“NO TWO POINTS ARE NEXT TO EACH OTHER.” That’s a very helpful idea, thank you.
I’m sorry, but why does everyone ignore this:
That is, the definition of a circle.
In favor of these:
Which, while technically correct, will only amplify the repeated premise by the O.P., repeatedly re-hashing Zeno’s parodox regarding the dichotomy between infinite segments.
As was referenced previously by the OP here: http://boards.straightdope.com/sdmb/showpost.php?p=20366495&postcount=64
I find that definition clear enough. Thank you.
Ahhhh…now at least I have partial understanding. 
As the resolution gets finer, you get closer and closer to some ideal state. At infinite resolution, you reach the ideal. You can’t ever actually write it out or draw it, and maybe it’s hard to visualize it, but it is all mathematically and logically sound.
1/2 + 1/4 + 1/8 + 1/16 + … gets closer and closer to 1. With an infinite number of terms, it is 1.
You can try to find the slope of a curved line by picking two points on the line and finding the slope of that segment. As the two points get closer and closer together, the slope approaches some number. At a single point, the slope is that number. (That’s calculus.)
As the number of sides in a polygon increases, the shape gets closer and closer to a circle. With an infinite number of sides, it is a circle.
The definition of a “limit” essentially says: “close enough counts”.
Working with the concept of doing something infinitely often (the real meaning of “infinity”) forces us to get a bit tricky in trying to define what we’re doing. We can’t actually DO something infinitely often. Thus, we can’t find all the digits of π in decimal expansion. But for some things, we can reach a point in our calculations where we can say, “We see where this is going, and while we won’t ever actually reach there, we know that’s where we would be if we could reach there. So we’ll give up and say that’s the answer.”
Calculus is the math of applying that idea, if you will.
The definition of a limit essentially says “If that’s not close enough for you, I can get it closer.”
Something from a non-mathematician, Big Picture guy, that I hope will be useful in all this:
it’s a common danger, that people who become aware of various things that people do with math to either work things out, or turn their understandings into functional products, or to create a useful mathematical model of a thing…
…can become confused, and forget that the thing, and the math which is used to describe the thing, are two DIFFERENT things.
In this case, it might be useful in some situations to describe a circle as being a perfect set of infinitely small single point straight line segments, but no matter how well you write out the math to do that, a circle is really still a circle.
Hm. The Platonism in your post is somewhat spooky, but there’s a lot of truth in what you say.
There are three kinds of knowledge which pertain to any mathematical object: The intuition, or what the object means in a high-level and somewhat hand-wavy sense; the definition, which is a complete list of very precise properties from which all of its behavior can be derived; and the implementation, or how you write it down, which in former times was just the notation but we have computers now.
It’s important to realize that a definition can be wrong, and can be changed without changing the intuition. A spectacular example of this was the crisis set theory went through when Bertrand Russell proved that the definition of sets mathematicians had been using up to that point implied a logical paradox, and so was unsound. The idea of a set as a collection of objects wasn’t destroyed, but a specific model of it was, and a new model had to be created to capture the intuition (or what of it could be saved) in a logically sound fashion.
It’s also important to realize that the same object can be implemented in multiple ways. This can be as simple as a change of base changing how an integer is written or a change of basis changing the column of numbers used to write a vector in component form, or as complicated as moving from Einstein notation to C or some other programming language when writing a tensor down. Yes, the programming language form is just as “mathematical” as any other: It meets the definition, but more importantly, it follows the intuition.
If you reify the intuition so far you begin to worship it, you get Platonism. But the intuition is still very important: It allows you to understand the point of the definition, what the rules are driving at, and it allows you to use the implementation as a convenient tool to guide thought, as any notation should.
Got it. 
Thank you, that’s very clear.
Also from a non-mathematician, I agree because it seems a circle is something ‘out there’ while a mathematical representation of it is ‘in here’ in our mental apparatus.