Infinite is easier than big | Hacker News Reader

Infinite is easier than big

johndcook.com

28 pointsfogus5 comments

As an infinite set atheist, I think it's important to distinguish people who say "infinite" when they want to talk about the limit of unbounded finite behaviors ("a Turing machine has an infinitely long tape") and people who say "infinite" when they want to talk about big collections ("the cardinality of the collection of real points between 0 and 1"). A smarter language would have different words for these.

What he's saying is that by "infinite," people usually mean "arbitrarily large". I've found that if you just say "arbitrarily large" when that's what you mean, it eliminates a lot of these misunderstandings.

Of course, I talk to a lot of engineers and mathematicians. You run the risk of sounding unnecessarily pedantic.

No he's not, he's saying that infinity is a useful abstraction for dealing with "arbitrarily large". That often the tools we have for dealing with infinity can give us a better idea of how things behave in arbitrary large cases than the tools we have for dealing with smaller, more manageable cases.

His point is that infinity and the tools we have for dealing with it are useful, not pointless abstractions.

I understand that point and I agree. What I'm saying is that upon reading the article closely, I noticed he was primarily promoting the "view infinite as arbitrarily large" technique for dealing with the appearance of infinity.

Example quote from article: "Interpret the problem as saying that the width of the capacitor is so large relative to its thickness that you don’t have to worry about edge effects."

There was a great post and a slightly crazy discussion about infinity and the Hilbert Hotel here a few months ago: http://news.ycombinator.com/item?id=1333424

The penultimate paragraph of the present post sounds like a conversation you might hear in one of the infinitely long infinitely many bars at the Hilbert Hotel:

Nervous Dude: When I went to grad school, my intention was to study functional analysis. Essentially this means infinite dimensional vector spaces.

Unimpressed Girl: That sounds terribly abstract and useless.

Nervous Dude: But it can be quite practical. My background in functional analysis served me well when I went on to study partial differential equations and numerical analysis.

Seriously Unimpressed Girl: Gotcha.

Reply on news.ycombinator.com