I'm not sure I'd describe the one you linked as "perfectly similar". At least to me, there's a couple obvious problems:
- Folding the corners of a rectangle an infinite number of times doesn't make it a circle, it just means it has an infinite number of corners.
- The folded corners always make right triangles, no matter how small they are. If you put the the non-hypotenuse legs of a right triangle against a circle, no matter how infinitely small the legs are, the corner of the legs will never touch the edge of the circle: an infinitely small triangle can't have all three points be the same point (or it's not a triangle). Which means the area of the folded rectangle will always exceed the area of the circle it's mimicking, even with infinite folds.
- As the folds become smaller and smaller, the arc of the circle (relative to the size of the triangles against it) becomes straighter and straighter. Which means each successive fold scrunches up more perimeter while becoming less and less circle-like.
There's also the intuition that the circumference of the circle must be less than the perimeter of the square, so if the perimeter of the polygon isn't decreasing as it gets closer to the circle, it doesn't approximate it better than the square itself.
I.e., the perimeter doesn't approach the circumference in value because it doesn't change.
It's an interesting thing to think through though, and maybe a good point about how arguments can seem intuitive at first but be wrong. On the other hand, I'm not sure that's any more true of visual proofs than other proofs.
I think your attempt to rebuke the proof is flawed too. The problem in your reasoning is mixing up "arbitrarily many" and "infinitely many".
There's no convergence after a finite number of steps. But at infinity, the canonical limit of this construction method is a circle. And because it is a circle, the circumference at infinity "jumps" to 2*pi. This is quite counterintuitive but perfectly legit in mathematical analysis. It's just one of many wacky properties of infinity.
I kind of ran into this when I was in high school and was introduced to limits.
For me the quandary was a "stair step" shape dividing a square with length of side "s" ("stairs" connecting two opposite diagonal corners). You could increase the number of steps—they get smaller—but the total rise + run of the stairs remains the same (2s). At infinity I reasoned you had a straight, diagonal line that should have been s√2 but was also still 2s in length.
At the very least you can say that the volume enclosed approached that of a right triangle (at infinity) but the perimeter stays stubbornly the same and not that of a right triangle at all.
This is indeed the common way most people encounter this. The proof of the difference in the limits for the perimeter vs. the area is in the first answer to the stack overflow question in the G(^n)P: https://math.stackexchange.com/a/12907
But imagine this was a domain you weren't familiar with, you didn't know that pi != 4, you didn't know that the proof was false going into it. Could you have come up with a list of problems so quickly?
For what it's worth, I'm not sure that problems 1 and 2 are actually genuine problems with the proof. You can approximate the length of a curve with straight lines by making them successively smaller. This is the first version of calculus that students learn. Problem 3 is the crux.
> But imagine this was a domain you weren't familiar with, you didn't know that pi != 4, you didn't know that the proof was false going into it. Could you have come up with a list of problems so quickly?
No, but if I didn't know anything about the domain, literally any proof (correct or incorrect) would seem fine. But then it's not really "proving" anything. Knowing enough for the proof to make sense but still unconditionally accepting assertions like "if you fold the corners an infinite number of times, it makes a circle" strikes me as odd.
> I'm not sure that problems 1 and 2 are actually genuine problems with the proof. You can approximate the length of a curve with straight lines by making them successively smaller.
But that's not what's happening here: the lines are straight, but you'd approximate the length of the curve with the hypotenuses, not the legs of the folds. Surely as you repeat this process you wouldn't think "wow, the circumference of this circle is actually equal to the perimeter of the original square." You'd have to disbelieve your own eyes and intuition and knowledge of circles to accept that this is true and hopefully you'd think "maybe I'm doing this wrong."
That's not to say 1 and 2 alone prove the visual proof incorrect, but they demonstrate that it is doing something wrong. Proofs that are correct don't have inconsistencies.
In math you have to disbelieve your own eyes and intuition an awful lot. Not in this case, I grant you. But there are plenty of counterintuitive results.
