B What could prove this wrong? I'm having a dispute with friends

[ducknumerouno]

TL;DR Summary

Pi = 4?
I know it theoretically never touches the circle, but does the circle ever really become a circle?

 

[mad mathematician]

How do you know that the diminished corners' square eventually converges to the circle?

 

[ducknumerouno]

Well… how do you define convergence — in shape, in perimeter, or in area?

and we could make that point that
does the limit of a sequence of functions always preserve properties like length?

 

[fresh_42]

The same trick can be used to "prove" $\sqrt{2}=2$ or $\pi=2.$ The problem is that you can not treat infinities like a number. You basically claim that $\infty \ \cdot \ 0=c\in \mathbb{Q}.$ The left-hand side is simply not defined. We have an infinite sum $\sum_{k=1}^\infty a_n=c$ and that does not vanish only because we cannot "see" the terms $a_n$ anymore.

 

[ducknumerouno]

If infinity isn't a number and behaves differently from regular numbers, how can we really understand and use it in things like limits? Can we even trust that it is useable in math then? Because I heard somewhere that some infinityies are bigger than others ( I don't know where I heard that some scientist proved it)

 

[fresh_42]

If infinity isn't a number and behaves differently from regular numbers, how can we really understand and use it in things like limits?

This depends on the particular occasion. Infinities in calculus are replaced by the epsilontic and topological principles, infinities in number theory are based on induction, which in turn is based on Peano's axioms.

Limits themselves, so they exist, are real numbers. Infinite limits are more a notational abbreviation for an increasing or decreasing behavior than actual limits.

Can we even trust that it is useable in math then?

Yes, because we have methods to cope with these notations and do not really use "infinity".

Because I heard somewhere that some infinityies are bigger than others ( I don't know where I heard that some scientist proved it)

You probably heard that there is a difference between countable many infinite numbers such as the integers, and uncountable many infinite numbers such as the real numbers. That makes a provable difference.

 

[FactChecker]

It's clear from the first figure that $\pi \ne 4$ with an error of $\pi-4$. Dividing that error up into a million small errors that still sum to $\pi-4$ does not change anything.
A simple proof that it is wrong is to physically measure the circumference of an actual circle.

 

[ducknumerouno]

I get that infinity is handled differently in different areas of math, but it still feels like we’re just ignoring how weird it is. Calling infinity a “shorthand” doesn’t really solve the problem—it just makes it easier to deal with, but doesn’t explain it fully.

 

[fresh_42]

I get that infinity is handled differently in different areas of math, but it still feels like we’re just ignoring how weird it is. Calling infinity a “shorthand” doesn’t really solve the problem—it just makes it easier to deal with, but doesn’t explain it fully.

As mentioned, it depends on the context. I cannot explain $\infty$ without knowing that context.

Even such a thing like the number $2$ depends on the context. It means something different in $2\cdot a,\, a/2$ and $a^2,$ and it means something different as an element of $\mathbb{Z}$ and as an element of $\mathbb{Z}_3.$

So if even the sign $2$ is context-sensitive, why do you demand $\infty$ to be used without context?

 

[ducknumerouno]

rue, infinity definitely depends on the context, but that just shows how confusing it can be to some people . If it can change meaning based on the situation, maybe it’s more of an issue than we know?

 

[fresh_42]

rue, infinity definitely depends on the context, but that just shows how confusing it can be to some people . If it can change meaning based on the situation, maybe it’s more of an issue than we know?

So my edit to the previous post.

 

[DaveC426913]

A slightly different way of intuiting the problem:

Imagine this square is one mile on a side and drawn on the ground.
You start walking the square at one mile per hour.
Assume it takes zero time to turn a corner.
It will take you four hours to walk a complete perimeter.

Now you remove the corners, subdividing the square.
You walk the new shape, turning left and right twice as often.
It still takes you four hours.

Now you subdivide it to step 3; It still takes you four hours because no matter how many times you change direction, you are always walking either across or down, never any other angle.

Even if you subdivided the square an infinite number times, it still takes you four hours.

But look at what that means: If you really were walking the perimeter of a circle, at one mile per hour, it would take you a mere 3.1416 hours to do a full circuit.

To the observer, you appear to be walking a circle (although you're vibrating left-to-right at an alarming rate) but - although your pedometer reads you walking at one mile per hour - your progress is only measurable as pi/4 mph.

You are moving slower. No matter how fine you make your turns, you never magically speed up from pi/4mph to 1mph - proving that you are not truly walking a circle.

 

[ducknumerouno]

hm., so we agree that infinity should not be agreed on yet?

 

[ducknumerouno]

I agree on the fact of the example given, it was very good and helped me understand more. Thank you. but now on the subject of infinity, should we belive it?

