What makes a good teacher? What is a formula?

[jlt]

AloneAgainstAll: my formula is just a paraphrase of the definition of the n-th prime number.

To compute f(5), search for the largest m such that if 1<a<b<c<d<e<m then one of the a,b,c,d,e is composite.

For instance, m=11 satisfies the property because if you try every family 1<a<b<c<d<e<11 of integers, you will find a composite one. It's a bit long to check because there are 126 such families but it's feasible.

But m=12 fails because you can take (a,b,c,d,e)=(2,3,5,7,11).

So f(5)=11.

[Bill Spight]

No, Greeks didnt defined functions at all (i am not sure when the first correct definition of function was provided, but it was well after the Greeks and even Gauss). Erastotenes knew algorithm to get all prime numbers, but algorith is not always function.

It doesn't matter whether the ancient Greeks had the concept of function, they knew how to find the nth prime, so f(n) = Pn is a well defined function. Your teacher knew how to find the nth prime, but that was not enough for her to say that there was a formula for it.

I asked you which expression you want me to explain. You didnt said, but keeps asking me to explain.

Sorry, I thought that you had only posted one expression, the one in #13 at viewtopic.php?p=253915#p253915 . That's the one I keep asking you to explain.

[Kirby]

So it’s just a matter of definition.

That's what I've said several times. The teacher has a definition of "formula". AloneAgainstAll has another definition in mind and concludes that his teacher was wrong, and is so bad that he wouldn't recommend her to his enemies. I found that judgment too hard.

Sure, I can see that. Personally, I think AloneAgainstAll‘s definition is a bit more reasonable. If you’re going to make the claim that “X is not possible”, it’s good to clear up ambiguity about what is meant by X.

So I don’t think this is necessarily grounds to call this person a “bad teacher”, but it would be better to be less ambiguous.

I’m reminded of an incident with my cousin’s wife, who is a second grade teacher. For a math test, she had a problem saying something to the effect of “color the circle” or something like that. Based on definitions of circles that I’d read, the circle is basically the border definition. So I would have prefer she wrote “color the area surrounded by the circle”. Because someone who colored only the border would be correct, in my opinion.

But there may be definitions of “circle”, which include the area and not only the border. But no definition of “circle” was given.

This doesn’t mean my cousin’s wife is a bad teacher. But personally, I think the ambiguity makes it a bad question.

[AloneAgainstAll]

So it’s just a matter of definition.

That's what I've said several times. The teacher has a definition of "formula". AloneAgainstAll has another definition in mind and concludes that his teacher was wrong, and is so bad that he wouldn't recommend her to his enemies. I found that judgment too hard.

Yes, you can make such a definition of formula that makes my "formula" not a formula anymore. I heard once from another teacher that 0 is not and odd number and not pair number. Its special number. Yeah, you can define pair numbers and odd numbers in such a way that 0 will not be not pair, nor odd. I used definition of formula that is generally accepted in mathemathics world. I even gave function, and if you go back to 2nd page you will see that this teacher denied existence of function (well, you too) :

This is qoute from you:

"The fact that "there is no formula for producing n-th prime number [f(n)=n-th prime number]" is correct, but the teacher could have said that f(n) ~ n log(n), maybe this would have fascinated part of his audience."

Its not a matter of life to be always right, sometimes you can be wrong. Me too.

Edit:

Some funny (and bit scary) story with a point:

In 1st year of elementary school teacher asked pupils to make drawing, topic : "my family after 100 years". One of childrens draw graves of his parents. Imagine that parents were asked to meet teacher, cuz teacher had suspicions that there might be something wrong with a child. Logic not always pays off in this world.

[Bill Spight]

I don’t understand why folks are so reluctant to call the given expression a formula.

Well, I was not particularly reluctant to call that expression a formula. I attempted to guess a definition of formula that was close to the one that the elementary school teacher had in mind, and which is also close to my sense of it. True, my definition was based upon my knowledge of linguistics and semantics, but consider the ways in which formula and related words like formulaic are used. I did not consider the expression at all in coming up with my definition. Afterwards I realized that the expression would not fit my definition, but I was prepared for it not to do so, and in that case to say that I was wrong.

[AloneAgainstAll]

AloneAgainstAll: my formula is just a paraphrase of the definition of the n-th prime number.

To compute f(5), search for the largest m such that if 1<a<b<c<d<e<m then one of the a,b,c,d,e is composite.

For instance, m=11 satisfies the property because if you try every family 1<a<b<c<d<e<11 of integers, you will find a composite one. It's a bit long to check because there are 126 such families but it's feasible.

But m=12 fails because you can take (a,b,c,d,e)=(2,3,5,7,11).

So f(5)=11.

Well, if so, then we can say F(n)=n-th prime number and thats formula for n-th prime number then. What is your definition of formula then? In my native language, formula and function are synonims, maybe in english they are not, and thats why in brackets i precisely expressed what teacher meant. But if you call what you described function, then i object, its not a function.

