Can the human mind comprehend/understand infinity?

When I was a child, I used to ponder the infinitness of space. Had trouble making sense of it, but it finally clicked when I was thinking through the basic paradox of 'if space would end in a wall, what would be on the other side? Another wall? And another and another...?'.
My head exploded. I got the most weirdest sensation ever, like I was about to disappear into some microscopicly tiny void, almost impossible to explain but I will never forget the feeling...

But yeah... I think we're easily able to understand the concept, but not sure if we can ever truely comprehend the ramifications.

In actual practice of mathematics, "infinity" is not a thing. In reality, the "infinite" involves a number of diverse mathematical procedures and really refers to conceptual algorithms, the various forms of limits, Cantor diagonalization, counting arguments, etc.

Once you get out of the mindset of being awed by The Infinity, and start to realize it's meaningless on its own, then you can start making progress. Mathematics is about transformation of things to other forms and their relations. "going to infinity" is part of those processes.

I will explain, totally in vein, once again, that there is nothing wrong with, odd about, or not real about infinity. It's a perfectly normal thing used in every-day life by mathematicians, physicists, and many other kinds of scientists. All of whom completely "comprehend/understand" it.

It's not "just a concept" or "a process" any more than other ordinary numbers are. There's nothing mysterious about it.

For example, the interval
0 < x < 1
does not include 0 or 1 inside of it. (Because 1 is not less than 1 and 0 is not greater than 0.) So, you can only get closer and closer to 1 and 0, but never get there.
So,
0.9 is in the interval,
0.99 is
0.999 is
0.9999 is
etc...

But 1 is not.

This is just like the interval
-infinity < x < infinity.
Numbers inside this interval can never get to infinity, they can only get closer to it.
10 is in this interval.
100 is
1000 is
10000 is
etc...

But infinity is not!

Things like this are called "limit points."

Consider also, for example, a circle. A perfectly ordinary circle. Now, draw a horizontal line below the circle, so the circle is sitting on top. Call this line "the number line" and let the point where it touches the circle be at 0, and to the right of it are the positive numbers, and to the left are the negative numbers.

Draw a line from the top of the circle to some point on the number line. Note how this line intersects the circle in only one place, no matter which point on the number line you picked. This relates each number on the number line to a point on the circle.

So, for example, the point 0 is the bottom of the circle. The line drawn to a very very large number intersects the circle near the top.

Where does the line drawn from the top to the point at infinity intersect the circle?
Where does the line drawn from the top to the point at negative infinity intersect the circle?

Only at the top of the circle!

Only at the top. So this relationship relates each point on the number line to a single point, except the top which is associated with the points at plus and minus infinity. (This is called stereographic projection, by the way!)

Here is a random picture I found that illustrates this:


Now imagine taking a string that has a length of 1 inch. This string is described by the interval
0 < x < 1.

Now, move the left and right ends of the string up to touch each other, so the string forms a circle.

This is the same thing as you did before: you identified the points 0 and 1 together as the point at the top of the circle.

This is because the intervals
0 < x < 1
and
-infinity < x < infinity
have the same kind of behavior.

This, by the way, is called a one point compactification, and is a very simple version of trick we use in string theory to help us solve problems.

Also see:
The definition of a limit
en.wikipedia.org...
en.wikipedia.org...
for more that's similar to thinking about infinity like this.

There are lots of other ways to understand it too, but it's not a hard concept to completely understand and use in your every day life as a scientist or mathematician!

reply to post by Moduli
There are lots of other ways to understand it too, but it's not a hard concept to completely understand and use in your every day life as a scientist or mathematician!

Infinity without a context is easier to understand than perhaps an infinite universe of possibilities. For when we try to visualize the combination of each and every possible universe coexisting together simultaneously things get much more complicated than just a mathematical trick.