[medvezhenok]

What is the color vision equivalent of a savant? Surely you don't believe that any person could (given infinite time & training from birth), match the intellectual performance of a savant on, say, multiplying 6 digit numbers in their head?

(I think a clear savant that has a untouchable ability in one small dimension of mathematics is a clearer example than, say, von Neumann - who was equally brilliant but across many domains and in a less obvious way)

[tptacek]

There is certainly variation in innate intellectual capability! There just isn't strong evidence of groupwide variation. Groupwide variation is the point of the "island" story.

[wizzwizz4]

> multiplying 6 digit numbers in their head?

This kind of thing is quite easy. Which mental delay-line caches you exploit depends on how your mind works, but multiplying 6-digit numbers in your head isn't a hard trick to learn, if you care.

There was a time when I cared about such parlour tricks, and I could pick them up quite quickly. I once spent a few days memorising the first 10 digits of pi. Once I'd figured that out, it was the effort of a few hours over the subsequent weeks to memorise to 36 digits. If I had cared, I could have learned twenty new digits a day, using the following scheme:

     3.14159
    265358979
     32384626
     43383
  2795028841
    9716
        93993
     75105820
     97494459
     23078164
     06286208
     99862803
     4825342
    117067
          982
     14808
          65132
       823066
     4...

See how it's lyrical? Just learn the poem. Except… I quickly found I didn't care, and at that point my motivation vanished, and I lost the "savant" ability. (Sure, if I wanted to, I could easily bootstrap the requisite intrinsic motivation – and I suspect I could learn a hundred digits a day thereby – but I don't want to.)

Despite my generally-absent enthusiasm, I'm still capable of aceing IQ tests, scoring highly in measures of cognitive ability that I do not possess, etc, because I have a certain stubbornness towards the idea that these tests truly measure anything important, which means I approach them sideways with the objective of breaking the tests, which means I break the tests. If anything of value hinged on my ability to quickly multiply 6-digit numbers in my head, I expect I could pick it up in… six months?

I do not say these things to brag: rather, the opposite. I don't think I am particularly exceptional: I never learned a thousand digits of pi, would probably take an hour to multiply 6-digit numbers in my head…

I am able to solve problems I've never encountered with computer systems I've never used, after half a second of thought, while concentrating on other things – but from the inside that doesn't feel exceptional at all: it's just a few tricks, well-practised. People who have memorised millions of digits of pi likewise claim to use a few tricks – and while those particular tricks don't always work for everyone, I don't think these people are innately exceptional.

[medvezhenok]

I understand the argument, but I think you're missing the nuance somewhat. There are a series of things that are learnable mental tricks; I have read Moonwalking with Einstein and am well aware about rhyme techniques, memory palace techniques, etc. I memorized ~250 digits of pi in the 6th grade, so I'm also aware of techniques for that. I wouldn't consider either of those a domain of savants.

(sidenote - I would be impressed with the people that could memorize millions of digits of Pi, given that the world record is either 70,000 digits or ~110,000 digits last time I checked (depends on the source), and it takes ~6 hours just to recite that many digits)

I'm talking specifically like things like Hypercalculia: https://en.wikipedia.org/wiki/Hypercalculia , which are documented feats that cannot be explained by "tricks". Usually people with savant syndrome also have co-occurring autism and other neurological conditions like synesthesia.

Here is a an ABC profile of one of these savants: https://abcnews.go.com/2020/autistic-savant-daniel-tammet-so...

I don't think you could learn a "trick" to compute 27^7 in a few seconds.

[wizzwizz4]

Funny you should give 27^7 as an example, because I actually did get good at powers of 2 and 3. 27^7 is 3^21 is 3×81^5, which is easy to calculate in your head if you're good at multiples and powers of 8: it's just binomial expansion with the next row after 1 4 6 4 1, i.e. 1 5 10 10 5 1. (I used to be able to directly recall 1 5 10 10 5 1 and 1 6 15 20 15 6 1, but this is literally the first time I'm doing non-trivial mental arithmetic in a decade.) Multiplying a power of 2 by 5 is the same as halving and multiplying by 10, which reduces the problem to a simple addition of digits of small powers of 2 (^0 = 1, ^2 = 4, ^6 = 64, ^9 = 512, ^11 = 2048, ^15 = 32768), then a multiplication by 3 – both of which are easy to perform in a streaming fashion, if you have a suitable delay line. (I use the auditory processing delay line ("echoic memory"), which would probably work better if I spoke a language like Mandarin, where all digits have the same syllable length – but I got by. Some find the mental abacus more reliable, but I have no training in this approach.)

I only memorised my powers of 2 up to 2^16, powers of 3 up to 3^5, and powers of 5 up to 5^5, because the part that made it fun was only memorising things I'd calculated myself, in my head, and this was only an occasional game. If my goal had been "develop the skill of quick arithmetic", I would've memorised the first 12 powers of every prime below 12, and my times tables up to 100×100 – but I resented times tables, so never really memorised them until I (briefly) got really into division.