Saturday, May 12
6:00 am
It is -1 °C, with a high forecast of +22 °C.
From the Environment Canada website:
Today Sunny. High 22. UV index 6 or high.
Tonight Clear. Wind west 20 km/h. Low 7.
6:30 am Mathematics
I am leaning toward a study of Groups rather than Linear Algebra, although I would like to keep my finger in both. I also find the Ash & Gross book more friendly than the Pinter book.
Chapter 2 Groups
Ash: "Mathematical sets can be very interesting and easy to define, but very hard to understand in detail. ... If a set can be endowed with extra structure, it can help our understanding. One very common kind of extra structure is a 'composition law' that turns the set into a group. The definition of a group is the most common way mathematicians have of formalizing the concept of symmetry." [p. 13]
Me: This is a perfect opening statement. The emphasis is on understanding and concepts, not algorithms and procedures.
Ash: "SO(3) stands for 'special orthogonal group in three dimensions." [p. 13]
Me: This is not a notation for manipulation, but simply a shorthand for describing a label. It represents the group consisting of rotations of a sphere.
Ash: "A group is a set along with a rule that tells how to combine any two elements in the set to get another element in the set. We usually use the word composition to describe the act of combining two elements of the group to get a third." [p. 14]
Me: This is a clear beginning.
Ash: "The key point about a group is the combination of any two elements to get a third." [p. 15]
Me: I like the emphasis.
Ash: "DEFINITION: A group G is a set with a composition defined on pairs of elements, as long as three axioms hold true:
- There is a neutral element e in G, so that x o e = e o x = x no matter what element of the group is substituted for x.
- For any element x of G, there is some element y in G so that x o y = y o x = e
- For any three elements x, y, and z in G, we have (x o y) o z = x o (y o z) " [p. 17]
Me: This notation may seem unnecessary at first, particularly the use of the small circle o to represent any operation, but this notation gives one the power to manipulate symbols without having to use language to describe what one is doing.
Ash: The group SO(3) is a mathematical idealization which can be used in certain circumstances to describe the rotation of the earth. This is an example of a mathematical idealization, where we ignore slight discrepancies such as the earth is not perfectly spherical. "It pays to ignore these things for our purposes, because if we included every bump on every log we would never see the forest for the trees. That is, we would not see any patterns that repeat. ... We could never see the regularities to abstract from the welter of reality. Mathematics would never be born." [p. 18]
Me: This is rarely made explicit, yet it lies at the heart of not only mathematics but psychology. A concept is an idealization. The concept of a cat is an idealization of all existing cats. This is the very essence of thinking.
Ash: "Any actual rotation should be buildable by integrating some infinitesimal rotation. This is the germ of the theory of Lie groups and Lie algebras, named for the Norwegian mathematician Sophus Lie (pronounced 'lee'). .. A group such as SO(3) hat has infinitesimal generators is called a continuous group. The other main type of group is called a discrete group. ... An example of a discrete group is the set of integers, where we 'compose' two integers by adding them together." [p. 19]
Me: This ends the second chapter. It is devoted totally to ideas, and simple ones at that. There is no actual manipulation of symbols to see what happens. First comes the idea. Using the idea comes second.
4:00 PM Life
Another perfect day. I visited the LFC for a few weights, then mowed the lawn and trimmed the mugo pine. We then relaxed in the back yard with a cold one and watched the flowers grow.
Plus a "To Mom" bouquet
6:00 PM Cooking
Dinner was our first attempt at replicating our dinner (seafood linguini) that we prepared while with friends in Australia. The big difference this time was that the seafood was all frozen rather than fresh. Nonetheless, it turned out rather well. This is a team effort!
We used a paella pan to make the seafood linguine and then found one bowl that would hold it all.
Cooking the veggies
Adding the seafood (prawns, mussels, scallops, tilapia)
Ready for eating
8:30 PM Literature
I have finished reading "Winter". It was the perfect book for me at the moment. I simply revelled in his descriptions of life during his first winter in northern Montana in a remote valley (so remote it didn't even have electricity). Nothing dramatic - just life, enjoyed to the fullest.