I don't see his way of viewing numbers as particularly efficient. It's very inn-efficient. It's an anomaly for sure but I would hesitate to call it a talent or super human ability.
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.
I relate to the OP on a fundamental level although the literal expression would be different for me. I do not think it has any relation to speed. It is not a deliberate step. It would be slower to mimic this behavior, but if you have it by default it's just kind of there.
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 fairly agree with this. I still wonder how the author visualizes irrational numbers, exponential functions, etc. and more importantly, proves some (even simple) theorems with this kind of visualization.
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).
You might be right about this particular version of synesthesia not being too useful but I have music->visual (shape, color, texture, distance, location) synesthesia that I can turn on and off (only when smoking even small amounts of pot) and itβs a huge advantage when trying to do anything music-related.
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.