 

[phinds]

... that just shows how confusing it can be to some people

Well, to you, clearly, but not to most of us. Study some math and you'll get over it.

 

[ducknumerouno]

But if infinity is always context-dependent and never truly defined, doesn’t that make it more of a philosophical idea than a mathematical thing?

 

[phinds]

But if infinity is always context-dependent and never truly defined, doesn’t that make it more of a philosophical idea than a mathematical thing?

No. MANY words in the English language have different meanings depending on context. Does that make them philosophical things rather than real words?

 

[phinds]

Look, @ducknumerouno, it should be clear to you from the responses in this thread that infinity is a mathematical concept that is quite useful. If you wish to view it as something else, feel free to do so, but do not expect the rest of the world to agree with you. Learn some math and you'll get over it.

You are complaining about something you don't understand. Instead of doing that, why not spend some time LEARNING about it.

 

[jedishrfu]

One can use this same faulty argument to prove the Pythagorean theorem wrong.

Consider a 3-4-5 right triangle, with steps replacing the hypotenuse. As you ascend the small steps, you'll measure 4 units horizontally and 3 units vertically, and conclude the hypotenuse is seven units.

However, if you measure the hypotenuse, you'll find it is actually five units, not seven.

 

[ducknumerouno]

Look, @ducknumerouno, it should be clear to you from the responses in this thread that infinity is a mathematical concept that is quite useful. If you wish to view it as something else, feel free to do so, but do not expect the rest of the world to agree with you. Learn some math and you'll get over it.

You are complaining about something you don't understand. Instead of doing that, why not spend some time LEARNING about it.

I understand your point, but just because infinity is useful in math doesn’t mean it’s fully understood or without its philosophical issues. It’s a concept that helps explain things, but the fact that it’s treated as something both undefined and context-dependent makes it weird. Learning about it doesn’t erase the questions, understanding something means recognizing that it’s not as clear as it seems?

 

[ducknumerouno]

Knowledge is a paradox. The more you learn, the more you realize how much you don't know.

 

[phinds]

Learning about it doesn’t erase the questions, understanding something means recognizing that it’s not as clear as it seems?

You only believe it's not clear because you refuse to actually LEARN about instead of just complaining.

Enough ! Either learn about it or stop complaining.

 

[ducknumerouno]

Maybe instead of simply learning about it, you should try to understand it and ask questions that can challenge you. There is no problem with learning, it just provides little understanding

 

[phinds]

There is no problem with learning, it just provides little understanding

Seriously ??? What do you think DOES provide understanding? Voodoo? Magic?

 

[fresh_42]

I understand your point, but just because infinity is useful in math doesn’t mean it’s fully understood or without its philosophical issues. It’s a concept that helps explain things, but the fact that it’s treated as something both undefined and context-dependent makes it weird. Learning about it doesn’t erase the questions, understanding something means recognizing that it’s not as clear as it seems?

$2\cdot 2=4$ and $2\cdot 2 =1$ are both true statements but, obviously, the twos represent something different in each statement.

How many meanings does the word light have?

Learning erases questions. I can understand the difference between a light weight and a light that brightens my room. The same is true if $\infty$ is used in a mathematical expression. The context enables me to describe what it means. If you want to discuss it philosophically, then it is as weird as love or consciousness.

Knowledge is a paradox. The more you learn, the more you realize how much you don't know.

True.

 

[ducknumerouno]

How exactly is 2 * 2 equal to one a true statement, and learning can erase questions but you are not deeply understanding, instead of saying "light weigh between light that brigtens my eyes" which proves you learned at some point that there was a difference between the two proves that you are thinking very simply mindedly why not say, "I can distinguish the meanings of the word light from the other meaning with the same spelling" instead of directly comparing them? also you could be asking yourself what are the other meanings and how can I compare this better? I appreciate the comparison but I don't get how this can relate to understanding rather than knowing the concept of infinity?

 

[fresh_42]

How exactly is 2 * 2 equal to one a true statement,...

It is true in the world of remainders by division by three. If we multiply two numbers, say $5$ and $14$ that both have the remainder $2$ then $5\cdot 14=70$ has the remainder $2\cdot 2=1.$

... and learning can erase questions but you are not deeply understanding ...

A deeper understanding starts with the insight that symbols mean different things in different contexts. There is no one and only infinity in mathematics. We use it as a symbol, and the context defines what it means. I could list quite a few, but that won't bring you closer to an understanding of infinity. It is precisely defined whenever we use it without having to refer to something infinite. That avoids the philosophical dimension in mathematics at the cost that we mean different things in different contexts if we use the symbol $\infty.$

Thus, there is no such thing as a deeper understanding to $\infty .$ I can explain to you what

• $\lim_{n \to \infty}a_n$ means,
• $\sum_{k=1}^\infty a_n$ means,
• $\aleph_n$ means, note that we do not even use $\infty$ if we describe sets with infinitely many elements,
• $\int_{-\infty }^\infty f(x)\,dx$ means,
• $\overline{\mathbb{R}}=\mathbb{R}\cup \{\pm \infty \}$ means,
• what a bump vanishing to infinity means,
• what various kinds of singularities mean,
• and probably some more.

