The lectures this week were interesting. I enjoyed the circular proofs of the three induction principles. However, I find a problem with the way we chose to prove them. We said that we would need to take any one of the as an axiom and can prove the other two that way. I have no problems with any of the principles being axioms as they just seem obvious, but if they are axioms then none of them should require a proof. It seems very arbitrary to say "pick one, the other two need proofs". In short, they should all either be axioms or require proofs using something other than a different principle of induction.
The assignment was pretty easy (we will see if this statement holds true when I receive my marks). The first question was structured in such a way that the answer was obvious, although I'm sure that was intended as it was the first question. The second question, about the rotating lunch orders, I found very interesting. The proposition of creating lunch menus that do not differ by more than 1 choice is very easy, yet I find that I wasn't thinking of the algorithm. I somehow naturally knew what the structure should be and only after writing a few different sized lunch combinations did I notice the algorithm. The proof was straightforward. The final question with the golden ratio had me fooled for a little as I tried to equate the hint to the formula, rather than plugging in n1, and n2, into the formula to transform it into the form that the hint was. However, after I reached the n2/(n1-n2) form it was smooth sailing from there.
1 comment:
The aesthetic with axioms is to have as few as possible, and derive the others. So Euclid's geometry starts with five.
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