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

[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 ...

 

[phinds]

... 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.

It also means "fits well" as in "That color will suit you well", and "is OK with me (or isn't OK with me)" as in "What you are suggesting does not suit me".


There are likely other uses that I'm not thinking of at the moment.

In others words, as we already agree:

Context is everything, as they say ...

SO ... @ducknumerouno, get with the program

 

[fresh_42]

I find it somehow telling that there isn't a mathematical definition for infinity. Even a definition of the form "A is infinite if ..." does not exist.

The most common examples are probably $\displaystyle{\lim_{n \to \infty}a_n}$ and $\displaystyle{\sum_{n=1}^\infty f_n(x)}.$ The first one does not use anything infinite in its definition, only arbitrarily high indices, and the second one is defined by the first, and so again, without using the term infinite.

The other natural association is the number of natural numbers. Infinities are noted by $\aleph_\alpha$ and are defined recursively. So $\aleph_0$ is the only occasion where the word infinite can, but does not need to be used. One can either define $\aleph_0=|\mathbb{N}|$ without using the term, or one can say that $\aleph_0$ is the least infinite ordinal number. However, even the infinity of natural numbers is defined by the simple statement that every natural number has a successor, again avoiding infinity.

In this sense, infinity isn't a mathematical concept. It is an abbreviation to make notation easier.

That's typical in mathematics. Logical problems like Russell's paradox are bypassed by more specific definitions instead of running into a wall. I suspect that infinities in physics are even more problematic and that physicists refer to mathematics instead of trying to define it. But there is no mathematical book that tells us what infinity is, only one that requires the property $s(n)=n+1.$ It remains a problem for philosophy and linguistics.

The mathematical answer to the original question is simply a statement about convergence
$$
4=\lim_{n \to \infty}\sum_{k=1}^n r_k \neq \lim_{n \to \infty}\sum_{k=1}^n c_k=L(\text{circle})=\pi.
$$
Crossing the limit is prohibited since the limit of the rectified curve is different from the limit of the parts of the curve.

 

[Klystron]

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?

Having read this thread and reread the OP's posts, the original poster often confuses symbol with referent, that which the symbol points or refers to. Suggest reading introductory texts on semiotics. The clarity gained from this knowledge will help understand puzzles such as 'squaring the circle' while providing a firm basis for communication.

 

[mathwonk]

That (post #1) is a beautifully simple example! For a similar, (but harder to visualize), example where area approximation of surfaces also fails, see Spivak, Calculus on manifolds, p. 129.

 

[DaveC426913]

try it.

Sure.

Quick question: What's epsilon?

Follow up question: what is 'a beautifully simple example'?

 

[mathwonk]

@fresh_42:
In reference to post #52, I don't know if this is satisfactory as a definition, but, as you no doubt know, a common characterization of infinite sets is of course:
A set is infinite if and only if there is a bijection from the set to a proper subset of itself.

 

[fresh_42]

@fresh_42:
In reference to post #52, I don't know if this is satisfactory as a definition, but, as you no doubt know, a common characterization of infinite sets is of course:
A set is infinite if and only if there is a bijection from the set to a proper subset of itself.

Yes, that's an infinite set. The object provides a context. And it doesn't even distinguish between countable and uncountable infinity, so even though there is a context, it is still insufficient. What I meant is what I said in post #35. There is no definition for infinity itself, stripped of any objects, other than greater or equal $\aleph_0$ for sets. https://ncatlab.org/nlab/show/HomePage lists 371 pages that have "infinity" in their titles. If we search for "infinity," then we get this list of 371 possible choices. The concept of infinity is void without further information. I meant that an infinite sequence is something different than the infinity of the continuum, or infinities as integration limits. We cannot use infinity without further specification.

Mathematics is of no help when it comes to philosophically explaining infinity. The original question about the convergence of area versus perimeter only shows how carefully the term infinite has to be used. For example, I think of infinite towers of modules when someone says infinite, probably something people had not in mind.

 

[mathwonk]

Ascending or descending towers? I guess one already has descending towers in the integers: Z, 2Z, 4Z, 8Z,..... In fact these are nice examples of bijectively equivalent subsets.

 

[fresh_42]

Ascending or descending towers? I guess one already has descending towers in the integers: Z, 2Z, 4Z, 8Z,..... In fact these are nice examples of bijectively equivalent subsets.

Lol, depends on whether I have an Artin day or a Noether day, I admire both. But I think more often about a tower of Lie algebras that are not included in each other.
https://www.physicsforums.com/insig...-representation-of-mathcalBmathfraksl2mathbbR

 

[Lnewqban]

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

Is impossible to create squares that follow the shape of the circle.
If you do, the line formed by the corners of the smaller squares never becomes a circle.