Thread: Mathematical Numbers are Theoretical; Not Proven

Originally Posted by Xei

Imaginary numbers are rotations of 90 degrees. Rotations of 90 degrees are real. :l

What imaginary concepts are you referring to? I'd say that multiplication of a negative by a negative has no striking physical counterpart.

Not really, numbers are idealised concepts. You will never be 100% accurate or certain about any physical number.

Mathematics can be any language with any set of particular rules. What I'm saying is that there is no 'world of mathematics', because different systems of mathematics contradict each other. It isn't really any different from any other language, for example, English; there is no 'world of English' in some other mystical plane. English is just a human invention, and resides only within human minds.

Both attempt to describe a 'world of truths' - that is to say, reality - but both are flawed and limited.

Then why bother making pi = 3.14...? Why not just make it 3, or 1... or 0? Wouldn't that make a ton of computations much easier? Why not change the quadratic formula to a + b + c = 3? Why not make the slope formula m = x1 + x2 + y1 + y2? Why not make the distance formula d = 1?

Originally Posted by Universal Mind

Then why bother making pi = 3.14...? Why not just make it 3, or 1... or 0? Wouldn't that make a ton of computations much easier? Why not change the quadratic formula to a + b + c = 3? Why not make the slope formula m = x1 + x2 + y1 + y2? Why not make the distance formula d = 1?

Because they attempt to to describe a 'world of truths.' These formulas don't even attempt to make sense.

But I'm actually on your side. Just pointing out holes in your argument.

Then why bother making pi = 3.14...? Why not just make it 3, or 1... or 0? Wouldn't that make a ton of computations much easier? Why not change the quadratic formula to a + b + c = 3? Why not make the slope formula m = x1 + x2 + y1 + y2? Why not make the distance formula d = 1?

If we called a dog a fish would it be able to breathe underwater?

pi, for example, has the value it does because it is a consequence of the axioms of Euclidian geometry (you can estimate pi with a series of fractions); it is defined by and only has meaning within that set of axioms. Euclidian geometry is almost perfect when dealing with things in human experience, so we can reliably say that the circumference of any physical circle will be pretty much pi times the diameter.

We don't do any of those things you mentioned because we prove that the real results are true (using the rules of our system). I can't imagine you've never proved anything before... all of those except the pi one are completely trivial.

Originally Posted by Artelis

Because they attempt to to describe a 'world of truths.' These formulas don't even attempt to make sense.

But I'm actually on your side. Just pointing out holes in your argument.

I don't understand the supposed holes. The real formulas do more than attempt to describe a world of truths. They pull it off successfully.

Originally Posted by Xei

If we called a dog a fish would it be able to breathe underwater?

pi, for example, has the value it does because it is a consequence of the axioms of Euclidian geometry (you can estimate pi with a series of fractions); it is defined by and only has meaning within that set of axioms. Euclidian geometry is almost perfect when dealing with things in human experience, so we can reliably say that the circumference of any physical circle will be pretty much pi times the diameter.

We don't do any of those things you mentioned because we prove that the real results are true (using the rules of our system). I can't imagine you've never proved anything before... all of those except the pi one are completely trivial.

The formulas are what they are because they describe reality accurately, and the phony ones I mentioned are flat out false. The system we have to work in is not our invention; the system is reality. We could not possibly rearrange our system to where pi = this many ***. Of course we can change the language, but we cannot change the realities that are represented by our language.

What do you mean you can't imagine I've never proved anything before? How are the other formulas trivial? Please explain.

What can I say? You're wrong, that's all. I've already clearly explained why. It has even been proved with maths that this whole single true mathematical system idea is wrong, and you don't accept it, so there's really nothing I can say.

What do you mean you can't imagine I've never proved anything before? How are the other formulas trivial? Please explain.

Just that the answer to your question - the 'reason' those things are true in algebra etc. - is that you can prove them to be true using the rules of algebra.

Solve ax2 + bx + c = 0 for example and you get the quadratic formula. It wasn't found heuristically or something.

Originally Posted by Xei

What can I say? You're wrong, that's all. I've already clearly explained why. It has even been proved with maths that this whole single true mathematical system idea is wrong, and you don't accept it, so there's really nothing I can say.

You are wrong.

Originally Posted by Xei

Just that the answer to your question - the 'reason' those things are true in algebra etc. - is that you can prove them to be true using the rules of algebra.

The rules of algebra are rules of reality. 2 apples + 2 apples = 4 apples. By the same completely realistic reasoning, 2x + 2x = 4x. In other words, 2 x's + 2 x's = 4 x's. 2 of anything plus 2 of that same thing equals 4 of that thing. That was true before there were humans. It is not some crazy idea somebody had for the purpose of writing a fiction story. There are completely legitimate reasons we use the systems we use. We use them because they are factual and logical. It is not a mere art project.

Originally Posted by Xei

Solve ax2 + bx + c = 0 for example and you get the quadratic formula. It wasn't found heuristically or something.

The quadratic formula was discovered, not invented. The value of x in your equation is

-b +/- square root of (b squared - 4ac)
2a

That is in fact the value of x, not something some person decided would be neat to merely call the value of x. If the quadratic formula were just some crazy thing somebody decided to invent and do strange things with, it would have been much easier to make it x = a + b + c. There was no choice on the matter. The quadratic formula is what it is, and there is nothing we can do about it even though we can invent a new language.

You are wrong.

Just read about Godel. If you don't understand it, that's your failing.

The rules of algebra are rules of reality. 2 apples + 2 apples = 4 apples. By the same completely realistic reasoning, 2x + 2x = 4x. In other words, 2 x's + 2 x's = 4 x's. 2 of anything plus 2 of that same thing equals 4 of that thing. That was true before there were humans. It is not some crazy idea somebody had for the purpose of writing a fiction story. There are completely legitimate reasons we use the systems we use. We use them because they are factual and logical. It is not a mere art project.

