This is a contribution I'd like to give to the "Math Support Gals" as soon as constituted. I'll revise it, but think it's of great importance.
The world, in this 2010, is "more mathematical" than we could imagine, for better or for worse.
If you use a credit card, the security of all your transactions relies on the assumption that it is very difficult to factor the product of two very large prime numbers.
In a very near future, you might wear a "state sensitive dress", holding sensors able to detect (for example) your hormonal balance, and changing color to inform yourself of an incoming flue or headache, or even more.
These are just little examples of the tens millions existing. The hard fact is, we are literally surrounded by an enormous amount of devices (mostly invisible) which open and close sliding doors, regulate airports, decide your career, protect you from harm, allow you to communicate, ... They all have one thing in common: a computer somewhere, and a more or less complicate set of rules telling it how to behave - and these rules are essentially mathematical. Designing and operating these devices confers an immense "practical" power, in large part passing unobserved, and which I wish we collectively become more aware of - that is, less passive.
Mathematics is not only the "big tool" allowing a large part of our artificial world to function. It's also a language used to communicate and convince. Think to statistics, for example: if you are about to vote, then you are likely to be bombarded by projections and exit polls apparently aimed at "informing" you, but as often used to manipulate your decisions. Statistics is routinely used as a powerful rhetorical weapon, by politicians and salespersons as well.
The World itself, as fuzzy and irregular as it is, can be better comprehended and, to some little extent, predicted in mathematical terms. Mathematics is a human creation, as any other form of culture (indeed, a fascinating one, once you master it a little bit) - so I will not arrive at claiming "the world is mathematical", as some people do (besides, I'm not "so" skilled to take a definite position). But mathematical models may arrive at level of predictive detail once unthinkable - let's get meteorological forecasts, as one (simple) example. And the immensely complex econometric models underlying the decision of "cutting half a point the dollar's discount rate", for another case.
Practical life, and dramatic decisions you may have to take, also have a deep mathematical basis.
Imagine this story. You live in a village with your beloved, and win your bread literally extracting it from land, cultivating a little field. One day an authoritative-looking man arrives on a flashy car, presents himself as a representative of a big food and agriculture multinational, and tries to convince you that it will be very convenient to use a certain kind of seeds and the treatments linked (maybe locked) on them.
Imagine another (this one has really happened to me). The usual authoritative-looking man arrives on the usual flashy car, presents himself as a bank consultant, and tries to convince you to change your pension plan to a more aggressive investment method.
What will you do?
These people will bombard us al with numbers and "hard facts", but then the decision relies on you. They're paid to defend their company interests, and might try doing so against yours. The decision might have a deep impact on your life, but also your children, spouse, overall community. You might find yourself in a good situation, but also in big troubles.
That decision will rely on confidence with numbers, and a form of mathematical reasoning.
All the same, when you will decide whether a book is good science or rubbish, a good is worth spending a given amount of money, the amount of your wage is fair, and so on.
In short, some mathematical knowledge is necessary, for us to be first class citizens of this world.
It's important, as an instrument to take more informed decisions.
It's useful, to see details of our world so often passing unobserved (and uncontrolled).
Mathematics can be a form of power, to claim, exert with responsibility, and acknowledge.
As any form of power, we can't give it away. In wrong hands can be too devastating. In the right ones, world changing for good to all.