Before understanding theorems in elementary Euclidean plane geometry

[LittleRookie]

Before looking at the proof of basic theorems in Euclidean plane geometry, I feel that I should draw pictures or use other physical objects to have some idea why the theorem must be true. After all, I should not just plainly play the "game of logic". And, it is from such observations in real life that result in the list of axioms. But I have great difficulty in doing so. In other words, I have great difficulty in understanding the natural behaviour in the real world that is described in the theorem. For example, below are two basic theorems from Euclidean plane geometry.

1. Given two lines cut by a transversal. If a pair of corresponding angles are congruent, then the lines are parallel.

2. (Basic proportionality theorem) If a line parallel to one side of a triangle intersects the other two sides in distinct points, then it cuts off segments which are proportional to these sides.
(and its converse)

Another example, 3. the sequence of 1/n (n is a positive integer) converges to 0. We can use the axiom of completeness to prove the Archimedean Property of R, which then can be used to prove the above statement. Basically, we set up the axiom so that statement 3 is true. In other words, we want 1/n to converge to 0. Again, I have difficulty visualizing that any positive real number must be smaller than some 1/n using physical objects.

What would you do for these 3 theorems? Feel free to use other basic theorems as examples in addition to these 3 theorems.
What should I do to overcome this problem?

Edit for further clarification, I'm asking a "non-mathematics" question. I'm not asking about proof technique, or how to prove the 3 theorems. I'm asking about the process before even thinking about a mathematical proof of the theorem.

 

[member 587159]

This goes both ways.

You start with some axioms: these are rules on which you agree that these definitely hold. You can start proving things with these rules and some of the theorems you proved will not be (immediately) intuitively clear. This is one of the reasons that math is so powerful: we can prove things we wouldn't otherwise notice!

In the other way, a good intuition can give rise to a good rigorous proof.

I think it is definitely a good idea to draw a lot of pictures. But sometimes a picture can also cloud your vision or does not cover all cases, so just use them as visual aid, never as entire proof.

 

[MidgetDwarf]

Before looking at the proof of basic theorems in Euclidean plane geometry, I feel that I should draw pictures or use other physical objects to have some idea why the theorem must be true. After all, I should not just plainly play the "game of logic". And, it is from such observations in real life that result in the list of axioms. But I have great difficulty in doing so. In other words, I have great difficulty in understanding the natural behaviour in the real world that is described in the theorem. For example, below are two basic theorems from Euclidean plane geometry.

1. Given two lines cut by a transversal. If a pair of corresponding angles are congruent, then the lines are parallel.

2. (Basic proportionality theorem) If a line parallel to one side of a triangle intersects the other two sides in distinct points, then it cuts off segments which are proportional to these sides.
(and its converse)

Another example, 3. the sequence of 1/n (n is a positive integer) converges to 0. We can use the axiom of completeness to prove the Archimedean Property of R, which then can be used to prove the above statement. Basically, we set up the axiom so that statement 3 is true. In other words, we want 1/n to converge to 0. Again, I have difficulty visualizing that any positive real number must be smaller than some 1/n using physical objects.

What would you do for these 3 theorems? Feel free to use other basic theorems as examples in addition to these 3 theorems.
What should I do to overcome this problem?

In general. Depending on the method of proof needed. Draw an image as general as possible.

It is important to note that no diagram constitutes a replacement of a properly written proof. Sometimes an image can not be drawn, the higher you go up the Mathematical food chain. Moreover, Euclidean geometry is an idealized version of the world we actually live in.

Based on previous post, I think you are not understanding fundamentally what Mathematics is, and how Mathematicians and say, physicist view it differently.

Going back to the drawing of diagrams. A good rule of thumb is to draw them as general as possible, and to do by the method of proof you are trying to use.

Ie., for the first one regarding lines cut by a transversal which forms corresponding congruent angles.

I would first prove that lines cut by a transversal which forms congruent alternate interior angles are parallel.

The method that comes to mind is proof by contradiction.

So I would draw my diagram such as the lines being cut by transversal which forms congruent alternate interior angles, BUT the diagram would have the lines intersecting somewhere, say far to the right. This choice is arbitrary.

But it was done so, because this diagram helps the proof writer think about the contradiction...

 

[MidgetDwarf]

You can overcome this problem by reading and doing more mathematics. It boils down to experience.

What level of education are you at? This would help a lot and what resource are you using to learn geometry?

Also. are you going about learning Euclidean Geometry, proving only those theorems which hold up in Absolute Geometry?

 

[MidgetDwarf]

@MidgetDwarf I have no problem with proof techniques and proving the 3 theorems given in my example. What I'm asking is not about formally proving them using the axioms. You misunderstood my post.

Re read my post. It answered your questions and how to think about diagrams when doing proofs for elementary geometry...

 

[MidgetDwarf]

@MidgetDwarf I quote my concern in the post, "In other words, I have great difficulty in understanding the natural behaviour in the real world that is described in the theorem. "

Thanks anyway for your reply, I'll be waiting for replies from other users.

I answered it by stating that Euclidean Geometry is an idealized model of the real world... That is the simplest answer one can give.
You also mentioned diagrams..
Which I answered and gave you an example of how this reasoning is applied...

Maybe the post went over you’re head. I would suggest to refrain from typing replies as in the post above myne. It’s borderline rude, especially when a member took time to give a detailed explanation.

 

[MidgetDwarf]

I answered it by stating that Euclidean Geometry is an idealized model of the real world... That is the simplest answer one can give.
You also mentioned diagrams..
Which I answered and gave you an example of how this reasoning is applied...

