2 HS students from NOLA shake up the world of math in a huge way... (1 Viewer)

[DCSaints_Fan]

The track analogy was to point out that your criticism was for a race they weren't trying to run, solving the theorem without using trig.

No, thats not what they did.

Many non-trigonmetric proofs of Pythagorean theorem exist.

What they are trying to show is a trigonometric proof of the Pythagorean theorem, that does not involve circular reasoning i.e., contain the Pythagorean theorem as one of its implicit assumptions.

 

[DCSaints_Fan]

Ok I'm going to go ahead and do a more in depth dive into this one. First I'll handle some preliminaries before I go into the details of the paper. So please excuse the long-windedness of this post - which I'm going to break up into several parts.

The claim in the paper (its right in the abstract) is they can prove Pythagorean theorem (a^2 + b^2 = c^2, where a, b, and c are the sides a right triangle), using trigonometry.

Part I - Preliminaries

What is a proof? In the mathematical context, it means showing a mathematical statement to be true, based on other true statements.

When constructing mathematics, you have to start somewhere - in other words you have to assume certain statements are true. These are usually called axioms, postulates and in a certain form definitions.

They are also terms which lack a definition. e.g. in plane geometry, point line and plane. Although we have an intrinsic conception of what these are (a point is a "dot", a line is a "inifite collection of points between two points that is contiguous and straight" ), since the language of math should be precise, if you can't do that its best to leave them undefined. "Number" is another example - we simply just "know" what a number is without having to provide a definition.

Also existing somewhat parallel to these axioms, are the "rules" of logic. These include ideas such as "if a statement P (aka predicate) is true, and P implies Q, then Q must also be true" and "if P implies Q (is true), and Q is false, then P is also false". Since these permeate every branch of mathematics, they are often simply assumed.

By starting with the axioms, together with rules of logic, we can construct new true statements, we didn't know before. This is called a proof. These new true statements are generally called theorems (sometimes lemmas or corollaries)

The theorems of geometry can be constructed assuming only five axioms, given by Euclid. I would note these postulates assume that certain terms like point, line, circle and angle are already understood.

Of particular interest in our case, is the third postulate having to do with the a circle. "Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center". Also the fourth postulate - "all right angles are concgruent". doesn't define right angle or even angle.

Now in terms out, you can define a circle as a "set of points equidistant from some center point".

Similarly, you can also define angle as something like the "length of the circular segment on a circle of radius 1 when intersected by two lines with a common origin at the center of the circle"

But in order to do that, we already have used the terms "length" and "equidistant". These are measures of geometric objects, in this case line segments. While not pure geometric ideas, in order to do anything much useful in geometry you need to introduce them.

So when we say something like the "length of the line segment AB is equal to the length of the line segment BC", what we mean is the number assigned to the measure of each of them is equal. For now assume the numbers obey the standard rules of algebra. e.g. if a = b then a + c = b + c, etc.).

(to be continued)

 

[Bleu Raeder]

Ok I'm going to go ahead and do a more in depth dive into this one. First I'll handle some preliminaries before I go into the details of the paper. So please excuse the long-windedness of this post - which I'm going to break up into several parts.

The claim in the paper (its right in the abstract) is they can prove Pythagorean theorem (a^2 + b^2 = c^2, where a, b, and c are the sides a right triangle), using trigonometry.

Part I - Preliminaries

What is a proof? In the mathematical context, it means showing a mathematical statement to be true, based on other true statements.

When constructing mathematics, you have to start somewhere - in other words you have to assume certain statements are true. These are usually called axioms, postulates and in a certain form definitions.

They are also terms which lack a definition. e.g. in plane geometry, point line and plane. Although we have an intrinsic conception of what these are (a point is a "dot", a line is a "inifite collection of points between two points that is contiguous and straight" ), since the language of math should be precise, if you can't do that its best to leave them undefined. "Number" is another example - we simply just "know" what a number is without having to provide a definition.

Also existing somewhat parallel to these axioms, are the "rules" of logic. These include ideas such as "if a statement P (aka predicate) is true, and P implies Q, then Q must also be true" and "if P implies Q (is true), and Q is false, then P is also false". Since these permeate every branch of mathematics, they are often simply assumed.

By starting with the axioms, together with rules of logic, we can construct new true statements, we didn't know before. This is called a proof. These new true statements are generally called theorems (sometimes lemmas or corollaries)

The theorems of geometry can be constructed assuming only five axioms, given by Euclid. I would note these postulates assume that certain terms like point, line, circle and angle are already understood.

Of particular interest in our case, is the third postulate having to do with the a circle. "Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center". Also the fourth postulate - "all right angles are concgruent". doesn't define right angle or even angle.

Now in terms out, you can define a circle as a "set of points equidistant from some center point".

Similarly, you can also define angle as something like the "length of the circular segment on a circle of radius 1 when intersected by two lines with a common origin at the center of the circle"

But in order to do that, we already have used the terms "length" and "equidistant". These are measures of geometric objects, in this case line segments. While not pure geometric ideas, in order to do anything much useful in geometry you need to introduce them.

So when we say something like the "length of the line segment AB is equal to the length of the line segment BC", what we mean is the number assigned to the measure of each of them is equal. For now assume the numbers obey the standard rules of algebra. e.g. if a = b then a + c = b + c, etc.).

(to be continued)

Is your last name Runnels, by any chance?

