A Mathematical Emergency.

[Qfwfq]

Originally Posted by Don Blazys

In post#57, you ended your first paragraph with: "Let's look at the fallacies in your argument".

Well, I already know that there are no fallacies in my argument, so the only possible reply from my point of view was and is: "There are no fallacies in my argument."

Please note that these forums have Rules which include something against this type of reply. The onus of proving your point when challenged is on you and I provided a clear argument against your claim about "reasonable implication" which you had not supported at all.

Originally Posted by Don Blazys

I suppose that I could also have mentioned that "my grasp of fundamental logic" is quite good, and that from my point of view, it's your grasp of fundamental logic that needs to be "straightened out", but I prefer not to indulge in such childish gibberish.

Which childish gibberish?

By definition, an implication is a true assert if there are no cases in which the consequent is false and the implicant true. Without this being the valid requistite, there would be no point in arguing out any implication. It is trivial that it holds if at least one of the following can be said about a given implication:

• there are no cases in which the consequent is false
• there are no cases in which the implicant is true

Now Don, I agree that at if least one of the above holds the implication is useless. Indeed this is the reason I don't see the point of your argument, based on being true. In the second case above, it is impossible to apply modus ponens and superfluous to apply modus tollens; vice versa in the first case. Where am I needing to be straightened out?

Originally Posted by Don Blazys

If the expression:

(1)^(0/0)

was "undefined", as you claim it is, then both (7) and (8) would become:

a^x+b^y=c= "undefined"

and

a^x=b^y=c^2= "undefined",

and that would make no sense whatsoever, because clearly, defined co-prime equations do indeed exist!

The properties of logarithms can not render all co-prime equations "undefined"!

Thus, it must be the case that:

(1)^(0/0)=1,

and

(1)^(0/0) is not "undefined".

Gosh I'm finally beginning to see that your argument is along the lines of:

is an identity (rather than as I say)

therefore, since 1 is defined even when then so must be. This appears to be the main source of disagreement.

Mathematics, as Gauss meant to imply in the sentence you quoted here weeks ago, is all a matter of: "Define, construct and work out the consequences!" Now Don, if you say the above is an identity rather than you imply that the expression defines a function that's equal to 1 and continuous even for the value and therefore, in order to adopt a non-standard definition of the terms function and continuous then you must state exactly what your definition of these terms is. If you can make the whole thing self-consistent it's an Alternative Theory, otherwise it's a Strange Claim and this case includes if you insist on your claims about implications but fail to support them.

[CraigD]

Let me first address a simple question Don asked about formal logic.

Originally Posted by Qfwfq

The implication is also formally a true one, what can you deduce from it?

Originally Posted by Don Blazys

I also can't see how you can possibly claim that the implication:

51/3=2 (implies) N*0=0

is "formally true". The equation 51/3=2 is false, while the equation N*0=0 is true. How can something false imply, "formally" or otherwise, something true?

Qfwfq is correct, because, in formal logic, “something false can imply something true”.

Like all formalism, formal logic is simply a collection of rules exactly defining the meaning of the symbols - operators and term - used in it. Unlike more well-known formalisms such as arithmetic, statements in ordinary formal logic evaluate to one of only two values: True and False.

The implication operator, , is defined as follows:


So the statement evaluates to True.

It's common to mix arithmetic statements with logical ones. Such a statement,

while intuitively silly-looking, is formally true.

On the less simple question of the Don's claim to have proven Beal's conjecture, I can offer only vague comment and advice.

I don't believe Don's proof is correct

Proofs are considered correct , categorizing roughly, for either formally or “sociologically”.

A formal proof is correct when each statement of a proof is “mechanically” produced according to a formal operation postulated to be correct for the formal system in question.

A “sociological” proof is “correct” when the intended audience of the proof believe it to be so. The importance of a sociological proof depends strongly on its audience. For example, for mathematical proofs, the acceptance as true by many professional academic mathematicians is more important than the acceptance, of, say, a gathering of friends.

By neither of these standards are Don's proofs correct.

My advice to Don is to take the various unusual phrases he's used in his proofs, and give them exact, formal definitions. “Cohesive term”, I believe, would be the best one to start with. State precisely the rule for transforming one expression into another given by the phrase. Don't attempt to prove anything, and avoid vague statements such as “cohesive terms are a consequence of the properties of logarithms”, and simply define the concept.

A final bit of advice: When people don't agree with you, don't simply state that you know you're right. Quoting from the site rules

Do not endlessly show us that *your* theory is the *only* truth. And don't follow this up by making people look stupid for pointing out that there are other answers, especially if they provide links and resources. It will get you banned!

