[RFC] finite and bounded

[Chris Smith]

Does anyone have a good reference for the definition of finite 0 < abs(x) < oo and bounded (abs(x) < oo) that they could add to https://github.com/sympy/sympy/pull/8043 ?

[Aaron Meurer]

You won't find a reference for finite being != 0 because no one uses
the term to mean that.

The real issue is that finite and infinite are more suggestive
adjectives for single numbers, whereas bounded and unbounded suggest a
function or a set, neither of which SymPy Symbols represent.

Aaron Meurer

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[Fredrik Johansson]

On Wednesday, September 17, 2014 6:54:54 AM UTC+2, Aaron Meurer wrote:

You won't find a reference for finite being != 0 because no one uses
the term to mean that.

I vaguely recall having seen "finite" meaning != 0 in scientific papers. And apparently, some dictionaries include this definition (http://dictionary.reference.com/browse/finite). But I would not use such a definition in a computer algebra system. It makes some sense if you have a notion of infinitesimals (finite = neither infinite nor infinitesimal).

Fredrik

[Aaron Meurer]

On Wed, Sep 17, 2014 at 6:51 AM, Fredrik Johansson
<fredrik....@gmail.com> wrote:
> On Wednesday, September 17, 2014 6:54:54 AM UTC+2, Aaron Meurer wrote:
>>
>> You won't find a reference for finite being != 0 because no one uses
>> the term to mean that.
>
>
> I vaguely recall having seen "finite" meaning != 0 in scientific papers. And
> apparently, some dictionaries include this definition
> (http://dictionary.reference.com/browse/finite).

I chalk this up to the fact that our natural language tends to
implicitly exclude trivial cases. If someone asks you "did it take you
time to get here?" and you answer "yes, it took me 0 seconds" you're
probably being snarky, but mathematically 0 seconds is a valid "length
of time".

Colloquially, if someone says something takes a "finite amount of
work" that understand that to mean, "a positive amount of work, but
small enough that it can be finished", that is, "finite" there means >
0 but also <= reasonable. But as I noted on an issue, in a computer
algebra system, we should use rigorous mathematical terminology, which
has precise definitions and doesn't require context or cultural
knowledge to understand.

> But I would not use such a
> definition in a computer algebra system. It makes some sense if you have a
> notion of infinitesimals (finite = neither infinite nor infinitesimal).

We are deleting the is_infinitesimal assumption, because it's not a
"real" infinitesimal (in the sense of non-standard analysis), and it's
essentially a duplicate of is_nonzero, except way more confusing and
harder to spell correctly.

Aaron Meurer

>
> Fredrik

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[Francesco Bonazzi]

On Wednesday, September 17, 2014 1:51:09 PM UTC+2, Fredrik Johansson wrote:

On Wednesday, September 17, 2014 6:54:54 AM UTC+2, Aaron Meurer wrote:

You won't find a reference for finite being != 0 because no one uses
the term to mean that.

I vaguely recall having seen "finite" meaning != 0 in scientific papers. And apparently, some dictionaries include this definition (http://dictionary.reference.com/browse/finite).

Etymologically, "finite" stems from "finitus" which is the past participle tense of the Latin verb "finio".

http://en.wiktionary.org/wiki/finio#Latin

The verb "to finish" is the English form of "finio", thus I would consider "finite" and "finished" as related words.

The prefix "in-" means "not" and it's related the English form "un-".

http://en.wiktionary.org/wiki/in-#Latin

This means, "infinite", which is the contrary of "finite", is a synonym to "unfinished".

To me, "finite" simply means "not infinite", I don't get the logic of those who argue that finite means != 0 or > 0

[Chris Smith]

Rather than coopting finite, the better word to use is "sensible". A sensible measurement is one that can be made because 1) there is something to measure (hence not zero) and 2) it is measurable (hence not infinite).

[Joachim Durchholz]

Am 20.09.2014 um 19:57 schrieb Francesco Bonazzi:
>
> I don't get
> the logic of those who argue that finite means != 0 or > 0

I think "not zero" comes from the classification
a) zero
b) not zero and not infinite
c) infinite
which is useful in many mathematical contexts. "finite" is just the best
available shorthand for "not zero and not infinite".

> "infinite" [...] is the contrary of "finite"

Given that there are many different behaviours that aren't bounded, "the
contrary of finite" isn't quite well-defined. Things could be positive
or negative infinity (ints and reals), "just infinite" (projective
geometry and complex numbers), all of these plus "divergent"
(sequences), and a hierarchy of uncounablenesses (set cardinalities).

IOW I think that saying "there is *the* contrary of finite" is a worse
sin (because it's unsound) than saying "finite means nonzero" (because
it's just terminology).
Since it's essentially all just terminology, everybody is invited to
think otherwise ;-)