# re: "Finite" For "Greater Than Zero"page 4

•  197
Aaron Davies: John Dean:

I don't see why you think "finite" here means "greater ... is not infinite. So why plump for the unusual meaning?

Because it's the fact that the angular diameter is nonzero that makes the durations of day and night asymmetric, whereas the angular diameter of an object cannot possibly be infinite.

In which case the criticism should be of the statement "since the sun has a finite angular diameter, the night and day durations are asymmetric" rather than of how the word "finite" is used.
John Dean
Oxford
example This odd usage has been around for quite some ... and author to author. I do not personally know a

I think it arose because people perceived a need for a word for "not infinitesimal". It was perhaps a bad choice, and for the original example "non-zero" would have done as well.

But in that use, 'finite' covers both ends of the spectrum: non-zero and not infinite. It's an economical way of saying that it's a number you could make definite if you had to, which you couldn't if it were indefinitely large or small.

john
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Aaron Davies turpitued:

I've heard people talk about "finite" probabilities before, when what ... is greater than zero i.e., that it can in fact happen.

Careful! "With probability zero" does not mean the same as "can never happen".

And "with probability one" doesn't mean the same as "must happen". Somehow.
If you pick (uniformly) real numbers between zero and one, the event of picking 0.5 has probability zero. So does the even of picking 2. I've never been able to wrap my head around the supposed difference. (And, indeed, outside of this forum have been led to believe that that's the whole point of saying that they are both probability zero events.) The nearest I can come is that one "can't happen" while the other "can happen in any given trial, but won't".
In the past times we've gone around on this, but I don't think I've seen anybody explain how one is supposed to distinguish the probability zero events which cannot happen from those for which it's merely the case that the expected (at any non-zero probability level) number of trials required to see them is infinite.

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We are in danger of drifting into maths again. Note that "infinitessimal" is not part of standard (*) maths.

Really? In high school calculus for me, quantities approaching zero in the limit were called "infinitesimal" (and "infinitesimals"). I've always assumed that it was standard terminology.

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If you search the group for older maths threads, you will find one about exponential growth. It is popularly used in ways that I am sure are not correct in any branch of maths.

But what does "popularly" mean?
There once was a thread in which I commented that a quantity with exponential growth could take a million years to grow by one percent. (See Message-ID:
(Email Removed)#1/1 , et seq.)

A response that seemed to be coming from a qualified scientist held that "exponential growth" is commonly understood to mean explosively rapid growth even among physicists who really know better.
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Aaron Davies turpitued: Careful! "With probability zero" does not mean the same as "can never happen".

And "with probability one" doesn't mean the same as "must happen". Somehow. If you pick (uniformly) real numbers between zero and one, the event of picking 0.5 has probability zero.

Must... resist... getting... sucked... in...
But can I ask what that "uniformly" is supposed to signify? I fail to see any use of random or randomly, so I wonder if this is supposed to modify the verb, as some sort of alternative to "randomly." Or does it modify "real," and so what does it mean to be uniformly real as opposed to just plain real? Or is it "nothing but" real numbers? And you're assuming "randomly" is part of the definition of "pick"?

Must... not... repeat... my... position... on... this. Not yet anyway.

Best Donna Richoux
If you search the group for older maths threads, you ... am sure are not correct in any branch of maths.

But what does "popularly" mean?

In this context, I meant outside mathematics. Was that not obvious?

There was a very long thread that you and I were involved in. I don't really want to rehash it but . . .
Yes, something that is growing exponentially could take a million years to grow by one percent.
But the special feature of true exponential growth is that it will eventually overtake most simple formulae (all polynomial ones). When mathematicians talk about functions growing quickly they usually mean that eventually they will overtake the slower ones. There is no limit to the eventually. Your example may take billions or trillions of years to overtake linear growth of 100% in the first year but, if it is truly exponential, it would sometime. And once ahead, it would stay ahead.
There are some simple formulae that grow even faster. n! (n factorial) is faster and n ^ n is faster still.
A response that seemed to be coming from a qualified scientist held that "exponential growth" is commonly understood to mean explosively rapid growth even among physicists who really know better.

Possibly because one example of real exponential growth is the runaway reaction in an nuclear explosion.
Most real examples of exponential growth do become noticeably fast in a short time.
I don't know about physicists but if a mathematician misused it, while discussing mathematics, I would doubt his abilities.

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Because it's the fact that the angular diameter is nonzero ... the angular diameter of an object cannot possibly be infinite.

In which case the criticism should be of the statement "since the sun has a finite angular diameter, the night and day durations are asymmetric" rather than of how the word "finite" is used.

Huh? What other possible grounds are there for criticism?
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And "with probability one" doesn't mean the same as "must ... and one, the event of picking 0.5 has probability zero.

Must... resist... getting... sucked... in... But can I ask what that "uniformly" is supposed to signify?

In this context, "uniform" means that the chance that the randomly chosen real number happens to fall in an interval (a,b) (for 0<= a < b <= 1) is equal to the length of the interval.
I fail to see any use of random or randomly, so I wonder if this is supposed to modify the ... you're assuming "randomly" is part of the definition of "pick"? Must... not... repeat... my... position... on... this. Not yet anyway.

J.