Once there was a demure young woman. So demure was she, in fact, that when a suitor first came to call, she sat on one end of the couch and made him sit on the other end. Each day he called on her, he could sit half as far from her as he had the last time.
A young mathematician called on her one day. When she explained the seating arrangements, he thought to himself, "It will take an infinite number of visits before our hands will meet," and did not return a second time.
He was surprised, then, to read in the paper a few months later that the woman was engaged to one of his acquaintances. He called the man up and said, "Didn't she make you start at the far end of the couch, then sit half the distance of your previous visit each time?"
"Yes, she did."
"But that series requires an infinite number of visits before it converges to zero. You can't have even held her hand yet!"
"Look, you're a mathematician, and your reasoning is flawless. But I'm an engineer. I only had to get close enough."
Life is not as formal as mathematics. As an educator and an intellectual, John Henry Cardinal Newman found this problematic:
Boys are always more or less inaccurate, and too many, or rather the majority, remain boys all their lives. When, for instance, I hear speakers at public meetings declaiming about "large and enlightened views," or about "freedom of conscience," or about "the Gospel," or any other popular subject of the day, I am far from denying that some among them know what they are talking about; but it would be satisfactory, in a particular case, to be sure of the fact....
There's no getting around the fact that we use language in a vague way. At the same time, there's no denying that, as a rule, we get close enough to each other's meanings in the ordinary exchanges of daily life.
One of the ways we do this is by being able to recognize how formally others are speaking, and adjusting our interpretations accordingly. "I'll call around 3" is a lot more vague than, "I'll call around 3:05." The precision of "3:05" implies a greater precision in the imprecise "around."
If we're going to successfully adjust our interpretations, though, we need to be able to adjust them properly; less tautologically, we need to be aware of the correct interpretation from among all possible interpretations.
Sometimes, though, the interpretation that needs to be given to "almost" is simply "not." If you're ready to jump out of an airplane, and they're almost done with the safety check, your take-away is that they aren't done with the safety check. If the new software database is almost compatible with the old software database, then it's not compatible.
People aren't always prepared to recognize this. We talk of "glass half empty" vs. "glass half full;" what happened to "glass not empty" and "glass not full"?
To understand the "first order meaning" of a statement like "we're almost there" as "we're not there" is something that may not occur to some people. It's a concept related to analogy, in which to say something is like another is to say it is also unlike another, and sometimes it's more important to understand how two analogous things are unlike.