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Two foundational mathematical thinking constructing constructs (if I may say): * (Existential-Universal) quantification * (Epsilon-Delta) language Once p
by rq1 5y ago
Two foundational mathematical thinking constructing constructs (if I may say):
* (Existential-Universal) quantification
* (Epsilon-Delta) language
Once people start to not only understand it, but actually think that way, they start to be scientists and mathematicians.
It opens the door of refutable and constructive thinking.
I've seen also a lot of people struggle with definitions.
- ifbutthen 5y agoOne problem with epsilon-delta definitions is that they don't scale. In many topological spaces, there's no concept of distance and so epsilon-delta stuff has to be replaced with more general open sets. About the same reason why sequences have to be replaced with nets and filters.
- ronald_raygun 5y agoyeah there are like 10 million ways of defining cont. or limits in R^N that turn out to be the same
- ifbutthen 5y agoR^n is Euclidean and so epsilon-delta defns work just fine. I am talking about topologies like cofinite topology or cocountable topology where the picture doesn't look like a grid to which you can apply distance related notions of epsilon- room here and delta-long there.
- rq1 5y agoYes of course, I didn't mean that we should stick to metrizable spaces. Though it's intuitive for humans and its limits are quite broad, if I may say :) Eg. we usually don't start defining the derivative with distributions. I didn't mention geometry neither btw though it's foundational too imo.