These are obvious problems to someone who has studied enough math/geometry/calculus to know how one form of "adding boxes together gets a curve" and another "adding boxes together does NOT get you a curve".
Are visual proofs meant for someone who hasn't studied math? I wouldn't expect them to prove anything to someone who hasn't. Any proof that's incorrect could reasonably fool a layman.
This error can be made for calculating the length of any curve. If you add the deltas in only one dimension, then you end up with a bounding box length measurement that doesn't follow the contours of the curve. It's a misuse of calculus, that can be done with or without the visualization.
On of the first things my geometry teacher emphasized in 9th grade was that a drawing (even a very carefully measured one) didn't prove anything. Proof had to be derived from axioms and other proven facts.
These are neat! I guess you have to be comfortable with geometric proofs for them to really pop as obvious visual proofs, and certainly Archimedes was. I would have just started summing until it got close to 1/3, which is brutish by comparison to these beauties.
A proof (visual or otherwise) shows "how" some statement is true, as in how it is built by the preceding truths. But I always wanted to know "why" something is true. For example, a biological cell grows and division happens. I could find tons of literature which talks about "how" this happens, but not "why" this happens. What's the motivation or goal? And why that goal is pursued? What is the force behind seeking of that goal?
You can't anthropomorphise a cell, just like you can't anthropomorphise a lawnmower, or a Larry Ellison. It's just an entity harnessing an entropy gradient.
On the topic of biology specifically, you might like The Selfish Gene by Richard Dawkins.
He argues/explains how evolutionary forces become dominant, with much more focus on the why. Why it has come to be that living things grow, multiply, and over time changed in ways that out-succeeded the prior ones, down to the level of DNA--and that these driving forces are manifested by individual genes.
I've been thinking about cancer. Maybe systems of replicators are prone to overdoing it by nature. The idea was that any universes compatible with life will also have spontaneous cancers because that's just what those universes do.
And then I learned the theory that many cancers are caused by undiscovered DNA-based viruses which tamper with the cell cycle to activate the replicative machinery that they need to make copies of their genome (HPV does this, and several others too). So then it was a switch: not an immutable feature of the universe, but caused by an agent.
But it's starting to look like viruses emerged independently more times than expected, so maybe it is more like "the universe just does that," and viruses are just cancers with a space program. Back to where I started.
I suppose some would see these loops as unproductive. "First principles" people. Descartes, etc. But I think that unresolvable why's like this are what understanding is made of.
This seems the most concise and precise answer to the question of "why". At some stage, somehow, self replicating molecules appeared in the primordial soup*. Everything since then has just been improvements (or maybe complete overhauls) to that process.
*How (or why) this step occurred is another intriguing topic.
Try talking about biological operations without invoking “function”. Claiming it’s “convenient” to do so doesn’t cut it: convenient for what?
Why do acorns become oak trees? They must be causally ordered toward that end. That’s telos.
Even efficient causality presupposes telos. Why does striking a match against a matchbox consistently produce fire? Because the match has a causal ordering toward that end. Otherwise, you could not explain why fire consistently results as opposed to random things like a flock of seagulls or a BMW 7 Series…or nothing at all.
Telos is not necessarily a matter of some external purpose or Paley-style watchmaker. That’s mechanistic metaphysics appealing to a watchmaker to explain a purpose things would - under that metaphysics - inherently lack. It is a matter of causal order and directedness.
> Try talking about biological operations without invoking “function”.
If you had a strong vendetta against mistaking map for territory, you could very well talk in terms of past survival and statistics. It's just not necessary for regular biological talk. It becomes relevant only when you start going to the boundaries.
> Why does striking a match against a matchbox consistently produce fire?
Because you wouldn't call these objects a "match" and a "matchbox" otherwise.
This is a good, succinct unpacking of the metaphysical stakes. Nonetheless I am curious for the world where striking a match results in a shower of seagulls.
When I was 10 years old, I asked my maternal grandfather, "why does anything exist at all?"
