Maybe this isn't the exact right place to bring it up, but is related. But there is a lot of self-paced learning things coming up between schools and alternative to schools. But an issue I have is that this self paced learning is really limited in scope. Why can I take classes so easily for an MS in computer science or IT, or get an MBA or take data analysis courses, but I can't easily get a degree / certificate for say a bachelors in Mathematics? Or literature? There seems to be a big split where so much is focused on job training but very little is being done on education.
The answer is that mathematics is a thing that you can actually assert that someone is doing incorrectly, but for which it will not be obvious to non-mathematicians. If a programmer writes a bad program, the user can tell. If a mathematician writes a bad proof, basically it takes a mathematician to tell. It doesn't take an incredible mathematician, but it does take one.
If you want to say that you learned mathematics, you have to do more than this student did. you may have chosen to give him the benefit of the doubt, but I did not. I checked his work, and its mostly wrong. The reason that it is mostly wrong is because the practice of mathematics requires some training to get to the point where you can be relied upon to know if you are lying, and the author of this blog did not reach that point.
Sure, but I will limit myself to one, which is fairly exemplary.
The question is to show that R2 is second countable. This means that it contains a countable base for the topology. The usual way to do this is to pick open balls with rational centers and rational radii, and use a little finesse to find one around any point in an arbitrary open ball.
The Author's answer was to take the set of all open balls, and pick only those with natural number radii. This is neither countable nor a base for the (usual) topology on R2. This answer has a check mark on it.
The oft cited refrain is that colleges/universities serve several purposes aside from education:
1.) Verification, standardization, and authentication of work being done and of knowledge being gained.
2.) Networking opportunities and social/professional development and maturation in a a limited-stakes playground surrounded by others in the same social and economic class, and
