Science Forums 2

[Don Blazys]

To: Modest,

Quoting Modest:

Your attitude and tone of demanding whoever you're conversing with
(and, indeed, all of mathematics) bend to your way of thinking here comes off as arrogant and demeaning.

Thanks for the constructive criticism.
I sincerely apologize for any of my posts that come off as "arrogant" or "demeaning".
In my heart of hearts, I know that I am nothing but a tired old
fuddy duddy with a cantankerous disposition and a fading memory,
which at the very least qualifies me as being "fundamentally humble".

(Perhaps not quite to the point of being "Modest" , but humble nevertheless.)
Anyway, I am well aware of the fact that I do get a little "too blunt" or "abrasive"
while defending my arguments and positions, but I still prefer to leave those parts in,
if only to make for a more lively and entertaining read.
(Believe it or not, there are some people out there who think that math is "boring"!)
Truly, it's all in fun, and if I really believed that those who I communicate with are
somehow not able to reason at my level or above, I would simply stop communicating.
The important thing here is that I will always admit when I am wrong,
(for me, it's part of being a man) which is, unfortunately,
more than I can say for many others in the "math community".
(If you want to get an idea of just how bumbling, yet pompous
many in the math community really are, just read some of the letters that they wrote to
Marilyn vos Savant after she gave the correct answer to the "Monty Hall Problem".)

Quoting Modest:

Yes, you are allowed to cancel the T's on the right.
It is basic arithmetic. At most you are offering a reason to choose
not to cancel the T's which is a valid choice. But, your statements about
the "algebraic impossibility" of canceling the T's in your term or canceling
(T/T) in the lhs of your identity are wrong.

"Crossing out" the 's on both sides of the equation:



would result in:



Now, if you or anyone else can explain what on Earth the "expression":



is supposed to mean, then I will concede that I am wrong in my assertion that it is
algebraically impossible to "cross out" or "cancell out" the 's on the left hand side.
(By the way, an alternate view of "crossing out" or "cancelling out"
is to "divide each and every by ,
but that doesn't work either because it results in "division by zero".)

Now, as for the term on the right, if we cross out those 's only,
then we will be left with:



which is "problematic" in several respects. For one thing, the term without the 's
shows no evidence that the greatest possible factor was extracted, then cancelled,
which means that it does not necessarily represent a prime as does the term on the left.

Also, there is now no evidence that the term on the left was derived from the term on the right,
which means that the equation can no longer be viewed as an "identity".

Most importantly however, the 's on the right tell us that it must be possible
to let , which litterally forces us to let
which in turn allows us to "cross out" the expressions involving logarithms,
so that all we are left with is:



where we may now let , or simply "cross out" the remaining 's so as to have:



which, if you think about it, makes perfect sense, because if the greatest possible factor
was extracted, then cancelled, then the terms must represent the same prime number,
and prime numbers do not have exponents!

In other words, if it weren't for the "Blazys term" on the left, then any extraction and
cancellation of the greatest possible factor would be utterly meaningless,
because there would be no logical argument that would result in the elimination of the exponent .

The "Blazys term" represents a "higher order of logic", and is thus a "template" for all other
terms.

Thus, if we are confronted with the term:

,

we can simply "invoke" the "Blazys term" to show that



Your example:



still contains as a factor, and is thus an incomplete
(and therefore "undefined") extraction and cancellation of the factor T.

eliminating it gives us:



Notice how we let and "cancelled out" or "crossed out"
the logarithms before we let .
Most importantly, notice how "Blazys terms" correctly show that
any incomplete extraction and cancellation of factors
is indeed "undefined"! Isn't that awesome?

You see Modest, all I am doing is correcting a "weakness"
in how factors and common factors are both represented and eliminated.
I should think that the math community would be overjoyed at the
prospect of correcting something that is not only meaningless,
but embarassing as well!

Don.