things like have homosexual relationships, giggle when tickled, and understand basic rules of economics... biologists are learning to phrase it as "until we prove animals do _X_ we cannot be sure if animals do _X_", which is much safer.
(Also, there are trillions of lines in nature - WTF? Squares are somewhat rarer, except on the ground in wombat territory...)
Only a couple brief mentions about how chess piece moves are lines and transforms of lines. Other than that the author never establishes the title.
I was actually looking for some insight about chess and did not get any.
Kings have Chebysev geometry while Rooks have taxicab geometry: https://en.wikipedia.org/wiki/Taxicab_geometry#See_also
It's left as an exercise for the reader to figure out the geometry of the remaining pieces.
I would've also appreciated a discussion of how intuition of geometry might apply to chess playing abilities and how it might not be sufficient for playing chess well.
As a side note, I appreciated the small typos as a further signal that this was written by a human.
https://www.goodreads.com/book/show/30231743-are-we-smart-en...
(And if you care about measuring artificial intelligence, you should definitely care about what we've learned from trying to measure animal intelligence...)
And the wiki page is a little limited but this subject can get complex but still very cool.
I've beaten over 2500 ELO in Crazyhouse on Lichess (2518 to be exact). Currently rated around 1900.
Am I missing something?
Not sure if he just recognizes the shapes as they appear or tries to make them appear, would be nice if he came here to answer.
[0] https://www.chessprogramming.org/Chunking
[1] https://pmc.ncbi.nlm.nih.gov/articles/PMC4361603/
> Shapes are hypothesized to be formed by a programming language in the brain.
And what does this even mean? What does it mean for there to be a "programming language" in the brain?
I see what the author did there.
I've got a kid so "what the sigma" and "six seven" are a thing.
Type "six seven" in Google search and you should get the screen wobbling ; )