1. Re: Abel Prize

Originally Posted by 7777777
furthermore, he was, in some sense, right, as right as the rest of you, that does not exist, it does not exist
as the form of a number, just like . He was right because the proof of these things was not given at that moment and it will not be given
"Infinity does not exist" for division by zero, because zero is a point, not a length, and the real numbers (even the integers) requires a "distance"; i.e., a metric. Without a metric, the number line doesn't exist (is irrelevant; if there are elements, it is an unordered set. Zero only exists as the midpoint of all possible lengths (or as the endpoint of all possible negative or positive lengths)

John is wrong because he does not understand the concept of independent variables; e.g., f(x,y) = x/y ; that is, two dimensions in Cartesian coordinates. Since he doesn't understand that, he doesn't understand curvature, in terms of which is defined as the ratio of the circumference to the diameter =(2r)/(2r). Since this is a fraction, it requires two dimensions for definition.

And his definition of slope is FLAT wrong, unless m = 0, even as an misguided attempt at differential geometry or finite element analysis... And even if m = 0, the slope is correct, but a line (a chord) is all he's got - the derivative is the rate of change along a curve at an arbitrary point x, where the curve is a function of x, and the chord is arbitrarily small. That is, by the mean value theorem, one knows that there will be a parallel line a line specified by the chord, but it will not be known at which point x the chord touches the function at a single point, since for the derivative, the function can be arbitrary.

"Existence" of infinity as a mental concept is all in the mind, like the existence of god - one can either believe or not, but an intellectual justification is a fools' errand; a discussion among idiots.... (well, ok, the mathematically challenged...

2. Re: Abel Prize

Originally Posted by BuleriaChk
"Infinity does not exist" for division by zero, because zero is a point, not a length, and the real numbers (even the integers) requires a "distance"; i.e., a metric. Without a metric, the number line doesn't exist (is irrelevant; if there are elements, it is an unordered set. Zero only exists as the midpoint of all possible lengths (or as the endpoint of all possible negative or positive lengths)

John is wrong because he does not understand the concept of independent variables; e.g., f(x,y) = x/y ; that is, two dimensions in Cartesian coordinates. Since he doesn't understand that, he doesn't understand curvature, in terms of which is defined as the ratio of the circumference to the diameter =(2r)/(2r). Since this is a fraction, it requires two dimensions for definition.

And his definition of slope is FLAT wrong, unless m = 0, even as an misguided attempt at differential geometry or finite element analysis... And even if m = 0, the slope is correct, but a line (a chord) is all he's got - the derivative is the rate of change along a curve at an arbitrary point x, where the curve is a function of x, and the chord is arbitrarily small. That is, by the mean value theorem, one knows that there will be a parallel line a line specified by the chord, but it will not be known at which point x the chord touches the function at a single point, since for the derivative, the function can be arbitrary.

"Existence" of infinity as a mental concept is all in the mind, like the existence of god - one can either believe or not, but an intellectual justification is a fools' errand; a discussion among idiots.... (well, ok, the mathematically challenged...
you just keep on babbling. but you don't have a proof. you have a result but you don't know how you arrived at the result.
for example you have

without knowing how it is possible

3. Re: Abel Prize

Originally Posted by 7777777
you just keep on babbling. but you don't have a proof. you have a result but you don't know how you arrived at the result.
for example you have

without knowing how it is possible
I didn't quote that result (Leibnitz's formula), you did. Since you are claiming it is possible, then you must believe infinity "exists" ....

The only claim I would make is the result might be useful in some contexts - if it is indeed a good approximation (it isn't, since it converges very slowly) ..

(Of course, I would note that since it is a fraction expansion, it involves two dimensions....)

Nevertheless, for those interested in this sort of thing: Pi expansions

(These are usually approximated in such series by using... wait for it... trigonometric functions. That is, curves.... in two dimensions....)

4. Re: Abel Prize

What an interesting example you chose, 7777777, considering it reminds the reader of "CircleSquarers" and the futility of arguing with them.

5. Re: Abel Prize

Originally Posted by 7777777
He got the right definition of the derivative without using limits:

but he ignored the case m=n=dx:

because he rejected the infinitesimals, but he is not worse than the rest of you. You all denied the infinitesimals.
I accept infinitesimals with the caveat that I'm now using hyperreal numbers and the "standard part" concept. Infinitesimals are quite useful conceptually and people like them way too much to abandon them.

John Gabriel wrote a nonsensical statement with the formula:

The left-hand side says that the right hand side is a function of one variable but then I look to the right hand side and I see three variables. The notation is bad from the start.

