Thread: Mathematical Numbers are Theoretical; Not Proven

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.

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.

Lol, I really want to see this argument. It seems like common sense...

General relativity? :l

It has been empirically proven that space is not Euclidian, ie. in the real world, the opposite angles of a parallelogram are almost never equal.

Originally Posted by Xei

General relativity? :l

It has been empirically proven that space is not Euclidian, ie. in the real world, the opposite angles of a parallelogram are almost never equal.

Please explain. I want you to tell me how the square and the other parallelogram I posted do not have congruent opposite angles. I am not asking for a reference to something else. I am asking you to tell me in your own words how in the world those angles do not have equal measures.

Like I said, parallelograms are two-dimensional. Also, Euclidian geometry is a required subject in practically every high school in my country and probably yours. Engineers and astronauts depend on the truth of it.

Prove that they're equal.

You'll find that you reduce the problem to a few starting facts which you cannot actually prove - you just have to say that they're 'obvious' (these are Euclid's postulates). One of these is the 'obvious' truth that two lines must be parallel if you don't want them to ever meet. This is in fact just wrong. Reality doesn't work like this. It just approximates to it on human scales.

As a result, everything that you prove with Euclid's axioms is also not necessarily true about reality. This includes the idea that the interior angles of a triangle add to 180 (they don't) and that opposite angles of a parallelogram are equal.

You definitely need to read about Einstein's General Theory of Relativity if you want to understand the basics of this. The formal theory is extremely complicated and requires several years of study to understand, but it is so important that it has been covered extensively in layman's terms. Just Google it. As I said, it has been empirically verified many times.

Ironically astronauts depend on Euclidian geometry not being correct. If we tried to use it, satellites would not stay in orbit. It's wrong.

Originally Posted by Xei

Prove that they're equal.

Take two parallel lines and make them perpendicular to two other parallel lines. All four interior angles created will be right angles. Then lean the first two parallel lines the same amount. The opposite angles will change exactly the same amount. That is because exactly the same thing is being done to them.

Now you... (This is about the fourth request.)

Originally Posted by Xei

You'll find that you reduce the problem to a few starting facts which you cannot actually prove - you just have to say that they're 'obvious' (these are Euclid's postulates). One of these is the 'obvious' truth that two lines must be parallel if you don't want them to ever meet. This is in fact just wrong. Reality doesn't work like this. It just approximates to it on human scales.

Skew lines never meet and are not parallel. That is because they are not on the same plane. In one plane, only parallel lines never intersect. If you can explain otherwise, it will be the most interesting explanation I have requested.

Originally Posted by Xei

As a result, everything that you prove with Euclid's axioms is also not necessarily true about reality. This includes the idea that the interior angles of a triangle add to 180 (they don't) and that opposite angles of a parallelogram are equal.

I don't see how in the world they would not. Also, please tell me why pretty much every high school and engineering and architecture school in the world teaches that they do. Can you tell me that?

Originally Posted by Xei

You definitely need to read about Einstein's General Theory of Relativity if you want to understand the basics of this. The formal theory is extremely complicated and requires several years of study to understand, but it is so important that it has been covered extensively in layman's terms. Just Google it. As I said, it has been empirically verified many times.

I have read about it. When you get the 4th dimension, time, into equations, bizarre things happen, but they do not contradict the postulates and theorems of two-dimensional geometry.

Originally Posted by Xei

Ironically astronauts depend on Euclidian geometry not being correct. If we tried to use it, satellites would not stay in orbit. It's wrong.

That is not true. The Pythagorean Theorem is used to determine vectors, for example.

Now please explain what I have requested, especially why Euclidian geometry is a requirement at practically every high school in the world and why architects and engineers have to know it and use it.

