A valuable method of learning math is to start at the beginning of recorded history and read the math-related texts that were produced by the people who made important contributions to the progression of mathematical understanding.

By the time you get to Newton, you understand the basic concepts of everything and where it all comes from much better than if you had just seen them in a textbook or heard a lecture.

Of course, speaking from experience, reading page after page of Euclid's proofs can be exhausting to continue to pay enough mental attention to actual understand them before moving on to the next one. :)

Still, it does help tremendously to be able to place the knowledge in the mental context of people who actually needed and made the advances.

"God, say the religious fundamentalists, is the source of all morality; there can be no morality without a Judge who rewards and punishes."

I suppose this may be a true position for some southern baptists or the like, I won't claim to know the normal religious arguments of every sect or region, but I've never heard it stated from anyone religious, only the "formerly religious" or the non-religious. So it seems like a bit of a strawman argument to me.

From my own "religious fundamentalist" position, a contrasting argument would be:

1. God is all knowing.
2. God loves us.
From which follows:
3. God knows what moral rules are best for us to follow in order to benefit ourselves and wants us to follow those rules.
As a result:
4. In order to maximise our collective benefit, we should follow god's commandments.

TGGP,

Sorry, forgot to answer your additional question in my reply. Whenever Thomas Sowell and Walter Williams get together in a small group (as they do occasionally), within that small geographical region darker skin color is a great indicator of higher intelligence. Makes all these divisions seem just a little more arbitrary, doesn't it?

On a more serious note, Queens, NYC, NY has a higher than normal proportion of black West Indies immigrants. In that geographic region, blacks have an average IQ higher than those with lighter skin. Other US locations include Pembroke Pines, Florida, Rialto, California and Brockton, Massachusetts. They also have a higher median income, are more likely to live in a two parent household, own their own home, etc...

Gareth, I don't believe I specified a quantitive amount by which they would differ, just that they would differ. You're right, normally (pun intended), the groups wouldn't differ by much. That's part of the point, isn't it. Why care that they differ at all? There isn't a useful reason to care that group X has a different average IQ than group Y. Does a particular group of dark skin people have a lower intelligence due to their skin color? Not likely. Other factors are much more significant. There's no causual relationship between the two factors. The only reason it's mentioned at all is because some people have a hang-up about defining themselves or others by the color of their skin.

If you took two groups. One group with high intelligence and a second group with low intelligence and genetically modified them to change only their skin color, would their intelligence change drastically as well? The point is that skin color is not a causal factor in determing intelligence, so it's meaningless to use as a filter when presented with a sample size of one random individual with skin color X.

For a bunch of reasons, recent black immigrants from the East Indies in NY have a much higher average intelligence than blacks in NY with a longer US family pedigree (and a higher average intelligence than your average white in NY, for that matter) Does it matter to their intelligence what color their skin is? Apparently other factors matter a lot more. My point is that racial grouping is more arbitrary than most other groupings as a basis for making judgements and is only really used by people out of historical inertia. The groupings of people into "race", whether done by individuals or by governments tends to be pretty damn unscientific compared to say, geneticly-based groupings. I don't know first-hand about the UK's classification schemes, but have you seen the racial categories governments in the US use?

TGGP,

Dividing the human population into two groups, male and female, does in fact result in one group having a higher average intelligence than another. Depending on your actually used definition of intelligence, you may decide that one group has a higher average than the other and which group that is may change over time. I suppose if you modify your definition of intelligence based on the actual level of that quality in males and females you might be able to come up with a definition that for an instant in time made the two groups equal, but the actual population changes quickly enough that your definition would also quickly become outdated and the two groups would no longer be equal in your proposed definition of intelligence.

On the question of if there is a usefully measurably large difference between the groups (as opposed to an actual difference), then I agree that it's not useful to use male/female group membership as a filter or test for an individuals estimated intelligence, regardless of which of the popular definitions of intelligence that you subscribe to. The variance is way too high within the population to make it a useful indicator for practical purposes.

Again, that's why the fact of there being differences should be as unremarkable as it is useless as a guide to decisions and policy.

Take a step back. Theoretically if someone divided the world population into two groups by randomly assigning each individual the letter "A" or the letter "Z", you would have two groups that on average have a miniscule difference in intelligence, or athletic ability, or whatever you want to measure.

But practically speaking, knowing the results of the "A" and "Z" groups and also knowing which group your individual belongs to is useless as compared to a factor that actually has a casual relationship to what you are trying to measure that individual for.

No offense to the sociologists among us, but individual characteristics matter far more than group characteristics when making any sort of judgement or decision.

Assumptions:

1. Intelligence describes some quality that individuals have. 2. Individuals have different aamounts of or ability in that quality. 3. Larger groups of can be created by grouping these same varying individuals and this group can be said to have a group average in terms of intelligence.

Some conclusions that follow from these assumptions: 1. Any group of people, divided by almost any arbitrary measurement into a large enough sample will differ in the average amount of intelligence each particular subgroup has. 2. Arbitrarily dividing the worlds population into "races" based on their skin pigmentation will create subgroups that also have a differing average intelligence.

Therefore, different racial groups have a different average intelligence. This logic doesn't depend on what exactly you define as intelligence, as long as something is defined as intelligence. This logic doesn't imply that any particular racial group must have more intelligence than any other particular racial group. It just implies that by definition, individuals are different from each other and thus groups of individuals will also tend to differ in their different characteristics.

Now for the fun part.

Does it matter that different races have varying "average" intelligences? Only if you are racist and judge people by what arbitrary "race" you've mentally classified them as. To me, if you insist on classifying people based on their physical appearance, it would make more sense to classify people by height, or eye color, or their weight. Those attributes tend to make more of a difference, genetically.

However, perhaps you find race an easy way to filter individuals into categories. In that case, based on the extensive evidence, in some geographic regions, people with darker skin have a lower average intelligence as judged by the best tests and definitions of intelligence available. What's interesting is that in some other geographic regions, people with darker skin have a higher average intelligence.

So used as a bayes filter, skin color isn't going to typically give you the ability to improve your judgement of intelligence unless you have other demographic information that has a much better correlation.

In actual fact, if you want to judge a person's intelligence based on some quick rules of thumb, there are much better demographic methods available to you than skin color. National origins, wealth levels, education obtained (for older individuals), etc...

On the other hand, if you really care that much about it, just ask them to take whichever intelligence test you prefer for you.

But to say that races differ in intelligence? That statement should be as non-controversial as a statement of fact as it is useless as a guide to actions and policy.