Is mathematics discovered, developed, or both?

[robvalue]

Developed, often with modelling reality in mind. But not necessarily.

Each piece of mathematics is an abstract system. To say that it has been "discovered" is to treat all possible abstract concepts as pseudo-existent; and that thinking about it is "finding" it.

The applications of mathematics upon reality can be discovered.

[Edwardo Piet]

It's all developed from the logical absolutes. Numbers are just symbols for counting but the logic behind it is all based off of the framework of the logical absolutes. As is all knowledge including our own self perception ("I think therefore I am" implies A=A).

[Chas]

All of math was developed, even the bits we "discovered". There was a thread a few years ago about whether Pi was discovered or invented.

Its a bit of both. Pi as a number was discovered, however the way it's defined (as the ratio of a circle's diameter to its circumference) is entirely invented. Circles are a human invention; they don't appear in nature.

If two bodies of equal mass are orbiting each other, the orbit is a circle.

[Tiberius]

If two bodies of equal mass are orbiting each other, the orbit is a circle.

It's not though. It looks very close to a what a circle looks like, but it will always be an approximation. If you got those bodies to leave a trail behind, and zoomed in on that trail, at some zoom level, you would find lots of straight lines at very small angles to each other, showing that the orbit was actually a many-sided polygon. A circle only has 1 side, and is made up of an infinite amount of points, which you can't get in the physical universe.

[Excited Penguin]

If two bodies of equal mass are orbiting each other, the orbit is a circle.

It's not though. It looks very close to a what a circle looks like, but it will always be an approximation. If you got those bodies to leave a trail behind, and zoomed in on that trail, at some zoom level, you would find lots of straight lines at very small angles to each other, showing that the orbit was actually a many-sided polygon. A circle only has 1 side, and is made up of an infinite amount of points, which you can't get in the physical universe.

That's fucking fascinating. Is it really true though, or is it like the infinitely halved distance across the room thing ?

[Tiberius]

That's fucking fascinating. Is it really true though, or is it like the infinitely halved distance across the room thing ?

Is what really true? That circles have an infinite number of points? That's true and proved. The equation for a circle is: x2 + y2 = r2, where x and y are points on a graph, and r is the radius, and the center of the circle is at (0,0). There are an infinite number of solutions to that equation, each solution being a point on the circle.

[Tiberius]

If you want a proof you can do yourself, think of it like this.

Pick a value for r, say 10. That means r2 is 100.

So we want values of x and y where x2 + y2 = 100.

Pick a value for x. Let's go with 10. 102 is 100, so our equation looks like this: 100 + y2 = 100.

Re-arrange: y2 = 100 - 100 = 0. So y = sqrt(0).

Now divide x by 2. 52 = 25, so our equation looks like this: 25 + y2 = 100.

Re-arrange: y2 = 100 - 25 = 75. So y = sqrt(75).

Keep dividing x by 2, and rearrange the equation, and you'll keep getting points for y. Plus, you can't ever reach 0 by dividing by 2, so no matter how small you make the value of x, you'll always be able to get a value for y.

[Tiberius]

I should also add that since we're talking about squares, for most points you'll find two solutions.

With the example where x is 5, and y is sqrt(75), there is also a solution where x is 5 and y is negative sqrt(75).

[Excited Penguin]

I can hardly keep up, myself, but the bits that shine through are still interesting, so thanks.

I'm more fascinated with the fact that infinity is an actual concept that can be proven, if only in mathematics, even if it seems simple on its face. I do wonder what it says about reality, though, if anything. Where does the similarity between math and real world break down? Are we biased to accept mathematic reasoning? Is it a self justifying explanation of the world where another one would work just as well or better? Granted, I'm not a mathematician by any stretch, not even amateurishly,  so these questions might very well sound like nonsense to those of you that are better acquainted with the subject. If so, then I apologize in advance for my naivete.

[Tiberius]

Mathematics at a basic level can help describe the world and how it works. For instance, we know what 2 apples looks like, and we know what 3 apples looks like, and via mathematics we can say that 2 apples + 3 apples is 5 apples.

What we can do with mathematics is pose questions (and find answers) that do not work in the real world. For instance, we cannot do 2 apples - 3 apples in the real world, but in mathematics we can ask the question "well, if we could, what would it be?" and the answer is -1 apples.