Principles of Mathematical Analysis

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After more than a week of working at this book, I finally managed to move on to Chapter 3, “Numerical Sequences and Series”. Chapter 2, which was titled “Basic Topology”, was quite intense and only basic in hindsight. Even though none of the individual theorems seem all that difficult, achieving a syncretic understanding of the material can be challenging. The problem is not logic, but language.

For the uninitiated, learning higher mathematics, which focuses on theorems that can be derived from definitions and not on computations, is like trying see when before one was blind. The picture is hazy and often the shapes do not hold their form. Just the concepts of limit points, isolated points and interior points took more than a day to latch on. It is not natural to think of sets as open or as closed or to think that they could be both open and closed. Compactness, which Rudin (author of the book) tells the reader is important, is even more removed from one’s natural and intuitive understanding of numbers. Had I not read this book, there is no point in my existence where I would have ever considered the need to think about or to define an open cover of a set never mind to regard its finitude.

Even though I do not understand everything, and Rudin’s proof of the Heine-Borel theorem is still a little slippery in my mind, but every ray of light brings with it that little bit of joy. Nor will I ever put to use the mathematical concepts that I am pushing myself to learn. But utility, especially when it is defined relative to personal preference, should always be looked at more broadly. So what if the math I use never moves beyond grade school arithmetic? Would the pursuit of pure logic be wrong?

Onwards, onwards, the stupid goes.
To the light, to the light.
Where one is lost,
In the embrace of nothingness.

Mathematics is just as elegant and beautiful as art and can be treated as such. You don't need to study mathematics so that you can use it, study it for it's own sake.

Or get a job in a uni where you will use all this stuff every day like me.

Interest in mathematics has never been particularly strong. People think that it is hard when it really is not. The issue is the monkey doodle that kids in school are forced to go through. Working on computational problems is annoying and it does not teach people how to think but only to imitate. For most people, even those involved in what might be regarded as disciplines heavily slanted towards mathematics like engineering, mathematics for them remains alien.

I certainly do subscribe to the idea that we can and really we should study mathematics for its own sake. The study of mathematics is perhaps the best way for us to develop our cognition as far as we can. At the very least you will learn to appreciate the power and limits of logic. But having embarked on my own self study of mathematics, I can understand why few mathematicians do well outside of the academy. People who choose to study mathematics tend to have lopsided brains that afford them powerful reasoning faculties but poor communication skills. The study of proofs that are tersely written do not help develop anyone’s abilities to read and write in any meaningful way. However correct you may be, if you are unable to explain and persuade, then your value to society will never be fully realized.

There are of course brilliant mathematicians who communicate beautifully, but the average math major tends to have poor verbal skills. I actually scouted the web pages of various math departments to get a sense of how I might approach my own math education, and some universities are quite upfront about the poor language skills of their math students. It is a known problem.

I suspect a correspondence between the mathematical mind and the autistic mind. Freed from emotions, or perhaps able to subsist in a state of unfeeling, along with a native reliance on logic, autistic individuals are built for mathematics. Anecdotally at least people who have autistic children also tend to be those who excel in logical endeavors like mathematics (of course a full generalization is false).

Less controversially, if anyone cares, Chapter 3 on Sequences and Series is turning out to be a lot easier than Chapter 2. I suspect it is a theme that will continue throughout the book.

Perhaps talking about it might help me break the Gordion knot. If every infinite subset of a set K has a limit point in K, then K is closed. I think about it this way. If K is not closed, then there is a limit point of K that is not in K. We can then select an infinite series leading up to this limit point which are isolated and thus do not have a limit point in K and therefore violating the hypothesis. I understand Rudin’s proof (the algebra at least) of this last and most important theorem but do not see the intuition behind it. He begins the process by selecting points which are less than 1/n distance from the limit point and then goes on to prove that the set of these points have no limit points. hmm...

Autism is not marked by a lack of emotions, but by an inappropriate expression of emotions. Hence the people you describe could fit the autistic mold. We are often thought of as difficult or tiring to engage as we do not understand intuitively the boundaries of human interaction and sometimes carry on long after it is clear that the other party wants to stop. Using logic to help ourselves fit into the human matrix actually ends up making us seem less human or we come across as badly behaving ones. Hence we fail in society.

Non-autistic people rely almost exclusively on intuition to find their way in human societies. They have a neural template that instinctively tells them what they need to do. Because everyone in any particular social and linguistic area tends to have the same neural template, we end up with distinctly human traits like culture. But reliance on intuition means that they do not develop their logical faculties. So despite being seemingly smart in the ways of their fellow human beings, they have trouble with mathematics.

As mathematics is purely logical, I claim that there is equivalence between the autistic mind and the mathematical mind. It does not mean that every mathematically inclined person is autistic, meaning that they would be diagnosed as such, but that they may simply possess just a small dose of autistic traits; enough to be free from being prisoners of their own emotions and thus free to pursue purely logical endeavors, without any of the more socially damaging effects of autism.