Here's a link to the site, again: http://mathworld.wolfram.com/Formula.html



So I suppose the claim is that "f(n) = f(n)" is a fact that is expressed in terms of mathematical symbols. Typically, the "f" mathematical notation here is used to represent a function (http://mathworld.wolfram.com/Function.html). What is a function?



So I suppose the fact being represented mathematically in "f(n) = f(n)" would be that "a function on n is a function on n". I suppose that's OK. Just not useful. Like saying that "5 = 5".

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be immersed

As I said, I have no definition of a formula. If I want to be precise, I say "a mathematical expression involving symbols among some list". Now, in that list, do you allow summation symbols? Integration symbols? Infinite series? Limits of sequences? Quantifiers? Sequences defined by induction?

Given the enthusiasm with which you have been in search for the formula for the n-th prime, it seems that your teacher has managed to inspire you with the thirst for knowledge. At the least he didn't obstruct it. His claim that something didn't exist made you wonder. That seems more like a success.

A teacher doesn't have to be 100% correct, not even about what is true or known. In fact, you'll be hard pressed to find even among the brightest scientists people who never made a wrong claim and passed it on.

Here's an interesting anecdote (I hope).

In my math curriculum, I had two opposite teachers. One was didactically great. She had a very well structured syllabus on topology. The other was a mumbling piece of chaos, with no syllabus, whose classes on Lie Algebra I came home from with incomprehensible notes. So I went to the library and set up a work group to find order in the chaos. Through self study and hard work, and long evenings with one fellow student, we finally found that order and because I had done such hard work, I mastered the subject far better than the well devised, easy to digest topology course. It may also be that Lie Algebra was just more my piece of cake than topology, but it makes you wonder ...

I note and welcome your own hesitation about that in your subsequent paragraph, but I still think it needs to be stressed that that's the sort of over-defining thinking that gets maths a bad name among most people, in my experience. Ordinary people want to say things like "the searchlight cast a circle of light" and expect you to know they don't mean a rim, but if they ask a group of you to form a circle, they don't expect you to cluster as a blob in the centre of the room. Common sense rules, OK?

In any case, although I'm way out of my depth, to talk about a border strikes me as incorrect, and not in line with the website pointed to - to use in a common way just two of the relevant words. The boundary in the definition is a set of points. As far as I know - very basic Euclid - a point is 0-dimensional. And if a line is given as a set points it only becomes 1-dimensional. Which can't be coloured in, no? Unless you invent 1-dimensional colours, or the like, and if you do the eyes of ordinary people just roll up heavenwards.

Ultimately - again in my experience - the real problem is that too many definition mavens - a tiny minority - can't accept that ordinary people - a huge majority - even exist. Haven't they got eyes? (Just to keep it on the topic of go )

I thought this was a discussion about characteristics that make a good go teacher.
I've been teaching people to play go forty years. It's only recently, say, ten years ago, I realized I was doing it all wrong. So I get points for being objective enough to realize I scared more people away from go than I recruited.

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David Bogie, Boise ID
I play go, I ride a recumbent, of course I use Macintosh.

It's true, I did not state it in the form of a definition. To be clear, let me take some time to build up to it.

We start with language. There is a common pattern in statements, a formula, if you will, where you state what is known, understood, or given by your hearer, and then say something new about it or previously unknown to them.

Suppose that we have a right triangle with hypotenuse C of length c, and two sides, A of length a and B of length b, and we know a and b. Then we have the formula for the length of the hypotenuse, c = √(a²+b²), the formula being the expression on the right side. Even though √(a²+b²) = c, c is not a formula. If we knew a and c we could write a formula for b. Note that the formula is in terms of what we already know, what is given. It tells us something that we don't know, that is not given.

Suppose that we know the first n-1 primes. Perforce, we also know whether a natural number less than the n-1th prime (P_n-1) is composite or prime. That we can take as given, if we want to know the nth prime. What about numbers greater than P_n-1? Can we take them as given? That's a question. Obviously we cannot take P_n as given, it is what we are trying to find. Note that AloneAgainstAll's expression includes P_n in the sum up to 2ⁿ, so it includes more than is given, and it is not a formula, in the intended sense. I would go further and say that we cannot include any number greater than P_n-1 in the formula for P_n.

Back to language. If I tell you I have three children, but I actually have four, am I lying? Logically, no. If I have four children then I also have three. But I am not supposed to hide the fact that I have the other child. So if I am given the nth prime, I supposedly know about the numbers less than P_n, but not necessarily about numbers greater than P_n, and if I do, I have to say so. Usually when we talk about formulas, we do not explicitly say what is given, but because of the ambiguity, I think we have to in this case. Anyway, here is my stab at a definition for a mathematical formula.

A mathematical formula for an unknown is an expression in terms of what is given which when evaluated yields the value of the unknown.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Your formula is indeed good soulution to our problem (to give formula for n-th prime) - you just need to add definition of f function, and then it works (you can use function i provided).

