Question

we know exhaustive check of conjecture is no sufficient for become a theorem, like Riemann hypothesis that is checked true for first 10,000,000,000,000 cases, but why? math is like pattern. someone know of an example once thought true after exhaustive check but turned false after analytical check? thx

Answer 1

hey cool glasses from where did u get? did they give them to u?

Answer 2

some questions are two long to check anayaltyical. For example prime numbers, no one has a formula to find all prime numbers. We could check 10,000,000,000,000 before finding the next prime number.

Answer 3

i guess my question is, "why can't rely on computer proof?" is there example of why not?

Answer 4

it does not work for prime numbers, no anayltyical method has been found. No one knows the next prime number. They have checked thousands of numbers and not found the next one, but we still believe there are other prime numbers out there.

Answer 5

i guess my question is, "why can't rely on computer proof?":

The thing is, this is not a computer proof. It doesn't enumerate all of the possible cases and verify the conjecture is true. An example of a computer proof that does that is for the 4-color problem, where a computer proof verified the result for the 1,936 cases. (See http://en.wikipedia.org/wiki/Four_color_theorem )

Rather, this is a numerical verification for what feels to us like a lot of numbers. So it seems reasonable to think the Riemann Hypothesis must be true. But on the other hand, the computer calculations have not even verified the result for 0.01% of all possible cases, so how strong is this evidence really?

Mathematicians have very high standards for proof, higher than in the sciences, higher than in Physics. In Physics, for example, we believe in the Superposition Principle, that we can add up waves. For example the sound waves you're hearing now propagate in the air as compression of air molecules. The Superposition Principle says the wave of sound from the radio can be added up with the wave from the ambulance outside with the sound of your friend talking to you. The Superposition Principle is used all the time in Physics, not only in sound, but most importantly in electromagnetism and quantum mechanics.

Is there any logically compelling theoretical reason for the Superposition Principle? No. But has it been confirmed in literally millions and billions of experiments? Yes. (For example, when you turn the radio on, you don't stop hearing the ambulance.) So as a matter of what is called inductive reasoning, the Superposition Principle is reasonable.

But for a mathematician, that isn't enough. Mathematicians insist on a deduce argument, starting with accepted axioms and theorems to derive the next result.

You might argue the experimental results for the Riemann Hypothesis have given us a good inductive argument to accept it. As I noted above, there's a cynical way to deflate that idea; the "less than 0.01%" argument. But even if you set that aside, for mathematics, experimental evidence just isn't enough.

Answer 6

many thanks for explanation mr james i understand now what you say ''less than 0.01%'' is very true if i think to infinity so it could to be false for some number with ten trillion digits who knows


thank u again