● Can we determine the value of infinity?

Our Maths teacher used to say, you can determine the value of infinity if you have a reference. Say, you are standing in front of a parallel railway track. It seems that they

rail lines are meeting each other. He says, that point is infinity with respect to you. (Parallel lines meet at an infinite distance)
Now the question is...Can we really determine the value of Infinity?
By definition, parallel lines are always same distance apart, no matter what. This is math, not physics, and definition, not theory or theorem. Thus, they do not.
The railway lines are (optical) illusion, based on physics, not math. Although true, the do look to be closer is valid. Also a highway going straight for long distance will appear to have sides meet, to where it shrinks to a point. That is optics. Physics. Not math. Perhaps teacher was goading students for such reply (get you grade of A+ ??).
A more modern view of some, is more relevant to the infinity term. A line (or more) going on for ever and ever -- (if it had a beginning) will meet itself over that infinite distance. Thus if you got on that highway, and travelled straight forever, you'd eventually arrive at the place you began. What a stretch, huh? I think this one, while sounding more like physics than math, began a century or more ago inside concept of non-euclidian geometry (math) or curved_space (now astrophysics?). If any continuing interest, you might want to try those on as search terms.

1. You can never determine the value of infinity because the term intrinsically has no numeric value or maximum.
2. If the lines are parallel then they are parallel, whether you measure them after one yard, one mile or any distance then they must remain separated by the same distance. That is what parallel means.

3. The apparent meeting of the lines at a distance is an optical illusion and has nothing to do with their mathematical relationship to each other.
4. If you can accept that the term infinity means 'never ending' then the lines will never meet.
Infinity can simply be beyond the limit of our understanding. If we take a child who has never left their village, then infinity can be as far as the next village. If we take a man who has never left his country, then infinity can be the next country. This may not be an entirely accurate description of infinity, but it does fit to a degree.
We consider that a modern explanation of infinity to be further than the furthest star. Basically up and out. The "outside". But today we know now that we will know more tomorrow. So what ever label, description, idea, measurement or theory we assign to the term "infinity" WILL, one day, be exceeded.
What else?
"To infinity, and beyond!" (Buzz Light year).
This sort of supports my idea that infinity is a boundary that is constantly moving. We have beliefs in what we cannot know and theories and proofs to support what we do know. Is infinity just another belief system yet to be defiled by proof? Buzz has faith in his abilities when he is only (and I apologize for the spoiler) a toy.
Infinity is simply a boundary of our own imagination. A much harder term, I feel, is "forever". Time vs distance?
To put the discussion on a scientific basis, the concept of infinity stems from that of innumerability. That is, whatever we are discussing, angles, equations, distances, and so on, we equate each of our "objects" with an integer number. When we run out of numbers for our counting we talk of an enumerable quantity. This is on first consideration, infinite. But if we try counting real numbers (those with decimal places) by association with integers we discover that there are more reals than integers. This is a second infinity which is not equal to the first.
It was Gregor Cantor who investigated these things at the turn of the twentieth century.
So, I'd go back to your maths master and ask him, which infinity he is talking about!
These two definitions of infinity were first proposed by Aristotle. He called the first one (running out of numbers to count) the infinity in length and the second one (infinity in decimal places) infinity in divisions.
As we are talking about parallel lines meeting at infinity, we ar talking about infinity in length.
In a more practical way, any finite number can be considered to be infinite if it is much greater than another one which you are comparing. For example, when calculating the orbits of small planets, such as the earth, you may consider that the sun is at a fixed point and doesn't move because it has such a great mass compared to earth. But this would only be totally true if the sun had infinite mass. So you can say, as an approximation, that the sun has infinite mass compared to earth. This kind of approximation is very useful in physics, and yields good results if used correctly.
This approximation is what your teacher used when he talked about the rail tracks. The distance between the tracks (lets say, 1 meter) is much smaller then the length of the rail track (100's of kilometers). So when you try to see the whole track, due to the change of scale, they seem to be together, because its length is much greater than the the width. When you are close to the track (lets say 2 meters close) you can see they are not connected. But if you look far away (10 kilometers) the width is so small that you cant resolve the two tracks anymore.
The rail track infinity is an approximate infinity. It is an open question today for physicists if real infinities do happen in nature. General relativity and quantum mechanics predict the existence of such true infinities (sometimes called singularities). But its not clear if they really exist of if they are a flaw in the theory. Anyway, physicists must work their way around them, since you cant make calculations or measurements with infinite values.