A limits problem (fractal initiator & generator)....

[micromass]

If series a+b+c+d+...=X, then we are forced also to conclude that 2X=X√2.

Why don't you prove rigorously why $a+b+c+d+...=X$ implies $2X = X\sqrt{2}$. Showing a picture is not a proof.

 

[Look]

Why don't you prove rigorously why $a+b+c+d+...=X$ implies $2X = X\sqrt{2}$. Showing a picture is not a proof.

Dear micromass, since $a+b+c+d+...$ is defined by "zig-zag" lines that all of them have the constant length $2X$, then $a+b+c+d+...=X$ implies $2X = X\sqrt{2}$.

If you disagree with this implementation then please show that $a+b+c+d+...=X$ doe not imply $2X = X\sqrt{2}$.

Without loss of generality you can use the case $X=1$ (in that case 1/2+1/4+1/8+1/16...=1 which implies $2 = 1\sqrt{2}$).

If you agree that 1/2+1/4+1/8+1/16...=1, then the diagram is a proof without words that 1/2+1/4+1/8+1/16...=1 implies $2 = 1\sqrt{2}$.

 

[fresh_42]

Dear micromass, since $a+b+c+d+...$ is defined by "zig-zag" lines that all of them have the constant length $2X$, then $a+b+c+d+...=X$ implies $2X = X\sqrt{2}$.

If you disagree with this implementation then please show that $a+b+c+d+...=X$ doe not imply $2X = X\sqrt{2}$.

Without loss of generality you can use the case $X=1$ (in that case 1/2+1/4+1/8+...=1 which implies $2 = 1\sqrt{2}$).

The point is: You seemingly want to establish deduction principles that differ from the framework, that mathematics provides. Even in mathematics there are frameworks that differ from the mainstream. However, in these cases it is carefully and rigorously said to what extend and which are the distinct principles. You haven't done this at all. What you are trying to do is, to engage people arguing within the standard mathematical framework where you simultaneously use an obscure and undefined system of logic without telling it. This cannot match by construction. Drawn pictures and off-label usage of standard mathematical terminology are not appropriate means to overcome this lack of fundamental agreement - it only tries to disguise it.
[And I refuse to participate in this kind of debate.]

 

[micromass]

Dear micromass, since $a+b+c+d+...$ is defined by "zig-zag" lines that all of them have the constant length $2X$, then $a+b+c+d+...=X$ implies $2X = X\sqrt{2}$.

That is not a rigorous proof, you just made an assertion.

If you disagree with this implementation then please show that $a+b+c+d+...=X$ doe not imply $2X = X\sqrt{2}$.

I can't prove it. It has been proven by Gödel that we can never proof that there is no proof. So while it's true that I cannot prove that an even number is odd, nobody can prove that it cannot be proven. This is coined in usual language "one can never prove a negative".

So because you made the claim that $a+b+c+d+...=X$ implies $2X = X\sqrt{2}$, I would like to see a rigorous proof of that.

Without loss of generality you can use the case $X=1$ (in that case 1/2+1/4+1/8+...=1 which implies $2 = 1\sqrt{2}$).

Definition:
In $\mathbb{R}$, we define $\sum_{n=1}^{+\infty} a_n = a$ if for all $\varepsilon>0$, there exists an $N\in \mathbb{N}$ such that for all $N'\geq N$ holds that $\left|a - \sum_{n=1}^{N'}\right|<\varepsilon$.

Part 1: If $k$ is a natural number, then we define $x_k = \sum_{n=1}^k \frac{1}{2^n}$. This is a real number since the sum is finite. Note that
$$2x_k = 2\sum_{n=1}^k \frac{1}{2^n} = \sum_{n=1}^k \frac{1}{2^{n-1}} = 1 + \sum_{n=0}^{k-1} \frac{1}{2^n} = 1 + x_k - \frac{1}{2^k}$$
Hence by rearranging, we have
$$x_k = 1 - \frac{1}{2^k}$$

Part 2: Choose $n\in \mathbb{N}$ arbitrarily with $n\geq 1$. We show that $n<2^n$ by induction.
Induction step: $1<2^1 = 2$.
Assume that $n<2^n$. Since $1\leq n$, we have in particular that $1<2^n$. So then $n+1< 2^n + 2^n = 2\cdot 2^n = 2^{n+1}$.