With some salvaging, that formula is just a central finite difference approximation for the first derivative but is not equal to the first derivative without letting m and n go to 0. Buleria's been confused on this issue for quite some time.

6. Re: Abel Prize

Originally Posted by mathnerd

Buleria's been confused on this issue for quite some time.
WTF? The SLOPE is wrong. Do you understand what a slope is?
What issue are you talking about. TRY to be clear....

Do you understand the diagram yet? Obviously not. Do you need to review Cartesian coordinates?
Why on earth are you confused? Ask me a specific question and I'll try to answer, but don't tell people I am confused without understanding my analysis. You're just quoting buzzwords without content.

Your comment is absurd (and misleading) for anyone that has taken (well, ok, passed) a course in pre-Calculus high school Analytic Geometry....

Sheesh!

Here's a diagram of John Gabriel's characterization of the derivative (using Analytic Geometry):

The equation for John Gabriel's slope (A*) is obviously wrong. The correct x to be used in calculating the slope A is shown in green). John's x is shown in red, and is too long because of m. The correct is shown in green. One then replaces n with h and takes the limit on h. (Which is the same thing as taking the limit on n, if n is a real variable so it can express the limit at a point on the curve of an arbitrary function. If n remains an invariant, all one has is a slope of the line with b = 0; y = Ax + b (look up the equation of a line), not a derivative....

Setting m = 0:

The length of the blue line is: f(x + n) - f(x)
The length of the green line is: (x + n) - x = n

Exercise for the beginning student:
1. What is the result of dividing the length of the blue line by the length of the green line?
2. What is the common name of this mathematical relationship?

This mathematical relationship is characterized by the line between the points (x.f(x)) and (n, f(x+n))

Hint: Ok, it is called the slope, sometimes designated by A in the equation of the line y = Ax + b
(where b = 0 above b is the y intercept)

That is, it is a line, not an arbitrary function intercepted in two points by a chord.
Why is this so difficult for you?

To the reader: see the "Toast" link in my signature for a full analysis...

7. Re: Abel Prize

Originally Posted by 7777777
He got the right definition of the derivative without using limits:

but he ignored the case m=n=dx:

because he rejected the infinitesimals, but he is not worse than the rest of you. You all denied the infinitesimals.
Incorrect, we do not deny infinitesimals in hyperreals or surreals, we state that they don't exist in real numbers and that is a provable statement.

8. Re: Abel Prize

Originally Posted by BuleriaChk
WTF? The SLOPE is wrong. Do you understand what a slope is?
What issue are you talking about. TRY to be clear....

Do you understand the diagram yet? Obviously not. Do you need to review Cartesian coordinates?
Why on earth are you confused? Ask me a specific question and I'll try to answer, but don't tell people I am confused without understanding my analysis. You're just quoting buzzwords without content.

Your comment is absurd (and misleading) for anyone that has taken (well, ok, passed) a course in pre-Calculus high school Analytic Geometry....

Sheesh!

Here's a diagram of John Gabriel's characterization of the derivative (using Analytic Geometry):

(diagram snipped)
I'm still looking for someone to volunteer and explain that diagram to me. I've given up on it.

An expression that looks like:

is an approximation for the derivative at x when h is small. Gabriel just decided to let one of the h's be called m and the other n. It's just a central finite difference approximation.

9. Re: Abel Prize

Originally Posted by mathnerd
I'm still looking for someone to volunteer and explain that diagram to me. I've given up on it.

An expression that looks like:

is an approximation for the derivative at x when h is small. Gabriel just decided to let one of the h's be called m and the other n. It's just a central finite difference approximation.
If you can't understand the diagram, you will never understand straight lines, much less calculus....
(but sheesh, didn't they teach you about map coordinates in the Cub Scouts...?

In case you're confused about slope: SLOPE

In fact, without two dimensions, you'll never understand the meaning of function...

The above is just a relation of algebraic (actually just arithmetic) symbols, with no further mathematical content.

What is the difference between that expression and

????

Many are called to (try to) understand calculus, and few are chosen........

The following table contains a few particularly important types of real-valued functions:

A linear function

f(x) = ax + b. f(x) = ax2 + bx + c.
Discontinuous function Trigonometric functions

The signum function is not continuous, since it "jumps" at 0.

The sine and cosine functions.

Roughly speaking, a continuous function is one whose graph can be drawn without lifting the pen.

Edit: I should have provided a link to the definition of a function:

Here it is: FUNCTION (click on the link)

(One has to scroll down for the graphs, but hopefully can assimilate some of the preceding concepts... However, in the case of mathnerd and others, I am not hopeful

10. Re: Abel Prize

BUleria, are you a damn engineer?

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