My grandpa explained it in layman terms which even I could understand. He said, "If nothing should exist because it is simpler state to be in for everything, a sort of Primordial Law. Then what is the mechanism by which this law is enforced. Who or what is ensuring that Law is implemented everywhere for eternity. If we assume that such a mechanism must exist, then we have just proved that something must exist."
That's a really bizarre and oddly Platonist take on things. Your grandfather was viewing laws of nature as rules imposed onto reality by some outside force, and which therefore need some "mechanism" to be in place to "enforce" them.
But I think a more reasonable understanding of natural laws is that they're our attempt to describe the cause-and-effect patterns observable within reality itself. They're not being enforced, they're simply manifest.
Construing "nothing can exist" as a rule that has to be enforced, and not just the absence of any patterns of causality that would produce something that exists, seems to be an error. It actually seems to be a more sophisticated version of reifying the concept of "nothing" such that "nothing exists" would be interpreted as describing the positive existence of an entity called "nothing" rather than merely describing the absence of any such entities within the context.
> In a reality containing nothing, there are no things as such — at least no material things. But in such a nothing, there is an abstract thing: zero.
> Zero reflects the number of material things to count. But how many abstract things are there to count? There is at least one. The one number that exists to define the number of material things is zero.
> But if we have one number and it is one thing to count, now another number exists: one. We then have zero and one together as the only numbers. But now we have two numbers. Now two exists…
Your grandfather's explanation seems to echo this in terms appropriate for a 10-year-old - there is something inherently unstable about nothingness.
Null set as 0 and then successor method of defining new numbers.
But your way of putting it is like these successor function could be considered as edges of graphs or references or signposts
Imagine a number system with 3 distinct types of Null Sets and they meet at number P after applying successor function for 10, 42, 135 times respectively.
This is a category error. A cell is not a thing that has a goal. To imagine it has one is pure anthropomorphism. The religious may have other views of course.
Just because things interact doesn't mean there was a goal end state.
But, life is different. Life that survives better, out competes life that doesn't survive as well. So biology becomes incredibly fine tuned for one goal: survival.
Not survival of the individual, although that is part of it. But survival of the thing that makes the choices of how to survive. I.e. our genes. They evolve to enhance their own survival.
Which is why we care about our family more than others. Genes for caring about family have an advantage over genes that don't. Because many of our genes will also be in our other family members.
So from that perspective, each of our cells cares about doing its part to help ensure our genes survive. So it cares about itself surviving, but it also cares enough (when working properly) to sacrifice itself when it is too damaged. Because the cell doesn't last forever no matter what, but the individual it is in can pass on its genes if it sacrifices for that individual.
> So biology becomes incredibly fine tuned for one goal: survival
Again, this is a category error. It works well as a metaphor, but evolution has no goal. Evolution is a description of a simple fact of the universe: things that are good at making copies of themselves become more prevalent. Hence things that become better at making copies of themselves prevail more than ones that don't.
The problem with visual proofs is that there are perfectly similar looking proofs that are false: https://math.stackexchange.com/questions/12906/the-staircase...
They’re great and cool for things you already know to be true, but they can be tricky.
I'm not sure I'd describe the one you linked as "perfectly similar". At least to me, there's a couple obvious problems:
- Folding the corners of a rectangle an infinite number of times doesn't make it a circle, it just means it has an infinite number of corners.
- The folded corners always make right triangles, no matter how small they are. If you put the the non-hypotenuse legs of a right triangle against a circle, no matter how infinitely small the legs are, the corner of the legs will never touch the edge of the circle: an infinitely small triangle can't have all three points be the same point (or it's not a triangle). Which means the area of the folded rectangle will always exceed the area of the circle it's mimicking, even with infinite folds.
- As the folds become smaller and smaller, the arc of the circle (relative to the size of the triangles against it) becomes straighter and straighter. Which means each successive fold scrunches up more perimeter while becoming less and less circle-like.
