jeff76

The way you were arguing makes it sound like you are against beginners having access to solutions, or that you completely misunderstood my initial argument. I am not arguing about anything other than that. If you want to address that in particular sure. I am not saying anything about people skipping material and getting bitten by it later.

Think you missed my entire point of including solutions for the beginner, i.e., someone that lacks math maturity and the reasoning of how easily such a person could fool themselves into thinking their thought process is valid.

Yes, which is why you include "several proofs of this could include..."

Disagree. Some proofs are very straightforward and standard. Those are easily checkable with a solution guide.

The issue isn’t just that the person can’t write a valid proof, but that the person thinks they have a valid proof, but it is actually invalid and they don’t have feedback to know it.

When you’re doing basic proofs, showing solutions are often simple enough for unconfused verification. There are often standard ways to go about proofs, solutions could provide those standard solutions with sufficient detail written out.

The point isn’t we need to do formal verification to test correctness. It’s that you probably wouldn’t write a program first pass and assume it is but free. So, why would you assume beginners that don’t know how to write proofs would write correct proofs without feedback to check?

I disagree with the notion that if you question your solution then your solution isn’t correct.

A beginner could easily fool herself onto thinking she has correct understanding, but actually her proof is buggy. Without feedback, she wouldn’t know.

Any books in: Introduction to proofs, Abstract algebra, Topology, Linear algebra

I understand but sometimes there are books targeted for the “self-learner” that provide 0 solutions and are the type of books that save off major theorems to be proven in the exercises.

That’s completely fine. Don’t advertise the books for self-learners or “programmers”.

I would like to see actual books targeted at these demographics that provide solutions and aren’t concerned about being adapted in a classroom.

There is a market for those that graduated university a long time ago and would like to learn math outside of a classroom.

These sort of people don’t want to see solutions because they have to turn their work in for a grade, but because they want to check their work and get feedback. Outside of having to consult with TAs or professors they don’t have access to.

I don’t understand why people recommend books like “a programmers introduction to mathematics”.

They’re advertised as books for “programmers” or people that know little math, yet they hide solutions from the reader.

The reader that doesn’t know proper proofs or deep mathematics likely isn’t the same ones that know if their solutions are correct.

Programmers like to write code with test cases. We don’t like to write code once and trust there are no bugs. So, why would we want to write mathematics any differently? How is a beginner math student even going to know if their answers are correct? I’m sure someone reading this will say “you’re robbing the reader” if you provide solutions. I don’t agree with that. That’s a bit of gatekeeping because not everyone has access to a TA or professor and we’d really like to learn this stuff and know if we’re on the right track.

Are there any actual math text with solutions that are better than the one advertised in this list?

Any books on how to do proofs, that also includes solutions?

This answer makes sense.

I suppose my question is more along the lines of, if someone is specializing in deep learning in a PhD program then shouldn’t they at the very least be able to implement models and also know optimization tricks?

In other words shouldn’t they be able to develop enough skills to go deep in one area but also know enough to be dangerous in the other three domains?

Great reply. My question based on this comment,

>My advice is assuming you’d like to be a person that trains/deploys ML models to solve problems in industry. This is much different than an ML Engineer, who’s implementing algorithms in low level languages and squeezing out efficiency. Obviously that would require a much deeper understanding of SWE. And a totally different person is an academic researcher that’s developing theory or technique. It’ll be hard to do that without a PhD

Can one not only train/deploy ML models, but in addition to that be able to implement the algorithms in low level languages and also be able to develop theory?

I’d imagine these are all skill sets that someone in PhD program could pick up.

If they could do all three, what kind of job should they be looking for?

This is interesting. What are the prerequisites to understand Univalent Foundations and how active is it?

This brings up a question I have. Is there a way we can code up proofs in a computer language and have it checked for correctness? If so, how? What resources can I follow up on? Can current technologies check really complicated proofs like this one?