Final proof:
Choose $\varepsilon>0$ arbitrary. By the Archimedes axiom, we can choose an $N\mathbb{N}$ such that $N\geq 1$ and $N\geq \frac{1}{\varepsilon}$. If $N'\geq N$, then by part $2$, we have
$$\frac{1}{\varepsilon} \leq N\leq N' < 2^{N'} = |2^{N'}|$$
Thus
$$\left|\frac{1}{2^{N'}}\right|<\varepsilon$$
Hence
$$\left|1 - \left(1 - \frac{1}{2^{N'}}\right)\right|<\varepsilon$$
By using part $1$, we have
$$\left|1 - \sum_{n=1}^{N'}\frac{1}{2^n}\right|<\varepsilon$$
This is what we had to show by the definition.

 

[Look]

The point is: You seemingly want to establish deduction principles that differ from the framework.

Dear fresh, since you agree that 1/2+1/4+1/8+1/16...=1, then the diagram is a proof without words that 1/2+1/4+1/8+1/16...=1 implies $2 = 1\sqrt{2}$.

 

[micromass]

Dear fresh, since you agree that 1/2+1/4+1/8+1/16...=1, then the diagram is a proof without words that 1/2+1/4+1/8+1/16...=1 implies $2 = 1\sqrt{2}$.

A proof without words doesn't exist. You need to show rigorously why $1/2 + 1/4 + 1/8 + ... = 1$ implies $2 = \sqrt{2}$.

 

[Look]

That is not a rigorous proof, you just made an assertion.

Dear micromass, since you agree that 1/2+1/4+1/8+1/16...=1, then the diagram is a proof without words that 1/2+1/4+1/8+1/16...=1 implies $2 = 1\sqrt{2}$ (which is a rigorous implementation of implication, without loss of generality).

 

[micromass]

Dear micromass, since you agree that 1/2+1/4+1/8+1/16...=1, then the diagram is a proof without words that 1/2+1/4+1/8+1/16...=1 implies $2 = 1\sqrt{2}$ (which is a rigorous implementation without loss of generality).

You demand from us a rigorous proof from real analysis? But when we ask you the same thing you just call it a "proof without words"? Very dishonest from you. I provided a rigorous real analysis proof. Now it's your turn to prove rigorously that $2 = \sqrt{2}$.

 

[Look]

A proof without words doesn't exist.

Please see https://en.wikipedia.org/wiki/Proof_without_words:

Such proofs can be considered more elegant than more formal and mathematically rigorous proofs due to their self-evident nature.

 

[micromass]

Please see https://en.wikipedia.org/wiki/Proof_without_words .

I know they exist. But they're not rigorous. Mathematicians will never accept them as actual proofs.

 

[micromass]

But sure, I can provide a proof without words that $X = a+b+c+d+...$ too, you know.

 

[Look]

I know they exist. But they're not rigorous.

They are mathematically rigorous.

 

[micromass]

They are mathematically rigorous.

Not according to 100% of the mathematicians. Sorry.

 

[Look]

They are mathematically rigorous.

Are going to ignore the following? :

Such proofs can be considered more elegant than more formal and mathematically rigorous proofs due to their self-evident nature.

 

[Look]

Not according to 100% of the mathematicians. Sorry.

Realy?, please support this claim.

 

[micromass]

Are going to ignore the following? :

Your quote just confirms that I'm right by contrasting such proofs with "more formal and mathematically rigorous proofs". Meaning that proofs without words are not rigorous.

 

[micromass]

Realy?, please support this claim.

Do you know the definition of a proof and the definition of a theorem?

 

[Look]

Not according to 100% of the mathematicians. Sorry.

Again, you simply ignore the following quote, taken from Wikipedia ( https://en.wikipedia.org/wiki/Proof_without_words ):

Such proofs can be considered more elegant than more formal and mathematically rigorous proofs due to their self-evident nature.

 

[micromass]

Again, you simply ignore the following quote, taken from Wikipedia ( https://en.wikipedia.org/wiki/Proof_without_words ):

I just addressed it: it implies that actual proofs are more mathematically rigorous than proof without words.

