One of the major points that this book is trying to convey is that your intuition is wrong. You need to burn it down, learn the formalism, and maybe develop new intuition based on that. Even then you need to recognize that intuition can be wrong.
An N-dimensional vector space is not like R^3. A monad is not like a burrito, and >>= is not like chipotle forgetting to put chicken in your burrito, opening it up, adding chicken and wrapping in a new tortilla.
Math is a new thing rather than the same old thing with new syntax. It needs to be understood on it's own terms. If you are serious you'll learn it.
I agree with your sentiment that math is important, but it is also true that math is not taught well. Just briefly glancing at chapter 2 the first part, the author presents the law of large numbers out of the blue, then just goes on to present proofs of it. There is no clear discussion of why this law is important, when you can use it and so on. If you contrast how wikipedia explains it:
> In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
This book:
> If one generates random points in d-dimensional space using a Gaussian to generate coordinates, the distance between all pairs of points will be essentially the same when d is large. The reason is that the square of the distance between two points y and z ...
I always get a eerie feeling for people who excessively stress on formalism. It almost goes to the point where I feel they wouldn't want to share their knowledge or intuition.
They are trying to help you get a correct intuition. If you have never approached things this way, there are almost certainly very large holes in your understanding. Not to say that everyone needs to be formal all the time, but you need to sometimes.
> If you have never approached things this way, there are almost certainly very large holes in your understanding.
When anyone tells me that X is the ONLY way to do it, almost exclusively I have found them wrong - beyond data science. Formalism is a means of communication, not the end. You can always communicate ideas without formalism.
FYI, I am a Ph.D Computer Science and a practicing Data Scientist for many years now.
You are essentially saying - the way I think of math is how it should be learned. Sorry but if you're serious, you will find a way to explain your mathematical intuition.
The problem is detailed in this wonderful essay by Paul Lockdhart
We can explain mathematical intuition. We just can't explain it as quickly or in as interesting way as you seem to want.
Here's a simple concept: all polynomials with coefficients in a number system (field) have a solution (possibly in a larger field).
Do you know how hard it is to convey what exactly this means and what the consequences are? I like Lockhart's essay. It resonates with me. However, I don't see his ideas being useful in terms of learning advanced topics. At some point one has to get their hands dirty and slog through the material. Intuition will come with experience.
It is shared. It's just that everyone has to go through to mental work to truly understand. It takes me a semester to convince my students that 3x+5x=8x and that this is true because of the distributive property. And this is why ax+2x = (a+2)x. And the reason we can't simplify further is because our language does not have a word for (a+2) but it does have a word for (3+5). But I know that after a semester of teaching this most still don't actually understand the distributive property. It takes a lot of effort on the part of students for the concept to click.
An N-dimensional vector space is not like R^3. A monad is not like a burrito, and >>= is not like chipotle forgetting to put chicken in your burrito, opening it up, adding chicken and wrapping in a new tortilla.
Math is a new thing rather than the same old thing with new syntax. It needs to be understood on it's own terms. If you are serious you'll learn it.