A Mathematical Emergency.

[Qfwfq]

Originally Posted by Don Blazys

I simply meant that it is easier to evaluate the expression involving derivatives at x=1 than it is to evaluate the original expression at x=1.00000000001 or x=.99999999999 or some such approximation to 1.

You might notice I did not do such a pointless thing, I did something else instead. Which one is quicker is a matter quite beside my point and depends much on specific case.

Originally Posted by Don Blazys

Now, the question is, at x=1, does 0/0=1.25?

No. It does not determine a value. Any value multiplied by 0 gives 0; this goes for 1.25 but also for countless others. Therefore it makes no sense to assert the = sign.

Originally Posted by Don Blazys

I say yes,

You are even going against your previous contention.

Originally Posted by Don Blazys

In other words, from my point of view, if one method of evaluation at x=1 yields 0/0, while another, equally valid method of evaluation at x=1 yields 1.25, then maintaining consistency actually requires that the two results be regarded as "interchangeable" in that particular case.

The other(s), which yield(s) 1.25, is not equally valid; it is a different thing. It is a computation of the limit but not of any value.

Originally Posted by Don Blazys

Therefore, I would not present the above facts to students who have no interest in my research.

Thank goodness!

Once has been constructed from and then and then and , I really don't see why we should only give a single-symbol name to natural numbers such as 823. Is it really a single symbol before we choose to call it a?

[Don Blazys]

To:Qfwfq.

Well, I guess that we will just have to agree to disagree that the left hand side of your equation in post #29 is "discontinuous" at x=1.

I will maintain that it is "continuous" at x=1.

By the way, I'm not alone in my view.

If you Google search:

(limits and continuity "we discuss a number of functions"),

then you will find a discussion (example 4.2, #4) about the similar expression:

(x^2-a^2)/(x-a)

and why it is reasonable to view it as "continuous" at x=a.

I will also maintain my point of view that any non-negative integer such as 823 constitutes essentially "one symbol" because it is "purely numerical", while a number such as -823 constitutes essentially "two symbols" because it is comprised of both an operation (-), and a pure number (823),... and that since any independent variable such as "x" and any non-negative integer such as 823 share the property of constituting essentially "one symbol", independent variables are "naturally suitable" for representing non-negative integers only.

You know, even though we mathematicians disagree on these particular issues, there are other issues that we do agree on.

For instance, we both agree that students who are not interested in my research should not be forced to learn the mathematics that it involves. However, thanks to good folks such as yourself who are interested enough in my work to post on my topics, my topics are now recieving a lot of attention in this and in other forums. (I even have a "record breaking" math topic on the Marilyn vos Savant forum!) As a result, many students (and teachers) are becoming very interested in my research, my website is getting many thousands of visitors, and I am now getting more e-mails from around the world than I can possibly answer.

Some of those e-mails are quite interesting and entertaining.
For instance, one college student bet six of his friends and classmates ten dollars each that he could re-write the term:

(T/T)a^x

so that it is algebraically impossible to "cross out" the T's. At first they didn't take the bet because they thought that it must be some kind of "silly trick" (such as writing the term on a piece of paper, then flushing it down the toilet before anyone else can get their hands on it), so he assured them that the result would be legitimate and invited them to search the internet for a way to make "crossing out" the T's algebraically impossible. The next day, four of them did take the bet, (two of them being math majors who, after doing an exhautive internet search, were now convinced that such a result couldn't possibly exist). He then wrote:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1)),

and even though it took a couple of weeks for them all to agree that the identity is true, finally got his money. More importantly however, he then decided to change his major from political science to math!

So you see, my research is worth it.

Don.

[Qfwfq]

Originally Posted by Don Blazys

then you will find a discussion (example 4.2, #4) about the similar expression:

(x^2-a^2)/(x-a)

and why it is reasonable to view it as "continuous" at x=a.

Well, it seems you meant #3, anyway notice that is defined in such a way as to be continuous.

While the expression does not define any value int the case, the expression does. The judicious reader may easily see that is defined by one or the other expression according to the value of . It's as simple as that, it's the reason why discontinuity of third species is (loosely) called eliminatable; strictly it isn't the same function with or without the discontinuity.

Originally Posted by Don Blazys

I will also maintain my point of view that any non-negative integer such as 823 constitutes essentially "one symbol" because it is "purely numerical", while a number such as -823 constitutes essentially "two symbols" because it is comprised of both an operation (-), and a pure number (823),...

Would you also maintain a point of view that 823 not shorthand for ?

