Sacks' Autistic Twins and Prime Testing

[qspeechc]

Hello everyone.

[Firstly, I didn't know if this belongs here or in General; please move if appropriate].
<Moderator's note: moved to GD>

I was reading this paper on the AKS primality test (undergraduates can understand it, highly recommended!), and on page 7 the author brings up the story of the severely autistic twins in Oliver Sacks' book The Man who Mistook his Wife for a Hat:

"...in which he tells us of a pair of severely autistic twins with a phenomenal memory for numbers and a surprising aesthetic. Sacks discovered the twins holding a purely numerical conversation, in which one would mention a six-digit number, the other would listen, think for a moment and then beam a smile of contented pleasure before responding with another six-digit number for his brother. After listening for a while, Sacks wrote the numbers down and, following a hunch, determined that all of the numbers exchanged were primes.

"The next day, armed with a table of primes, Sacks butted into their conversation, venturing an eight-digit prime and eliciting, after a short pause, enthusiastic smiles from the twins. Now the twins kept on going, increasing the number of digits at each turn, until they were trading (as far as Sacks could tell) twenty-digit prime numbers. So howdid the twins do it? Perhaps we will never know, since the twins were eventuallyseparated, became “socialized” and forgot their amazing algorithm!"

I don't know whether this is a question for mathematicians, neuroscientists or psychologists, but what do you think about this story? To me it seems a bit suspicious, especially how Sacks conveniently adds the twins became separate and eventually lost the skill.

On the other hand, the author of the paper, Andrew Granville, is a number theorist, and says the following:

"The advent of the AKS algorithm makes me wonder whether we have missed some such algorithm, something that one could perform in a few minutes, by hand, on any enormous number."

He seems to suggest the twins may have found a relatively simple algorithm for testing primes.

What does PF think of this story of the twins? I leant towards scepticism, but I'm nobody.

Further reading

 

[tnich]

Hello everyone.

[Firstly, I didn't know if this belongs here or in General; please move if appropriate].

I was reading this paper on the AKS primality test (undergraduates can understand it, highly recommended!), and on page 7 the author brings up the story of the severely autistic twins in Oliver Sacks' book The Man who Mistook his Wife for a Hat:

"...in which he tells us of a pair of severely autistic twins with a phenomenal memory for numbers and a surprising aesthetic. Sacks discovered the twins holding a purely numerical conversation, in which one would mention a six-digit number, the other would listen, think for a moment and then beam a smile of contented pleasure before responding with another six-digit number for his brother. After listening for a while, Sacks wrote the numbers down and, following a hunch, determined that all of the numbers exchanged were primes.

"The next day, armed with a table of primes, Sacks butted into their conversation, venturing an eight-digit prime and eliciting, after a short pause, enthusiastic smiles from the twins. Now the twins kept on going, increasing the number of digits at each turn, until they were trading (as far as Sacks could tell) twenty-digit prime numbers. So howdid the twins do it? Perhaps we will never know, since the twins were eventuallyseparated, became “socialized” and forgot their amazing algorithm!"

I don't know whether this is a question for mathematicians, neuroscientists or psychologists, but what do you think about this story? To me it seems a bit suspicious, especially how Sacks conveniently adds the twins became separate and eventually lost the skill.

On the other hand, the author of the paper, Andrew Granville, is a number theorist, and says the following:

"The advent of the AKS algorithm makes me wonder whether we have missed some such algorithm, something that one could perform in a few minutes, by hand, on any enormous number."

He seems to suggest the twins may have found a relatively simple algorithm for testing primes.

What does PF think of this story of the twins? I leant towards scepticism, but I'm nobody.

Further reading

Having had a similar experience, I believe the story. When I was 18, a total stranger challenged me to a game of Go. I had barely heard of the game, so he explained the rules and we started playing. Within a few moves, I realized that I understood the game, and a few minutes later when I won, my opponent was incensed. He refused to believe that I had never played it before. At the time, I didn't understand why. Only later did I learn that Go is considered even more difficult to master than chess.

I had a deep intuitive understanding of the game, but I had no algorithm. I wasn't playing n moves ahead, it was just obvious to me where I needed to play to win. When several years later I had the opportunity to play the game again, I discovered to my dismay that I no longer had that ability.

Later on I came to understand that I had been - for lack of a better term - a savant, that in some way that I was not conscious of, my mind processed the positions on the board and came up with the winning moves.

I suspect that the twins in Sacks' story experienced something similar. If you had asked them how they knew a number was prime and they had been able to answer, I think they would have told you it was obvious.

 

[BWV]

Here is sort of a reverse analogy. Face blindness -the inability to recognize faces between people of similar appearance - is fairly common with autistics. No one without this condition has ever given a conscious thought on how it is that they can instsntly recognize, say, the face of their mother from another woman of similar appearance. So recognizing faces for someone with this condition is as perplexing to them as I find the twin’s ability to recognize primes. What is the algorithm for the brain to recognize faces? You can train a computer to do it with a neural network, but how much similarity is that to what happens unconsciously in our brains? Certainly no autistic person with face blindness could train themselves to recognize people using machine learning algorithms. We have no clue how the brain actually recognizes faces, why should we have any better understanding of how these twins recognize primes?

 

[qspeechc]

I'm still not convinced. It would be easy to explain why one face was different from another, you could say one person has a bigger nose or whatever. Recognising faces would be driven by evolution, and so the lack of it in autistic persons would, presumably, have been an evolutionary disadvantage in our history as a species. Also, I don't know anything about Go, but wikipedia says it's about pattern recognition, which I'm guessing our brains also had to evolve. But determining whether 20-digit numbers are prime or not? Our brains can do amazing things the best computers can't, but so too can other animals, it's natural when considered from an evolutionary standpoint. And what's this business of the twins losing their ability after been separated and 'socialised', what does that even mean?

 

[BWV]

Fair point - neural networks have been surpassing humans in pattern recognition (witness the dominance in Go) but googling around, they seem to be incapable of identifying primes. But did not Ramanujan have a similar ability?