How "learnable" are mathematics?

[Mecill]

In thinking about how I abstract things, most of it is based on something I've read or heard someone say and connecting it with another related idea. The more I hear about abstractions the more natural it becomes to think that way. At first some types seem very unnatural but others seem very natural to me because they go along with things I have previously learned. Lol, I guess typing this is an abstraction itself, though.  

[Suicidesoldier#1]

Intelligence and math skills are not ethnically related. As a matter of fact, "race" is only determined by two out of nearly three billion genomes; the gene from your mother, and the gene from your father. "Races" only began emerging 40,000 years ago when people began to migrate from the Nile into areas like Asia, Europe, and according to some scientists even into the Americas. Because of this, humans simply have not had enough time for strong characteristics to develop in isolated communities, if we exclude the neanderthal's and pygmies that we killed into extinction. Because ethnicity has such a small impact on the human genome, it is very unlikely, if not simply impossible for traits to be related to certain 'races'. In reality, the nature of people's environments and the culture causing the development for children is much more important for deciding things like intelligence, on a general scale.

So no, there is no distinct culture or 'race' which has an inherent ability for math or learning skills; although there are specific, isolated people, sometimes at random, who are gifted which extremely powerful mathematical and learning skills, and those who are at a disadvantage.  

[Layra-chan]

Um...

Quote:

In reality, the nature of people's environments and the culture causing the development for children is much more important for deciding things like intelligence, on a general scale.

And

Quote:

So no, there is no distinct culture...which has an inherent ability for math or learning skills

There seems to be a logical compatibility error with these two statements. The first one states that culture can have the kind of effect that we are speaking of, but the other denies it.

Culture is not genetic, and genetics do not tie into discussions regarding culture. Ethnicity, ultimately, is not genetic, although biological heritage is a factor. While it is usually taboo to suggest that biology plays a factor in determining intelligence (and studies support the lack of a biological bias), culture and ethnicity, being primarily mental and social constructs, can have an effect on someone's mental development and social decisions.

There is often a gut reaction to statements regarding disparities and race, and unfortunately culture, ethnicity and race are tied together closely enough that talking about one often leads to that gut reaction firing up. This is a problem in that it leads to an inability to objectively (or even logically, as demonstrated) consider possible correlations between purely mental and social constructions that currently have racial connotations.

At the moment, culture and race are heavily linked. But there is more to being Asian than yellow skin, and there are certainly enough widespread non-biological differences between Asians and Westerners that cultural contrasts between these two ethnic groups can be made.  

[StandingOnTheMoon]

Suicidesoldier#1

Intelligence and math skills are not ethnically related. As a matter of fact, "race" is only determined by two out of nearly three billion genomes; the gene from your mother, and the gene from your father. "Races" only began emerging 40,000 years ago when people began to migrate from the Nile into areas like Asia, Europe, and according to some scientists even into the Americas. Because of this, humans simply have not had enough time for strong characteristics to develop in isolated communities, if we exclude the neanderthal's and pygmies that we killed into extinction. Because ethnicity has such a small impact on the human genome, it is very unlikely, if not simply impossible for traits to be related to certain 'races'. In reality, the nature of people's environments and the culture causing the development for children is much more important for deciding things like intelligence, on a general scale.

So no, there is no distinct culture or 'race' which has an inherent ability for math or learning skills; although there are specific, isolated people, sometimes at random, who are gifted which extremely powerful mathematical and learning skills, and those who are at a disadvantage.

Um, I thought I addressed this sort of thing in my original post. Something about populations and a bell curve.  

[Mecill]

Layra-chan

Also, one of my mathy friends a while ago speculated on the existence of a personal "abstraction limit", wherein a person finds a kind of mathematics too far removed from reality to be comprehensible. I'm pretty sure that logic/model theory is beyond my abstraction limit, and for certain subjects I'm just going through the motions without any real understanding of why what I'm doing works.

Actually, I hadn't thought about how highly this has influenced me but my freshman year I took a philosophy course in logic for humanities credit. We defined the operators, learned a few theorems and then had practice translating some simple sentences into symbolic form. You would assign a letter to represent a subject and just go from there. It may not sound very interesting to some but I thought it was a really amazing, powerful concept!

Some of it I had seen before, for example, the structures of deductive and inductive arguments I learned in AP English in high school. But I don't think it is conceptually very difficult to learn if you get it presented to you in a class like that.  

