Kind of reminds me of the course this thread has been taking.
People usually cheat in this thought experiment. Knowing full well such a process of division never ends, they unconsciously say: skip it! Let’s just assume we got to the end. What do we get? We get points! But geometric points come with a whole suite of paradoxes. This happens anytime you allow for something paradoxical in the first place, like skipping a whole infinity.
Yes. Infinity means there is no end. How can you assume there is an end?
When you stop an infinite process what you do is you turn it into a finite process. In fact, what happens is you realize it wasn’t infinite in the first place.
When you say that a circle is an infinite number of points that are equidistant from some fixed point all you are saying is that it can be ANY number of points that are equidistant from some fixed point and that it is BETTER if the number of points is LARGER. Nothing else.
As long as there are no straight sides.
This is a circle James.
You can be absolutist all you want. You’re still wrong fundamentally.
All shapes have sides. A circle is a shape. Therefore circles have sides. Your mind loses track after 1000 or 1,000,000. Just because you can’t perceive the imperfections, doesn’t mean they’re not there, and finally, doesn’t discount that the shape is a circle. It is.
James you have backed yourself into a corner of “only perfect circles are circles”, which is absurd. Just admit being wrong already.
Not hardly.
First, that is a “spot”, not a “circle”.
But regardless, anything that has a straight side, is not a circle.
That represents a circle … no straight sides.
There is no absolute difference between straight sides and curved sides.
To sum things up:
There is no such thing as a perfect circle.
Sure, you can pick any shape you want and call it a perfect circle.
For example, you can pick the one that passes the million-points test and call it a perfect circle.
But if the shape that passess the million-points test is the perfect one, how would you call the one that passess the billion-points test? Super-perfect?
No matter how perfect your shape is, it is only perfect in relation to what you already know. In other words, perfection is subjective i.e. it depends on experience. Outside of the context of experience, there is no such a thing as perfection.
Shapes that pass the million-points test would appear perfect to those who have no experience with shapes that pass the billion-points test.
What this means is that circles can only be MORE OR LESS perfect.
There is no UPPER BOUND and there is no LOWER BOUND.
There is no THE MOST PERFECT circle and there is no THE LEAST PERFECT circle.
The process is infinite in both directions.
But how can we decide which shapes are circles and which aren’t if there is no perfect circle against which we can make comparisons?
The answer is context.
The way we test whether some shape is a circle or not is by choosing a number of points on its boundary – I call these “key points” – and then measuring the distance of these key points from the center of the shape. If every key point is at the same distance from the center, the shape passes the test and we say the shape is a circle.
In effect, we determine which shapes are circles and which aren’t by measuring how CIRCULAR they are.
The number of key points we choose depends on our need.
If we choose a smaller number of key points, we are testing for a lower degree of circularity.
If we choose a larger number of key points, we are testing for a higher degree of circularity.
Only key points are expected to be at the same distance from the center.
All other points can be at any distance from the center.
This means that polygons, and pretty much all other shapes that satisfy the above criterion, can be considered circles.
It also means that circles are allowed to have straight sides.
The official definition does not forbid it.
A shape that passes the test of lower degree of circularity (fewer key points) can fail at the test of higher degree of circularity (greater number of key points.)
This shouldn’t make us think that the shape is NOT a circle.
Instead, we should adopt the attitude that it is a circle of lower degree of perfection.
Remember, whenever we test whether any given shape is a circle or not what we are doing is we are testing whether it has a specific degree of circularity.
Nothing more than that.
You see, that is the problem. When you can’t distinguish a square from a circle, you have a problem.
The problem is that you’re autistic. You take things out of context. You de-contextualize them.
If there is no ABSOLUTE difference between straight sides and curved sides that does not mean there is no DIFFERENCE between them.
In other words, it does not mean we cannot distinguish between the two of them.
It simply means the difference is not absolute.
When we look at a hectagon from a distance, we do not see a hectagon, we see a circle.
This indicates that circles are hectagons simplified – simplified by our brain.
The circle we are looking at DOES have straight sides it’s just that our brain is not seeing them because they are VERY SMALL.
Some one with the obvious learning disabilities that you have suffered all your life really shouldn’t be trying to denigrate others.
Yes, I do have a disability, but this disability isn’t a learning disability, it is a disability to agree with something that is wrong, in this particular case, the disability to agree with what James made up in order to imagine himself as some kind of God who understands everything about the universe.
You have embarrassed yourself in every single post in this thread but noone is expecting you to accept that considering how much you lied to yourself over the years.
Consider that I have nothing at stake – almost nothing to lose – which means that I have no problem accepting that I am wrong and no problem to learn something new.
You, on the other hand, have a lot at stake.
Unlike you, I never told myself I am a God.
If we combine that with the fact that you’re old – and we all know that older people are more rigid than younger people – then what do you think, my friend, is it more probable that I am the one who finds it difficult to learn or is it perhaps more probable that it is you who finds it difficult to learn?
You’re nothing more than an arrogant moron.
You are full of shit.
James you’re demanding absolutism for definitions which is wrong. Absolutism is theoretical, doesn’t exist in nature. Absolutism is pure idealism, within the mind but not reality. In reality, many “straight” lines are curved, as I mentioned in carpentry and how wood warps or changes over time. If you want to be exact, like an engineer or rocket scientist, using autocad and other advanced measurement, then they will agree with me anyway. Calculus compensates for infinity. Calculations are made derivative of infinity. This happens all the time.
