Seventh Week BeyondResearch Insights, Year 1

Hi! Welcome to another one of my BeyondResearch Insights! This time coming from the Abstract Analysis side of the programme.

Last week, we started off the session as usual with a reflection on our problem attempts. This led us to a discussion on the importance of knowing the purpose of each assignment.

For abstract analysis, apart from writing summaries of our bi-weekly sessions and completing problem sets, the assignments also include independent reading. Since the beginning, we have been slowly going through Analysis I by Amann and Escher and started the Lean E-book on Logic and Proof not long ago. The purpose of this was for us to learn the technical knowledge of a mathematician. But what does that entail?

For one, we must get familiar with different types of proofs. The simplest one is the direct proof, where we start with assumptions and then use logic and known theorems. There is also indirect proof, where we infer that our claim is true from the fact that its negation is false. However, importantly, this only works in classical logic which uses the law of excluded middle, about which I wrote in my first ever BeyondResearch insight, so check it out!

The indirect proof is easily confused – and I’m saying this from personal experience – with another proof technique called negation introduction. There we prove the negation of our original claim, which is in fact valid in non-classical (intuitionistic) logic.

In addition, there is proof by induction and its generalization structural induction. Mathematicians also have certain higher-level methods in their arsenal, like homology theory, which we have been dabbling into in search of a proof that a circle cannot be deformed into a line or vice versa without cutting or glueing. But this will have to wait for another time, as we have not arrived at the full solution yet.

Lastly, to be able to do proofs effectively, we must build them on previous results. Therefore, we must learn, understand, and know where to find various theorems and lemmas, which is the core of technical mathematical knowledge.

This also gives away the reason, why we must learn the above in the first place: so that we can prove theorems! A trained mathematician (and hopefully us in the future) is actually able to – using their technical knowledge, creativity, and intuition – come up with and prove new theorems and thus push the boundaries of mathematics further!

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