Re:Math Foundations from Scratch
by ipnon on Tuesday October 19, 2021 @04:26PM
(#28923313)
There are two schools of machine learning education: Math first like Andrew Ng and code first like Jeremy Howard. That both of these leaders have found academic and professional success shows you should approach machine learning in whatever way comes more naturally to you. Later on, you can approach it from the other angle to improve your understanding.
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To clarify, I'm not arguing that math isn't important. You need your linear algebra, your probability theory, and so on. That's non-negotiable. But set theory? Type theory (as someone below suggests)? I would avoid unless you're separately interested in mathematics for its own sake.
Its not immediately practical... but spending a lot time learning a science like math in depth makes learning various applications very easy later.
I think sometimes educators have to let people be motivated by some particular application to teach them something more important. Perhaps that is what the author is up to?
I’m fairly certain they’re not suggesting you need to know type theory for ML, but that theorem provers should have more ML libraries. My wife does some ML as a postdoc at Stanford and I doubt she’d even know what type theory is.
I agree, however I'm not sure of how far you can get in probability theory without set theory. For instance, I recall my stats prof defining a lot of sets in class, e.g. sigma algebras.
Another point of clarification: I'm all for telling people what sets are. I'm less warm toward discussing things like ZFC, esoteric cardinality issues, and certain paradoxes (and large cardinals, forcing, etc.), which is typically what mathematicians mean when they say "set theory."
Re:Math Foundations from Scratch
by actually_a_dog on Tuesday October 19, 2021 @06:19PM
(#28924486)
Right. What most people need to learn is "algebra of sets," essentially.
How do you even do serious linear algebra or probability without set theory?
I have a PhD in physics and studied graduate level mathematics. Naive set theory is sufficient for basically everything except a few instances, where you have to bite the bullet and invoke the axiom of choice. But this happens rarely and typically is only relevant for instances where there are more constructive but less general approaches.
For the most part sets are used as a notational device, you almost never use any axioms about sets explicitly. If you asked most professional mathematicians to give you the ZFC axioms (or whatever) they would need to look them up.
yes but most of the math-based approaches to ML start from real (multivariate) analysis, or from probability theory. i haven't heard anyone claim that Peano axioms are important. as much as I will defend the notion that applied mathematics is extremely important for ML research, peano axioms are unnecessary and will just cause confusion