[Kirby]

I split these threads away from the Shin Jinseo thread, since it's a bit of a topic in its own right. It's also in the "Off Topic" forum, now.

[Kirby]

Well, I was not particularly reluctant to call that expression a formula. I attempted to guess a definition of formula that was close to the one that the elementary school teacher had in mind, and which is also close to my sense of it.

By the way, what *is* this definition? I can understand the one I posted from Wolfram, which seems consistent with AloneAgainstAll's interpretation. Given that we have an alternate definition in play here, and that we can't ask AloneAgainstAll's teacher what their definition, would it be possible for you to express your intuitive definition in a precise way?

[AloneAgainstAll]

You mean to explain what is under attachment? You ask me to explain this formula? How this formula works? Or what?I still dont get it, really.

If you ask how formula works, it works (by works i mean "produce n-th prime number) beacuse:

The sum contains three factors which we multiply k, and 2 complicated fractions. We sum from k=2 to k=2^n, but
k is always different from 0, first fraction is different from 0 only when k is prime, and 3rd factor is equal to 1 only when there is exactly n prime numbers below k(otherwise is equal to 0). Thanks to this, this big sum, has terms always equals 0, except 1 time, when its n-th prime number.

Why we summing from k to 2^n? beacuse we need to have n-th prime number between, thanks to Chebyshew rule, which says that between x and 2x there is prime number, which follow that n-th prime number is lower than 2^n (and thats what we needed).


You need to try to examinate first this (its element of this function) to get how this works.For given a, this function is equal to 1 only when x=a, otherwise is equal to 0 - look at F1 attachment.

Then you need to examinate next funtion, which is equal to 1 if k is prime number, otherwise is equal to 0 - look at F2 attachment.

If you get this 2 steps, and examine my formula, you will see how it works. I hope that this close case. I am exhausted really.


I am not the best explaining mathemathical things in english, i hope i didnt make any mistakes.

[jlt]

Well, if so, then we can say F(n)=n-th prime number and thats formula for n-th prime number then. What is your definition of formula then? In my native language, formula and function are synonims, maybe in english they are not, and thats why in brackets i precisely expressed what teacher meant. But if you call what you described function, then i object, its not a function.

• A function is not the same as a formula. A function from N to N is a subset F of P(NxN) such that for all x there exists exactly one y such that (x,y) belongs to F.
• Whatever your definition of "formula", a function is not the same as a formula. There are uncountably many functions from N to N, but countably many formulas.
• The word "formula" is not a mathematical term, as far as I know there is no universally accepted definition of "formula".
• In my last few messages, I showed that according to Kirby's definition of a "formula", I could produce dumb "formulas" that "compute" the n-th prime number.
• The formula you gave is less dumb than mine, but doesn't really have much more mathematical content.
• Last, I don't know what your native language is, but if I don't speak it, there may be a translation problem here as well.

[Bill Spight]

I’m reminded of an incident with my cousin’s wife, who is a second grade teacher. For a math test, she had a problem saying something to the effect of “color the circle” or something like that. Based on definitions of circles that I’d read, the circle is basically the border definition. So I would have prefer she wrote “color the area surrounded by the circle”. Because someone who colored only the border would be correct, in my opinion.

But there may be definitions of “circle”, which include the area and not only the border. But no definition of “circle” was given.

Gee, Kirby, haven't you heard of the area of a circle? (That is, instead of "the area bounded by a circle".)

[AloneAgainstAll]

Well, if so, then we can say F(n)=n-th prime number and thats formula for n-th prime number then. What is your definition of formula then? In my native language, formula and function are synonims, maybe in english they are not, and thats why in brackets i precisely expressed what teacher meant. But if you call what you described function, then i object, its not a function.

• A function is not the same as a formula. A function from N to N is a subset F of P(NxN) such that for all x there exists exactly one y such that (x,y) belongs to F.
• Whatever your definition of "formula", a function is not the same as a formula. There are uncountably many functions from N to N, but countably many formulas.
• The word "formula" is not a mathematical term, as far as I know there is no universally accepted definition of "formula".
• In my last few messages, I showed that according to Kirby's definition of a "formula", I could produce dumb "formulas" that "compute" the n-th prime number.
• The formula you gave is less dumb than mine, but doesn't really have much more mathematical content.
• Last, I don't know what your native language is, but if I don't speak it, there may be a translation problem here as well.

Thats why in brackets i expressed precisely what teacher meant. What is your definition of formula then?

[jlt]

As I said, I have no definition of a "formula". I think that sentences like "find a formula" are too ambiguous and should not be used if we want to avoid misunderstandings.

[AloneAgainstAll]

How can you know then that there is countable many formulas fron N to N?

[jlt]

In my mind, a formula is necessarily a finite sequence of mathematical symbols (but the converse is not necessarily true). The set of mathematical symbols is finite, and for every finite set S, the set of finite sequences with values in S is countable, so whatever your definition of a formula, there are countably many of them.