Can a Conjecture Be Proved Empirically?

[mhill]

can a conjecture be proved by 'empirical' means (observation) ??

i mean let us suppose that exists some functions named $$f_{i} (x)$$

so $$\sum _{n=0}^{\infty} = \sum _{p} f(p)$$

then an 'empirical' method would be to calculate the 2 sums and compare the error , let us suppose that the error made in the equation above is less or equal than 0.001

so $$|\sum _{n=0}^{\infty} - \sum _{p} f(p)| \le 0.001$$

then , would this be simple coincidence or a fact that our conjecture is true ? , for example physicist and chemists work this way , as an approximation of a theory to our observed reality.

 

[mathman]

It would help if your statements could be clarified. Your first sum is n=, but there is nothing being summed. Your second sum is for a function over p, without anything said about what p is.

 

[mhill]

Uh.. excuse me , the sum on the left is made over f(n) , the sum on the right is over all 'primes' p

$$\sum _{n=0}^{\infty}f(n) = \sum _{p} f(p)$$

 

[frb]

of course it is not a proof, it can give an indication that the theorem might be true and give you a reason to find a formal proof.

for example
the difference of the partial sums might eventually be smaller than some epsilon, but it might also always be larger than some lower bound

some ideas seem very plausible but might have very pathological counterexamples.
http://en.wikipedia.org/wiki/List_of_disproved_mathematical_ideas