It's simple maths. They wouldn't teach it and make it required if it was false.

|----------------|
| a b |
| |
|c d|
|----------------|
a=90
b=90
c=90
d=x

a+b+c+d=360 (Proof of 360, try it with a protractor.)
90+90+90+x=360
270+x=360
-270 -270
=========
x= 90

Take two parallel lines and make them perpendicular to two other parallel lines. All four interior angles created will be right angles. Then lean the first two parallel lines the same amount. The opposite angles will change exactly the same amount. That is because exactly the same thing is being done to them.

Now you... (This is about the fourth request.)

Sorry, that's not a valid geometric proof. But you've illustrated my point all the same; you had to resort to 'obvious' facts which you could only justify by your experience.

Skew lines never meet and are not parallel. That is because they are not on the same plane. In one plane, only parallel lines never intersect. If you can explain otherwise, it will be the most interesting explanation I have requested.

I'm not talking about skew lines, I'm talking about lines in a plane in hyperbolic geometry, of which there are many non-parallel lines which do not intersect.

I don't see how in the world they would not. Also, please tell me why pretty much every high school and engineering and architecture school in the world teaches that they do. Can you tell me that?

Once again; it's an extremely good approximation on human scales, but on larger scales, it is completely wrong.

Architects don't build things larger than solar systems, remember?

I have read about it. When you get the 4th dimension, time, into equations, bizarre things happen, but they do not contradict the postulates and theorems of two-dimensional geometry.

It sounds like you're getting confused with special relativity.

And yes, for the nth time, reality does contradict those postulates (there's no need to restrict them to two dimensions though). Seeing as you've repeatedly refused to look up any of this (it's extremely easy to find out about and is covered in the introduction of both the general relativity and Euclidian geometry pages on Wikipedia), here's independent verification:

For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein's theory of general relativity is that Euclidean geometry is a good approximation to the properties of physical space only if the gravitational field is not too strong.

That is not true.

Wrong.

Originally Posted by Xei

Sorry, that's not a valid geometric proof. But you've illustrated my point all the same; you had to resort to 'obvious' facts which you could only justify by your experience.

I resorted to logic. You reject the geometric proofs, so I gave you another way to look at it. Can you counter the logic I used?

Originally Posted by Xei

I'm not talking about skew lines, I'm talking about lines in a plane in hyperbolic geometry, of which there are many non-parallel lines which do not intersect.

Explain. (!!!!!!!!!!)

Originally Posted by Xei

Once again; it's an extremely good approximation on human scales, but on larger scales, it is completely wrong.

How?

Originally Posted by Xei

It sounds like you're getting confused with special relativity.

I was just taking a guess at what you could possibly be talking about since you are not telling me.

Originally Posted by Xei

And yes, for the nth time, reality does contradict those postulates (there's no need to restrict them to two dimensions though). Seeing as you've repeatedly refused to look up any of this (it's extremely easy to find out about and is covered in the introduction of both the general relativity and Euclidian geometry pages on Wikipedia), here's independent verification:

That is not an explanation. It is just one more assertion. That is all you are giving me.

Originally Posted by Xei

Wrong.

Wrong. Are you saying NASA does not use the Pythagorean Theorem to determine vectors? What do you think they use?

Why is Euclidian geometry taught as FACT in practically every high school, school of architecture, and school of engineering in the world????????????????

Once again; it's an extremely good approximation on human scales, but on larger scales, it is completely wrong.

For fuck's sake...

Originally Posted by Xei

For fuck's sake...

They are not taught as approximations. They are taught as FACTS. Why??????????????????????????????????????????????? ?????????????????????

Why do they work so well as "approximations"?

Explain your assertions when you think you are ready.

An implication of Einstein's theory of general relativity is that Euclidean geometry is a good approximation to the properties of physical space only if the gravitational field is not too strong.

.

Originally Posted by Xei

.

That is an appeal to an appeal to authority, not an explanation. You are playing some major dodgeball in this discussion.

The formal theory is extremely complicated and requires several years of study to understand, but it is so important that it has been covered extensively in layman's terms. Just Google it. As I said, it has been empirically verified many times.

.