There's also the intuition that the circumference of the circle must be less than the perimeter of the square, so if the perimeter of the polygon isn't decreasing as it gets closer to the circle, it doesn't approximate it better than the square itself.
I.e., the perimeter doesn't approach the circumference in value because it doesn't change.
It's an interesting thing to think through though, and maybe a good point about how arguments can seem intuitive at first but be wrong. On the other hand, I'm not sure that's any more true of visual proofs than other proofs.
I think your attempt to rebuke the proof is flawed too. The problem in your reasoning is mixing up "arbitrarily many" and "infinitely many".
There's no convergence after a finite number of steps. But at infinity, the canonical limit of this construction method is a circle. And because it is a circle, the circumference at infinity "jumps" to 2*pi. This is quite counterintuitive but perfectly legit in mathematical analysis. It's just one of many wacky properties of infinity.
Does it jump? I feel like it's a "fat perimeter".
I kind of ran into this when I was in high school and was introduced to limits.
For me the quandary was a "stair step" shape dividing a square with length of side "s" ("stairs" connecting two opposite diagonal corners). You could increase the number of steps—they get smaller—but the total rise + run of the stairs remains the same (2s). At infinity I reasoned you had a straight, diagonal line that should have been s√2 but was also still 2s in length.
At the very least you can say that the volume enclosed approached that of a right triangle (at infinity) but the perimeter stays stubbornly the same and not that of a right triangle at all.
This is indeed the common way most people encounter this. The proof of the difference in the limits for the perimeter vs. the area is in the first answer to the stack overflow question in the G(^n)P: https://math.stackexchange.com/a/12907
Well, yes, it is false, hence there are problems.
But imagine this was a domain you weren't familiar with, you didn't know that pi != 4, you didn't know that the proof was false going into it. Could you have come up with a list of problems so quickly?
For what it's worth, I'm not sure that problems 1 and 2 are actually genuine problems with the proof. You can approximate the length of a curve with straight lines by making them successively smaller. This is the first version of calculus that students learn. Problem 3 is the crux.
> But imagine this was a domain you weren't familiar with, you didn't know that pi != 4, you didn't know that the proof was false going into it. Could you have come up with a list of problems so quickly?
No, but if I didn't know anything about the domain, literally any proof (correct or incorrect) would seem fine. But then it's not really "proving" anything. Knowing enough for the proof to make sense but still unconditionally accepting assertions like "if you fold the corners an infinite number of times, it makes a circle" strikes me as odd.
> I'm not sure that problems 1 and 2 are actually genuine problems with the proof. You can approximate the length of a curve with straight lines by making them successively smaller.
But that's not what's happening here: the lines are straight, but you'd approximate the length of the curve with the hypotenuses, not the legs of the folds. Surely as you repeat this process you wouldn't think "wow, the circumference of this circle is actually equal to the perimeter of the original square." You'd have to disbelieve your own eyes and intuition and knowledge of circles to accept that this is true and hopefully you'd think "maybe I'm doing this wrong."
That's not to say 1 and 2 alone prove the visual proof incorrect, but they demonstrate that it is doing something wrong. Proofs that are correct don't have inconsistencies.
In math you have to disbelieve your own eyes and intuition an awful lot. Not in this case, I grant you. But there are plenty of counterintuitive results.
These are obvious problems to someone who has studied enough math/geometry/calculus to know how one form of "adding boxes together gets a curve" and another "adding boxes together does NOT get you a curve".
Are visual proofs meant for someone who hasn't studied math? I wouldn't expect them to prove anything to someone who hasn't. Any proof that's incorrect could reasonably fool a layman.
They’re often (incorrectly) touted as such, with the idea that they’re somehow more intuitive or similar.
This error can be made for calculating the length of any curve. If you add the deltas in only one dimension, then you end up with a bounding box length measurement that doesn't follow the contours of the curve. It's a misuse of calculus, that can be done with or without the visualization.
On of the first things my geometry teacher emphasized in 9th grade was that a drawing (even a very carefully measured one) didn't prove anything. Proof had to be derived from axioms and other proven facts.