I didnt gave definition of formula cuz for me its synonim of function. But thats aside.

And btw - by your definition of formula, some functions are not formulas. If i would make my own definition of formula, it would be wider than function - i could come with for example recurence definition of n! - its not a function (but define a function) but for me it should be counted as formula - haha, thats more and more offtopic .

[f(x)]^2=x is not a function but would be formula for me. Another example.

@Knotwilg

That was not my teacher - she taught in my elementary school, but not me. I came across this "formula" completely by accident. But i admit that if teacher inspires and motivate, its great.

I heard a story about being too precise in math:

Elementary school.
Pupil: I have 2 sandwiches for breakfast.
Teacher : You should explain precisely. You should say "i have set of sandwiches containing 2 elements".

That joke was invented to counter exactly what was pointed above - common sens rules should apply, teaching chilrens at basic level in kind of super precise manner is wrong . After i laughed hardly i realised that there is a catch - what if sandwiches were identical?

But Wolfram turns around and uses a different definition in this sentence without providing it.

The area of a set of points equidistant from a given point is 0, thank you very much.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

If you define formula that way you cant provide formula for sum of 1+2+3...+n because it contains n+1, and n+1 is not given.

Also sometimes you have problem (thats example) - you have 3 constans a,b,c (which descibe sth, whatever). Find sth else. And "formula" for it is x=1+abc. According to you, 1 is not given so you cant use it.

Another example:

You have ball with raduis R. Find volume. You cant give formula for volume cuz it contains both 4/3 and Pi which are not given before (only radius is given).


Well, ofc someone can say 1 is not given but when a is given i can make a/a and voila. If i have numbers 1,2,3 up to n i can do same thing and get 2^n.

The first line of the circle definition is here:


So, the "circle" with radius R does not include points on the plane that have distance less than R from O, i.e., it doesn't include the area surrounded by the circle. In the real world, if you were to try to draw the border of a circle, the width of the points you are drawing would be positive. So there would be a colored area that is not exactly R distance from O. But that's a different problem - it's not possible to be infinitely precise in the real world. If you have a problem that says to draw a line segment that's exactly 5cm, it'd be impossible to verify that it was correct to infinite precision. But anyway, having the person fill in the entire area surrounded by the circle is clearly wrong.

But that's exactly the reason I don't like the wording of the question. It could be worded in a less ambiguous way: Fill in the area surrounded by the circle with radius R. Or maybe better yet, fill in the area surrounded by the circle with radius R with X precision (to account for the real world problem that, in practice, you might have a non-zero amount of fill exiting the boundary of the circle, if you were to try to color it.

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be immersed

If you have no formula, you cannot say that the teacher was correct.

What I strongly suspect here is that the teacher had a meaning in what they were trying to convey: they meant that there is no non-constant polynomial to express the primes in question. This has a more constrained definition, and is not open to this debate. Casually saying that there is no such function is wrong, because there are common uses of the word function for which the statement would not be true.

My takeaway is that the teacher had the right idea, but was too vague in terminology.

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be immersed

As I said, we have to make what is given explicit. Different givens allow different formulas, which makes "formula" ambiguous.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Hmm, this would depend on the mathematical definition of "area", which admittedly, is a bit lacking from the page at Wolfram: http://mathworld.wolfram.com/Area.html


and "surface" is defined here: http://mathworld.wolfram.com/Surface.html

But actually, it's arguable whether the set of points is even a "surface", per say. So maybe we need a more clearly specified definition for a set of points, which is not part of a surface?

---

It seems this is somewhat debated, not only for circles, but for polygons:
http://mathworld.wolfram.com/Polygon.html

There, it explicitly calls out that some definitions of polygon include the area which they surround, whereas some do not.

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be immersed

I believe that AloneAgainstAll's formula does this. If the 17th prime is not known, you can evaluate the expression on the right hand side, and obtain it.

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be immersed

Why you assune (we didnt made any assumptions) what is given? Logically then nothing is given, so its impossible to give any formula. You cant give formula for volume of ball (if i express it that way radius is not given), you cant give formula for area of square.

If anybody ask "give formula for n-th prime number" we must assume that given is whst is needed to define prime numbers. Actually it contains set on natural numbers, so we must assume that set of natural numbers is given.

Kirby, he means that i can use only numbers up to 16th prime number then (and only primes, completely dont know why, but whatever).In my formula you need to use number above 16th prime number.

? I don't get it.

If this is the definition:


We can use the expression on the right to get the prime you want. What's the big deal? Why can you only use certain numbers?

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be immersed

He assumed that we have all prime numbers up to n and want to make formula to compute (n+1) th prime number. So we can use only this all prime numbers up to n-th prime number. At least i got it that way.

The trouble being that the unknown is in the expression.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

In what way? Knowing nothing about primes, you can calculate one from the expression, no?

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be immersed

Because the teacher certainly knew about the Sieve of Eratosthenes, and obviously did not mean that you could test each number up to the next prime. She apparently had a restricted set of numbers in the givens.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.