Thread: Mathematical Numbers are Theoretical; Not Proven

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.

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.

Uh.

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

Uh?

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

Perhaps the bit I posted below it.

It's a universal truth within Peano arithmetic. Which is a created system.

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?

Okay, put it like this: Euclidian geometry and hyperbolic geometry are two different mathematical systems with different sets of axioms. Which one of these is real?

Appealilng to supposed authority is not going to cut it.

Oh Lord...

Look, why are you here? If you care about this issue, why won't you even read into it? There's absolutely no way you can get into a discussion about mathematical philosophy without first knowing about Goedel for goodness sakes.

You cannot seriously be saying that appealing to a mathematical proof is a logical fallacy...

Originally Posted by Xei

Oh Lord...

Look, why are you here? If you care about this issue, why won't you even read into it? There's absolutely no way you can get into a discussion about mathematical philosophy without first knowing about Goedel for goodness sakes.

You cannot seriously be saying that appealing to a mathematical proof is a logical fallacy...

My point was that saying, "Well, this other guy says otherwise," does not explain to me how this many ** ** is not this many ****. I can read the book in time, but we will probably be pretty far away from this discussion by then.

Originally Posted by Xei

Okay, put it like this: Euclidian geometry and hyperbolic geometry are two different mathematical systems with different sets of axioms. Which one of these is real?

I have only studied Euclidean geometry. It says the opposite angles of a parallelogram are congruent and other such undeniable facts. Does hyperbolic geometry say otherwise?

Originally Posted by Xei

Uh.

Uh?

You are backing down to my challenge? Make it where 714 times any number equals the number. Show me how any number plus 38 equals the number. Demonstrate this for me and tell me how real it is. My eyes are open.

My point was that saying, "Well, this other guy says otherwise," does not explain to me how this many ** ** is not this many ****. I can read the book in time, but we will probably be pretty far away from this discussion by then.

It's not a book, it's a proof... it's certainly one of the more famous proofs in history, and has an extremely important place in the history of mathematics and philosophy in the 20th century. Just Google it, the very informal version is quite easy to grasp and I've posted it in this thread (it's also explained as a part of many famous philosophical books such as Goedel, Escher, Bach and The Emperor's New Mind).

I have only studied Euclidean geometry. It says the opposite angles of a parallelogram are congruent and other such undeniable facts. Does hyperbolic geometry say otherwise?

Yep, in hyperbolic geometry, two lines at angles to each other don't have to intersect. The interior angles of a triangle add up to less than 180.

It's ironic that you refer to those rules as 'undeniable', because actually, in this universe at least, they're wrong. This is just what I mean by basing axioms upon nothing more than common experience.

You are backing down to my challenge? Make it where 714 times any number equals the number. Show me how any number plus 38 equals the number. Demonstrate this for me and tell me how real it is. My eyes are open.

You could do that if you wanted, but the resultant system would evidently be pretty useless.

However, changing other axioms needn't have a degenerative effect. For example, you can do away with Euclid's parallel axiom, and you still get a consistent system.

Originally Posted by Xei

It's not a book, it's a proof... it's certainly one of the more famous proofs in history, and has an extremely important place in the history of mathematics and philosophy in the 20th century. Just Google it, the very informal version is quite easy to grasp and I've posted it in this thread (it's also explained as a part of many famous philosophical books such as Goedel, Escher, Bach and The Emperor's New Mind).

Like you said, it's in books, like Godel, Escher, Bach. I am familiar with Godel and the fact that he has a "proof". I just don't think merely mentioning its existence suffices for a counterargument.

Originally Posted by Xei

Yep, in hyperbolic geometry, two lines at angles to each other don't have to intersect. The interior angles of a triangle add up to less than 180.

Is it because of different language or because placing angles congruent to the angles of a triangle next to each other would not form a line? They in fact form a line. Any system that says otherwise is fiction.

Originally Posted by Xei

It's ironic that you refer to those rules as 'undeniable', because actually, in this universe at least, they're wrong. This is just what I mean by basing axioms upon nothing more than common experience.

Opposite angles in a parallelogram are not congruent?

Originally Posted by Xei

You could do that if you wanted, but the resultant system would evidently be pretty useless.

However, changing other axioms needn't have a degenerative effect. For example, you can do away with Euclid's parallel axiom, and you still get a consistent system.

Are you going to take me up on the challenge or not? It gets to the very heart of our discussion.

Why would the resulting system be useless? Is it because it would be out of line with reality?

It would be useless because every number would equal every other number, so you wouldn't be able to derive any meaningful results. Most of the time we only care about mathematical systems which model reality, like that of arithmetic for macroscopic experiences, although sometimes they are studied for their own sakes.

Opposite angles in a parallelogram are not congruent?

Nope. Space isn't Euclidian. That hypothesis is outdated by about a century now.

Originally Posted by Xei

It would be useless because every number would equal every other number, so you wouldn't be able to derive any meaningful results. Most of the time we only care about mathematical systems which model reality, like that of arithmetic for macroscopic experiences, although sometimes they are studied for their own sakes.

So it has to jive with reality to at least some extent? It has to jive with reality to the full extent.

Originally Posted by Xei

Nope. Space isn't Euclidian. That hypothesis is outdated by about a century now.

Outdated? WTF???? It wasn't that long ago that I was teaching it. I still help high school students with it sometimes. It is still taught in practically every high school in the U.S. and probably the world.

Parallelograms are two-dimensional while space is three-dimensional, but the second dimension does exist. Opposite angles of a parallelogram are congruent. I never dreamed I would be debating somebody on that some day.



A square is a parallelogram. I challenge you to do your absolute best and argue that the opposite angles of the square below do not have the same measure. Try to even convince me that all four angles of a square do not have the same measure. I really want to understand this.