hmmm... here's a question for ya!

[ELorenz]

When speaking about mathematics you have to be very careful about how you say things. When you say that f and U both have a square relationship it is true, they could be in a set of functions where that is the parameter. However since I defined f such that f(x)=x^2 the only function that I can think of off the top of my head that is truly equal in different terms is just writing it out. f(x) = x^2 = x*x, this is true. However it is not true that x^2 = 1/2*k*x^2, unless you specifically define that k is always equal to 2, which it isn't. In the way that I have defined f it might not even be a function on real numbers, it could be polynomials. I was actually thinking of real numbers when i wrote it though so I'll concede that point.

If you look at this link http://www-groups.dcs.st-and.ac.uk/~his ... paces.html it states that vectors actually first appeared in geometry in the early 1800's. And yes I did mean magnitude when I said quantity.

It's late at this point and my brain hurts, but I would take a closer look at your statement about coordinate space and euclidean space. The coordinate space is the product space over an n dimensional field, (yes i had to look that up), which the reals and complex number systems are fields and vector spaces. Therefore R^n and C^n would be coordinate spaces, and the inner product space is defined on both of those fields. You might be thinking of the Euclidean metric rather than Euclidean space. Euclidean space is something that is discussed within geometry and physics relating to the existence of a rectangle in some respects, long discussion, but anyway, the Real and Complex fields have the Euclidean metric, if I'm remembering right then that property allows them to be ordered in ascending order.

I however am rambling, I would like to hear more about what you meant by that statement if I'm misinterpreting it.

[JMas]

ELorenz,
Thank you for your reply.

When speaking about mathematics you have to be very careful about how you say things.

I agree.

However it is not true that x^2 = 1/2*k*x^2.

I agree, I never claimed otherwise.

What does f(x) = x^2 describe in nature

I interpreted this question as follows:
"Is there any relationship in nature where there is a dependence on the square of the independent variable ?"
In the context of the topic being discussed, I thought that is what you were after.

it states that vectors actually first appeared in geometry

I didn't get that impression.

The first sentence in the book, "Linear Algebra Thoroughly Explained", Milan Vujicic writes:

The idea of a vector is one of the greatest contributions to mathematics, which came directly from physics.

a point itself could be a vector in mathematics.

How do you represent a point in real space? Normally by an n-tuple of real numbers representing its location from a known reference point. This n-tuple of numbers is (sometimes) called a real coordinate space (As you haven't specified otherwise, I assumed a real space).

The definition of Euclidean space I was thinking about can be found in "Linear Algebra", on page 50,

A finite dimensional vector space over R with an inner product defined on it is called an Euclidean space

By inner product, all I (and the definition above) mean is a map that takes a pair of vectors to a real number (and satisfies other properties). I thought that this would be clear from context given that your first example was of physical vectors where the inner product is the vector scalar product.

are fields and vector spaces.

Fields satisfy the axioms of a vector space, but not vice versa.

You might be thinking of the Euclidean metric rather than Euclidean space.

No, I am not. A metric is a generalization of a "distance" between two members of a set. This is clearly not what I was referring to. In fact, this is a completely different topic altogether.

Please note, I have not considered/discussed vector spaces over C.

Apologies for going off-topic.

[ELorenz]

As summer school is in session, and getting to the latter part of it, I am losing the ability to communicate well, lol. I'm actually glad you got involved in the discussion.

I will take a book definition over a web article any day.

My ability to intelligently discuss C is very limited as I haven't used to do anything, just learned some of its' properties. Not that my last post did much to raise the intelligence flag for me. Classic case of not following ones own advice of being very careful about what you say in mathematics. lol

I was confusing the Euclidean metric with something else also.

[whatabrother]

The study of mathematics becomes more extensive every time an unsolvable problem arrises. All mathematical equestions can solve a problem in nature regardless of weather or not the problem actually exists. f(x)=x^2 most obviously provides the answer to the area of a square. Any function you can come up with describes the behaviour of some idea found in nature. Sometimes the idea is not always possible, but nevertheless it can be described.

[-Ege-]

I think math is invented and that it is a language. For example no one invented the fact that 1+1 is 2. They discovered that fact and invented mathematics to describe it.

[nosidius]

but who says that they didn't event it as a problem instead of a fact? as it is i believe mathematics to be a man made thing..... it's not perfect, nor will it ever be

you cannot define something as discovered as it was always there. While this may just be my opinion, i believe i to be true in its own sense


however, feel free to discuss this, i'm not closed to new ideas...........

[JSturgesJr]

ELorenz,
I think I know the point you're trying to make. Mathematics is a language that can be used to model nature, though, this is not its sole responsibility/endeavour (this is a huge topic).

As for your f(x) = x^2, well, this relation describes one of the more important natural laws. The Harmonic Potential:

U = 1/2 * k * x^2

where;
U = Harmonic Potential
k = a positive constant
x = displacement

This has been used in a lot of models, for example, an approximation of lattice vibrations in crystals (small amplitude), called the "Harmonic Limit" or "Harmonic Approximation" and the Quantum harmonic oscillator (V. Important) amongst others.

don't forget the trajectory of an apple shot by a cannon.

[Akumetsu]

I think discovered more so, well math is included in everything no matter how you try to avoid it. Numbers imo are in EVERYTHING Measurements, weights, ways to analyze things, time, space, etc.

MATH IS POWER!!

[5c71p10v37]

Is math discovered, or invented?

IDK AN REALLY IDC i like math without math they would be no value of money now heres a question for yall 2+2=4 so the square root is 2 an the square root of 2 is 1 in the square root of 1 is .5 an the point of .5 is 0.25 an the root of that is ???

[cilpolir]

please let this topic die....
/getting my shotgun to kill the zombies