I animated these and related 'proofs' some time ago
https://evanberkowitz.com/images/2014-03-15-quarters/SquareA...https://evanberkowitz.com/images/2014-03-15-quarters/Triangl...
https://evanberkowitz.com/images/2014-03-16-eighths/EighthsA... https://evanberkowitz.com/images/2014-03-17-ninths/NinthsAni...These are neat! I guess you have to be comfortable with geometric proofs for them to really pop as obvious visual proofs, and certainly Archimedes was. I would have just started summing until it got close to 1/3, which is brutish by comparison to these beauties.
A proof (visual or otherwise) shows "how" some statement is true, as in how it is built by the preceding truths. But I always wanted to know "why" something is true. For example, a biological cell grows and division happens. I could find tons of literature which talks about "how" this happens, but not "why" this happens. What's the motivation or goal? And why that goal is pursued? What is the force behind seeking of that goal?
You can't anthropomorphise a cell, just like you can't anthropomorphise a lawnmower, or a Larry Ellison. It's just an entity harnessing an entropy gradient.
On the topic of biology specifically, you might like The Selfish Gene by Richard Dawkins.
He argues/explains how evolutionary forces become dominant, with much more focus on the why. Why it has come to be that living things grow, multiply, and over time changed in ways that out-succeeded the prior ones, down to the level of DNA--and that these driving forces are manifested by individual genes.
I've been thinking about cancer. Maybe systems of replicators are prone to overdoing it by nature. The idea was that any universes compatible with life will also have spontaneous cancers because that's just what those universes do.
And then I learned the theory that many cancers are caused by undiscovered DNA-based viruses which tamper with the cell cycle to activate the replicative machinery that they need to make copies of their genome (HPV does this, and several others too). So then it was a switch: not an immutable feature of the universe, but caused by an agent.
But it's starting to look like viruses emerged independently more times than expected, so maybe it is more like "the universe just does that," and viruses are just cancers with a space program. Back to where I started.
I suppose some would see these loops as unproductive. "First principles" people. Descartes, etc. But I think that unresolvable why's like this are what understanding is made of.
> unresolvable why's like this are what understanding is made of
Certainly the deeper into the why chain one can get personally, the greater understanding one has.
Cells that didn't grow were outcompeted. Cells that didn't replicate were outcompeted.
This seems the most concise and precise answer to the question of "why". At some stage, somehow, self replicating molecules appeared in the primordial soup*. Everything since then has just been improvements (or maybe complete overhauls) to that process.
*How (or why) this step occurred is another intriguing topic.
> What's the motivation or goal? And why that goal is pursued? What is the force behind seeking of that goal?
There's no force and there's no goal. These things happen because every moment is a direct consequence of the previous one.
> there's no goal
Try talking about biological operations without invoking “function”. Claiming it’s “convenient” to do so doesn’t cut it: convenient for what?
Why do acorns become oak trees? They must be causally ordered toward that end. That’s telos.
Even efficient causality presupposes telos. Why does striking a match against a matchbox consistently produce fire? Because the match has a causal ordering toward that end. Otherwise, you could not explain why fire consistently results as opposed to random things like a flock of seagulls or a BMW 7 Series…or nothing at all.
Telos is not necessarily a matter of some external purpose or Paley-style watchmaker. That’s mechanistic metaphysics appealing to a watchmaker to explain a purpose things would - under that metaphysics - inherently lack. It is a matter of causal order and directedness.
> Try talking about biological operations without invoking “function”.
If you had a strong vendetta against mistaking map for territory, you could very well talk in terms of past survival and statistics. It's just not necessary for regular biological talk. It becomes relevant only when you start going to the boundaries.
> Why does striking a match against a matchbox consistently produce fire?
Because you wouldn't call these objects a "match" and a "matchbox" otherwise.
This is a good, succinct unpacking of the metaphysical stakes. Nonetheless I am curious for the world where striking a match results in a shower of seagulls.