[Layra-chan]

Basic symbolic logic is very nice, clean, and simple. It quickly becomes quite terrible, though, once you get into type theory and orders and all sorts of machinery for describing what you can express.
For example, one statement you probably encountered was "((p→q)&(q→r))→(p→r)". This is a nice, basic statement: if p implies q and q implies r, then if p is true then q is true, and thus r is true. If all dogs are mammals and all mammals have hair, then all dogs have hair. No question about this, right?
Well, what if we have an infinite string of such propositions?
(a0→a1)&(a1→a2)&(a2→a3)&... forever. Suppose that a0 is true. Can we say that all of the ai is are true, for all i? Naively, yes. More rigorously, well, it depends. It depends on a lot. In fact, even being able to formulate this question within a given system depends on a lot. In propositional logic we can't even reference all the statements ai at the same time. We can talk about them individually, but never as a whole.

And of course, the big question in logic is, when given a bunch of statements p1, p2, etc, are they consistent with each other? In other words, if we can derive statement p from those statements, can we also derive the statement not p? Can we show that we can never derive both a statement and its inverse from the initial set of statements? For the basic set of axioms upon which most mathematics rests (ZFC), the answer to this last question is: no, we need to go outside the system in order to be able to show that the system defined by ZFC is consistent. The existence of what is known as a weakly inaccessible cardinal would show that ZFC is consistent, but we cannot show that such a cardinal exists using the ZFC axioms alone. Nor is the existence of such a cardinal guaranteed by systems larger than ZFC; there are systems with such cardinals, and there may be systems without them, but these systems themselves cannot be shown consistent without going to yet larger systems, and so on.  

[VorpalNeko]

Infinitary logic is possible, but it's interesting that much of it is incompatible with ZFC. For example, for the proposition
p : (∀x)(∃y)(∀x')(∃y')... (x,y,x',y',...)∈X
The statement "for any set of sequences of natural numbers X, p or not p" is equivalent to the axiom of determinacy, which contradicts the axiom of choice. So that sort of thing is more natural in ZF+AD (ZFD?) rather than ZFC.

Layra-chan

The existence of what is known as a weakly inaccessible cardinal would show that ZFC is consistent, but we cannot show that such a cardinal exists using the ZFC axioms alone.

Thank goodness for that. A ZFC proof of the consistency of ZFC would actually show that ZFC inconsistent.  

[Mecill]

As a more practical example of some infinitary logic I have seen a problem in circuits where you have an infinite series of resistors and you calculate the equivalent resistance of the whole thing. But I'm guessing this is using a naive/non-rigorous version.  

[Layra-chan]

That's not really logic; that's a mixture of physics and calculus.
An infinitary logic is one in which your statements can be infinitely long; no physics or any connection to the real world, since statements don't have a physical manifestation.  

[Mecill]

... I think it is somewhat difficult for the average person in the US to learn math. Not because math is too difficult, but because our method of teaching it is not very effective. If you want to learn how to use math you definitely should try to study it on your own in addition to what is required for a class.  

[zz1000zz]

The education system in the United States has always sucked.  

[Mecill]

zz1000zz

The education system in the United States has always sucked.

Education systems in general tend to suck. People complain about formal education being too boring, and alternative methods often don't cover enough topics... US isn't so bad, but getting good resources and teachers can be difficult.  

[zz1000zz]

The United States education system is strange. I find it is more appropriate to call it a "training system." The fascinating thing about the education system in the United States is it has suffered constant criticism for at least the last century. Because of this criticism, it has undergone nigh-constant change for the entire period. Despite the amount of effort put into changing it, it has effectively remained the same.

I became interested in the education system because I wound up bouncing between so many different schools. I don't know much about the education systems of other countries, but I do know the education system in the United States is insane.  

[Mecill]

I think one of the things I value in education is the presentation of a somewhat more liberal worldview. For that reason, from what I've heard about the systems in the UK and Australia, for science, they sound appealing. However, the US system is not without its advantages.

I don't know much about the systems in non-English countries. Maybe a little about Japan, from watching a Japanese TV show once, but I'm not sure how realistic it was. But in that show it seemed like they all were very respectful of the prof, and sat in a very orderly fashion. If they weren't paying attention, they hid it better. When I go to classes here sometimes I see people arriving late, and not paying attention blatantly. Maybe its just about cultural differences though. Sometimes profs will start a conversation with students in a friendly way. And usually American students do honestly respect teachers that they like. I also think they should respect teachers they don't like, if merely to be sympathetic to the fact they are attempting to teach.