Then what about the true algebraic facts that you can't work out with algabraic rules?

The 'obvious' rules of arithmetic often don't apply on the quantum scale, for example. They're only obvious because the only things we've ever experienced are those in immediate experience.

The quadratic formula was discovered, not invented. The value of x in your equation is

-b +/- square root of (b squared - 4ac)
2a

That is in fact the value of x, not something some person decided would be neat to merely call the value of x. If the quadratic formula were just some crazy thing somebody decided to invent and do strange things with, it would have been much easier to make it x = a + b + c. There was no choice on the matter. The quadratic formula is what it is, and there is nothing we can do about it even though we can invent a new language.

That is pretty much exactly what I said so I don't really know what you're going on about. Like I just said, you just solve ax2 + bx + c. You do understand how to do that, right..?

It was something which was 'discovered' in a system with particular rules - rules which were invented. The rules are those of Peano arithmetic, which include,

1. ∀x, y, z ∈ N. (x + y) + z = x + (y + z), i.e., addition is associative.
2. ∀x, y ∈ N. x + y = y + x, i.e., addition is commutative.
3. ∀x, y, z ∈ N. (x · y) · z = x · (y · z), i.e., multiplication is associative.
4. ∀x, y ∈ N. x · y = y · x, i.e., multiplication is commutative.
5. ∀x, y, z ∈ N. x · (y + z) = (x · y) + (x · z), i.e., the distributive law.
6. ∀x ∈ N. x + 0 = x ∧ x · 0 = 0, i.e., zero is the identity element for addition
7. ∀x ∈ N. x · 1 = x, i.e., one is the identity element for multiplication.
8. ∀x, y, z ∈ N. x < y ∧ y < z ⊃ x < z, i.e., the '<' operator is transitive.
9. ∀x ∈ N. &#172; (x < x), i.e., the '<' operator is not reflexive.
10. ∀x, y ∈ N. x < y ∨ x = y ∨ x > y.
11. ∀x, y, z ∈ N. x < y ⊃ x + z < y + z.
12. ∀x, y, z ∈ N. 0 < z ∧ x < y ⊃ x &#183; z < y &#183; z.
13. ∀x, y ∈ N. x < y ⊃ ∃z ∈ N. x + z = y.
14. 0 < 1 ∧ ∀x ∈ N. x > 0 ⊃ x ≥ 1..
15. ∀x ∈ N. x ≥ 0.

There are plenty of systems which use contradictory or separate axioms.

And as I keep trying to communicate to you, there are many true facts about arithmetic which can't be proved in the above system. So your 'obvious facts' are really completely arbitrary and limited.

Originally Posted by Xei

Just read about Godel. If you don't understand it, that's your failing.

Appealilng to supposed authority is not going to cut it.

Originally Posted by Xei

Then what about the true algebraic facts that you can't work out with algabraic rules?

What about them? I didn't say all mathematical facts can be proven. It is not an issue.

Originally Posted by Xei

The 'obvious' rules of arithmetic often don't apply on the quantum scale, for example. They're only obvious because the only things we've ever experienced are those in immediate experience.

They definitely apply in the reality we are now in. If what you have said is true, then maybe you should redesign our reality's math to where it does apply on the quantum scale. How would that work?

Originally Posted by Xei

That is pretty much exactly what I said so I don't really know what you're going on about. Like I just said, you just solve ax2 + bx + c. You do understand how to do that, right..?

Yes, and the quadratic formula works every time. It is a universal truth. What is your point?

Originally Posted by Xei

It was something which was 'discovered' in a system with particular rules - rules which were invented. The rules are those of Peano arithmetic, which include,

1. ∀x, y, z ∈ N. (x + y) + z = x + (y + z), i.e., addition is associative.
2. ∀x, y ∈ N. x + y = y + x, i.e., addition is commutative.
3. ∀x, y, z ∈ N. (x · y) · z = x · (y · z), i.e., multiplication is associative.
4. ∀x, y ∈ N. x · y = y · x, i.e., multiplication is commutative.
5. ∀x, y, z ∈ N. x · (y + z) = (x · y) + (x · z), i.e., the distributive law.
6. ∀x ∈ N. x + 0 = x ∧ x · 0 = 0, i.e., zero is the identity element for addition
7. ∀x ∈ N. x · 1 = x, i.e., one is the identity element for multiplication.
8. ∀x, y, z ∈ N. x < y ∧ y < z ⊃ x < z, i.e., the '<' operator is transitive.
9. ∀x ∈ N. ¬ (x < x), i.e., the '<' operator is not reflexive.
10. ∀x, y ∈ N. x < y ∨ x = y ∨ x > y.
11. ∀x, y, z ∈ N. x < y ⊃ x + z < y + z.
12. ∀x, y, z ∈ N. 0 < z ∧ x < y ⊃ x · z < y · z.
13. ∀x, y ∈ N. x < y ⊃ ∃z ∈ N. x + z = y.
14. 0 < 1 ∧ ∀x ∈ N. x > 0 ⊃ x ≥ 1..
15. ∀x ∈ N. x ≥ 0.

All discovered. If you disagree, make 38 the identity element for addition and 714 the identity element for multiplication. I would love to see that.

Originally Posted by Xei

There are plenty of systems which use contradictory or separate axioms.

And as I keep trying to communicate to you, there are many true facts about arithmetic which can't be proved in the above system. So your 'obvious facts' are really completely arbitrary and limited.

Then create those rules I brought up. Try doing that and telling me you have invented a system of reality and not fiction.