In short, when encountering disagreement, back up steps in your argument until you encounter agreement, then, at the first step at which you encounter disagreement, support your claim with links and references to accepted texts, and discuss, being prepared to acknowledge that you may be wrong.

[Don Blazys]

To:Qfwfq and Craig D,

You know, in my previous post, I made it absolutely clear that those pesky "indeterminate forms" don't even exist if we do the algebra correctly and "cross out" the logarithms at z=1 and z=2.

In other words, the beingness of the pesky expression: (1)^(0/0) is "zip", "zero", "nada" at z=1 and z=2.

Now, if they don't exist, then they are a non-issue and we are really going off topic in discussing them!

The proof works precisely because z=1 and z=2 are the only possible values of z that allow us to "cross out" the logarithms preventing T=c.

That's all there is to it. No point in discussing "indeterminate forms" that either can, or must be "crossed out" before they even exist and are therefore perfectly consistent with the result:

(1)^(0/0)=1,

just as the "counterintuitive" expression: (p)^(0), is perfectly consistent with the result:

((p)^(q))/((p)^(q))=(p)^(q-q)=1.

Take another look at my post #60 and consider this fact: If it were not the case that the expression:

(1)^(0/0)=1,

then logarithms themselves would have to be viewed as an "inconsistent construct" because their properties would not allow T=c in (3) and (4), but would allow T=c in (5) and (6).

We would then have to banish logarithms from mathematics altogether!

Don't you see, we simply can't eliminate logarithms from mathematics, so we absolutely must conclude that:

(1)^(0/0)=1,

because it's not just the only logical and consistent conclusion, but the only possible conclusion as well! I simply can't accept your view that the properties of logarithms are somehow "bogus".

We must keep in mind that new discoveries often show us new things and therefore bring about new points of view.

I discovered "cohesive terms" about a decade ago, and showed how they are derived and defined apart from "non-cohesive terms" throughout my website. As their inventor, I had to give them a name, and chose "cohesive term" because they are the first and only algebraic terms in the history of mathematics that actually prevent cancelled factors or cancelled common factors from "falling off" and getting "lost".

I introduced them here in this forum because that property alone is nothing less than astonishing!

Now, let's get back on topic and discuss whether or not the equation:



is telling us that, in principle, multiplication and/or division by unity automatically results in division by zero.

Letting T=1 sure seems to indicate that it does!

Is this a "strange claim"?

Perhaps, but that's a good thing because in this case, it is obvious that the truth is indeed stranger than fiction!

Don

[Nootropic]

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

is telling us that, in principle, multiplication and/or division by unity automatically results in division by zero.

Don, have you studied ring theory? Field theory? Abstract algebra of any sort? Quite honestly, what this implies is that the way mathematics has been done is a sham. I really don't even have to read the previous posts to know that if you claim that your argument implies this, then it is false. Stranger than fiction? I think not. If you knew how about constructions and completions of fields, it would help to understand that this statement is simply wrong. If this were the case, mathematics would be reduced to trivial statements. Mathematicians may have pondered such questions as these at one point (perhaps even ones that would seem more obvious) but, I stress, multiplication by unit does NOT result in division by zero. Do you need the entire mathematical community to ensure you about this?

[Don Blazys]

To: Nootropic,

I'm almost 60 years old, so I have had plenty of time to familiarize myself with all those particular branches of mathematics... and then some. How about you?

I never said, or even implied that "the way mathematics has been done is a sham". Those are your words, not mine.

Mathematics must adapt to and be consistent with new discoveries if it is to remain both both vital and dynamic. Therefore, new discoveries must be disseminated rather than "swept under a rug" for fear of change.

Courage is required!

Most importantly however, new discoveries must be studied so that all of their consequences and ramifications are properly, which is to say logically, interpreted.

Initially, there may be disagreements with regards to various possible interpretations, (such as the disagreement between Newton and Liebnitz on the subject of limits) but in the end, the truth will prevail, and mathematics will grow, thrive and flourish rather than stagnate.

Your fear that my equation would "reduce mathematics to trivial statements" is without foundation. My equation happens to be true, and the truth can never render the robust and sublime body of knowledge called mathematics "trivial".

Now, let's muster up all of our courage and ask ourselves the following questions:

Given the true equation:



if our variables are to represent non-negative integers, then:

(1) Which term has the better defined variables?

(2) Keeping in mind that unity is not an actual common factor but a "trivial" or "degenerate" common factor, which term is more suitable for representing actual common factors?

(3) Keeping in mind that what we do to one side of an identity, we must also do to the other, can we "cross out" the cancelled T's the way we were taught in school?

(4) What occurs at T=1 and what does it imply?

When answering these questions, please remember that the above equation is absolutely new to mathematics ( I discovered it only a decade ago) so there is, as of yet, no "general consensus" on what it actually means.

