What makes a good teacher? What is a formula?

[Kirby]

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?

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.

[Bill Spight]

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

[Kirby]

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

A circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area.

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

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

The area of a surface or lamina is the amount of material needed to "cover" it completely

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.

[Kirby]

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

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.

[AloneAgainstAll]

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.

[Kirby]

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:

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

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?

[AloneAgainstAll]

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.

[Bill Spight]

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

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.

The trouble being that the unknown is in the expression.

[Kirby]

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

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.

The trouble being that the unknown is in the expression.

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

[Bill Spight]

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:

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

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?

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.

[Bill Spight]

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

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.

The trouble being that the unknown is in the expression.

In what way?

In the sum up to 2ⁿ.

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

But then why not allow the Sieve of Eratosthenes? See above.

[AloneAgainstAll]

If she assumed such a restriction she should express it beforehand, and thats true for any restrictions.

And such a restriction is nonsense, if we assume such a restriction we cant solve problem: "using Erastotens algorithm find all primes below 100" because only number 100 is given! And if you somehow get number 1, 2 and others, you can get same way numbers 101, 102 and all others natural numbers.

As i said - we must assume that whole set of natural numbers is given, cuz its needed to define prime numbers.
Sieve of Erastotenes is not a formula nor function, its algorithm. Very big difference.

Transforming algorithm to function is sometimes very hard.

[Kirby]

In what way?

In the sum up to 2ⁿ.

A sum is a sum. We know how to sum.

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

But then why not allow the Sieve of Eratosthenes? See above.

If you can express it with math symbols like that, why not allow it? I don't see a problem. It's a formula.

[Kirby]

I'll try to summarize my stance:

1. "Formula" often doesn't have a rigorous definition
2. Folks often use "formula" casually, and can be just some way of expressing some rule or fact mathematically
3. Other things like "polynomial" have much stricter definitions. There are constraints in there that you don't always have when using the word "formula"
4. The teacher was probably thinking of a particular constrained definition of "formula" when it was being used
5. It would have been less ambiguous to use a less ambiguous term, which expressed the constraints that teacher was trying to convey

If you just use "formula", you have folks like me and AloneAgainstAll, which find the statement to be false. If you're going to make a sweeping statement like that, you should use a precise definition.

This is not to say that the teacher is a bad teacher. I like Knotwilg's point that, if nothing else, this has caused us all to think about the problem on our own!

[AloneAgainstAll]

Kirby he means that we can sum only up to given number by his assumption that only first n-th prime numbers ar given, and only those we can use in our formula. Its sad that if we assume that we cant give formula for volume of ball (and other standard formulas), even if we have radius, but thats his definition.

For me, formula is equal to function. If we assume Jlt definition we have problems i showed. If we assume Bill's definition we have even more problems.

But whatever we beat from this "formula" discuss its completely offtopic - i expressed explicitly what formula meant for teacher in the problem. Maybe we should stop beating this dead from the beginning horse?