Jerzy Karczmarczuk jerzy.karczmarczuk at unicaen.fr
Mon Sep 14 18:17:08 UTC 2015

```The NeverEnding story never ends. Good!

Le 14/09/2015 04:17, Richard A. O'Keefe a écrit :
> Without taking a stand, I note that Clean has one class
> per "arithmetic" function.
This should not be understood too rigidly, we can combine various
sectors under one name, e.g.,

class Determinant a | *, -, fromInt a

-- as exemplified in the Clean book on FP, page 85. This is a shortcut,
but replaces Haskell classes with many members.

> I also note that Money / Money -> Number (not Money)
> and if Money * Money -> something, it certainly isn't Money.
> Similarly, DateAndTime ± Duration -> DateAndTime,
> and DateAndTime - DateAndTime -> Duration, but
> DateAndTime + DateAndTime does not make sense.
> (DateAndTime and Duration come from Smalltalk.)
>
> I've chosen these examples because they are not "metaphorical"
> E.g., now + (now - now) = now, now + (dur*2) = (now+dur)+dur.
> (Assuming exact numbers underneath.)
>
> Sounds like multi-parameter typeclasses are going to be vital
> for a restructuring of the numeric classes.

I am afraid that multi-parameter classes will not "save" anything
without some precision concerning types themselves. As I say from time
to time, physicists live with similar questions for centuries, in the
context of physical units.

Money*Money is strange. But what about  time*time? mass*mass? The
gravitational constant G has the dimension m^3 kg^(-1) s^(-2). But only
in SI, and related system.

You find all possible powers, of dimensional quantities, depending on
your choice of *fundamental* unities. I learnt in school the
MKS/electrostat system, where an electric charge had the dimension of
the square root of force, m^(3/2) kg^(1/2) s^(-1). Later, studying
quantum electrodynamics, I assimilated the "truth", that electric charge
is just a number, dimensionless. Why?

Because physicists have no reluctance to use UNIVERSAL constants as pure
numbers, which redefines meters, seconds, etc. : the speed of light is
1. The Planck constant is 1 as well. And the electric charge becomes
dimensionless.

Cosmologists add G=1, and all dynamical quantities become dimensionless,
no more meters, seconds, kilograms. Just numbers. They write formulae,
where some quantity is proportional to the logarithm of time. This may
disturb even those who accept non-integral powers of length. Is it
"cheating", implying log(t/t_0), where the denominator reduces the time
unit?

What about fractal objects with abominable dimensions, say
m^(1.2654132089) ?
And such things exist in physics, they are called, for example,
anomalous dimensions, measurable.

If we decide that there exists universally One Currency to Rule Them
All, say, Renminbi,  then
money + money*money + 1/money + exp(-money) becomes computable.

Of course, you will not measure your waist in Planck length units
(although for myself, year after year, it becomes less crazy), but all
this is not a trivial issue.

==

This is another reason (the first was torsors) why I don't believe in
too simplistic playing with straitjackets, forbidding some operations in
the name of "coherence".  DateTime vs Duration is another version of the
point/vector discussion, or vector vs. affine spaces. Thousands of texts
covered all that...

Jerzy Karczmarczuk

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