Consider the Bolzano-Weierstrass theorem, which states that a real bounded sequence contains a converging sub-sequence.
There was a time when it didn’t appear to me that the BW theorem was true, and I didn’t believe in it. Now, it appears to me that it is true, and I believe it is.
The transition from non-belief to belief in the BW theorem wasn’t based on appearance, although it coincided: it was based on presentation of rigorous argument (which appeared valid) whose inevitable logical consequence was that the BW theorem was true.
My belief in the BW theorem grew first from authority, then internal acceptance of the argument, then I cached the thought and now may use it without recalling the proof.
I suspect that a large part of mathematical teaching is giving the students an internal appearance of the validity of the arguments (as in, it appears to their introspected reason, regardless of teacher authority or contextual clues).
I wonder WHICH proof you were taught. Wikipedia has two proofs, of which the second one seems to be more intuitive. It works by repeatedly splitting the interval into subintervals, then noticing that a subinterval has infinitely many points and using this fact. Receiving “an internal appearance of the validity of the arguments” is NOT how math is supposed to work, it is supposed to rely on rigorous proofs.
Consider the Bolzano-Weierstrass theorem, which states that a real bounded sequence contains a converging sub-sequence.
There was a time when it didn’t appear to me that the BW theorem was true, and I didn’t believe in it. Now, it appears to me that it is true, and I believe it is.
The transition from non-belief to belief in the BW theorem wasn’t based on appearance, although it coincided: it was based on presentation of rigorous argument (which appeared valid) whose inevitable logical consequence was that the BW theorem was true.
My belief in the BW theorem grew first from authority, then internal acceptance of the argument, then I cached the thought and now may use it without recalling the proof.
I suspect that a large part of mathematical teaching is giving the students an internal appearance of the validity of the arguments (as in, it appears to their introspected reason, regardless of teacher authority or contextual clues).
I wonder WHICH proof you were taught. Wikipedia has two proofs, of which the second one seems to be more intuitive. It works by repeatedly splitting the interval into subintervals, then noticing that a subinterval has infinitely many points and using this fact. Receiving “an internal appearance of the validity of the arguments” is NOT how math is supposed to work, it is supposed to rely on rigorous proofs.