All these deal with infinity in different ways. They demonstrate what mathematicians understand if they use the symbol $\infty$ not what infinity is. Their explanations do not require to speak about infinities other than in a common sense.

Everything beyond that isn't a question of mathematics or physics, only a philosophical one. I once had a professor who said that everything real is finite and enumerable because we can only see a finite number of particles in the world. Well, before this discussion derails about what this professor said and meant, please note that I varied the quotation a bit to match the context here. It was actually about the difference between discrete and continuous. Even the word real is subject to philosophy. In mathematics, it is simply the topological completion of rational numbers.

I can't tell you who among the many philosophers dealt with "what is real", I assume almost all of them, or who dealt with the concept of infinity. Mathematicians only solved what affects them, and this is context sensitive. If you want to study the role of language, I recommend Wittgenstein.

... instead of saying "light weigh between light that brigtens my eyes" which proves you learned at some point that there was a difference between the two proves that you are thinking very simply mindedly why not say, "I can distinguish the meanings of the word light from the other meaning with the same spelling" instead of directly comparing them? also you could be asking yourself what are the other meanings and how can I compare this better?

I did not use a comparison; I only mentioned two (among even more) possibilities.

I appreciate the comparison but I don't get how this can relate to understanding rather than knowing the concept of infinity?

It should have shown that there is no single concept of infinity in mathematics. I even doubt that there is a single concept of infinity in philosophy. Hence, if you search for a deeper understanding, you have to either learn all the different uses and gain a deeper understanding in each case, or study philosophy.

Infinity itself is no mathematical concept.

 

[DaveC426913]

I don't get how this can relate to understanding rather than knowing the concept of infinity?

Just because you can't operate on infinity with arithmetic as if it were a number does not mean you can't learn how we do work with it.

You didn't learn your times tables in a two paragraph forum post, but you did learn them. It just takes a little study.

There's no magic bullet here. No secret password we can give you that instantly unlocks your full understanding of the concept of infinity.

It's just another concept for which you need to learn the rules of operation before you can use it, like anything else.

 

[PeroK]

TL;DR Summary: Pi = 4?
I know it theoretically never touches the circle, but does the circle ever really become a circle?

View attachment 360769

You have a sequence of shapes, $S_n$ say, which converge to some shape, $S$. In this case $S$ is a circle.

Each of shapes $S_n$ has a total perimeter of length $4$. Your assumption is that the shape $S$ must have perimeter of length $4$.

This assumption is false, as this example shows.

In general if a sequence of shapes each has some property, then the limit shape need not have that property. Again, as this example shows.

Nothing to do with infinity, per se, IMO.

 

[gmax137]

I would not say the square with knocked off corners "converges" to the circle. The steps might get too small to "see," but they're still there.

 

[DaveC426913]

I would not say the square with knocked off corners "converges" to the circle. The steps might get too small to "see,"

Well, the point of the demo is that, eventally, they become infinitely small.

 

[phinds]

Infinity itself is no mathematical concept.

Huh? You lost me there.

 

[gmax137]

Well, the point of the demo is that, eventally, they become infinitely small.

So what? They're still orthogonal steps, they don't converge to the circle. You're never following a hypotenuse.

 

[PeroK]

So what? They're still orthogonal steps, they don't converge to the circle. You're never following a hypotenuse.

The concept of convergence requires some way of measuring the difference between two shapes. You could measure the total area between them, for example. In that sense the shapes converge to the circle.

This is no different from the concept of Convergence of a sequence of functions. In fact, defining the shapes as functions is another way to define convergence.

 

[fresh_42]

Huh? You lost me there.

You cannot speak about infinity itself in mathematics, i.e., without mentioning the domain. Infinity stripped of any context is not defined in mathematics. In this sense, mathematics cannot contribute any insights to the philosophical question.

What I mean is this (nLab):

etc.

 

[phinds]

You cannot speak about infinity itself in mathematics, i.e., without mentioning the domain.

Ah. Now I'm back with you. Thanks.

 

[PeroK]

TL;DR Summary: Pi = 4?
I know it theoretically never touches the circle, but does the circle ever really become a circle?

View attachment 360769

In fact, you can use this method to calculate pi, but you need to consider the area and not the perimeter.

The total area of all those rectangles is greater than the area of the circle. As you increase the number of rectangles and reduce their width, you get a better estimate of the area of the circle.

