What exactly is "exact" (was Clean Singleton Docstrings)

[Ben Finney]

Rustom Mody <rustompmody at gmail.com> writes:

> AIUI…
> There are two almost completely unrelated notions of exact
>
> 1. ⅓ in decimal cannot be exactly represented though 0.3 0.33 etc are
> approximations. We could call these inexact forms of ⅓

Better would be to use the term already used: 0.3333 is an inexact
*representation* of ⅓.

> 2. Measurement and observation produces numbers. These are inexact
> inherently.

What is “those”? The measurement is imprecise, the observations are
inexact.

It makes no sense to say that a number is inexact. Exactness is not a
property of a number. It has one value, which is its identity; it is a
singleton.

> Scheme's notion of exact is towards capturing the second notion.
> According to which
> “There were 20,000 people in the stadium” would be an inexact integer
> [Yeah note Inexact INTEGER]

The number 20 000 is an integer. It has exactly that value.

The number of people may differ (say, 19 997 people), and “20 000
people” is then an inexact representation of the number of people.

> whereas √2, e, π are all exact.

Exactly what?

A number either is π, or it is not. The number π is not exact or
inexact; it simply has one value.

> IOW numbers picked off from the real world are just naturally wrong

The measurement can be wrong. We know that the chances are very high
that the measurement will be imprecise.

How can the *number* be wrong?

You will be able to express yourself much more clearly on this topic
when you cease conflating a number with measurements of that number, or
conflating a number with representations of that number.

-- 
 \     “I know you believe you understood what you think I said, but I |
  `\         am not sure you realize that what you heard is not what I |
_o__)                                     meant.” —Robert J. McCloskey |
Ben Finney