# Thread: Understanding formulas, shapes, algorithms etc

1. but the prob is, that even maths contains conflicts between facts derived directly from axioms...

And brewbuck, I did not understand your questions, sorry

2. We are creating laws that work in certain circumstances... But possibly, not in all...

3. The 1/3 * s * h formula can be proven without calculus.

Consider the unit cube, with vertices (0,0,0), (0,0,1), through (1,1,1).

There are three faces that don't touch the origin. Connecting them to the origin forms three congruent cones (or pyramids, since they have square bases) with base area 1 and height 1 that fill up the volume of the cube.

Any other cones are formed by linear transformations and rearranging the bases' shapes. (The rearranging the bases' shapes part can be proven without going through all of calculus, too, but that depends on what your definition of 'area' and 'volume' are. And on how radically you're willing to rearrange bases' shapes.)

4. Originally Posted by hardi
but the prob is, that even maths contains conflicts between facts derived directly from axioms...
Can you give an example?

And brewbuck, I did not understand your questions, sorry
My question is about the quantifiable nature of reality. The only way we can really quantify anything is to measure it via some measuring apparatus. If the quantity being described is unmeasurable, can it really exist?

The real question is, "What does it mean to measure something?"

5. Originally Posted by hardi
but the prob is, that even maths contains conflicts between facts derived directly from axioms...
That's if you choose axioms that are inconsistent. Why would you do that?
Whenever such contradictions occur, mathematicians generally like to fix the axioms. See naive set theory vs. Zermelo-Fraenkel set theory.

6. Originally Posted by brewbuck
Can you give an example?

My question is about the quantifiable nature of reality. The only way we can really quantify anything is to measure it via some measuring apparatus. If the quantity being described is unmeasurable, can it really exist?

The real question is, "What does it mean to measure something?"
Umm, i think of it as getting the most exact amount as possible...

7. I hate proofs with a passion.

8. I think this is an example(though i am not sure, maybe my definitions are errenous).

Given a straight line and a point, there is only one line that goes through that point and is parallel to the other line. (I think this is one of the axioms...).

In case we are talking about a sphere, then there can be no such thing... (ofcourse, nothing is derived here). Actually, i don't know of any good examples about colliding derivations. And maybe I missed out a part of the definition...

9. Originally Posted by hardi
Given a straight line and a point, there is only one line that goes through that point and is parallel to the other line. (I think this is one of the axioms...)
Trying to apply such an axiom to the surface of a sphere is nonsensical. For one thing, you haven't defined "line" and "parallel" in the context of a sphere. A line on the surface of a sphere, when considered in a Euclidean space, isn't a line at all.

You see contradictions because the axioms have been (over) simplified.

10. very possible - i just need to recheck them

11. If you're doing geometry on the surface of a sphere, you're working with a different set of axioms. There's nothing contradicting, you're just working with a different set of axioms. That's like defining your own addition operator on the integers and saying there are conflicts in the axioms because what you've defined implies 2 + 2 = -4 while another definition of addition implies that 2 + 2 = 4.

12. yup, that axiom just doesn't work on a sphere.