You can also consider rectangles enclosed within the circle. Their total area must be less than the area of the circle. Again, as you increase the number of rectangles, you get a better estimate.

The important point is that the upper estimate and lower estimate converge to the same number. That number is pi.

This was similar to the way pi was originally estimated by the ancient Greeks.

In fact, this technique is the basis of integration from first principles. The area of any shape can be calculated using this technique.

It doesn't work with perimeter lengths.

 

[gmax137]

The concept of convergence requires some way of measuring the difference between two shapes.

Yes!

You could measure the total area between them, for example. In that sense the shapes converge to the circle.

I think saying, "converge to the circle" by itself, is a vacuous statement. If you say, "as the number of sides increases, the area of the pixellated figure converges to the area of the circle" then that has meaning and can be verified.

 

[PeroK]

I think saying, "converge to the circle" by itself, is a vacuous statement.

Not at all. It's simply a sequence of functions converging to a limit function. Those square-wave functions meaningfully converge to a circle. This is also the basis of Fourier analysis, where an infinite series of functions converges to the required limit function.

The convergence in this case is both pointwise and uniform.

 

[PeroK]

It doesn't work with perimeter lengths.

PS I should have said it doesn't necessarily work with perimeter lengths. If we use regular polygons with increasing number of sides, for example, outside and inside the circle, then we get an estimate of pi. I think that was the original method of estimating pi.

 

[fresh_42]

It is not necessary that the upper and lower bounds of the rectangles converge to the same number. The procedure already works if we consider only one of them. Each of which converges to $\pi$ independently of what the other one does.

The process also works with perimeters if we rectify the curve, possibly with a uniform convergence. Every differential can be approximated by decreasing differences.

 

[Filip Larsen]

I have not read all posts in this thread so I may repeat someone adding what I usually add when this problem is mentioned, namely that you can also understand what happens by applying the concept of fractal dimension. The curve remains at length 4, but by applying the prescribed folding it will end up with a fractional dimension above 1 and thus "squeezed" it can now follow the shorter circular curve infinitely close.

 

[Dullard]

The initial problem strikes me as sleight-of-hand. Doubling the number of 'sides' while halving their individual lengths results in no change in the total length. No matter how many times you do it (up to and including an infinite number). The problem (basically) tries to trick you into talking about perimeter while showing you converging area.

 

[DaveC426913]

The initial problem strikes me as sleight-of-hand. Doubling the number of 'sides' while halving their individual lengths results in no change in the total length.

That's true. I mean, the initial square is composed of the exact same infinite number of line segments. All the problem does is mix them up.

Square: x,x,x,x,x,x,x,x,x,...,y,y,y,y,y,y,y,y,y,...
Beveled polygon: x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,...
where x and y are infinitesimally short line segments.

In the first case, we have no problem seeing the square has a perimeter of 4. But for some reason, when we interweave x segments with y segments, we struggle with seeing the beveled polygon as having a perimeter of 4.

Heck, we can make an inifinite variety of shapes, all with perimeter 4:
Wobbly, stellated star: y,y,y,y,y,y,y,x,y,y,y,x,x,y,x,x,x,y,x,x,x,x,...


Ask your friends what they think the perimeter of this object is:

 

[BWV]

By the same token, if the OP's perimeter method works for circles should it not also work for any polygon inscribed in the square? (but it obviously does not, for example a dodecahedron would have a perimeter of 3.106)

 

[PeroK]

By the same token, if the OP's perimeter method works for circles should it not also work for any polygon inscribed in the square? (but it obviously does not, for example a dodecahedron would have a perimeter of 3.106)

Which is getting close to pi.

 

[Office_Shredder]

A far stupider example is to just take a line and draw a squiggle that goes up and down at a diagonal. 1=sqrt(2) can get you to a lot of contradictions really quick

 

[SammyS]

By the same token, if the OP's perimeter method works for circles should it not also work for any polygon inscribed in the square? (but it obviously does not, for example a dodecahedron would have a perimeter of 3.106)

( Dodecagon )

 

[Nugatory]

But if infinity is always context-dependent and never truly defined, doesn’t that make it more of a philosophical idea than a mathematical thing?

The word “infinity” is used to name several different things, but that’s a problem with the language not the concepts being named; these are precisely and unambiguously defined mathematical concepts.

It would be nice if we had a different word for each of these concepts, just as it would be nice if the English language didn’t use the same word “suit” to describe the thing I wear to my friend’s wedding, the thing lawyers file in courts, one of the four categories of playing cards…. But that’s just not how natural languages evolve.

 

[Filip Larsen]

it would be nice if the English language didn’t use the same word “suit” to describe the thing I wear to my friend’s wedding, the thing lawyers file in courts, one of the four categories of playing cards

Context is everything, as they say ...