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There is already way too much "bad blood" on this God forsaken planet,
and I refuse to contribute any more. Please close my threads.
Don
I apologize if post #75 was in any way viewed as "rebuking the poster for post #74".
I am being as polite as I know how, but if that is insufficient for this forum,
then please close this thread, along with my other thread here and I will politely go away.
Don.
The question in post #73 involves a paradigm shift.
http://en.wikipedia.org/wiki/Paradigm_shift
I know it's a "hard question", but I believe that the human mind
is capable of answering it with a simple yes or no .
Don
It's not a demand.
It's simply a polite request.
Those who are unable to answer with a simple yes or no
are free to post whatever they want, like or desire...
as long as they are having fun.
Don
Please just answer yes or no without any commentary whatsoever.
Given the identity:
can we substitute
for ?Please just answer yes or no without any commentary.
Don
Quoting anonimnystefy:
The substitution cannot be done because the of the restrictions that the lofarithms pose.
You see folks, that so called "substitution axiom of equality",
which states that we can always and in all cases substitute
We can not allow that utterly ridiculous "axiom" to be shoved down our children's throats!
Don
Quoting anonimnystefy:
The axioms of equality do not say we can do that.
Of course they do! Take for instance the substitution axiom of equality which states that
if two quantities are equal, then one can be replaced by the other in any equality or expression.
Well, the two quantities
and are indeed equal,then we quickly find that it's quite impossible, because clearly, the above identity has thoroughly and irrevocably
negated the closely related symmetric axiom of equality which states that if
Now, this is not the first time that a faulty axiom has been negated by a perfectly logical mathematical construct.
We must not forget that the negation of Euclids fifth axiom (parallel postulate) ultimately resulted in many vastly
superior "non-Euclidean" geometries, one of which even allowed Einstein to formulate his theories of relativity!
Don.
Quoting anonimnystefy:
The identity doesn't hold when a=b.
That's what I said in the opening post of this thread!
So, do we now agree that given the identity:
Don
Quoting anonimnystefy:
This is the flawed step.
Quoting anonimnystefy (one post later):
It is an identity when a<>b.
Well, if it's an identity, then there can be no "flawed step".
Don
That's an identity, pure and simple. You are wrong.
Yes, I comprehend it perfectly. It shows that you are wrong.
Don.
Which of course proves my point!
That your posts aren't even in LATEX shows that you can't
even take the time or trouble to articulate yourself clearly!
In other words, it implies that you are not serious.
cmowla's graphs clearly show that my equations are both true and correct.
Why you can't handle that, I don't know.
Don.
Your posts aren't even in LATEX. cmowla's "Mathematica" results are clear. You are simply wrong!
I wish there was a kinder way to say that, but... this issue is just too important for us to "beat around the bush."
Don.
To: cmowla,
Thanks for the extended derivation and for articulating the stipulation "AND a and b are non zero".
Most of all, thanks for making it abundantly clear that the equations are both true and correct.
It should now be obvious to at least some of our more advanced readers that the axioms of mathematics
are indeed badly flawed and need to be revised immediately.
Math is supposed to be fun, but how can students have fun when their reasoning power has been compromised
by their having been indoctrinated into a system of logic that has, as its foundation, "axioms" that are badly flawed?
Indoctrinated minds can't think. They can only "parrot" what they have been taught.
You have seen first hand the damage that has been done. Just look at all the hassle that you had to go through
in order to get this argument even partially understood....... if that!
Why do you and I see the gist of this issue so clearly, while others are still struggling with it ?
I will venture to say that it's because you and I had teachers who did not force us to believe in
a bunch of gibberish and actually encouraged us to think for ourselves!
Now, the only question that remains is what are we going to do about it?
In another month from now, most students will be back in school, and many more will be indoctrinated into those
same badly flawed "axioms".
Are we going to let that happen?
Don.
So clearly, we all agree that division by zero is strictly disallowed.
Therefore, we must all agree that given the equations in this post:
Quoting myself from post #1:
The "foundations of mathematics" are its axioms, which are defined as "self evident truths".
So, let's have some fun with them. Let's "shake" those foundations a little and see what happens!Consider the "symmetric axiom of equality" which states that "if
, then .Well, if
where ,and the properties of logarithms allow
where ,then clearly, that so called "symmetric axiom of equality" is neither self evident, nor always true!
Don.
it's not always true that "if
then ",Don
To: noelevans,
Quoting noelevans:
We can certainly define a function f(x)=x/x by f(x)=1 if x<>0 and f(x)=a (a any real number) if x=0. And it is certainly nice to define this as 1 since this is the limit of the function as x approaches 0.
But this is not to say that the actual number 0 divided by itself (0/0) is one. That would be equivalent to saying that zero has a multiplicative inverse, which is precluded in the field axioms.
I agree.
We can not assign a specific value to the indeterminate form
In and of itself,
can be any number becauseFrom my point of view, since
implies a true statement,Don.
Hi,
Quoting anonimnystefy:
There is no logarithmic law that allows you to transform (3*log(a)/log(b))/(log(a)/log(b)) into (3*log(a)/log(b) -1)/(log(a)/log(b) -1).
Two functions are same if and only if they map the same domain into the same range in exactly the same way. When you remove the singularity at 0 you change the domain of the function sin(x)/x from R\{0} to R.
Can you post the above in LATEX? I'm sure that our readers will appreciate it!
Thanks,
Don.
Quoting anonimnystefy:
Your "identity" isn't correct. You cannot subtract 1 from the denominator and the numerator
of a fraction and say it is the same fraction. x/y<>(x-1)(y-1) in the general case.
Please look carefully.
We are not "subtracting 1 from the denominator and the numerator of a fraction".
We are applying the properties of logarithms.
Quoting anonimnystefy:
Yes, we could define another function partially, so that it has no singularities,
but that wouldn't be the same function we started with.
That's kind of like saying that after somebody "pops a zit", they don't have the same face they started with.
I tend to view it as the same function but in the light of a higher order of logic.
The important thing is that my identity has a non-removable singularity at
Therefore my identity presents a much stronger argument for eliminating the symmetric and substitution axioms of equality.
However, if you want to join my crusade to eliminate those shoddy axioms using MrButtermans much weaker equations,
then I still welcome your support because really, those axioms have got to go.
Don
To:anonimnystefy,
Quoting anonimnystefy:
No one defined sin(x)/x to be 1.
Quoting the article "Removable singularity" from Wikipedia:
...the function
has a singularity at .
This singularity can be removed by defining ,
which is the limit of as tends to .
Please note the phrase "removed by defining".
Quoting anonimnystefy:
But, either way, you didn't derive the formula correctly.
It's not a formula. It's an identity, and it's correct.
To see that it's correct, apply the property: ln(a/b)=ln a - ln b
before the change of base. It works out the same.
Don.
In complex analysis, if we can't define
as being at ,then neither can we define
as being 1 at .Let's all Google the phrase "removable singularity" and find out!
Don
To: bobbym,
Quoting bobbym:
In arithmetic operations as you are doing care must be taken not to use 0 / 0 as 1.
I agree. Care must be taken and we can't just let the indeterminate form
.We must first know the details of how it occured in order to give it a specific value.
For instance, in the expression
Don.
To: anonimnystefy,
Hi,
Quoting anonimnystefy:
This step is wrong:
All the steps are correct, including that one.
Don.
To: TheDude,
Quoting TheDude:
Can you show the steps you took to get from
to
It's not obvious to me how you do that.
The "Blazys identity" is derived as follows:
Note that it is not possible to derive this identity if
the coefficient of the first term is either
Don.