You might choose to give all natural numbers an arbitrary name such as klurbfampheryqusdeckpghanmegg, not composed of tokens like eighthundredandtwentythree is. Couldn't the same be done for minuseighthundredandtwentythree as well? No law would dictate it would have to be minusklurbfampheryqusdeckpghanmegg rather than pakdangleflvsaquourpgoose or something...

Originally Posted by Don Blazys

For instance, one college student bet six of his friends and classmates ten dollars each that he could re-write the term:

(T/T)a^x

so that it is algebraically impossible to "cross out" the T's.

Define what exactly you mean by "it is algebraically impossible to 'cross out' the T's" because I'm not sure exactly. Would you say the same in a case such as:



for example?

[Turtle]

I have no mathematical opinion on Don's expression, but I do endorse the mathematical exploration that led to it.

On the matter of 823 and a single symbol, one only has to change the base to achieve that. Any base over eight-hundred-twenty-three necessarily assigns a single symbol to every value less than that. The numeral is not the number.

[CraigD]

Originally Posted by Don Blazys

Thus, if we want our math to be "perfectly rigorous", then we absolutely must insist that our independent variables represent non-negative integers only, and that all other "reasonable" numbers (negatives, fractions, irrationals, transendentals, imaginaries, complex, and so on) then be derived by performing operations on them, and them alone.

As I think Qfwfq has explained in this thread, restriction “perfectly rigorous” math to use only positive integers () is both unusual and unnecessary. In any formal system, it’s only necessary that theorems – true statements – may be generated from a collection of postulates – given true statements. Formal systems do not even need to involve numbers – any unambiguously defined collection of symbols may be used.

The mention that all of the variables in your writing represent to members of is, however, helpful. I strongly recommend, Don, that you include such mention in all your writing, as without such, most readers will assume that variables not explicitly typed represent real numbers.

Originally Posted by Don Blazys

With this in mind, please check out my proof of the Beal Conjecture on my website (donblazys.com).

Even knowing that , and ignoring the suspicious and unsupported claims between (3) and (4) in Don’s paper, and the lack of any mention of the to-be-proven conjecture’s requirement that and are coprime, the proof appears incorrect for the following reason:

The choice of the constant “2” in equation (1) is arbitrary. One could typographically replace “2” with any positive integer (eg: “7”) without otherwise altering the proof, in which case equation (6) would read:

rendering it a disproof of claim that, given , and are coprime, only

or

are true.

Don, what statement in your paper formally prohibits this choice of constants? There appear to me to be other errors in your proof, but this one strikes me as possibly easier to discuss, being as it requires the acceptance of no more than the generalization of a true statement you made in post #22 which I wrote in post #26,


In short, I believe Don’s shown us an interesting identity that illustrates some useful properties of logarithms, but that he’s trying to apply it to prove conjectures for which it’s unsuitable.

However, Don, if despite our criticism, you’re convinced your proof is correct or can be successfully corrected, I recommend you contact Daniel Mauldin as described at http://www.math.unt.edu/~mauldin/beal.html. There’s a $100,000 prize for a correct proof or counterexample of the conjecture, and I’m confident that if any of the prize committee members see even slight promise that your approach is useful, they’d be more than willing to help you publish your work and claim the prize.

Of course, you can continue trying to convince your fellow hypographers, but speaking for myself, I just don’t see any way to use your identity to prove Beal’s conjecture or Fermat’s last theorem.

[Don Blazys]

To: Qfwfq.

Karl Gauss once said: "There is much in mathematics that depends on ones point of view".

Now, let's consider the equation:

(x^2)/(x)=x.

Both sides can be viewed as "functions".

Thus, from my point of view, for you to claim that the left hand side is "discontinuous" at x=0 and that both sides are therefore "different" is simply not consistent with the fact that the x's cancel on the left hand side, and what we really have is simply x=x where neither side is "discontinuous" at x=0.

To put it another way, if we begin with the function:

x,

which is not discontinuous at x=0,

then multiplying that x by x/x=1 results in:

(x^2)/(x),

which should not "magically" present us with a "discontinuity" at x=0, because all we really did was multiply by one!

As for my conviction that any "one symbol" independent variable is "naturally indicative" of any "one symbol" non-negative integer, well, Turtle's most astute observation that given the possibility of number systems with arbitrarily large bases, non-negative integers are the only numbers that can possibly be represented by "one symbol" is very helpfull here. (Thanks Turtle!).

You know, Leopold Kronicker once remarked: "God created the integers, all else is the work of man", so I'm not the only one who prefers to give them special treatment and consideration.

