Physics - Math

People think we understand many fundamental aspects of the natural universe. I think it would be more correct to say we understand more than we did, but what we don't understand is immense.

Math can be thought of as a language for describing the physical world. As such, understanding and applying it is valuable and interesting.

Visual Complex Analysis, by Tristan Needham, is a math text which attempts to facilitate this.

I've decided that one part of what makes me want to keep plowing ahead is that the author keeps giving tantalizing hints that parts I'm finding obscure get handled in a different way later on. I think that problem he has in this book is that his final results are on the opposite side of a whole complicated structure of our existing math training.

My only guess at an example of such a thing are stories of crazy yoga things. So the guy who could consciously cause one side of his hand to be warmer, then switch the sides. I suspect we can't normally do that because the autonomous neurons go through a second nerve plexus between the brain and their final site, unlike skeletal nerves. We're used to a particular specific way of causing actions like moving a finger. Due to the second nexus, creating a specific effect takes a very different type of stimulation. But the final result is a specific activity from a specific motivation.

Anyhow, back in less strange terms, the descriptions and examples seem to be an effort to lead the reader through understanding what assumptions they have been making and how to not do it. So you get to the same point but in a very different way. But the amazing thing is that this technique seems to bring many disconnected items from calculus and math education under the same umbrella. So sine, cosine and their hyperbolic counterparts are more similar than they are different. Fourier transforms and power series are strongly related. He has hinted that integration and differentiation are more similar than not... These are all things that aren't clear from my recollection of my education.

Even ellipses become a natural side-effect of simple motions and transforms instead of their rather frustratingly limited presentation in basic math. Which is very interesting given his mention of orbits of planets which led me to reading about astronomy yesterday. It seems truly odd that orbits would be elliptical with the sun at one focus and nothing at the other. It would seem that a more ovoid shape would make more sense.

I like the book because it's tying together many things and requiring inquiry into assumptions. Plus I like the occasional mention of things like "it's easy enough to show but involves a lot of un illuminating algebra". Very true. Maybe it won't happen, but I get my hopes up that this will lead me into a way of concepualizing problems in a useful fashion. So like with some of my electronic fantasies, I have a feel that there is something very right about my approach, but I can't turn it into anything concrete because I lack some tools. Plus, I have some hope that this will help for finding a way to turn around how I would solve a problem. I could define a solution then work it backwards to fit the problem.

Anyhow, it's a little dense but not so dense that I have any desire to quit working on it. And I guess I have some hope that at some point it would be possible to draw pictures and do some hand waving to accurately and effectively describe complicated problems.

Ultimately, if math serves to model the world around us, we should be able to start with a familiar model and build from there. It wouldn't get rid of the complexity of a rigorous description, but it would perhaps provide a better symbology and best of all lead people into curiosity about how they might go about understanding more of an interesting problem.