To: Modest,
When you wrote that I "accidentally missed all major points of contention and critisism",
you forgot that I am addressing the many points that were made, one at a time,
in the order that they were made, because that's the best way to avoid further confusion.
Please be patient. I will get to each and every point, I promise.
Anyway, now that we have established that the "math community"
is in a state of confusion as to whether or not unity constitutes a
"proper factor" or "proper divisor", let's move on to the next point,
which is one of those "points of contention".
Quoting Modest:
Quote:
|
One is a factor of 28 once.
|
Quote:
Either I'm missing something big or there's nothing about
number theory that precludes multiplication by unity.
Since multiplication by unity is included in the axioms of natural number arithmetic,
I'm pretty confident that it isn't precluded.
|
Well, I agree that
is a factor of
only once.
But, here is the real question:
If we have the multiplication:
,
then did we begin with
and increase it by a factor of
,
or did we begin with
and "increase" it by a factor of
?
You see, while it's clearly possible to increase unity by a factor of twenty-eight,
it's simply not possible to "increase" anything by a factor of unity,
so "multiplication by unity" can't possibly exist, and something that
can't possibly exist, can't possibly be defined!
Thus, if unity occurs in a factorization only once,
then it is possible to interpret that factorization "meaningfully",
because unity can then be viewed as "strictly a multiplicand".
However, if unity occurs in a factorization twice,
then it is no longer possible to interpret that factorization "meaningfully",
because unity would then have to be viewed as both a "multiplicand",
and a "multiplier".
In other words, if unity occured in a factorization twice,
then we would, in fact, be lying, because the "multiplication":
can only be interpreted as:
"We began with
, then "increased" it by a factor of
".
As I mentioned in post #97, in number theory, multiplication is strictly and stringently
defined as "repeated addition", so that the multiplication:
is viewed as either:
or
,
where the number itself is called the "multiplicand",
and the number of times that it occurs is called the "multiplier".
Now, if we apply this definition to the multiplication:
, then we get either:
or
,
Notice that the representation:
does not qualify as a "repeated addition", or, for that matter, a "repeated" anything!
This tells us unequivocally that unity is definable only as an operand.
In other words, "multiplication of unity" can be defined as a "repeated addition",
but "multiplication by unity" can not be defined as a "repeated addition".
Applying this "repeated addition" definition of multiplication to the multiplication:
yields:
,
and it is now painfully obvious that the representation:
does not "repeat", and that the representation:
has nothing whatsoever to do with multiplication as it is defined in number theory!
Now, the only reason that "multiplication by unity" has been tolerated for so long,
is because up until a decade ago, mathematicians had no way of precluding it.
In other words, until I invented "Blazys terms" about ten years ago,
mathematicians had no idea as to how to write an algebraic term
that is perfectly consistent with the actual definition of multiplication
as it applies to unity!
"Blazys terms" verify the above observation that "multiplication by unity" is undefined,
and since they are nothing more than a logical and therefore unavoidable consequence
of the properties of logarithms, it is clear that any notions whatsoever regarding
"multiplicative identities" or "identity elements" can not possibly qualify as "axioms",
because any such notions are clearly inconsistent with both the definition of multiplication,
and the properties of logarithms.
We have already shown that the "math community" has a very poor understanding of unity.
If it can't even agree on whether or not unity constitutes a "proper factor",
or even a "proper divisor", then how can it be trusted with the subtle nuances of
establishing it's axioms !?
When it comes to science, we must not be like lemmings, blindly following each other off a cliff!
We must be men (and women) who actually use our God given ability to reason!
Don.
