Re: apples and oranges

["Donald E. Simanek" <dsimanek@eagle.lhup.edu>]

On Mon, 26 Oct 1998, Ed Eckel wrote:

> Kind folks:
>
> In a recent discussion on mathematically representing the
> relationship between mass and acceleration for a given
> force, the following issue arose.  We have been taught from
> early grades that we can't add apples and oranges.

This is one of those "lies my teacher taught me." Of course you can add
apples and oranges. 5 apples plus 4 oranges = 9 pieces of fruit. 4 marbles
plus 3 marbles = 7 marbles. But what if some of the marbles are red, some
blue? Does it matter? For some purposes it does matter, for others it does
not. The crux of the question is how we decide to relate numbers to real
things and real measurements.

But if someone suggests adding apples and temperature; well, that's
physically absurd. Why?

> And we have been taught that multiplication is a shorthand for
> repetative addition. 

Another half-truth. Sometimes multiplication can be thought of as
repetitive addition, but sometimes that interpretation gets you into all
sorts of difficulties, as in the example you cite. Why do teachers
continue to lie to us? Because they don't know better?

> How come we can multiply apples and oranges and get something
> reasonable?  That is, why is it that mass times acceleration yields
> something useful when addition of mass and acceleration does not? 

Now there's the horn of the dilemma. Is _ma_ to be thought of as mass
multiplied by itself _a_ times? Or as acceleration multiplied by itself
_m_ times? Either interpretation is absurd. In PV = nRT, is it pressure
multiplied by itself _V_ times? And why should that give us something
which is proportional to the temperature?

However, a person bent on saving a worthless hypothesis might devise a
calculus intreptation with limits and all that which could force this into
a form which looks like repetitive addition. But to what purpose? We need
to address the fundamental question of why the albebra of unit labels
works, and why adding acceleration to mass makes no physical sense.

> I am at a bit of a loss on this one and would appreciate
> some suggestions on how to deal with this (interesting)
> issue constructively.
>
> Ed Eckel
> Georgetown Day School
>  

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

......................................................................
Dr. Donald E. Simanek                            Office: 717-893-2079
Professor of Physics                                FAX: 717-893-2048
Lock Haven University of Pennsylvania, Lock Haven, PA. 17745
dsimanek@eagle.lhup.edu                 http://www.lhup.edu/~dsimanek
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