# Re: What are units-of-measure? e.g., what's a "kilometer"?

```Alaric B Snell scripsit:

> A kilometer is just a thing, like a potato, or a person. It's not a
> solid object, but it's an amount of *space*, and space certainly exists,
> even if you only perceive it as a "nothingness" in which to put stuff.

The trouble with that view is that if a kilometer is a quasi-physical
object, then there are literally uncountably many of them.  You have
to represent each kilometer is a 6-tuple of real numbers: the coordinates
of the start (relative to some arbitrary physical origin) and the
coordinates of the end (ditto).  And to say that the Yangzi is 6300 km
long, you have to construct a sequence of 6300 of these 6-tuples
and claim that this sequence is the object of the "length" property
as applied to the Yangzi.  (This neglects the whole question of the length
of a river being in fact fractal, so saying its length is 6300 km
requires an indication of the scale, explicit or implicit.)

And as you imply, the 6-tuple view handles only straight kilometer objects:
the wiggly ones require reifying over a complete mapping between the
real numbers, and there are even more of those (aleph_2) than there are
real numbers (aleph_1).

All this is massive overkill compared to the simpler view that
"kilometer" is inherently a relation that maps an abstract object
called a "distance" onto a pure number.  See my previous posting.

> This enforces "dimensional consistency". The s=d/t equation is
> dimenstionally consistent because speed is measured in "m/s",
> displacement is measured in metres, and time is measured in seconds; if
> you take all the variables to be one of their particular unit the
> equation balances.

Note that dimensional consistency is not enough for physical consistency:
force and torque both have units of kg m / sec^2, but an equation that
purports to equate a force with a torque has no physical meaning.

> Saying that the river has a length property of "X kilometres" is pretty
> similar to stating that Alaric has an "owns" property of a set including
> (but not limited to ;-) the value "six pencils". The differences being
> that I can lay my six pencils down on a table and they are distinct
> objects, while all kilometres are indistinguishable and interchangeable;

That being so, there is no point in reifying them.

--
John Cowan      jcowan@r...        http://www.reutershealth.com
"Not to know The Smiths is not to know K.X.U."  --K.X.U.
```

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