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Edited on Thu Nov-05-09 06:25 PM by Boojatta
Introduction:
Specifically, I'm thinking about statements that are officially classified as "self-evident", and used as a basis for deductions. Students are likely to make some false positive errors. In other words, they are likely to classify as "self-evident" some statements that aren't self-evident, and that aren't even true. After some study, students are likely to be persuaded that some statements that initially seemed "self-evident" aren't true.
However, even if all people who express agreement with a statement later adamantly deny it, we cannot necessarily conclude that the statement is false. People can be indoctrinated. They may simply recognize a conflict between official doctrine and a particular statement. If they accept official doctrine, then on that basis they may reject a particular statement.
The main point of this thread:
Suppose that many students, when consulted, express sincere doubt that a particular statement is self-evident, even though it has been officially classified as "self-evident." In this situation, it might be possible to find alternative statements that all or almost all students accept as self-evident, and to show students how the alternative statements can be used to prove that the officially "self-evident" statements are at least true.
What is the nature of the learning process or screening process to become a math professor that ensures that, when people who work as professors of mathematics assert that some particular statement is "self-evident", it actually is a self-evident statement?
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