PaulW, you are correct about parallel lines never meeting. I do not know the origin of the concept that they meet at infinity. It might be a "slang expression" used instead of saying that they never meet. Perhaps it is equivalent to saying that parallel lines meet when Hell freezes over. As far as I know this is a totally erroneous "common notion" rather than a formal statement about geometry. Stating that 1 / 0 = infinity is a common notion that cannot be called erroneous, but it does not lead to any usable mathematical ideas.

Dealing with these concepts is confusing and counter-intuitive enough without introducing erroneous notions like parallel lines meeting at infinity (see below).

Among other statements, you said
Mathematics is a tool - simply a meta-language to describe the real world. Serious mathematicians include those who apply mathematics (such as engineers) as well as professors in ivory towers.
I cannot fault your statement, but I do like to think that mathematics has esthetic value and is more than a tool. By the way, I do not think that 1 / 0 = infinity is a concept used by practical engineers. While above is correct, the professors in ivory towers do their best to make sure that the tools used by the engineers have no hidden flaws.

I have been on both sides of the theoretical fence, and have actually used computational techniques to validate numerical methods not considered provable by the academic community, something considered a sin by some of my college professors.

When mathematical analysis (as opposed to modern set theory) deals with infinity, it uses the concept of limits or limiting values.

I repeat what I said about how mathematicians deal with infinity and limits. They really do avoid statements like "1 / 0 = infinity" and "A curve touches its asymptote at infinity." They prefer to make statements like "1 / X grows without bound as X approaches zero" or the curve X * Y = 1 approaches the X-Axis as X grows without bound." It is not considered a punishable blasphemy to say "1 / X = infinity when X equals zero" or the "The curve X * Y = 1 touches the X-Axis when X equals infinity." It is merely considered bad form or sloppy language to be avoided in formal literature.

Claiming that a curve meets its asymptote at infinity has some merit. This is not analogous to claiming that parallel lines meet at infinity.[/B] Consider the straight line Y = 1 and the curve X * Y = 1.
  • For X = 1, 2, 3, 4 . . . one hundred million . . . Y = 1 on the straight line. Nothing seems to change as X grows without bound.
  • For X = 1, 2, 3, 4 . . . one hundred million . . . on the curve, Y = 1, 1/2, 1/3, 1/4 . . . .00000000001 . . ..
Considering the above, the statement that the curve touches the X-Axis at infinity has some justification. There is no justification for stating that the line Y = 1 ever touches the X-Axis.

There are good reason for mathematicians to be nit-picky about language. Sometime in the last half of the 19th century, it was discovered that some erroneous concepts had been introduced into mathematics. They discovered provable paradoxes (an oxymoron made up by me) due to sloppiness in the way theorems were proved. In an effort to eliminate past and future erroneous concepts, they became very fussy about language.