 

[livefromDC]

No, thats not what they did.

Many non-trigonmetric proofs of Pythagorean theorem exist.

What they are trying to show is a trigonometric proof of the Pythagorean theorem, that does not involve circular reasoning i.e., contain the Pythagorean theorem as one of its implicit assumptions.

You're really struggling with that analogy. Read it again, that's exactly what I said. Of course it's not what they did or what they were setting out to do. You criticized them for doing a proof using trig by comparing it to the people who proved the theorem without using trig.

Based on your earlier post, it seems you're just catching on to that fact and trying to change your tune. At any rate, I have no faith you will correctly interpret their proof.

 

[DCSaints_Fan]

You're really struggling with that analogy. Read it again, that's exactly what I said. Of course it's not what they did or what they were setting out to do. You criticized them for doing a proof using trig by comparing it to the people who proved the theorem without using trig.

Based on your earlier post, it seems you're just catching on to that fact and trying to change your tune. At any rate, I have no faith you will correctly interpret their proof.

That wasn't the main thrust of the criticism. That point there was that I don't see how using more complex concepts (trig) to prove simpler ones (Pythagorean identity) gives you a whole lot of insight.

I suspect they still have the Pythagorean theorem embedded in their assumptions, but I believe involves a degree of high level of geometric formalism as I was alluding to in Part I. In the paper they provide an alternative definition of the sien and cosine function based on the unit circle. Now the problem is do you define circle as a set of points equidistance from a point, which requires a definition of distance ...

 

[livefromDC]

I'm really failing to see the real impact. There have been hundreds of proofs, many of which are much more intuitive and do not involve trig, only geometric constructions and algebra (which they also use)

That wasn't the main thrust of the criticism. That point there was that I don't see how using more complex concepts (trig) to prove simpler ones (Pythagorean identity) gives you a whole lot of insight.

Which indicates that you're still missing the point. The Pythagorean Theorem doesn't need to be proven. It doesn't need another proof. The entire point is to use a method that hadn't been done before. There's no reason to use try to prove the theorem other than to attempt to do something people haven't been able to do. Math for math's sake.

 

[guidomerkinsrules]

You're really struggling with that analogy. Read it again, that's exactly what I said. Of course it's not what they did or what they were setting out to do. You criticized them for doing a proof using trig by comparing it to the people who proved the theorem without using trig.

Based on your earlier post, it seems you're just catching on to that fact and trying to change your tune. At any rate, I have no faith you will correctly interpret their proof.

let him cook
while recognizing that 'cook' is an undefined term

 

[livefromDC]

let him cook
while recognizing that 'cook' is an undefined term

Using a paper submitted by 2 high schoolers to try to show off your math prowess is unreal, but I guess on par for 2024.

 

[guidomerkinsrules]

but I guess on par for 2024.

What really is a number?

 

[DCSaints_Fan]

I can see this thread has devolved into personal attacks and there is a lack of interest in discussing the paper itself. As such I don't really find any purpose in continuing to post in this thread.

 

[Sun Wukong]

I can see this thread has devolved into personal attacks and there is a lack of interest in discussing the paper itself. As such I don't really find any purpose in continuing to post in this thread.

Unfortunate, as I was looking forward to your analysis on this. Always lame when actual discourse gets derailed by childish outbursts.

 

[DCSaints_Fan]

Unfortunate, as I was looking forward to your analysis on this. Always lame when actual discourse gets derailed by childish outbursts.

Thanks, I'll continue, probably this weekend.

I admit myself was derailed by a discussion of the media

So I will totally focus on just the paper itself now that its out. When this story first broke, I refraned from saying anything because there was no paper, but now that its out, there is alot more substance to discuss

 

[livefromDC]

Unfortunate, as I was looking forward to your analysis on this. Always lame when actual discourse gets derailed by childish outbursts.

I'm under no delusion that there's more than one adult here that wants to see these girls work "taken down" for whatever miserable excuse they can conjure.

I guess this desire to put these girls in their place is just so high you're also willing to overlook that he originally got the premise of their paper entirely wrong. Then when he got called out for that he started this middle school level, rudimentary explanation of geometry AS IF Trigonometry is not a subset of geometry and therefore well in line with the purpose of their proof.

But let's overlook all of that or the fact that the childish outburst here was him comparing their work to nonsense in an attempt to discredit it. He's been trying to downplay their achievement since the first page of the thread. And oh look, you're right there with him on page 2.  So even after their paper was peer reviewed and published in American Mathematical Monthly, which has been printed since 1894, a respected math journal, he was still trying to discredit it. He's the real victim here. Poor guy just wanted to pick on a paper written by a couple girls for trying to do math good.

I'm not sorry for pointing out utter bias and crap when I see it. I doubt some guy on SR that's just mad that these girls are getting attention for writing a paper proving the Pythagorean Theorem using Trigonometry is going to find something that peer review and a Mathematical Journal fiunded when Grover Cleveland was President didnt find. Especially when his preliminary breakdown is basically, "so they're using geometry..."

If he genuinely was trying to explain the paper in terms most could understand it would be one thing. If he was truly embracing their achievement and trying to understand it, it would be one thing. His tone and his words since page one have been dismissive. He hasn't hidden his skepticism and contempt from page one. I don't care if his feelings got hurt after being called out for attacking these girls' work. He should be ashamed.

 

[guidomerkinsrules]

Math Fight!!