Archive for category mathematics
Space for mathematicians is an interesting set. Or we can say a set with structure. Trying to get used to this concept, I am listing a few spaces, with some explanation, here. By the way, members of such sets are usually called points.
- Vector Space: members of such a space can be scaled and added. Here the definition is dependent on properties of the members.
- Topological Space: in this space points have neighbourhoods. Or, the topological space is a set on which a collection of open sets (closed under unions and finite intersections) is defined.
- State Space: This means different things to computer scientists, physicists, and control engineers. For control engineers, state space is the space in which members are tuples of state variables.
- Metric Space: a metric or distance is defined between members of such a space.
- Euclidean Space: for most of us, this is the space acquired by age eighteen.
Dimension is not easy to understand. At the turn of the century it was one of the major problems in mathematics to determine what dimension means and which properties it has. And since then the situation has become somewhat worse because mathematicians have come up with some ten different notions of dimension: topological dimension, Hausdorff dimension, fractal dimension, self-similarity dimension, box-counting dimension, capacity dimension, information dimension, Euclidean dimension, and more. They are all related. Some of them, however, make sense in certain situations, but not at all in others, where alternative definitions are more helpful. Sometimes they all make sense and are the same. Sometimes several make sense but do not agree. The details can be confusing even for a research mathematician.
Everybody firmly believes in it because the mathematicians imagine it is a fact of observation, and observers that it is a theory of mathematics.
So, which is it? To apportion the blame, I suggest that the fact of observation be that “errors” are independent and identically distributed with finite variances. Observers may as well vote for Lyapunov’s set of conditions. I am sure that mathematicians are more than happy to shoulder the responsibility for the resulting Central Limit Theorem.