With that being said, trying to think a different way for the challenge of it is definitely interesting. Reading through some of the other comments here and trying to taste words or replicate other people's minds is a weird, fun exercise :)
https://sites.google.com/site/steveyegge2/math-every-day
https://www.cantorsparadise.com/the-unparalleled-genius-of-j...
I would argue his way of thinking of numbers makes him slower at doing calculations.
When you create a 2D visual representation of a number system you want to choose a shape that has the same properties as numbers. Namely the shape must be monoidal under composition. This allows you to keep one type of shape
For example (int + int = int). When you compose two triangles together you get a parallelogram, so triangles are actually kind of bad as you would need to classify several different types as numbers. (triangle + triangle = parallelogram) The only shape that I can think of that is monoidal under arithmetic composition is rectangular quadrilaterals with at least two parallel sides.
Examples: Rectangles, parallelograms, and trapezoids each can be composed to form another shape in its own class. With rectangles likely being the most efficient representation as they are fully symmetrical (to compose two trapezoids to form a new trapezoid one trapezoid has to be inverted, this does not happen with rectangles).
So his even number visual representation is quite good (it uses blocks) but his odd number representation is all over the place and seems arbitrary. Just look at 9. It involves "orange peeling" another number just to shove it into the little dent. His system involves mutating, rotating and changing the shape of each "number" in order to perform composition. This costs more "brainpower" to do and is the main reason why I don't classify his ability as a "gift".
It's highly inefficient. I think many HNers are mistaking it for a super human ability. I don't agree. This is more of an interesting ability then it is a talent.
But that's just a guess. Would actually like to see a quantitative measure of how fast he is at adding numbers under his system. This would definitively answer the question.
Certain calculations are actually faster because i begin to have faith in my feeling of the math over doing an actual calculation - with the same type of confidence i have when recalling a times table for example. Still, it usually doesnt get me all the way to an answer
There are certain mathematical rules that you can probably identify that are related to my internal expressions and how they "fit" together. For example, I do not know without calculating what "25 x 15" is, but I have an idea of what the answer feels like. anything below 100 or over 1000 feels outright OCD level out-of-place. Numbers like 114, 201, etc, feel dirty and incomplete. we can identify in this scenario that the shape / feeling of the answer for me is related to an intuition for the mathematical principle that the product of two numbers that are divisible by 5 is also divisible by 5 - but at no point did I deliberately evoke that rule when conceiving of a possible answer. Also this is a simple example, this intuition runs beyond my knowledge and ability to formally explain the principles. In reality, many such principles (learned or inferred) come together at once to feed my internal expression of the answer. A calculator says 375 is the answer, though 325 and 475 feel about the same
I do not think it makes me better at getting correct answers, but it does help me accept an answer as being correct when looking at it also feels right. It's most useful when identifying errors. There is a big help when you see "15 x 25 = 356" and without thinking you can feel internally like something is out of place, dirty, needs attention (this applies to advanced topics as well). As I said above though, more than the correct answer can have the same or similar feeling - so it is prone to false negatives
With something like math, intuition based guess work that has room for false negatives is hardly that useful overall. So maybe the only real edge it can provide is in working with novel concepts where you have to guess a direction to explore and hope you uncover something useful. That is an unfounded hypothesis though.
I have a similar impression when reading posts elsewhere about categorical structures in programming: they are repetitive and mostly trivial (actually, the category theory without context is trivial).
For example, a string concat can be understood as an addition operation:
1 + 0 = 1 (identity)
1 + 1 = 2
1 + 2 = 3
2 + 1 = 3 (communitive)
"a" + "" = "a" (identity)
"a" + "a" = "aa"
"a" + "b" = "ab"
"b" + "a" = "ba" (non-communitive)
There's this whole intuition about addition itself that can be applied to something other than integers, and being able to reason about that is applicable to how you design software, particularly function interfaces.
Just as a note, my mother made me memorize the multiplication table when I was a kid, and I had ended up memorizing additions just through sufficient practice. I was able to intuit what additions and multiplications meant, but for the purpose of taking tests in school or doing homework, additions just pop out as answers because of the memorization. It wasn't until much later in life that I started encountering ideas such as, what if you were adding something other than numbers.
It's hard to imagine having only one instinctive visualization for integers.
Q: What's purple and commutes?
A: An abelian grape.
First number is 1, so its table of 1. Then x as multiplier sign. Then a count from 1 to 10. Then = sign. Then the result. We kids are supposed to write each line in left to right direction, then move to next line.
We use paper with square tables or graph on it. Most of the time, kids simply write 1, move to next line, again write 1, all the way till 10th line. Then we move to next column, write x, then move up, x all the way till 1st line. Then 1,2,3, in next column, = in next column coming up. Then the answers going down.
edit: But actually after that I used something called "Math-U-See", which used physical blocks to develop intuition. That was pretty cool.
OP does what they do in number blocks.
I guess with enough practice they are both fine for solving known problems. I think our way is better for programming, and his way is "better" for physical building.
I also think there’s no need for people to feel like they need to be some math or grammar prodigy to get by in life. It’s perfectly fine to outsource your mental functions, including memory to a calculator, notebook or PKM system like Obsidian.
I am a successful software developer and I’m terrible at math. To me, 6+3 is not an interaction between two different anything, rather, it’s a key in a hash table where I’ve stored “9” as the value. All arithmetic is rote memory recall for me. I work with complex numbers by just breaking them down into multiple steps.
Now I’m wondering if I should challenge my brain to do this differently.