Universal goal of life: make more life.
When I was 10 years old, I asked my maternal grandfather, "why does anything exist at all?"
My grandpa explained it in layman terms which even I could understand. He said, "If nothing should exist because it is simpler state to be in for everything, a sort of Primordial Law. Then what is the mechanism by which this law is enforced. Who or what is ensuring that Law is implemented everywhere for eternity. If we assume that such a mechanism must exist, then we have just proved that something must exist."
That's a really bizarre and oddly Platonist take on things. Your grandfather was viewing laws of nature as rules imposed onto reality by some outside force, and which therefore need some "mechanism" to be in place to "enforce" them.
But I think a more reasonable understanding of natural laws is that they're our attempt to describe the cause-and-effect patterns observable within reality itself. They're not being enforced, they're simply manifest.
Construing "nothing can exist" as a rule that has to be enforced, and not just the absence of any patterns of causality that would produce something that exists, seems to be an error. It actually seems to be a more sophisticated version of reifying the concept of "nothing" such that "nothing exists" would be interpreted as describing the positive existence of an entity called "nothing" rather than merely describing the absence of any such entities within the context.
I don't disagree with you. You might like this video.
A Bubble of Absolute Nothing - Sixty Symbols
https://youtu.be/t8QonEChDGY
This is interesting, it reminds me of the chain of logic from this article:
https://alwaysasking.com/why-does-anything-exist/
> In a reality containing nothing, there are no things as such — at least no material things. But in such a nothing, there is an abstract thing: zero.
> Zero reflects the number of material things to count. But how many abstract things are there to count? There is at least one. The one number that exists to define the number of material things is zero.
> But if we have one number and it is one thing to count, now another number exists: one. We then have zero and one together as the only numbers. But now we have two numbers. Now two exists…
Your grandfather's explanation seems to echo this in terms appropriate for a 10-year-old - there is something inherently unstable about nothingness.
Null set as 0 and then successor method of defining new numbers.
But your way of putting it is like these successor function could be considered as edges of graphs or references or signposts
Imagine a number system with 3 distinct types of Null Sets and they meet at number P after applying successor function for 10, 42, 135 times respectively.
Here's a video lecture of Graham Priest
Graham Priest - "Everything and Nothing" (Robert Curtius Lecture of Excellence)
https://youtu.be/66enDcUQUK0?si=nAZjkauxg75lvuZm
basically: "life, uh, finds a way"
really? when you were 10?
when you sleep under clear skies in the night, these questions are normal.
This is a category error. A cell is not a thing that has a goal. To imagine it has one is pure anthropomorphism. The religious may have other views of course.
Then, who or what has the goal that is driving the cell division?
What is the goal of a rock falling off a ledge?
Just because things interact doesn't mean there was a goal end state.
But, life is different. Life that survives better, out competes life that doesn't survive as well. So biology becomes incredibly fine tuned for one goal: survival.
Not survival of the individual, although that is part of it. But survival of the thing that makes the choices of how to survive. I.e. our genes. They evolve to enhance their own survival.
Which is why we care about our family more than others. Genes for caring about family have an advantage over genes that don't. Because many of our genes will also be in our other family members.
So from that perspective, each of our cells cares about doing its part to help ensure our genes survive. So it cares about itself surviving, but it also cares enough (when working properly) to sacrifice itself when it is too damaged. Because the cell doesn't last forever no matter what, but the individual it is in can pass on its genes if it sacrifices for that individual.
> So biology becomes incredibly fine tuned for one goal: survival
Again, this is a category error. It works well as a metaphor, but evolution has no goal. Evolution is a description of a simple fact of the universe: things that are good at making copies of themselves become more prevalent. Hence things that become better at making copies of themselves prevail more than ones that don't.
The what is evolution and again, it is a category error to assume it has a goal. Why do you presume a goal is necessary?
I think this Veritasium video might speak to your questions: https://youtu.be/XX7PdJIGiCw?si=5lwB3rsFNKuXyMfA