[christiangenco]

I've had the same sort of difficulty with phrases like "most" or "almost all" or "hardly any"—I crave for these to map to unambiguous numbers like the probability yardstick referenced in this article.

I spun up a quick survey[1] that I sent out to friends and family to try to get some numbers on these sorts of phrases. Results so far are inconclusive.

1. https://www.vaguequantifiers.com/

[SAI_Peregrinus]

"Almost all" is an interesting one, because it has family of mathematical definitions in addition to any informal definitions. If X is a set, "almost all elements of X" means "all elements of X except those in a negligible subset of X", where "negligible" depends on context but is well-defined.

If there's a finite subset of an infinite set, almost all members of the infinite set are not in the finite set. E.g. Almost all integers are not 5: the set of integers equal to five is finite and the set of integers not equal to five is countably infinite.

Likewise for two infinite sets of different size: Almost all real numbers are not integers.

Etc.

[mannykannot]

The more precisely they are defined, the less frequently will you see them used correctly.

[jbaber]

"Almost all" in math can mean "except at every integer or fraction" :)

[JadeNB]

> "Almost all" in math can mean "except at every integer or fraction" :)

I am a mathematician, but, even so, I think that this is one of those instances where we have to admit that we have mangled everyday terminology when appropriating it, and so non-measure theoretic users should just ignore our definition. (Similarly with "group," where, at the risk of sounding tongue-in-cheek because it's so obvious, if I were trying to analyze how people usually understand its everyday meaning I wouldn't include the requirement that every element have an inverse.)

[tejtm]

I would expect almost NO numbers are rational (integer or fraction) with an infinite number of Reals between each.

[JadeNB]

> I would expect almost NO numbers are rational (integer or fraction) with an infinite number of Reals between each.

You're right (technically correct, which is the best etc.)! That is why "almost all" can mean everything except rational numbers.

[noqc]

in between any two real numbers, there is a rational number, and vice versa.

[concordDance]

And yet somehow there are infinity times more reals than rationals...

Very hard to get your head around!

[noqc]

An fun, immediate, horrifying, consequence.

If I remove n elements from R, the remainder has n+1 connected components.

The complement of Z in R has |Z| connected components.

The complement of Q in R has |R| connected components.

[layer8]

The semantics are almost always reasonable: https://en.wikipedia.org/wiki/Almost_all

[dullcrisp]

Sure but that’s because 100% of real numbers, by any standard measure, aren’t integers or fractions. It bothers me if it’s used to mean 95% of something though.