Re: apples and oranges

["Rick Tarara" <rtarara@saintmarys.edu>]

This will probably be not be *good* because I'm not so wise, but the
question at hand certainly helps explain why mathematicians avoid 'real
world' examples like the plague.  I think Donald has hit upon the essence of
the answer--the _rules_ and _definitions_ from our math courses really
aren't universally valid.  But let me suggest that another problem is in
thinking of Newtons as _something_ akin to apples.  Newtons is simply a
label for an abstract quantity that we have devised to help us understand
how the world works.  In fact, we really can't _just_ add Newtons like we
can apples or oranges (unless we're very careful to add only the 'horizontal
Newtons' to horizontal Newtons).  The only way we can even get the
product/ratio of Mm/R^2 to identify itself as a force with units of Newtons,
is to saddle the appropriate proportionality constant (G) with a mish-mash
of units (N-m^2/kg^2).  The units or labels really have a separate algebra
from the numerical values and the result is meaningful only if the resulting
algebra leads to a physically meaningful (or at least recognizable) set of
units.
While we can multiply one acceleration times another, the algebraic result
gives us a quantity with units unrecognizable as any physical quantity.
This is why we work so hard in lab to have students carefully carry the
units through all calculations and especially in slope determinations when
plotting and fitting measured quantities.  We also teach unit analysis in
problem solving to help be certain that our algorithms and equations lead to
meaningful answers.

{We just did a lab where we essentially _discover_ Newton's second law
through analysis of lots of data obtained using air tracks as modified
Attwood's machines,  but where we 'pretend' we don't know the
proportionality constant between falling mass and force.  Ultimately, a
detailed analysis will obtain the acceleration due to gravity, but only if
one recognizes that the slope of the particular data plotted has the units
of cm/s^2.}

Perhaps another aspect of the 'trap' we fall into with _multiplication is
repeated addition_ comes from the computer world where indeed integers are
often multiplied by repeated addition (at least in a way).  For those of us
who have looked at binary adder circuits, this way of thinking about
multiplication is operationally simple.  However, by the time one has to
deal with converting the numbers to logarithms and then adding the logs and
converting back, the purity of the operation gets lost.

Yet another problem may be the 'New Math' that teaches such definitions.
Back in 'the old days' (when we walked to school and back home in the snow,
uphill both ways) when you simply memorized the multiplication tables and
drilled, drilled, drilled, one didn't work about the underlying 'essence' of
multiplication.  Hence I don't have such a deep seated understanding of
multiplaction as addition but rather just think of multiplication as
multiplication, and I *think* I know what it means to combine two quantities
through this operation without breaking it down into addition.

Anyway, to my mind, the algebra of the units (labels) is separable from the
algebra of the numerical values and separable from the *pure* mathematics of
arithmetic operations.

Rick


From: Donald E. Simanek <dsimanek@eagle.lhup.edu>

> >
>
> I look forward to a *good* answer from the wise folks on this list. So
> far I haven't seen one. You have made my day with the best question
> striking at the heart of the fundamentals of physics which I've seen
> here in a while. Though we may think we understand this, writing a good,
> complete and coherent answer will be mighty difficult. If we can't
> answer this one, all our high-sounding talk about black holes, string
> theories, synchronicity, relativity and quantum mechanics may be suspect
> as little more than moonshine.
>
> -- Donald