It's proper interpretation is perhaps the greatest challenge that the mathematical community has ever faced! Try answering those four simple questions and see for yourself!

Don.

[Nootropic]

Do you not realize that if multiplication by 1, in, say the real numbers, results in division by zero then let x be a nonzero real number (note this is is also finite because we are not working in the "extended real number system" [ie. the closure of the real numbers]) x = 1*x = x/0 (which is again, not defined). And then by the density of the rationals in the real numbers, we have some sequence of rational numbers converging to x. So basically what this is saying is that NO sequence of rationals converges to any real number and density can be proven use elementary properties of the real numbers without resorting to topology. So basically this apparent "discovery" contradicts the entire field of real analysis. Now what is correct: Some "amazing" discovery or a field built upon almost 200 years of solid foundation?

If we are talking about division rings with an identity (unity), then how exactly do we deal with division by zero? This would make division rings (and hence fields) virtually unusable. Personally, I find that you have an extremely disturbed view of what the term "variable" means, how mathematics is done, and mathematics in general.

You really don't realize that such a discovery attempts to invalidate numerous fields of study. It basically makes the real analysis classes undergraduates and graduates take each year useless, because apparently the entire field of real numbers is unusable. One sign of crackpot mathematics is its implications and you fail to realize the implications your apparent discovery imposes.

[Don Blazys]

To: Nootropic,

We are, at present, discussing algebraic terms as they apply to common factors.

You are conjuring up constructs involving "extended reals" that have absolutely nothing to do with my discovery and are invoking a completely different branch of mathematics that has no bearing on this discussion whatsoever!

Also, please back up your claims. If you think that my equation is "crackpot mathematics" then show us where the error lies.

If you can neither point to a mistake in my equation, nor answer those four simple questions, then all you are really doing is avoiding the issue.

Please stay on topic and answer the questions!

Don.

[Nootropic]

You are conjuring up constructs involving "extended reals" that have absolutely nothing to do with my discovery and are invoking a completely different branch of mathematics that has no bearing on this discussion whatsoever!

I am not "conjuring up any constructs". Read a book on analysis, preferably Rudin's Principles of Mathematical Analysis. I am arguing on the basis that this apparent discovery implies things we know to be false (ie. multiplication by one results in division by zero). If someone came up with a proof of the Riemann Hypothesis that implied 2 = 1 (in the integers of course), then what are we going to accept? 2 = 1 or this proof? It doesn't take a great amount of intellect to realize what's right and wrong. CraigD has already pointed out that it is an identity for an exponential function, but it does not imply multiplication by 1 results in division by zero.

I was not working in the extended real number system. What I'm saying is that the "x/0" is not defined. Basically what you are saying is that a sequence that converges to a real number does, in fact, not converge. I can pick any concrete example, multiply it by one, and then still find a sequence that converges to it. I don't know why you are so insistent on this ridiculous statement.

We are, at present, discussing algebraic terms as they apply to common factors

Yes, and the real numbers are an algebraic object--a field. This is basic field theory arithmetic.

[Don Blazys]

To: Nootropic,

Again, you are way, way off topic.

If you want to discuss "analysis" or "fields", then start your own thread on those subjects.

Here, at this thread, we are discussing "elementary number theory" where the concept of a common factor requires that any common factor T>1.

Now, can you, or can you not point to an error in my equation and answer the four simple questions that I asked in post #65 ?

Don.

[Nootropic]

Once again, you utterly fail to realize the connections that different areas of studies have. Clearly, if this equation implies multiplication by one implies division by zero, then, as the integers are contained in the real numbers, we have n = n/0 as an undefined expression in the real numbers. This is hardly an off-topic matter. And for the tenth time, it does not matter what the equation says (others have analyzed this). It matters that if this "true equation" holds, then multiplication by one results in division by zero. False. The middle-schoolers at the Art of Problem Solving Forum had more than enough background to dispel such ridiculous arguments.

No fallacies in your arguments? Hah, show some modesty. There were even flaws in Wiles and Perelman's arguments! Mathematics is a thing that must be widely agreed upon by professionals. You wouldn't find one professional mathematician with an upstanding reputation who would back this. If there are no fallacies, send it in to a professional journal and see what they say.

I would hardly call fundamental logic "childish gibberish". Mathematics rests upon it and a solid knowledge of it should be every mathematician has. You apparently have a very weak knowledge of it.

Answering your questions, Don, is apparently only possible if you answer them because either no one else's mathematics is "correct" to you or no one knows what you mean by your terms you have yet to define, for example, what do you mean "better defined variables"?

Don, I suggest you go back and read others posts and take your equation for what it means, not that it is some utterly "profound statement".