- irrefutable is "cannot be denied or disproven" - it means absolutely proven. Very different from untestable (which means you can't verify or prove it).
This is precisely the distinction we've been calling out from the original - it's a demonstration that works for some cases. The diagrams fail as a proof because they can be refuted by the negative/0 cases where they don't work.
- testing or verifying with a set of data is also very different from a proof. This "checking" or "demonstrating" provides some assurance of correctness or utility for the test domain.
- demonstrating and checking against known facts is not sufficient for a proof - "I've tested my division function for millions of positive and negative integers and real numbers! I even verify by multiplying the quotient by the divisor and I've proven it is correct!" did you happen to test a divisor of 0? (dividing by 0 can also invalidate proof attempts that do not exclude 0 as a divisor)
It needs to be proven to hold for all cases, not just a sampling of cases (though it is valid to define the range of a proof - the example could have stated, "this is a proof of the equation for positive values of a and b, and were b < a" maybe that could constitute a visual proof for that domain?)
- irrefutable is "cannot be denied or disproven" - it means absolutely proven. Very different from untestable (which means you can't verify or prove it).
This is precisely the distinction we've been calling out from the original - it's a demonstration that works for some cases. The diagrams fail as a proof because they can be refuted by the negative/0 cases where they don't work.
- testing or verifying with a set of data is also very different from a proof. This "checking" or "demonstrating" provides some assurance of correctness or utility for the test domain.
- demonstrating and checking against known facts is not sufficient for a proof - "I've tested my division function for millions of positive and negative integers and real numbers! I even verify by multiplying the quotient by the divisor and I've proven it is correct!" did you happen to test a divisor of 0? (dividing by 0 can also invalidate proof attempts that do not exclude 0 as a divisor)
It needs to be proven to hold for all cases, not just a sampling of cases (though it is valid to define the range of a proof - the example could have stated, "this is a proof of the equation for positive values of a and b, and were b < a" maybe that could constitute a visual proof for that domain?)