A good introduction to math would turn a person's innate interest in thinking and logic into an interest in math. After all, at its base, math is really just a way to approach logic in an organized and systematic fashion. I think at least a basic understanding of some more abstract math--especially formal logic--is as valuable to a well-rounded person as an appreciation of literature or knowledge of basic history. Sure, the basic person on the street doesn't need to be an expert on algebraic topology, but they don't need to be experts on romantic literature or 17th century Belgian history either. This doesn't mean that they shouldn't be well versed in some--and probably a fair bit--of literature and history, and, in the same way, they should be well versed in at least basic mathematics.

Now, one of the problems with math education is that what they teach is not really basic math. Rather, they teach subjects that are readily applicable at a fairly superficial level. I think a good grounding in formal logic and set theory, for example, is far more valuable in general than a thorough understanding of differential calculus. And yet it's the latter that is considered basic and widely taught, probably because it is immediately useful to engineers and physicists.

Oh, this is so wrong. Did you like Avatar? Then you like math. Avatar is one long, beautiful mathematical expression, from beginning to end.

> This is like trying to become smarter by listening to classical music.

Granted the problem with the basic idea, this has it all over trying to become smarter by listening to Country & Western music.

> Is it ironic that those suggesting these ideas can't separate correlation from causation?

Math is neither a cause nor an effect -- it is both.

I could argue this point in detail, but instead I will get Bertrand Russell argue it for me: "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry."

Richard Feynman said, "To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature."

I can't think of a place that equals America for complete and willful misunderstanding of mathematics.

Can you elaborate? To me, the interesting part of mathematics is the process of manipulating expressions, not the expressions themselves. For example, the number pi would be pretty boring if it weren't for the theorems related to it.

I think most mathematicians and physicists would disagree. I think such people wouldn't want distinguish between generating equations and studying them and their effects, especially with physical equations that must reflect reality to have any value.

For example, I can make elaborate predictions about a lot of physical reality with:

f = G M_1 M_2 / r^2 (gravitational force)

And if I want to study tidal forces, all I need to do is take the first derivative of the above:

f = 2 G M_1 M_2 / r^3 (tidal force)

I have just "manipulated" an equation, but with an awareness of the equation's physical consequences and effects, and I can now use the second equation to make predictions about ocean tides, the future position of the moon, and the temperature of the volcanoes on Io (which arise from the energy of tidal forces). So for me, the equation itself is immensely valuable for modeling reality -- it's more than a case of manipulating terms.

> For example, the number pi would be pretty boring if it weren't for the theorems related to it.

Pi, as it turns out, may serve as a source for random sequences of digits. So apart from its geometric meaning, Pi stays interesting.

Right, but the fun part (for me) is finding out how and why pi can be used as a source of random sequences of digits, not the fact itself.

> I think most mathematicians [...] would disagree.

I would be surprised to meet a mathematician who enjoyed learning theorems/proofs more than finding theorems and proving things on their own. That's why I said the interesting part is the process of manipulating expressions rather than the expressions themselves.

Based on your gravity/tides example, I think you would take the same stance (Do you?). I just wasn't clear enough in my previous comment.

One must crawl before one can walk. Reading the work of others is quite enjoyable. And not everyone yearns to reinvent the wheel -- for many problems, understanding the prior work gives one more than enough satisfaction.

> That's why I said the interesting part is the process of manipulating expressions rather than the expressions themselves.

A finished equation can stand alongside the finest art, and garner the same kind of appreciation. People still read Einstein's relativity equations, and Maxwell's electromagnetic equations, with a deepening appreciation of their beauty, quite apart from the degree to which they describe reality.

Also, there is the interesting task of applying equations to real-world problems. I don't need to rewrite the gravitational and tidal equations to discover new things while applying them.

> Based on your gravity/tides example, I think you would take the same stance (Do you?).

Not really. It would be like asking someone whether they prefer reading, or writing. Obviously the full experience of literacy involves both.

