On Mathematical Intelligence and How It Grows
appliedsentience.com
appliedsentience.com
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...
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.
[edit: see other replies saying the same thing below]
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").
I fail to see how IQ is "defined to follow a bell curve" as you wrote. The tests were designed so the answers distribution is a bell curve?
In my statement above, just replace "Intelligence" by "IQ", I believe that's what being discussed.
In any case, "Intelligence" can takes a thousand forms that I don't think IQ tests or even our vocabulary can really pin down.
"When current IQ tests are developed, the median raw score of the norming sample is defined as IQ 100 and scores each standard deviation (SD) up or down are defined as 15 IQ points greater or less"
Raw test scores are transformed so the test results distribute on a bell curve. Can't find a more precise definition though.
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.
What are the independent and identically distributed random variables that are being averaged in this case?
"Identically distributed" is a not a requirement in CLT, it's just the simplest variant to prove.
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.
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
"The distribution of the measured IQ very closely follows a bell curve." would have been the correct way to formulate.
If it's "non-technically correct" it is ill-frased.
This debates started by people stating wrong things but still wanting others to understand them.
The word "technically" is used with "correct" as a rhetoric tool. It seems like they are attacking the truthness of the statement, but in fact they are trying to put the statement into a bigger context. So many people misuse it as such that some even understand it as an attack of the truthness of the statement.
No, they use it as a device to point out that there is more context around the discussion than whether or not something is true or false (hence my point about 2-valued logic).
I could say race is defined by genetics, and while that's technically correct, it adds nothing to a discussion about race relations.
This is the context in which people use this term.
When people say things like "you're technically correct, the best kind of correct", it's a tongue in cheek way of pointing out the observation is true, but lacks contextual information in order to be useful. This is why it tends to be a joke, in general everyone implicitly understands this.
I could say race is defined by genetics, and while that's technically
correct, it adds nothing to a discussion about race relations.
Genetics define physical race. That is technically correct, hence it's plain correct and it's true.
If a statement doesn't add to a discussion, then it's misusing of language to say that the statement is "technically correct" instead of clearly saying that the statement is not relevant.
It could be conventional to use "technically correct" to mean "true but important context is missing", so everybody understood it. Some define a language to be correct the way it is used commonly. However, I don't agree and I'd still consider it misusing language.
If context is missing, then why talking about the correctness instead of supplying missing context?
People using "technically correct" like you described indirectly devalue correctness. That follows the lines of valuing more the consequences of a statement than whether it's true or not.
I'd rather see the world like it is than how I'd like it to be. Therefore, correctness is very important to me and I think it shouldn't be subtly devalued.
How about you?
You'll outgrow that attitude eventually, you should revisit that last post when you do.
The random variable is simply whether you answered question N on the test correctly or not. For every question there is a probability Pn that any given test taker answers it correctly, and that probability is relatively uniform per-question. (so scores on any test, where the same test is administered to a large numbers of test takers, will be normally distributed. Yes, even if the questions are nonsensical and the answers random)
So an IQ score simply indicates the probability, relative to the general population, whether a test taker will answer simple abstract mathematical questions correctly (where correctly should be understood to have cultural biases).
But taking 2 variables that correlate 80% or 20% or even more will not change the outcome. It's just very near the edge that it totally breaks down, or if the number of variables that correlate is really too high, or if they're really chaotic variables, or ...
So one the one hand you're right. But it's not a point that a statistician would make, only a logicist or a rather pedantic mathematician. In the graphs, even if things correlate it will "look okay". Therefore the statement that intelligence is normally distributed along a bell curve is technically correct, but good look finding a large piece of data that doesn't look like a bell curve.
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 (read top paragraph, note that n will be a relatively large 1000) [2] http://en.wikipedia.org/wiki/Binomial_distribution#Normal_ap...
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
I think that one is easy:
In fact, @waps has a bag of M questions, where M > 1000. To compose a new WQ test, he draws 1000 questions, without replacement, from his bag. Then the series of tests has some similar or even same questions. In such a case the scores must be correlated.
This is not too far off-base. Especially when you consider that many questions are superficially different, but really the same. E.g., those "name the next number in the series" questions.
But you did answer my question on why IQ is normally distributed: the IIDs being averaged are the test questions themselves! (although I don't fully understand if those questions are bernoullis?)
Did I model that correctly?