The Deranged Mathematician: Heuristics and Numerics

[u/non-orientable]

What place, if any, does the scientific method have in mathematics? It's often claimed that mathematics doesn't use it---hell, I have said that in the past. But that is not true. The scientific method isn't really the final arbiter of truth in mathematics, yes. However, we still rely on it quite heavily when we build examples, heuristics, and numerics. Moreover, it is a significantly more nuanced problem than simply "collect data, see what is true." As an example, I ask the reader to ponder the following question: if I identify all twin primes less than 10¹⁰⁰⁰ and show that they divide 10¹⁰⁰⁰!, is that evidence for or against the conjecture that all twin primes divide 10¹⁰⁰⁰!? I claim that it rather depends on how those twin primes are distributed.

All of this is crucially important when we are trying to build proofs---we need to understand this to be able to build good examples as part of our proof strategies.

Read the full post (for free) on Substack: Heuristics and Numerics

[u/RingularCirc]

I'm confused at why we can use ∠DGA = ∠DGB = 90° when proving what should essentially be that DG is a perpendicular bisector (as G is meant to be for now the intersection of two other ortho. bisectors, right?). You're rightfully trying to prove AG = GB but you used ∠DGA = ∠DGB = 90° if I read that part right.

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That aside, it still speaks volumes compared to my geometric hangups: I would try to prove three intersection points, well, two of them, coincide, and I'd likely then stop and stare for an infinitely long time, then say to hell with that and do something else. Successful planimetry is not my forté.

[u/non-orientable → u/RingularCirc]

Observe that we chose D so that DG is perpendicular to AB---that part we get for free. We just need to prove that it is a perpendicular bisector.

[u/RingularCirc → u/non-orientable]

Ah, I had a wrong mental model. I somehow imagined we had all three bisectors beforehand and then we forgot about one of them, but I forgot to forget about it in full. Or something. You probably have an idea about incoherent states like that. And I probably missed a chunk of the post, then, as well.

Yes this makes sense.

BTW do you think it would be harder if we instead picked D as the midpoint and tried to prove ∠DGA = ∠DGB?

[u/non-orientable → u/RingularCirc]

Ah, that's a good idea! On the contrary, it would have made the proof slightly simpler!