Maybe the post went over you’re head. I would suggest to refrain from typing replies as in the post above myne. It’s borderline rude, especially when a member took time to give a detailed explanation.

I also asked what your education level was, so that I could better address you with a more appropriate response that you can gain some insight from...

You seem to be more interested in philosophizing about mathematics, and have a misunderstanding of mathematics in general. This is apparent in your thread in regards to mathematics as a deductive system.

 

[Mark44]

Another example, 3. the sequence of 1/n (n is a positive integer) converges to 0. We can use the axiom of completeness to prove the Archimedean Property of R, which then can be used to prove the above statement. Basically, we set up the axiom so that statement 3 is true. In other words, we want 1/n to converge to 0. Again, I have difficulty visualizing that any positive real number must be smaller than some 1/n using physical objects.

Why do you need physical objects? It should be easy enough to understand that for a given n and its reciprocal 1/n, then 1/(2n) will be smaller.

If you absolutely have to have a physical object to think about, imagine a pie cut into n equal portions, with each portion amounting to 1/n of the whole pie. If you then cut one of these pie sections in half, you have a portion that is smaller than 1/n.

 

[LittleRookie]

Why do you need physical objects? It should be easy enough to understand that for a given n and its reciprocal 1/n, then 1/(2n) will be smaller.

If you absolutely have to have a physical object to think about, imagine a pie cut into n equal portions, with each portion amounting to 1/n of the whole pie. If you then cut one of these pie sections in half, you have a portion that is smaller than 1/n.

We can also say we want to have a number x that is less than 1/n for any positive integer n. Then, people in the past would have come up with a different standard list of axioms for real numbers (because the current one contradicts this idea). However, we do not want to have x in our real number system. This is what I meant by "In other words, we want 1/n to converge to 0". Here's the actual question I've asked in my post (and nobody seems to catch on it, maybe I'm bad at expressing myself), "Why wouldn't we want such a number x to exist in our real number system?" Since Mathematics (I can say at least the most basic high school part of mathematics) set up systems with the aim of mathematizing natural observations, I should try to observe (or a better word, visualize?) 1/n with n increasing infinitely, and hopefully come up with the conclusion that given a physical object U representing the unit (i.e. 1), and no matter how small another physical object V is, we can eventually cut U into sufficient parts, whereby one part is smaller than V. And my problem is, I can't.

 

[Mark44]

We can also say we want to have a number x that is less than 1/n for any positive integer n. Then, people in the past would have come up with a different standard list of axioms for real numbers (because the current one contradicts this idea).

This is not clear. What is the "standard list of axioms" that you're talking about, and how does the current list of axioms contradict what you're saying about x being less than 1/n?
I've already given you an example of a pie being cut into n equal pieces. If you cut one of these pieces in half, the resulting slice will be smaller than 1/n of the pie.

However, we do not want to have x in our real number system.

Why not?

This is what I meant by "In other words, we want 1/n to converge to 0". Here's the actual question I've asked in my post (and nobody seems to catch on it, maybe I'm bad at expressing myself), "Why wouldn't we want such a number x to exist in our real number system?"

Yes, maybe you aren't being clear. I can't think of any reason why such a number should not exist. I.e., a number x such that x < 1/n. In the example I gave, we have 1/(2n) < 1/n.

Since Mathematics (I can say at least the most basic high school part of mathematics) set up systems with the aim of mathematizing natural observations, I should try to observe (or a better word, visualize?) 1/n with n increasing infinitely, and hopefully come up with the conclusion that given a physical object U representing the unit (i.e. 1), and no matter how small another physical object V is, we can eventually cut U into sufficient parts, whereby one part is smaller than V. And my problem is, I can't.

The only problem with cutting a physical object like a pie into ever smaller pieces is that you eventually get down to pieces consisting of a single atom, which we presumably can't split by ordinary means.

 

[MidgetDwarf]

i think that maybe constructing the real numbers starting from the Peano Axioms would illuminate things.

I believe most of OP questions would be answered by more exposure to mathematics. Particularly an Intro Analysis Course and a book discussing Geometry at a higher level then the average high school book.Here is a book that is accessible to high school students without skimping on the mathematics and may illuminate what mathematics is and what it entails/concerns.

Moise/Downs: Geometry.

I would suggest purchasing this book and reading it. The author has a more advanced text, “ Elementary Geometry From an Advance Standpoint.”.

I would hold from studying this book. It’s kinda “advanced” and meant for a University Geometry Course.

He does talk about models...

So maybe reading parts of this book can also be illuminating.

 

[Infrared]

Yes, maybe you aren't being clear. I can't think of any reason why such a number should not exist. I.e., a number x such that x < 1/n. In the example I gave, we have 1/(2n) < 1/n.

The statement is that there is no $x>0$ such that $x<1/n$ $\textit{for all}$ natural numbers $n$. This is of course true.

It is worth noting that there are non-Archimedean ordered fields. For example, the field $\mathbb{R}((x))$ of formal Laurent series in $x$ can be given the lexicographical order (compare with coefficients of the smallest power of $x$, and if they are equal, move to the next, etc.). This order is compatible with the field operations, and $x<1/n$ for all natural $n$.

 

[Mark44]

We can also say we want to have a number x that is less than 1/n for any positive integer n.

The statement is that there is no $x>0$ such that $x<1/n$ $\textit{for all}$ natural numbers $n$. This is of course true.

Yes, certainly. I interpreted the OP's sentence (quoted above) as meaning for any specific positive integer n.