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We need formalism, and we need motivation, examples, and pictures. One isn't better than the other. In particular, formalism isn't about hazing (though I think

We need formalism, and we need motivation, examples, and pictures. One isn't better than the other. In particular, formalism isn't about hazing (though I think some people abuse it this way). Books that claim to appeal more to intuition or rely on visual arguments certainly help a lot with motivating ideas and establishing context, but at a certain point we need to be clear about what exactly we are talking about. When learning a new topic in math, I personally prefer to start with a more formal, terse text. For me, the texts that focus too much on examples and motivation tend to be chattier -- I have to read an entire paragraph to understand what it is they are getting at, making more challenging the process of chunking the information into digestible bits that I can hold in mind as I shower or go for a walk (and generally less fruitful). Compare to say working through Rudin or Kolmogorov, where I can read a distilled sentence where each word is carefully chosen that I can easily recall and munch on. Part of it is that I have ADHD -- texts that are more formal and less chatty make it easier for me to focus. That said, I do think Courant had a point with > Mathematics presented as a closed, linearly ordered, system of truths without reference to origin and purpose has its charm and satisfies a philosophical need. But the attitude of introverted science is unsuitable for students who seek intellectual independence rather than indoctrination; disregard for applications and intuition leads to isolation and atrophy of mathematics. It seems extremely important that students and instructors should be protected from smug purism. Basically, math doesn't exist in a vacuum, and I think part of the reason there are so many people with low emotional intelligence in math is this belief that it can. But that's another story

Jugurtha 6y ago
>When learning a new topic in math, I personally prefer to start with a more formal, terse text. For me, the texts that focus too much on examples and motivation tend to be chattier Soviet books (MIR Publishers) are notoriously terse and to the point. This is what we used in college, and it has made it difficult for me to read a different style, because the content tries my patience. For example, I couldn't consume Andrew Ng's Coursera ML course, but I appreciated the CS229 recorded at Stanford because he dove directly into the maths part. I didn't want to see slides, I needed to see problem statement, pause the video, work through gradient descent, play the video, and check I got it right. Side note: doing this [pause, working on it, play] has its advantages, it helped avoid a confusion the instructor had in the course with notation, for example, which you had to be sorted out in the later part of the course. As you said, it is a spectrum of content with different styles for different needs and people, but even for the same person, one might need a style at a certain point and the other at another point. I remember in my third year I used some MIT OCW resources because I felt my brain was shot and I needed to be spoon fed on a topic I was so far behind on.

Jugurtha 6y ago

>When learning a new topic in math, I personally prefer to start with a more formal, terse text. For me, the texts that focus too much on examples and motivation tend to be chattier Soviet books (MIR Publishers) are notoriously terse and to the point. This is what we used in college, and it has made it difficult for me to read a different style, because the content tries my patience. For example, I couldn't consume Andrew Ng's Coursera ML course, but I appreciated the CS229 recorded at Stanford because he dove directly into the maths part. I didn't want to see slides, I needed to see problem statement, pause the video, work through gradient descent, play the video, and check I got it right. Side note: doing this [pause, working on it, play] has its advantages, it helped avoid a confusion the instructor had in the course with notation, for example, which you had to be sorted out in the later part of the course. As you said, it is a spectrum of content with different styles for different needs and people, but even for the same person, one might need a style at a certain point and the other at another point. I remember in my third year I used some MIT OCW resources because I felt my brain was shot and I needed to be spoon fed on a topic I was so far behind on.