The answer to your last question is pretty straight forward. If:

(T/T)a^x=T(a/T)^(xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1)),

then we can neither "cross out" the cancelled T's and make them "disappear", nor can we let T=1. Moreover, the logarithms themselves cancel if and only if x=1.

The last expression in your last post contains ln(0), so you will have to fix it before I can respond.

Don.

[Don Blazys]

To: Turtle.

Great explanation! Thanks again.

Don.

[Qfwfq]

Originally Posted by Turtle

On the matter of 823 and a single symbol, one only has to change the base to achieve that. Any base over eight-hundred-twenty-three necessarily assigns a single symbol to every value less than that.

Yup, that's what I did by making up such kxuwrdgfaptlaquishious single names for the numbers in question...

Originally Posted by Turtle

The numeral is not the number.

Exactly! And of course the same goes for -1, -2, -3 etc...

I was going to fully address Don's points but my time is running out and sorry for my blunder due to scarce time yesterday...

You could even let the numerator just be T and explain more exactly why, in your reply, you say "we can neither 'cross out' the cancelled T's and make them 'disappear'", also please try to get the hang of latex.

[Don Blazys]

To: CraigD.

The only "mechanism" we have for ensuring that a particular number is either positive, negative, rational, irrational, transendental etc. is through the use of symbols. Thus, if we allow the "one symbol" variable "c" to represent the "multi-symbol" irrational number (2^(1/3)), so that c=(2^(1/3)), then, as you wrote in one of your previous posts, the equation:

1^5+1^4=(2^(1/3))^3

can indeed be represented as:

a^x+b^y=c^z,

and we are doomed to failure before we even begin!

Mathematics itself is then rendered "hopelessly ambiguous" and therefore "inadequate", and hardly worth studying!

Now, "cohesive terms" are absolutely consistent with the notion that if a variable is not specifically symbolized as irrational, then it can't be irrational.

That alone makes them superior to non-cohesive terms.

Please continue your study of my proof by reading version number 4 ("The Beal Conjecture In A Nutshell") first. Then move on to versions 3, 2 and 1 in that order. I promise that I will get to your questions on it in my next post.

Thanks for staying interested, and for your good advice.

By the way, I contacted Dan Mauldin, and he told me to send it to a refereed journal, so I sent it (electronically) to the American Mathematical Society, and the editors at Princeton University sent me an e-mail to the effect that it is being processed. They have had it now for well over two months, which seems to me an inordinately long time to referee a one page proof. Is there a time limit on how long they can take? Their rules state that I am not allowed to submit it to another journal while they are "considering it for publication".
Andrew Wiles (the mathematician who solved FLT) is the head of their math department! I hope that he is an honest and ethical person.

Don.

[Don Blazys]

To: CraigD. (This is a continuation of post#39)

When you say that "any unambiguously defined collection of symbols" can be used to create a self consistent axiomatic and/or postulational system of logic, you are, of course, absolutely right.

But, is that what we are doing when we let a particular independent variable such as "x" represent "any real number"? In other words, are we practicing what we preach?

From my point of view, the "definition" x=(Any real number) is the height of hypocracy and the epitome of ambiguity because non-negative integers, negative numbers, fractions, irrationals, transendentals etc. all have different properties, and are therefore, fundamentally different from each other. Thus, letting them all be represented by "x" is kind of like letting the word "violin" mean "vacation", "vermin", "vinegar", "volcano", "vulgar" and all the rest of the words that begin with the letter "v".

This is not to say that it is always wrong to let x=(Any real number), because for most practical, everyday problems, it really doesn't make any difference. However, for problems that require us to separate and isolate numbers of a particular type or classification, requiring that x=(Any non-negative integer) is absolutely essential.

Now, the term on the right in the equation:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

is called a "cohesive term" because unlike the term on the left from which it was derived, the cancelled T's don't "fall off" and get "lost". They are, in effect, "glued" to the rest of the term and can't be eliminated by "crossing them out" or by letting T=1.

Astonishing as it is, the "cohesive term" is really just a logical extention (and thus an unavoidable consequence) of the properties of logarithms. Thus, the above identity must be true for all values of the variables:

T={2,3,4...}, a={1,2,3...} and x={0,1,2...},

and here we should note that each and every variable is perfectly defined by it's own unique domain, and that the "non-cohesive term" on the left does not have this property. Thus, it should be expected that the "cohesive term" represents a more accurate model of the general composition of numbers.

It can also be shown that the "cohesive term" is the only term that perfectly reflects the properties of non-negative integers while actually requiring that it's variables represent only non-negative integers or "unsigned numbers".

I will get to that, and to the proof itself in my next post.

Don.