Passage 1

One very interesting thing about math is the most fundamental and
simplistic considerations of things like addition and multiplication. The
distinction between multiplication and addition may be regarded as being a
question of order and disorder. In addition quantities of varying sizes are
all lumped together to yield a sum. In multiplication, you have quantities
which all have equal sizes, and instead of lumping them all together as in
addition you simply multiply. The only reason that multiplication is
possible is because it is possible to have a collection of numbers of equal
magnitude in the first place.

It does sound strange to think that addition works on "disordered
collections" of things, and multiplication works on collections of things
which inherently contain "more order" because they are all identical.
Admittedly, this last statement is very poorly written, but I think that the
idea is clear even if we (or maybe just I ) lack a suitable nomenclature to
explain it properly.

At any rate, if it were impossible to assemble a collection of objects
of equal magnitude then multiplication would likewise be impossible. And to
think that this might be related somehow to order and disorder, most people
simply dont think that way. I think that most mathematicians would not think
of it that way either. But the argument is daunting. Beautiful, and
profoundly simple.

So, what's the difference between addition and multiplication ? Order ?
Disorder ? Can it be thought of that way ? Does anyone already think like
that ? Does this seem so outrageous ? I dont think that it is outrageous at
all. Just very, very strange.

But, as we shall see, even more strange is the fact that if you are
given any number there is no way to determine if it is a sum or a product.
For any given number you therefore have a "duality", whether a given number
is dissected into a collection of identical constituents, or whether those
constituents all have different sizes, the sum and product are certainly
equal to each other and this is very mysterious. I think that it really is
mysterious, at least in terms of how it might be related to order and
disorder.




Passage 2

Elsewhere on these newsgroups I have been babbling ad nauseum regarding how
to justify treating length as being probabilistic. For brevity, much
material has been omitted here and we jump right into a couple interesting
considerations.

We want to use a "complex-like" number system where a is real and ~b is
taken to represent an uncertainty. Formally, I dont think that ~b is a
number, but it should obey certain rules of algebra as if it were.


Multiplication implies an everywhere probabilistic notion of length:
(a)*(~b)
| ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ |


Addition implies length which is well behaved everywhere except the
endpoint(s).
(a + ~b)
|---|---|---|---|---|---|---|---|---|---|---|---|---|---| + (~ ~ ~)


Using these notions of length we can now imagine a space which is
"indeterminately either continuous or discrete". It seems quite tempting and
reasonable to try to explaion the wave-particle duality as being a result of
this indeterminacy.



Passage 3
Conclusion.

Lastly, compare and contrast these ideas. The ideas in the first passage,
and the ideas in the second passage. Clearly, addition and multiplication
lie at the very heart of both situations. Am I the only one who finds that
just a little bit peculiar ?

Is it possible that the wave particle duality boils down to something as
simple as just this ? An oddity related to multiplication, addition and
order ? Wouldn't that be strange if it were ?

I have tried to present this as non-technically as possible in order to
avoid losing the central notion in a sea of rigor and technicalities.
Perhaps the reader will find the general idea to be a thing of beauty, and
that is my hope.

Would'nt it be funny if the world's leading physicists had been stymied for
over a hundred years over something as simple as the difference between
addition and multiplication ? Would'nt that be funny ? Will they _ever_ live
it down ? : )

Regards,
Tomoko Kanazawa