Continued from Part 2

I remember reading an blog post
which said that Von Neumann's incredible intellect was produced by a
large number of single-nucleotide polymorphisms. The implication was
that it was high IQ, high *g* alone which caused Von Neumann's
intelligence. In other words, like height, which is mostly thought to
be caused by a small number of genes, each of which add a little bit
more height, this post said that the same is true for intelligence.
This is true to some extent, but when we try to explain extreme outliers
by this method, it fails. It goes without saying that to be a world
class intellect you need high general intelligence, but I believe that
Euler and Von Neumann's unusual abilities can only be explained by interaction between
general and special intelligence.

To see why, consider a list of the 10 documented oldest people who have lived:

Jeanne Calment: 122 years, 164 days

Sarah Knauss: 119 years, 97 days

Kane Tanaka: 117 years, 357 days

Nabi Tajima: 117 years, 260 days

Marie-Louise Meilleur: 117 years, 230 days

Violet Brown: 117 years, 189 days

Emma Morano: 117 years, 137 days

Chiyo Miyako: 117 years, 81 days

Misao Okawa: 117 years, 27 days

Maria Capovilla: 116 years, 347 days

Notice
that even though we have some slight outliers with the top two, the
difference between the oldest and youngest is about 6 years and many of
these ages are very close together. Which suggests that the differences
between these people can be explained by *small* differences in
genetic and environmental influences. Most likely by a lack of
deleterious mutations causing unhealthiness. So we can model lifespan
as being caused by a large number of independent mutations which each
have a similar degree of influence. If we model lifespan in this way, we would expect to
see what we actually do see - a fairly smooth gradation
between the highest and lowest with most of the gaps filled in. We can
compare this to this distribution of sums of rolls caused by rolling
five 6-sided dice.

On
the other hand, if lifespan was caused by independent mutations each of
which had significantly different degrees of influence, we would expect
to see something like one person living 120 years, one 140, one 90 ... large gaps in
between the ages. We might compare this to rolling all 5 Platonic
solids: one 4 sided die, one 6 sided, one 8 sided, one 12 sided, and one
20 sided, which would have a completely different distribution from
rolling five 6-sided dice

I suspect that speed of calculation for most people does follow a Bell Curve with few gaps. In other words, the differences in speeds between ordinary people are not so great on an absolute scale. However, when we compare Von Neumann and Euler to an ordinary person, they are like someone who has lived 150 years. The gaps are too large.

To illustrate this, here is a statement paraphrased from Wikipedia:

Here is what Enrico Fermi said to Herbert Anderson:

"*You know, Herb, Johnny can do calculations in his head ten times as fast as I can! And I can do them ten times as fast as you can, Herb, so you can see how impressive Johnny is!*"

So, where are all the people who can calculate 9 times faster, 8 times faster, etc. We are talking about a geometric increase, not merely an arithmetic increase. So, I postulate that while obviously John Von Neumann and Euler had high general intelligence, their unusual traits must be explained by reference to special intelligence. Further, it is possible that certain types of special intelligences, when they occur together, produce a "multiplier effect" where each one individually has an effect, but taken together their effects work synergistically to become even more powerful.

**Update**: William James Tychonievich has shown that rolling a mix of polyhedral dice does in fact make a normal distribution. It turns out that was not the right metaphor to use. I think the problem is that it does not take account of multiplier effects.

Would rolling a mix of polyhedral dice really result in anything significantly different from a normal bell curve. I haven’t done the calculations, but my assumption is that it would not.

ReplyDeleteNever mind. I've checked it, and my assumption was totally wrong!

ReplyDeleteCan you post what you found? I was thinking more of the smoothness of the curve than its shape, but I would be curious to know how the mix of polyhedral dice distribution looks.

DeleteAlso, I realized after writing this that rolling dice doesn't take account of a multiplier effect, unless you add a rule such as, when all the dice are a multiple of 4, add 20 or something like that.

Sure. Turns out I was right the first time after all.

Deletehttps://narrowdesert.blogspot.com/2020/12/the-distribution-of-mixed-polyhedral.html

For a "multiplier effect," couldn't you roll several dice and take the product rather than the sum of the numbers rolled? For example, if you rolled 5d6, the results would range from 1 to 6^5 and would not look anything like a bell curve.

DeleteYes, that would work and it would create larger gaps. Here's what it looks like when we roll 2d4:

Delete1,2,3,4

2, 4, 6, 8

3, 6, 9, 12

4, 8, 12, 16

The comment did not preserve the spacing, unfortunately.

DeleteI was just saying to my wife the other day that mathematical and musical abilities seem more varied than anything else I know of.

ReplyDelete(And I think there is a surprising positive between music composition and maths.)

Most people are pretty close to zero! - and then the people who are good are So much better - and then there are some who are So much better than the people who are good...

But the distribution doesn't seem anything like a bell curve - more like something very positively skewed.