Saturday, March 5, 2011

Spinozian Calculus.

Epsilon-Delta level up.

Me and Tom had a really good understanding of limits and continuity already, much better than we ever had before starting this project, but the FTC pushed as again the limits of our models, so we went back digging and getting an even deeper understanding.

Here is a part of my newest conclusions, as usual the full discussion is at:
I went through Karl's Calculus pages again, the number system and limits parts. I am starting to live revising things even more. Things become clearer every time I read them, even if it is the same text.
I had a slight change of mental model after I did this, I will try to describe it.

Without thinking strictly about functions, limits are interesting places that really are related to a computation that is in seemingly impossible at infinite precision, at least at first sight.
They involve interesting places involving infinities, like the integral being a sum of an infinite large count of infinitely small numbers, or in a sequence of converging quantities, where we know we can get as close as we wish to some quantity yet to be found, or in derivatives, where we want to know if we can get as close as we like to some quantity to be found, starting with the ratio of 2 infinitely small quantities.
All interesting places.
Another point is that limits are about a relationship, the relationship between a domain (input), a range (output) and a function (a computational algorithm). I am finding it useful to visualize each of these separately instead of as a curve. The idea is that is that you take a piece of the domain, which is continuous by the very fact that we are dealing with reals and have a density (to be discussed!), and then taking at the piece of the range that the algorithm produces, again luckily, the reals make it even possible for that piece to be 'continuous' because the reals are closed under the operators found in the function, and check that piece of range out.
Is it connected just like the piece of domain is? is it not?
This is related to one other important point, it is not good to think of this as walking along the domain in some direction and looking at the range, specially not when we talking about limits around a specific point.
When we do this, the focus is on the point, so we start from that point in the domain and expand out, in the range, we start at the point's output and expand out, it is a subtle but important difference, specially that it made me realize how mixed together the multiple kinds of continuity in my mind. I always used to think that a function being continuous in calculus means continuous at every point, and whenever I was confronted with having to use this, I started thinking about some kind of induction logic to show that continuous at every point made all the points somehow connected. But there is also uniformly continuous, continuous at every point but not uniformly continuous, etc... "But first, let me answer a few questions that curiosity may be stuffing into your head at this very moment. In the example of u(x) we saw a function that was continuous everywhere except at one point. Is there a function that is continuous everywhere except at several points? Yes. Is there one that is continuous everywhere except at infinitely many isolated points? Yes. Is there one that is continuous nowhere at all? Yes. Is there one that is continuous only at a single point? Yes. Is there a function that is continuous everywhere yet cannot be graphed? Yes. And later on I shall give you examples of all of these. Click here if you want to skip ahead and see this optional material right now. Or you can continue reading and you will get to it eventually."

I have done some introspection and I came to some conclusions about what made me uncomfortable about epsilon-delta definitions.
Although I really did understand them they still felt strange in the sense that one would think that there must be a better, nicer, more usable way to express this, something that can be manipulated easily like algebra.
But after giving this some deep thought I came out even more convinced of Bishop's argument that it is common sense,
  1. They are a relationship between domain and range
  2. The relationship is TOTALLY dependent on the function, the algorithm that maps domain to range
This means, there is no general lazy way of having limits that solve themselves using mechanical manipulation, because it DEPENDS on the operator, this made me coin one term that I have been noticing the more I went into theory be it math, computer science, solving problems at work ... limits will always be in the category: among other categories.
The problem can be reduced to having to FIND a relationship between epsilon and delta that satisfies some constraints, or FIND a counter example. Like all other 'inverse' problems, where we know the goal but the path is still to be found, it is also in the category: .
Those are 2 uncomfortable categories, they do not allow for laziness, impatience, or lack of intuition and very solid understanding. It is a game...
But once this had crystallized in my mind like this, it suddenly began to feel much more comfortable.

Also, thinking about it like this, separating domain and range, looking at them separately, and looking at the problem in terms of sequence convergence for one point, started to convince that the epsilon-delta is the best definition because it does say in math exactly what we are saying.

I still have to wrap my investigate EVT and MVT and see how they are related to point continuity, uniform continuity, continuity in calculus. This will be the last piece of a level up.
I now see MVT and EVT as THE calculus formal tools that stem from the essence of epsilon-delta and can luckily be used instead of epsilon-delta in some cases to alleviate the need for hairy statements.
It is like this: you can build a basic airplane from pieces of metal and fuel, but if you are planning a war campaign you do not want to be planning on the level of detail of pieces of metal and fuel, you just start with airplanes, otherwize, while still possible, it becomes very hard very fast, ah, theorems are great building blocks, reducing everything to axioms will trash the caches of our tiny brains very fast. "

Not only does the rule make life easier, but without such rules, mathematics would be so thick with undergrowth as to make it virtually impossible to understand.""