Suppose you are a maths teacher and give the students an assignment of proving or disproving X.
One student rights what appears to be a proof of X. You have looked over it closely and can’t find any mistakes. Another student has written what appears to be a proof of not X. Again you can’t find any flaws. A few students have handed in nothing or rubbish Some have just asserted X or not X without proof.
You have no higher authority to appeal to. You don’t know whether X is true or false yourself. If you use your weak personal suspicion that X is slightly more likely to be true than false, the class plays a game of guess the teacher. The best you can do is to pass the student who “proved” X and the one who proved not X. A perfect mathematician would pass. A rock or coin toss would fail.
Suppose you are a maths teacher and give the students an assignment of proving or disproving X.
One student rights what appears to be a proof of X. You have looked over it closely and can’t find any mistakes. Another student has written what appears to be a proof of not X. Again you can’t find any flaws. A few students have handed in nothing or rubbish Some have just asserted X or not X without proof.
You have no higher authority to appeal to. You don’t know whether X is true or false yourself. If you use your weak personal suspicion that X is slightly more likely to be true than false, the class plays a game of guess the teacher. The best you can do is to pass the student who “proved” X and the one who proved not X. A perfect mathematician would pass. A rock or coin toss would fail.
You would have given your students what is called an “open problem,” and students will tell everyone to never take your courses.