Here's a comparison -- a common problem for student writers, usually pointed out by someone with more experience, is that they haven't read enough to be able to write effectively. There's a parallel in mathematics -- those who take the trouble to read enough mathematics, by so doing learn how to express themselves more efficiently.

For me, the two equations I posted earlier, one that describes the gravitational force, and the other the tidal force, the mathematically interesting thing is the relationship between them, not so much the equations themselves -- the fact that a simple derivative operation produces the second equation (in physical terms it's because the tidal force is felt by any two adjacent masses placed arbitrarily close together).

For physical equations, working with them means either imagining their consequences, or modeling them, usually with a computer. In that case, you don't manipulate the equations, you use them to model reality. So having an equation that's known to represent some aspect of reality is just the beginning of a research program that models the consequences of the equation and compares the model to reality.

Here's an example. Observations of Jupiter's moon Io revealed the possibility of a large, static tidal force on its mass. You may be aware that a static force doesn't require any energy expenditure (imagine a book lying on a table). Then someone pointed out that Io has an elliptical orbit, which means Io is constantly moving toward, and away from, Jupiter. This would have the effect of changing the tidal force, and a changing tidal force would perpetually change the moon's shape -- and changing the moon's shape would require energy. This would generate a lot of heat. Shortly thereafter, volcanoes were observed on Io, a moon too small to have the kinds of volcanoes we have here, and the explanation was the elliptical orbit and tidal force.

That's mathematics.

Your parent comment missed the point of his parent.

That might be a fair objection if they could be separated. If I say y = cos(x) * e^-x^2, when expressed in two dimensions I get this:

http://i.imgur.com/MePPy.png

And in three dimensions I get this:

http://i.imgur.com/9gr4u.jpg

But I'm still looking at y = cos(x) * e^-x^2 -- the picture only confirms the mathematical identity. So it is with Avatar -- when we watch Avatar, we're looking at math.

And much of nature is defined using math -- note that I said, not described, but defined. Here's an example -- there are species of locusts (cicadas, actually) that reproduce at 13-year and 17-year intervals. Until recently, no on knew why. It turns out that both 13 and 17 are prime numbers, and this ties into a survival strategy hatched by natural selection (quite by chance). All explained here:

http://arachnoid.com/prime_numbers/index.html#Mathematical_L...

In other words, the locust survival strategy is math speaking out loud.

If you read a book in which a seashore is described, do you argue that the description isn't germane to the thing being described? If the writer isn't skilled, or the reader is lacking in the capacity for visualization, then that is perhaps a legitimate objection, but for most people, words convey meaning. So does math. Math is a language in much the same way that words are a language.

The distinction between "loving math" and "loving things made of math" is dubious at best. When P.A.M. Dirac wrote his now-famous equation that describes how relativistic electrons behave, he noticed that it had two solutions -- sort of like a quadratic equation.

At first he doubted that there could really be two kinds of matter, as his equation suggested. But within a few years antimatter was observed, and Dirac forever afterward wished he had been willing to take his own equation at face value and predict antimatter himself.

My point? The math told him something about reality that no one knew, even him. To put it another way, the math was reality.

Do you mean the computer-graphics movie, or The Last Airbender? None of those really has any mathematical interest for the viewer -- though The Last Airbender rocks.

I meant Avatar, the computer-graphics movie, although I really like The Last Airbender, and I wish it had gotten more attention.

Avatar is nothing more or less than a very long mathematical expression, sort of like a computer plot conducted very carefully for aesthetic reasons. When they watch Avatar, viewers are looking at mathematics.

> None of those really has any mathematical interest for the viewer

People who watch Avatar, who find it interesting, are appreciating mathematics even though they may not realize it. All the lighting, the colors, the motions, are expressed by the mathematics of physical reality.

When a real tree falls, it's very mathematical. When the big tree fell in Avatar, that was also mathematical. As to the first, mathematics could be used to predict exactly how the real tree falls. As to the second, the falling tree was the result of a computer solving a differential equation that described the tree and gravity.

They're really the same -- mathematics predicting reality, and mathematics generating an imaginary reality.