
On Mathematical Intelligence and How It Grows - __Joker
http://appliedsentience.com/2015/02/10/on-mathematical-intelligence-and-how-it-grows
======
j2kun
While I don't care much for IQ as a statistic, I do believe that discipline
and hard work are important.

The author gives a quote from Euclid and two physicists, but for the aspiring
mathematician, or anyone who is bewitched by the myth of a genius
mathematician, I think a more appropriate quote comes from Terry Tao [1]:

> Actually, I find the reality of mathematical research today – in which
> progress is obtained naturally and cumulatively as a consequence of hard
> work, directed by intuition, literature, and a bit of luck – to be far more
> satisfying than the romantic image that I had as a student of mathematics
> being advanced primarily by the mystic inspirations of some rare breed of
> “geniuses”. This “cult of genius” in fact causes a number of problems, since
> nobody is able to produce these (very rare) inspirations on anything
> approaching a regular basis, and with reliably consistent correctness.

He goes on to describe how some of the most valuable traits in mathematics
include asking dumb questions, pateince, maturity, etc. I would add to that
the skill of being wrong. If you're wrong all the time and comfortable with
being wrong, you can identify more easily when you're wrong or right and work
to fix the wrong stuff and identify the keys to the right stuff.

[1]: [https://terrytao.wordpress.com/career-advice/does-one-
have-t...](https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-
genius-to-do-maths/)

------
david927
_[The bell curve] crops up a lot when talking about traits in a population,
such as intelligence. The reason for this is well understood (see central
limit theorem) and won’t concern us here._

No! The Central Limit Theorem states that the average of the values in the
distribution creates a bell curve -- _not_ that it starts out that way. Traits
in a population do not follow a bell curve and the CLT doesn't give any reason
why they should.

~~~
myg204
If one considers intelligence (or other features) as the sum of many random
sources, then it makes sense again to invoke the CLT. Maybe that's what the
article implied.

[edit: see other replies saying the same thing below]

~~~
leni536
"Intelligence" itself is not a well defined quantity like height or weight. IQ
is well defined but it is defined to follow a bell curve, it has nothing to do
with CLT.

~~~
gwern
Intelligence is a well-defined quantity like temperature, and it follows a
genetic architecture like height: thousands of genetic variants of small
effect size. Which setup does indeed give you a binomial or normal sort of
distribution, justifying the norming as more than a mathematical convenience.

~~~
leni536
_> Which setup does indeed give you a binomial or normal sort of distribution_

distribution of what? Still no quantity defined.

 _> Intelligence is a well-defined quantity like temperature_

As a physicist it sounds about right. Temperature is tricky to define in non-
stationary systems (aka. "real world").

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davidgerard
> Otherwise known as the standard normal distribution. It crops up a lot when
> talking about traits in a population, such as intelligence. The reason for
> this is well understood (see central limit theorem) and won’t concern us
> here.

No, the reason IQ follows a bell curve is that IQ is _defined_ as following a
bell curve. Results are adjusted to make a bell curve.

------
Synaesthesia
So the gist of the article is that advanced mathematical abilities are
attainable to anyone! It just takes some clerical work. This is an important
point, many people still believe in the myth that a mathematician, physicist
or programmer is born, not made. In reality it is both, but if you are not as
naturally endowed you can still compensate for that.

------
gwern
SAT is a bad example because SAT prep works very poorly. People like to cite
anecdotes (which are driven by the noise of retesting until an extreme score
is reached and selection effects) and prep companies like to cite how
'students who took our course scored 200 points higher than those who didn't',
but that's driven by the obvious confounding of who goes through with prep
courses; estimates from the randomized experiments tend to look more like 20
points (see
[https://en.wikipedia.org/wiki/SAT#Preparation](https://en.wikipedia.org/wiki/SAT#Preparation)
and reviews like
[http://libgen.org/scimag/get.php?doi=10.1111%2Fj.1468-2397.2...](http://libgen.org/scimag/get.php?doi=10.1111%2Fj.1468-2397.2011.00812.x)
).

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rawnlq
Can someone with more stats background explain how to invoke the central limit
theorem to see that intelligence should be distributed as a bell curve?

What are the independent and identically distributed random variables that are
being averaged in this case?

~~~
panic
Intelligence (at least as measured by an IQ test) isn't distributed as a bell
curve. Bell curves always include negative values, and intelligence can't be
negative.

~~~
thaumasiotes
> intelligence can't be negative

Asserted without proof. To show this, you'd need to interpret intelligence as
something that physically couldn't be negative. I feel safe in saying you
haven't done that.

In particular, "intelligence (as measured by an IQ test)" can easily be
negative, in the sense of yielding a negative IQ score. It would have to be an
unrealistically long test, and you'd need way more people than exist in the
world, but those problems apply to the tails of actual normal curves too.

~~~
rawnlq
I am really impressed by wolfram alpha's ability to parse my butchered syntax:
[https://www.wolframalpha.com/input/?i=probability+that+gauss...](https://www.wolframalpha.com/input/?i=probability+that+gaussian%28100%2C+15%29+%3C+0)

Probability of finding someone with an IQ (as defined as normal distribution
with mean 100 and std 15) less than 0 is 1.30839×10^-11

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cwhy
Just look at the SAT maths scores of different races. IMHO that the high
scores of Asian students was due to the Asian cultures which promotes hard
work.

------
waps
Here's the thing about IQ curves. Let's assume, for the moment, that we create
a new "WQ" (waps quotient). It is 1000 random questions that you either know
or don't know, which have absolutely nothing to do with eachother (who is the
beatles' lead singer, what is the 7th prime number, did Mr. Garrison ever kill
Kenny, is Spam searching related to Bayes' rule ? (nope, it's Bayes theorem),
...

For every question you have right you get +0.1, for every wrong answer you get
0. Obviously this proves nothing about you, except that you've at some point
looked up the beatles, or actually looked up history of Thomas Bayes, or ...
In other words it's a long list of "do you know trick X" bits.

What would the division of WQ in the population look like ? Exactly like the
division of IQ in the population. [1] [2]

Here's a supposition : the IQ stat is exactly that. It's measuring of how many
of the "little tricks" known by a test maker you know. The better you match
the test maker, the better your score (think mostly cultural, but also whether
your parents are academics or not, ...)

Although I must say, teaching kids "tricks" with numbers (e.g. how to tell if
a number is divisible by 9 by looking at individual digits) and symbols is a
good way to make them good at math over time.

[1]
[http://en.wikipedia.org/wiki/Binomial_distribution](http://en.wikipedia.org/wiki/Binomial_distribution)
(read top paragraph, note that n will be a relatively large 1000) [2]
[http://en.wikipedia.org/wiki/Binomial_distribution#Normal_ap...](http://en.wikipedia.org/wiki/Binomial_distribution#Normal_approximation)

~~~
gohrt
Your theory would also need to explain correlations between scores on
_different_ IQ tests taken by the same test-taker.

And knowing the answers to more or less questions than someone else is
suggestive of something, if the questions are broad enough. Especially if
someone else grew up _in the same culture as you_

~~~
Jach
Better to explain the correlation with job performance, and why the
correlation improves with more complex jobs.
[http://www.udel.edu/educ/gottfredson/reprints/1997whygmatter...](http://www.udel.edu/educ/gottfredson/reprints/1997whygmatters.pdf)