Which is why as a shape approaches infinity sides, it becomes a circle, as I stated from the onset. So you actually agree with me, despite all this pedantic back-peddling you’re doing.
However, commonly, people view the Chiliagon and intuit “yes that’s a circle”. Children can also draw circles in the sand or dirt. Those are circles too. You standing over them, driven to madness, yelling at them, “NO IT’S CIRCULAR IT’S NOT A CIRCLE!!!” isn’t going to change anything, other than make them scared of your psychosis. Give it up. You lost the point.
The Chiliagon is a circle. The more you cling to absolutes, the more out of touch with reality you become. Absolutism is the opposite of Pragmatism. You’re being unreasonable.
You’re wrong:
One could say that mathematically defined perfect shapes only exist as abstract entities not real ones
And then those that exist in nature are therefore not as perfect as the ones that exist in mathematics
No, that is just your excuse.
Your first problem is that you are ignoring the definitional issue of SIDES.
A circle cannot have any straight sides, else it is merely a circular shape and of course with children can be referred to as “a circle”, but then one used to be able to expect them to grow up rather than argue about Santa Claus being real all of their lives.
Your second problem was telling Arc that she was wrong when she properly told you that circles have no sides. Even if you wanted to include sloppy circular shapes into the definition of “circle”, Arc would still have been correct, circles have no straight sides.
She is wrong.
You are wrong.
There is no such thing as a shape without sides. And circles are shapes.
They don’t exist in mathematics either.
They quite simply don’t exist.
The purpose of abstract entities is to represent real entities or entities that can be real.
Everything else is meaningless.
A hectogon is a polygon with 100 straight sides.
However, when we process it globally, i.e. as a whole, we do not see a polygon.
We see no straight sides.
Instead, what we see is a shape without straight sides – we see a circle.
This is a hectogon:
upload.wikimedia.org/wikipedia/ … on_100.svg
It looks exactly like circle even without these indicators for key points (you can get rid of them by inspecting the document.)
Now, take a global look at a hexagon and what you’ll see is just that – a hexagon.
You’ll see a polygon, which means, you’ll see straight sides.
You can do that with polygons that have a small number of sides but you can’t do that with polygons that have a large number of sides.
If you didn’t know that the shape in the above picture is a hectogon you’d think it’s a circle.
There is no way to know UNLESS you process the image locally i.e. by zooming in very closely and by paying close attention to every little detail.
If James says that hectogons are not circles then what that means is that he favors local processing of information above global processing of information.
In other words, that he suffers from some sort of autism.
A hectogon processed locally is just that – a hectogon.
A hectogon processed globally is something else – it is a circle.
But . . .
A circle processed globally is just that – a circle.
And a circle processed locally might be something else – a hectogon, for example.
And even saying “processed locally” is ambiguous because it does not specify to what degree locally.
A circle can remain a circle even if processed locally.
And a polygon can remain a circle even if processed locally.
A chiliagon will remain a circle even if you zoom into it quite a bit.
Why?
Because you didn’t zoom into it enough.
There is no point in defending Arc because she took things out of context.
She didn’t follow the discussion.
Ant that applies to James too.
They take things out of context.
That’s their job description.
You can’t expect them to admit they did something wrong.
Because to them details are far more important than the big picture.
It’s not much of a mistake to call them Grammar Nazis.
Appears you’re mixing up abstract with concrete.
In math a circle is “the set of points in a plane that are equidistant from a given point”. So if even one point is out, you’ve not got a circle.
But you can’t run your test anyway, since both the shape you’re testing and your measuring equipment must be made of atoms, and atoms are bumpy. Atoms can’t do points, since a point is “a geometric element that has position but no extension”.
Hume would be the goto guy on relations of ideas vs matters of fact (and I guess on the OP).
btw first post, hello world
Magnus,
Tell me, Magnus, how do scientists go about determining the BIG picture as you say?
By ignoring the details?
There are at least two axioms and there may be more which point to the importance of details: God is in the details and the other is that the devil is in the details ~~ aside from the fact that scientists have shown us a big or bigger picture through both struggling and trying to look for and figure out the details.
I included I believe at least two hyperlinks to show, to suggest, to imply that a circle can be viewed as both having no sides, seemingly being infinite and could also be looked upon as having sides when determined through measuring degrees of such.
Another hyperlink discussed the nature of reality and perception which are two sides of the same coin in this discussion.
There isn’t just one reality in this world. Universally speaking, most would see as a circle as having no sides. That is a legitimate point of view.
I myself must be a bit autistic since i see both of these terms as being equally important and in harmony with one another.
Incidentally, I have no problem with admitting that I am wrong when I am wrong. Sometimes I actually thrive on it because it makes me realize just how much I do not know and it makes me want to take another look at something.
In the case of the circle, can you look at it and see that in a sense it is not so much a question of being right or wrong to see it as being infinite with no sides and at the same time to see it having sides when it is measured?
It is a question of being able to hold two different points of view in your mind and seeing both as valid.
If you cannot do that, then you become the Nazi - not James and myself.