A Mathematical Emergency.

[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.

[CraigD]

Originally Posted by Don Blazys

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



would result in:

If by “crossing out the Ts”, one means “removing every occurrence of the character “T”, this is true. However, this typographical is not a valid algebraic transformation, any more than “crossing out the ‘+’s” is a valid transformation of
“1+2+3=6” into “123=6”.

In algebra, the correct meaning “crossing out” like terms is “summing the exponents of like terms”. So

is rewritten

summing the exponents of like terms gives

which may be rewritten


I don’t doubt that there are many “mathematical crises” in the form of schoolteachers who understand Math so poorly they can’t distinguish between typographical and algebraic transformations. The solution, however, is IMHO better education of teachers and more careful selection of teachers by schools to prevent people who don’t understand fundamental math from teaching it, not new, informal, and unsupported ideas about algebra, like “Blayzs terms”.

Originally Posted by Don Blazys

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".

Ah, but you’re a very fun tired, cantankerous old fuddy duddy, and a valued member of hypography.

Humility, in my experience, is an easier character trait to claim than to attain. It’s easy to express humility by acknowledging ones shortcomings. It’s harder to express humility by recognizing that criticism of a cherished idea is correct, and the idea is wrong.

[modest]

Originally Posted by Don Blazys

Thanks for the constructive criticism.
I sincerely apologize for any of my posts that come off as "arrogant" or "demeaning".

Thank you, no problem, and I did indeed mean it as constructive criticism and it’s wonderful you took it as such. A respected member of our forums here sums up nicely where I was coming from in this post. It’s excellent advice. In fact, I'll quote it:

Originally Posted by Buffy

When introducing revolutionary breakthroughs, probably the worst approach is to say "everything you know is wrong, and I'm about to prove it to you." It may sound more sneaky, but you'll find that you have a much easier time if you introduce what you're talking about as a "slight improvement" on existing theory--something that you almost do up until you hit the excerpted quote!

Let someone *else* characterize your astounding breakthrough as such: when you say it yourself, it pretty much guarantees that it will be dismissed out of hand, even by those who should know better.

Successful Physics requires good Marketing,
Buffy

Originally Posted by Don Blazys

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



would result in:

Nice We can call this the whiteout method of canceling common factors... you just erase any mention of ‘em

Ok... seriously though... I think you’re probably making a play on the words “crossing out”, but you’re the only one using that phrase so there’s really no point in exaggerating its meaning. I’m saying “cancel” which is the appropriate mathematical term for eliminating a product that equals one. It is not algebraically impossible to cancel (T/T) in the RHS of this identity:

Every reason you give for leaving (T/T) is a reason to choose to leave it and not a reason to prohibit the operation. By the way, I believe the line following the identity properly sets your domain which is necessary to include for reasons I gave in my last post.

As far as the other thing I said “canceling the T's in your term”. I’m referring to this term:

In particular—the story you told about some student winning a bet because T could not be eliminated from this term using algebra. I contend (and I think I’ve shown) that T can be eliminated from the term so long at T isn’t zero. I realize that doing so goes against what you’re using this term for, but that does not make it algebraically impossible. It is possible. As I said before, it’s just as possible as canceling the T’s in this term:

I’m not saying you would do this to your identity. That would defeat any purpose of having the identity. I’m saying by normal rules of algebra, it can be done.

Originally Posted by Don Blazys

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.

Possibly a fine reason to *choose* to leave (T/T).

Originally Posted by Don Blazys

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".

No. Notice the Pythagorean trigonometric identity,

The right side of this identity shows no evidence that it was derived from the left yet it keeps its title “identity”. Examples abound.

Originally Posted by Don Blazys

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:

But T cannot equal a in your identity—it results in division by zero (if x=1 or not). So, I’m not sure where you’re coming from with that. Similarly the rest of your post goes more into that topic which I’ll have to think about to a greater extent.

~modest

[Don Blazys]

To: Craig D,

Quoting Craig D:

Ah, but you’re a very fun tired,
cantankerous old fuddy duddy, and a valued member of hypography.

Thanks. Hypography is an incredible science forum,
and being a valued member means a lot to me.
If I collect the "Beal prize", then I will have enough money to
be a generous sponsor.

Quoting Craig D:

Humility, in my experience,
is an easier character trait to claim than to attain.
It’s easy to express humility by acknowledging ones shortcomings.
It’s harder to express humility by recognizing that criticism of
a cherished idea is correct, and the idea is wrong.

I agree.

Now, why so many in the math community continue to cherish the demonstrably wrong,
totally ineffectual present day method of representing and eliminating common factors
using terms that idiotically allow trivial common factors to creep in is beyond me.
It doesn't match obsereved results, or even common sense and clearly needs fixing!

A worldwide show of humility on this matter is definitely in order!

As for me, well, I already proved my humility over ten years ago by admitting
that the method for representing and eliminating common factors that I was taught
and had gotten so accustomed to using, was not only wrong,
but was so blatantly wrong as to constitute a joke!
I also had the courage to do something about it,
so I developed a stronger, more logical algebraic term that
doesn't allow trivial common factors to creep in.

I still feel that algebraic terms that don't allow trivial common factors,
and whose variables are so well defined that they can actually be
refered to by their domains, are probably the only hope we've got
to make the representation and elimination of common factors meaningfull.

So let's explore this possible solution together.
If it turns out that my "Blazys terms" are also inadequate for
representing and eliminating common factors, then I will be the first to admit it.
(Only a fool would stand by a result that is demonstrably wrong.)
However, if that turns out to be the case, then the problem of
meaningfully representing and eliminating common factors will still be with us
and a worldwide show of humility on this matter will still be in order!

Don.

[Don Blazys]

To: Modest,

Quoting Modest:

As I said before,
it’s just as possible as canceling the T’s in this term:

The above term, as is, allows us to divide each and every by ,
which results in:



However, the "Blazys term" as is does not allow us to divide each and every
by . That's a big difference!

Quoting Modest:

The right side of this identity shows no evidence that it was derived from the left
yet it keeps its title “identity”. Examples abound

.



is how I prefer to view it.

Quoting Modest:

But T cannot equal a in your identity—
it results in division by zero (if x=1 or not).
So, I’m not sure where you’re coming from with that.
Similarly the rest of your post goes more into that topic
which I’ll have to think about to a greater extent.

If:



then letting results in:



where the entire exponent on the left "cancels out" using the "white out" method!
Thus, at , all we are left with is:



Now, and only now can we let .

You see, if were indeed true for all values of ,
then we would have the greatest mystery ever on our hands because
we would then be required to explain exactly why the properties of logarithms
should insanely preclude when both the term on the right
and common sense tell us that must be possible
under certain conditions such as

For me, it's a pretty safe bet that the properties of logarithms are not insane!

In short, if there is absolutely no condition under which my "Blazys terms" will allow ,
then I have discovered a huge flaw in the properties of logarithms because clearly,
must be allowable!
After all Mr. Napier was a great mathematician. He was not a "nut" , and his invention,
when used properly, could not possibly result in "nutty" preclusions!

So you see, it absolutely must be the case that
is indeed possible if and only if .

Don.