zermelo wrote:
AloneAgainstAll wrote:
You really think this is good paraphrase? I dont think so. There are things we cant "enclose" in mathemathical formulas (like results of integral of sin(x^2) cant be described by elementary functions, only infinite sum ) etc. So the fact that we can produce such a formula (even if its completely useless for computing) is mathemathically significant.
Ok, I define S(t) as the integral of sin(x^2) from 0 to t. Now I can write you a nice short correct formula for it, i.e. S(t). Yes, it does not use elementary functions only, but your prime formula with floor functions does not use only elementary functions either. The whole issue is really about what kind of formulas we allow.
I don't dispute that maybe you had a bad teachers, and maybe he/she could have discussed the nuances of formulas and algorithms better.
Its not "mine" formula, its Paulo Ribenboim formula i believe (or maybe someone other, not sure). If you look closely at your example you will find a small mistake there, but i know what you mean (thats completely sidenote, you dont need to improve your example, i trust you know the drill)
Yes this formula use floor function which is not elementary function. But that is completely another topic, i used this example with S(x^2) to explain (i guess i was not understood) that enclosing sth in formula is mathematically significant, even if formula is not usefull for computations.
You ask what formulas we allow? All that are well defined in ZFC (+AC or not, as you wish). If we would try to limit it, then why to make any formulas at all?
You really think this is about what formulas we allow? I dont think so. Now we are wise, we know that such a formulas exist (though they are not usefull for computations, not important), but can you provide formula which would produce nontrivial zeros for Riemmann-Zeta function? I guess no. But imagine that in 10 years mathematicians broke this, computed all nontrivial zeros and proved that there are no more. Now producing such a formula looks pretty different, isnt it?
She couldnt explain nuances of formulas and algorithms. She was elementary school teacher, and i was 10 years old then.