Also, relying on wikipedia to make your argument, really?? Refer to an actual math book if you want me to take you seriously.

 

[Look]

Do you know the definition of a proof and the definition of a theorem?

Do you know that proofs without words "can be considered more elegant than more formal and mathematically rigorous proofs due to their self-evident nature" ?

 

[micromass]

Do you know that proofs without words "can be considered more elegant than more formal and mathematically rigorous proofs due to their self-evident nature" ?

Yes, they are more elegant. They are also less formal and less mathematically rigorous which is stated by the very quote you give!

 

[Look]

I just addressed it: it implies that actual proofs are more mathematically rigorous than proof without words.

Wrong, by Wikipedia proofs without words "can be considered more elegant than more formal and mathematically rigorous proofs due to their self-evident nature"

Also, relying on wikipedia to make your argument, really?? Refer to an actual math book if you want me to take you seriously.

Wikipedia is a reliable source for mathematics, which is not less reliable than math books.

 

[micromass]

Wrong, by Wikipedia proofs without words "can be considered more elegant than more formal and mathematically rigorous proofs due to their self-evident nature"

I am not disagreeing that they can be considered more elegant. I'm saying that they're not mathematically rigorous. Your quote doesn't disagree with that assertion.

 

[Look]

Yes, they are more elegant. They are also less formal and less mathematically rigorous which is stated by the very quote you give!

A proof without words is a self-evident true that uses symbols without the need of further verbal-symbolic definitions.

 

[micromass]

A proof without words is a self-evident true that uses symbols without the need of further verbal-symbolic definitions.

For centuries, people thought it was self-evident that the Earth is the center of the universe. Self-evident doesn't mean true. There are many self-evident things in math that turn out to be false. This is why we need mathematical rigor.

 

[Look]

A proof without words is a self-evident true that uses symbols without the need of further verbal-symbolic definitions.

This is exactly the reason of why a proof without words is more elegant than verbal-symbolic-only proofs.

 

[micromass]

This is exactly the reason of why a proof without words is more elegant than verbal-symbolic-only proofs.

I own over hundred math books and I have read hundreds of research papers in math. Can you explain why exactly zero of them have proof without words?

 

[Look]

For centuries, people thought it was self-evident that the Earth is the center of the universe. Self-evident doesn't mean true. There are many self-evident things in math that turn out to be false. This is why we need mathematical rigor.

Come on, we are not talking about inductive "self-evidence" , but about deductive self-evidence.

 

[micromass]

Come on, we are not talking about inductive "self-evidence" , but about deductive self-evidence.

Proofs without words are exactly that: inductive. A picture can only show a special case. It can never show a general case.

 

[micromass]

Here's another fun proof without words:

 

[Look]

I own over hundred math books and I have read hundreds of research papers in math. Can you explain why exactly zero of them have proof without words?

Hundred math books or hundreds of research papers not cover math.

I think that you simply find what you search for (according to your philosophical view of math), no less no more.

Here is some stuff that you probably did not read:

http://www.maa.org/press/periodical...hout-words-and-beyond-proofs-without-words-20

http://link.springer.com/article/10.1023/A:1024312321077

http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Borwein897-910.pdf

http://link.springer.com/chapter/10.1007/1-4020-3335-4_2

http://web.stanford.edu/group/cslipublications/cslipublications/pdf/1575863243.pdf

https://www.researchgate.net/profile/Mateja_Jamnik/publication/7575141_What_is_a_proof/links/542bda5a0cf27e39fa91b493.pdf

etc...

 

[micromass]

Of course you'll find papers that discuss visualization in mathematics. All of the papers you cited are exactly on the topic of visualization. None of the topics you cited actually produce a novel result.

 

[Look]

Of course you'll find papers that discuss visualization in mathematics. All of the papers you cited are exactly on the topic of visualization. None of the topics you cited actually produce a novel result.

Did you read all of them in ten minutes, understood them and then concluded that non of them "actually produce a novel result" ?

You must be some kind of superman.

 

[micromass]

Did you read all of them in ten minutes, understood them and then concluded that non of them "actually produce a novel result" ?

Yes.

 

[Look]

Here's another fun proof without words:

My diagram is not only a picture, but it uses a linkage among visual AND symbolic reasoning.