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1. Rewriting systems are very useful tools. One of the things I learned from this post was about the existence of FullEquationalProof [1], which I think is pretty darn neat and super useful.
2. This post is imbued with a latent metaphysics that is somewhat common in formal mathematics and will invariably come out at relevant meetings/conferences after a few glasses of wine. Not this instantiation in particular, but a generally similar sort of metaphysics. I've always gotten church vibes from that sort of thing. (I never "got" church.)
I never got the "emergent properties have special aesthetic and nearly spiritual significance" or "everything is just an <insert structure here>" cognitive confusion that so many mathematicians (especially formalists) seem to have.
But the way that it happens does make sense. Intellectual curiosity is a valid work selection strategy and sometimes invention/discovery requires a leap of faith. Developing a predisposition to spiritual thought patterns while doing work that requires a leap of faith makes sense, I guess, but it's something I warn young mathematicians to guard against.
But then, I'm also not allergic to fava beans. Maybe math-as-spirituality and believing beans are evil are also just useful tools.
[1] https://reference.wolfram.com/language/ref/FindEquationalPro...
This might be the source of disconnect. I frequently encounter this perspective and worry there's a fundamental problem with how mathematics is taught if so many people walk away believing this. Whether or not humans ever mastered mathematics, what is and isn't mathematically true would not change. Humans can create notation and formalisms, but they do not invent the truths those mathematics represent.
It's a study which shows that pictures that people select to represent themselves publicly have features that indicate political leaning.
The software industry has its own share of problems, but from what I've seen the research community is still largely operating on an outdated software model that shuns open collaboration out of fear of being "scooped".